From 707a99d9d9a0dc62bd9a8fa52ac4caf5d4d739fe Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 19:10:03 +0330 Subject: [PATCH 01/51] feat: add first version of files --- .../mohammad_mahdi_abootorabi.jpg | Bin 0 -> 53290 bytes .../authors_images/yalda_shabanzadeh.jpg | Bin 0 -> 25798 bytes .../index.md | 335 ++++++++++++++++++ .../metadata.yml | 36 ++ 4 files changed, 371 insertions(+) create mode 100644 notebooks/20_markov_decision_processes_part_2/authors_images/mohammad_mahdi_abootorabi.jpg create mode 100644 notebooks/20_markov_decision_processes_part_2/authors_images/yalda_shabanzadeh.jpg create mode 100644 notebooks/20_markov_decision_processes_part_2/index.md create mode 100644 notebooks/20_markov_decision_processes_part_2/metadata.yml diff --git a/notebooks/20_markov_decision_processes_part_2/authors_images/mohammad_mahdi_abootorabi.jpg b/notebooks/20_markov_decision_processes_part_2/authors_images/mohammad_mahdi_abootorabi.jpg new file mode 100644 index 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+ + Markov Decision (from value iteration) + +
+ +

Table of Contents

+ +- [Introduction](#introduction) +- [Value Iteration](#value_iteration) + - [Convergence of Value Iteration](#vi_convergence) + - [Contraction](#vi_cont) + - [Error Bound ](#vi_error) + - [Value Iteration pseudocode](#vi_code) +- [Policy Iteration](#pi) + - [The idea of policy iteration](#pi_id) + - [Policy Evaluation](#pe) + - [Fixed Policy](#fp) + - [Policy Extraction (Improvement)](#pe) + - [Computing Actions from Values](#pi_cafv) + - [Computing Actions from Q-Values](#pi_cafq) + - [Policy Iteration Summary](#pis) + - [Policy Iteration pseudocode](#pi_code) +- [Conclusion](#conclusion) +- [References](#references) + +

Introduction

+ +In the previous lecture, you became familiar with what MDP is. As you know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on his current state. It’s better to know MDP before Reinforcement Learning. + +In this lecture, we are going to talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us. + + +--- + +

Value Iteration

+ + + +The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. + +Let $U_{i}$ be a utility function that $U_{i}(s)$ gives the utility value for state s at the $i$th iteration. The iteration step, called a **Bellman update**, looks like this: +
+
+$$ +U_{i+1}(s) = \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] +$$ +
+where the update is assumed to be applied simultaneously to all the states at each iteration. If we apply the Bellman update infinitely often, we are guaranteed to reach an equilibrium, In fact, they are also the unique solutions, and the corresponding policy is optimal. + +

Convergence of Value Iteration

+ + +Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{*} = B U^{*}$ that $U^{*}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$. + +>

Contraction

+ +The basic concept used in showing that value iteration converges is the notion of a **contraction**. An important fact about the Bellman operator is that it's contraction. An operator $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is said to be a $\alpha$-contraction if $\alpha \in (0,1)$ and +
+
+$$ +\forall x,y \in \mathbb{R}^{n}, \quad \Vert f(x)-f(y)\Vert \leq \alpha \Vert x-y \Vert +$$ +
+The **Banach fixed-point** theorem guarantees that if $f$ is a contraction, then $f$ has a unique +fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^* \in \mathbb{R}^{n}$ satisfying +$$ +x^* = f(x^*) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R}^{n} +$$ +
+where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. +
+
+Hence, two important properties of contractions are +
    +
  • + A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. +
  • +
  • + When the operator is applied to any argument, the output must get closer to the fixed point (because the fixed point does not move), so repeated application of a contraction always reaches the fixed point in the limit. +
  • + +
+ +We will use the **max norm** to measure distances between utility functions: +
+
+$$ +\Vert U\Vert = \max_{s} \lvert U(s)\lvert +$$ +
+Now, we want to prove that the Bellman operator is $\gamma -contraction$ in the max norm. To prove this, we use the +following Lemma: +
+
+$$ +\lvert \max_{x} f(x) - \max_{x} g(x) \lvert \leq \max_{x} \lvert f(x) - g(x) \lvert +$$ +
+To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then +
+
+$$ +\begin{aligned} +\lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\ +& \leq f(b) - g(b) \\ +& \leq \max_x (f(x) - g(x)) = \max_{x} \lvert f(x) - g(x) \lvert\\ +\Rightarrow \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & \leq \max_{x} \lvert f(x) - g(x) \lvert +\end{aligned} +$$ + +Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have +
+
+$$ +\begin{aligned} +\Vert B U_i - B U^{\prime}_{i}\Vert & = \max_{s} \lvert B U_{i}(s) - B U^{\prime}_{i}(s) \lvert \\ +& = \max_{s} \Big \lvert \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U^{\prime}_{i}(s^{\prime})] \Big \lvert \\ +& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U^{\prime}_{i}(s^{\prime})] \Big \lvert & (Lemma)\\ +& = \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \sum_{s^\prime}^{} P(s^{\prime}|s,a)[U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime})] \Big \lvert \\ +& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \Big (\sum_{s^\prime}^{} (P(s^{\prime}|s,a)\Big ) \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big )\Big \lvert \\ +& = \Big \lvert \gamma \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big )\Big \lvert \\ +& \leq \gamma \max_{s^\prime} \Big \lvert U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big \lvert \\ +\Rightarrow \Vert B U_i - B U^{\prime}_{i}\Vert & \leq \gamma \Big \Vert U_{i} - U^{\prime}_{i}\Big \Vert +\end{aligned} +$$ + +That is, the Bellman operator is $\gamma -contraction$ on the space of utility functions. The fixed point of the Bellman operator is $U^*$. Hence, from the properties of contractions in general, it follows that value iteration always converges to $U^*$ whenever $\gamma < 1$. + +>

Error Bound

+ +We can't have $\infty$ iterations to converges to $U^*$. If we view $\Vert U_{i+1} - U^* \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert < \delta$ we have +$$ +\begin{aligned} +\Vert U_{i+1} - U^* \Vert & = \max_{s} \lvert U_{i+1}(s) - U^{*}(s) \lvert \\ +& = \max_{s} \Big \lvert (U_{i+1}(s) - U_{i+2}(s)) + (U_{i+2}(s) - U_{i+3}(s)) + ...\Big \lvert \\ +& \leq \max_{s} \Big ( \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \Big ) \\ +& \leq \max_{s} \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \max_{s} \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \\ +& = \Vert B U_{i} - B U_{i+1}\Vert + \Vert B U_{i+1} - B U_{i+2}\Vert + ... \\ +& \leq \gamma \Vert U_{i} - U_{i+1}\Vert + \gamma^{2} \Vert U_{i} - U_{i+1}\Vert + ... \\ +& = \frac{\gamma}{1 - \gamma} \Vert U_{i} - U_{i+1}\Vert & \text{for } \gamma < 1\\ +& < \frac{\gamma}{1 - \gamma} \delta +\end{aligned} +$$ +Thus, if $\delta < \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} - U^* \Vert < \epsilon$. + +

Value Iteration pseudocode

+ +**function** Value-Iteration(MDP,$\epsilon$) returns a utility function
+  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,
+          rewards $R(s,a,s^{\prime})$, discount $\gamma$
+        $\epsilon$, the maximum error allowed in the utility of any state
+  **local variables**: $U$, $U^{\prime}$, utility functions for states in S, initially zero for all states
+                $\delta$, the maximum relative change in the utility of any state
+  **repeat**
+      $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$
+    **for each** state s **in** S **do**
+      $U^{\prime}(s) \leftarrow \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U(s^{\prime})]$
+      **if**   $\lvert U^{\prime}(s) - U(s)\lvert > \delta$   **then**   $\delta \leftarrow \lvert U^{\prime}(s) - U(s)\lvert$
+  **until** $\delta < \frac{1 - \gamma}{\gamma} \epsilon$
+  **return** U + +Now, we want to calculate the number of iterations N required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have +$$ +\begin{aligned} +U(s0,a0,s1,...) & = \sum_{t=0}^{\infty} \gamma^{t} R(s_{t},a_{t},s_{t+1}) \\ +& \leq \sum_{t=0}^{\infty} \gamma^{t} R_{max} \\ +U(s) & \leq \frac{R_{max}}{1-\gamma} +\end{aligned} +$$ + +--- + +

Policy Iteration

+ +In the previous section, we observed that it is possible to get an optimal policy even when the utility function estimate is inaccurate. +If one action is clearly better than all others, then the exact magnitude of the utilities on the states involved need not be precise. +This insight suggests an alternative way to find optimal policies. +Policy iteration is a different approach to find the optimal policy for given states and actions. + +

The idea of policy iteration

+ +Once a policy, $\pi_0$ (could be initialized random), has been improved using $V^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $V^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. + +Each policy is guaranteed to be a strict improvement over the previous one (unless it is already optimal). Because a finite MDP has only a finite number of policies, this process must converge to an optimal policy and optimal value function in a finite number of iterations. + +

Policy Evaluation

+ + + +>

Fixed Policy

+ +Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. + +
+
+ $V^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') + {\gamma}V^{\pi}_{k}(s') \Big]$ +
+ + + +**Idea:** Calculate values for the fixed policy: + +$V_0^\pi=0$ + +$V_{K+1}^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') + {\gamma}V^{\pi}_{k}(s') \Big]$ + +**Efficiency:** for each state we compute all the calculation which is the above equation. so order is $O(s^2)$ per interation. + +

Policy Extraction (Improvement)

+ + + +>

Computing Actions from Values

+ +Assume we have the optimal values $V^*(s)$. + +We need to do a mini-expectimax. + +$$\pi^*(s)=\arg\max {}_a \Sigma_{s'} {T(s,a,s')} \Big [R(s,a,s') + {\gamma}V^{*}(s') \Big]$$ + +This is called policy extraction, since it gets the policy implied by the values + + +>

Computing Actions from Q-Values

+ +Assume we have the optimal Q-values. + +This is kind of one-step expectimax with assumption that terminal values $V^\pi_k(s)$ are less or equal than result value from bellman equation. so we can conclude that new policy is better than the previous one. + +The action is completely trivial. + +$$\pi^*(s)=\arg\max {}_a Q^*(s,a)$$ + + +**Comparison:** +actions are easier to select from q-values than values. + +**Efficiency:** +for each state we compute all the calcaulations and take max on different actions. so order is $O(S^2A)$ in one-step. + +

Policy Iteration Summary

+ + + +The policy iteration algorithm alternates the following two steps, beginning from some initial policy $\pi_0$ + +- **Policy evaluation** + + Calculate utilities for some fixed policy (not optimal + utilities!) until convergence. + + For fixed current policy $\pi$, find values with policy evaluation: + +$$V_{K+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {T(s,\pi_{i}(s),s')} \Big [R(s,\pi_{i}(s),s') + {\gamma}V^{\pi_{i}}_{k}(s') \Big]$$ + + +- **Policy improvement** + + Update policy using one-step look-ahead with resulting + converged (but not optimal!) utilities as future values. + It is an one-step algorithm. + + For fixed values, get a better policy using policy extraction: + +$${\pi}_{i+1}(s)=max_{a}\Sigma_{s'} {T(s,a,s')} \Big [R(s,a,s') + {\gamma}V^{\pi_{i}}(s') \Big]$$ + +- **Repeat steps until policy converges** + + The algorithm terminates when the policy improvement step yields no change in the utilities. + + At this point, we know that the utility function ${U_i}$ is a fixed point of the Bellman update, so it is a solution to the Bellman equations, and ${\pi_i}$ must be an optimal policy. + + Because there are only finitely many policies for a finite state space, and each iteration can be shown to yield a better policy, policy iteration must terminate. + + +**Terminate condition:** + +When algorithm stops changing utility, because we know that the utility function ${U_i}$ is a fixed point of the Bellman update, we conclude that it is a solution to the Bellman equations, and ${\pi_i}$ must be an optimal policy. + + +

Policy-Iteratoin pseudocode

+ +**function** Policy-Iteratoin(MDP) returns a utility function
+  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,
+  **local variables**: $U$ , a vector of utilities for states in $S$, initially zero
+                $\pi$, a policy vector indexed by state, initially random
+  **repeat**
+    $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP)
+    $unchanged? \leftarrow$ true
+    **for each** state s **in** S **do**
+      **if**   $\max\limits_{a \in A(s)} \Sigma_{s^{\prime}} P(s^{\prime}|s,a) U[s^{\prime}] > \Sigma_{s^{\prime}} P(s^{\prime}|s,\pi[s]) U[s^{\prime}]$   **then**   **do**
+        $\pi[s] \leftarrow \arg\max\limits_{a \in A(s)} \Sigma_{s^{\prime}} P(s^{\prime}|s,a) U[s^{\prime}]$
+        $unchanged? \leftarrow$ false
+  **until** $unchanged?$
+  **return** $\pi$ + + +**Optimality:** +Policy iteration is optimal + +**Convergence:** +Can converge mush faster under some codition + +**Efficiency:** +$O(S^2A)=O(S^2)+O(S^2A)$ which are orders of policy evaluation and policy improvement. + +--- + +

Conclusion

+ +These two methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. + +In policy iteration, you start with a random policy, then in several steps, you update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, you must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. You find a better policy using a one-step look-ahead. If the policy doesn’t change, it means you reach the optimal answer. + +In value iteration, you start with a random value function. In each step, you improve the values and (implicitly) the policy together. We don't care much about policy apparently, but taking maximum action will improve it as well. + +The value iteration algorithm is more straightforward to understand as you only have to do one thing in each step. On the other hand, in practical terms, often policy iteration algorithm converges within fewer iterations and is much faster. Maximum selection in the value iteration is not here and it has a great impact on this fact. In theory, however, policy iteration must takes the same number of iterations as value iteration in the worst case. They differ only in whether we plug in a fixed policy or max over action + +Each of them has its cons and pros. Depending on the situation you can choose each of them, but policy iteration is more commonly used. + +In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s, a, s') and T(s, a, s') and we have to do some actions to find or estimate them. + + + + +

References

+ ++ AI course teached by Dr. Rohban at Sharif University of Technology, Fall 2021 ++ Russell, S. J., Norvig, P., & Davis, E. (4rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. ++ https://www.geeksforgeeks.org ++ http://ai.berkeley.edu ++ https://towardsdatascience.com ++ https://ai.stanford.edu/~gwthomas/notes/mdps.pdf diff --git a/notebooks/20_markov_decision_processes_part_2/metadata.yml b/notebooks/20_markov_decision_processes_part_2/metadata.yml new file mode 100644 index 00000000..d55c251f --- /dev/null +++ b/notebooks/20_markov_decision_processes_part_2/metadata.yml @@ -0,0 +1,36 @@ +title: Markov Decision Processes - Part 2 + +header: + title: Markov Decision Processes - Part 2 + description: An Introduction to Markov Decision Processes (From Value Iteration Till The End) + +authors: + label: + position: top + content: + - name: Mohammad Mahdi Abootorabi + role: Author + image: authors_images/mohammad_mahdi_abootorabi.jpg + contact: + - link: https://github.com/aboots + icon: fab fa-github + - link: mailto:mahdi.abootorabi2@gmail.com + icon: fas fa-envelope + + - name: Yalda Shabanzadeh + role: Author + image: authors_images/yalda_shabanzadeh.jpg + contact: + - link: https://github.com/yaldashbz + icon: fab fa-github + - link: mailto:yaldashabanzadeh@gmail.com + icon: fas fa-envelope + + - name: Amirreza Soleimanbeigi + role: Author + image: authors_images/amirreza_soleimanbeigi.jpg + contact: + - link: https://github.com/invisible0831 + icon: fab fa-github + - link: mailto:amirsoli80@gmail.com + icon: fas fa-envelope From 02ab63b4307bfc376c208110f90c0dc52a5ff95b Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 19:11:26 +0330 Subject: [PATCH 02/51] feat: add images --- .../images/MDP.png | Bin 0 -> 75339 bytes .../images/MDP_vs_RL.png | Bin 0 -> 188833 bytes .../images/MDP_vs_RL2.jpg | Bin 0 -> 87290 bytes .../images/RL.png | Bin 0 -> 122250 bytes .../images/fixed_policy.png | Bin 0 -> 41606 bytes .../images/fixed_policy2.png | Bin 0 -> 58071 bytes .../images/policy_evaluation.png | Bin 0 -> 60928 bytes .../images/policy_extraction.png | Bin 0 -> 100974 bytes .../images/policy_iteration.png | Bin 0 -> 112096 bytes .../images/value_iteration.png | Bin 0 -> 119705 bytes 10 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 notebooks/20_markov_decision_processes_part_2/images/MDP.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/MDP_vs_RL.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/MDP_vs_RL2.jpg create mode 100644 notebooks/20_markov_decision_processes_part_2/images/RL.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/fixed_policy.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/fixed_policy2.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/policy_evaluation.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/policy_extraction.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/policy_iteration.png create mode 100644 notebooks/20_markov_decision_processes_part_2/images/value_iteration.png diff --git a/notebooks/20_markov_decision_processes_part_2/images/MDP.png b/notebooks/20_markov_decision_processes_part_2/images/MDP.png new file mode 100644 index 0000000000000000000000000000000000000000..324e6fb0bd2394ff1ee1747aa73a162ad7efa435 GIT binary patch literal 75339 zcma&NRa9JEur-RiySoI35ZocSLxQ^#+}$mBaCc2`cN%whmnP7-yWY-s&iS8jkFiI~ zL+`aqR?V6(ZUEH{08ZwCM^z8Jw!*uqS;vn%LcbWU) zhTIH3mxGB6W9O}|B?Fr?H5Il0UW|BZDw_ZEiSGZ;r)O-}k9WU~4v&kBv)&N+XKF@< zHw~m?Iku(+*z>2KJzfUxFZXuYLOyugzBi~!*}PW~rz>@)kFPh=SF4tJLUp9Pju7k3 zcKxX=x+qiJ_8ULHS7;dTkEH$bD$96hbXrUj)A>lt$EK#)kp68ZFQz}5?{>DvK)YSRm!S zeQS#(@pGL(N<)niv5f%xnKnmT&;#`qiT|Ibz&Fr&yTb0zZo zzAXJ6_^f&}A{95?d=b?L_$hM12xqcMbo(Io839mW+Pq43$j1{X@6LxYj$k7|EJviZQ;ogy`=7jfwMo?F%>3@4T`}+ISJL!xm zD8Hpv%cT5UeQ|!ZnF=s0ny@#l!4vVSZ&%V_A^q#HtxY%uq7Q8eknw#h1IOcJiRBG% zq|zJB*5hogWE<*Sle1IRYMLc!JpSU`xlZHdu})N}?sE$i@9LxSWEg>8Avj zLxre4*eHA!i`MxQ6-A!pWLFb?{v_!P_Th7%P~I;0dvl|6M9OK%YmaTevVbr(*DwQP?K(JA~gP0!i#b((EDS#~;o&Tj3=g95&(OK`Kw@2tEGa>IX@WPq-bzV_Vq5k^a-%&ha-1xkjN6!-E76WSN-J_0NH9WRkZ^`uJu(P>LS)UbDn z4F5T1d``=7oF*UmM{&KmSAZYy9+Glo#4hY!$h&W7o_ITYx>Ed7ZdxAcm{uLUOTf`{ z&+Ry;3`#Py=rsC

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index 1b2133c0..43a61232 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -52,6 +52,8 @@ notebooks: kind: S2021, LN, Notebook - notebook: notebooks/16_deep_neural_networks/ kind: S2021, LN, Notebook + - md: notebooks/20_markov_decision_processes_part_2/index.md + kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook \ No newline at end of file From eb71ae202dafe1fcf81846be76d0d322d4456e06 Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 19:49:57 +0330 Subject: [PATCH 04/51] feat: add notebook to index --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 43a61232..374032cc 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -52,7 +52,7 @@ notebooks: kind: S2021, LN, Notebook - notebook: notebooks/16_deep_neural_networks/ kind: S2021, LN, Notebook - - md: notebooks/20_markov_decision_processes_part_2/index.md + - md: notebooks/20_markov_decision_processes_part_2/ kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ From 4994cb5022b1b5a62bea3cca9ee75a19e40f349a Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 21:10:24 +0330 Subject: [PATCH 05/51] feat: changes in index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 97536cc5..b3808d9c 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -36,7 +36,7 @@ In this lecture, we are going to talk about two different methods for solving MD

Value Iteration

- +
The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. @@ -187,7 +187,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle

Policy Evaluation

- + >

Fixed Policy

@@ -198,7 +198,7 @@ Back to expectimax tree, there was different actions for node s, so we had diffe $V^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') + {\gamma}V^{\pi}_{k}(s') \Big]$ - + **Idea:** Calculate values for the fixed policy: @@ -210,7 +210,7 @@ $V_{K+1}^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') +

Policy Extraction (Improvement)

- + >

Computing Actions from Values

@@ -242,7 +242,7 @@ for each state we compute all the calcaulations and take max on different action

Policy Iteration Summary

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a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -13,11 +13,12 @@ - [Contraction](#vi_cont) - [Error Bound ](#vi_error) - [Value Iteration pseudocode](#vi_code) + - [Time Complexity](#time_vi) - [Policy Iteration](#pi) - [The idea of policy iteration](#pi_id) - [Policy Evaluation](#pe) - [Fixed Policy](#fp) - - [Policy Extraction (Improvement)](#pe) + - [Policy Extraction (Improvement)](#pex) - [Computing Actions from Values](#pi_cafv) - [Computing Actions from Q-Values](#pi_cafq) - [Policy Iteration Summary](#pis) @@ -25,6 +26,8 @@ - [Conclusion](#conclusion) - [References](#references) +--- +

In the previous lecture, you became familiar with what MDP is. As you know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on his current state. It’s better to know MDP before Reinforcement Learning. @@ -40,67 +43,49 @@ In this lecture, we are going to talk about two different methods for solving MD The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. -Let $U_{i}$ be a utility function that $U_{i}(s)$ gives the utility value for state s at the $i$th iteration. The iteration step, called a **Bellman update**, looks like this: -
-
+Let $U_{i}$ be a utility function that $U_{i}(s)$ gives the utility value for state s at the $i$th iteration. The iteration step, called a **Bellman update**, looks like this: + $$ U_{i+1}(s) = \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] -$$ -
+$$ where the update is assumed to be applied simultaneously to all the states at each iteration. If we apply the Bellman update infinitely often, we are guaranteed to reach an equilibrium, In fact, they are also the unique solutions, and the corresponding policy is optimal. -

Convergence of Value Iteration

+

Convergence of Value Iteration

Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{*} = B U^{*}$ that $U^{*}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$. ->

Contraction

+>

Contraction

+ +The basic concept used in showing that value iteration converges is the notion of a **contraction**. An important fact about the Bellman operator is that it's contraction. An operator $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is said to be a $\alpha$-contraction if $\alpha \in (0,1)$ and -The basic concept used in showing that value iteration converges is the notion of a **contraction**. An important fact about the Bellman operator is that it's contraction. An operator $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is said to be a $\alpha$-contraction if $\alpha \in (0,1)$ and -
-
$$ \forall x,y \in \mathbb{R}^{n}, \quad \Vert f(x)-f(y)\Vert \leq \alpha \Vert x-y \Vert -$$ -
+$$ The **Banach fixed-point** theorem guarantees that if $f$ is a contraction, then $f$ has a unique fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^* \in \mathbb{R}^{n}$ satisfying $$ x^* = f(x^*) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R}^{n} -$$ -
-where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. -
-
+$$ +where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. + Hence, two important properties of contractions are -
    -
  • - A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. -
  • -
  • - When the operator is applied to any argument, the output must get closer to the fixed point (because the fixed point does not move), so repeated application of a contraction always reaches the fixed point in the limit. -
  • - -
+- A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. +- When the operator is applied to any argument, the output must get closer to the fixed point (because the fixed point does not move), so repeated application of a contraction always reaches the fixed point in the limit. -We will use the **max norm** to measure distances between utility functions: -
-
-$$ +We will use the **max norm** to measure distances between utility functions: + +$$ \Vert U\Vert = \max_{s} \lvert U(s)\lvert -$$ -
+$$ Now, we want to prove that the Bellman operator is $\gamma -contraction$ in the max norm. To prove this, we use the following Lemma: -
-
+ $$ \lvert \max_{x} f(x) - \max_{x} g(x) \lvert \leq \max_{x} \lvert f(x) - g(x) \lvert -$$ -
-To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then -
-
+$$ +To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then + $$ \begin{aligned} \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\ @@ -110,9 +95,8 @@ $$ \end{aligned} $$ -Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have -
-
+Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have + $$ \begin{aligned} \Vert B U_i - B U^{\prime}_{i}\Vert & = \max_{s} \lvert B U_{i}(s) - B U^{\prime}_{i}(s) \lvert \\ @@ -128,9 +112,9 @@ $$ That is, the Bellman operator is $\gamma -contraction$ on the space of utility functions. The fixed point of the Bellman operator is $U^*$. Hence, from the properties of contractions in general, it follows that value iteration always converges to $U^*$ whenever $\gamma < 1$. ->

Error Bound

+>

Error Bound

-We can't have $\infty$ iterations to converges to $U^*$. If we view $\Vert U_{i+1} - U^* \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert < \delta$ we have +We can't have $\infty$ iterations to converges to $U^*$. If we view $\Vert U_{i+1} - U^* \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert \leq \delta$ we have $$ \begin{aligned} \Vert U_{i+1} - U^* \Vert & = \max_{s} \lvert U_{i+1}(s) - U^{*}(s) \lvert \\ @@ -140,28 +124,30 @@ $$ & = \Vert B U_{i} - B U_{i+1}\Vert + \Vert B U_{i+1} - B U_{i+2}\Vert + ... \\ & \leq \gamma \Vert U_{i} - U_{i+1}\Vert + \gamma^{2} \Vert U_{i} - U_{i+1}\Vert + ... \\ & = \frac{\gamma}{1 - \gamma} \Vert U_{i} - U_{i+1}\Vert & \text{for } \gamma < 1\\ -& < \frac{\gamma}{1 - \gamma} \delta +& \leq \frac{\gamma}{1 - \gamma} \delta \end{aligned} $$ -Thus, if $\delta < \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} - U^* \Vert < \epsilon$. - -

Value Iteration pseudocode

- -**function** Value-Iteration(MDP,$\epsilon$) returns a utility function
-  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,
-          rewards $R(s,a,s^{\prime})$, discount $\gamma$
-        $\epsilon$, the maximum error allowed in the utility of any state
-  **local variables**: $U$, $U^{\prime}$, utility functions for states in S, initially zero for all states
-                $\delta$, the maximum relative change in the utility of any state
-  **repeat**
-      $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$
-    **for each** state s **in** S **do**
-      $U^{\prime}(s) \leftarrow \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U(s^{\prime})]$
-      **if**   $\lvert U^{\prime}(s) - U(s)\lvert > \delta$   **then**   $\delta \leftarrow \lvert U^{\prime}(s) - U(s)\lvert$
-  **until** $\delta < \frac{1 - \gamma}{\gamma} \epsilon$
-  **return** U - -Now, we want to calculate the number of iterations N required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have +Thus, if $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} - U^* \Vert \leq \epsilon$. + +

Value Iteration pseudocode

+ +**function** Value-Iteration(MDP,$\epsilon$) returns a utility function +  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$, +          rewards $R(s,a,s^{\prime})$, discount $\gamma$ +        $\epsilon$, the maximum error allowed in the utility of any state +  **local variables**: $U$, $U^{\prime}$, utility functions for states in S, initially zero for all states +                $\delta$, the maximum relative change in the utility of any state +  **repeat** +      $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$ +    **for each** state s **in** S **do** +      $U^{\prime}(s) \leftarrow \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U(s^{\prime})]$ +      **if**   $\lvert U^{\prime}(s) - U(s)\lvert > \delta$   **then**   $\delta \leftarrow \lvert U^{\prime}(s) - U(s)\lvert$ +  **until** $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$ +  **return** $U$ + +>

Time Complexity

+ +We want to calculate the number of iterations required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have $$ \begin{aligned} U(s0,a0,s1,...) & = \sum_{t=0}^{\infty} \gamma^{t} R(s_{t},a_{t},s_{t+1}) \\ @@ -170,6 +156,32 @@ U(s) & \leq \frac{R_{max}}{1-\gamma} \end{aligned} $$ +If we run for $N$ iterations, we have +$$ +\begin{aligned} +\Vert U_{N} - U^* \Vert & = \Vert B U_{N-1} - B U^* \Vert \\ +& \leq \gamma \Vert U_{N-1} - U^* \Vert \\ +& = \gamma \Vert B U_{N-2} - B U^* \Vert \\ +& \leq \gamma^{2} \Vert U_{N-2} - U^* \Vert \\ +& \leq \gamma^{N} \Vert U_{0} - U^* \Vert \\ +& = \gamma^{N} \Vert U^* \Vert & \forall s \text{ } U_{0}(s) = 0 \\ +& \leq \gamma^{N} \frac{R_{max}}{1-\gamma} +\end{aligned} +$$ + +if $\gamma^{N} \frac{R_{max}}{1-\gamma} \leq \epsilon$, then $\Vert U_{N} - U^* \Vert \leq \epsilon$. Thus +$$ +N = \Big \lceil \frac{\log{(\frac{R_{max}}{\epsilon (1-\gamma)})}}{\log{(\frac{1}{\gamma})}} \Big \rceil +$$ + +$N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value iteration converges exponentially fast but $N$ grows rapidly as $\gamma$ becomes close to 1. We can get fast convergence if we make $\gamma$ small. + +
+ +Figure shows how $N$ varies with $\gamma$, for different values of the ratio $c = \frac{2\epsilon}{R_{max}}$. + +Time complexity of each iteration of value iteration is $O(\lvert S\rvert^{2} \lvert A\rvert)$. Thus, time complexity of value iteration is $O \Big (\lvert S\rvert^{2} \lvert A\rvert \Big \lceil \frac{\log{(\frac{R_{max}}{\epsilon (1-\gamma)})}}{\log{(\frac{1}{\gamma})}} \Big \rceil \Big )$ + ---

Policy Iteration

@@ -179,70 +191,70 @@ If one action is clearly better than all others, then the exact magnitude of the This insight suggests an alternative way to find optimal policies. Policy iteration is a different approach to find the optimal policy for given states and actions. -

The idea of policy iteration

+

The idea of policy iteration

-Once a policy, $\pi_0$ (could be initialized random), has been improved using $V^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $V^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. +Once a policy, $\pi_0$ (could be initialized random), has been improved using $U^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $U^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. Each policy is guaranteed to be a strict improvement over the previous one (unless it is already optimal). Because a finite MDP has only a finite number of policies, this process must converge to an optimal policy and optimal value function in a finite number of iterations. -

Policy Evaluation

+

Policy Evaluation

- +
->

Fixed Policy

+>

Fixed Policy

Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. -
+
- $V^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') + {\gamma}V^{\pi}_{k}(s') \Big]$ + $U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big]$
- +
**Idea:** Calculate values for the fixed policy: -$V_0^\pi=0$ +$U_0^\pi=0$ -$V_{K+1}^{\pi}(s) \leftarrow \Sigma_{s'} {T(s,\pi(s),s')} \Big [R(s,\pi(s),s') + {\gamma}V^{\pi}_{k}(s') \Big]$ +$U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big]$ -**Efficiency:** for each state we compute all the calculation which is the above equation. so order is $O(s^2)$ per interation. +**Efficiency:** for each state we compute all the calculation which is the above equation. Thus order is $O(\lvert S\rvert^{2})$ per interation. -

Policy Extraction (Improvement)

+

Policy Extraction (Improvement)

- +
->

Computing Actions from Values

+>

Computing Actions from Values

-Assume we have the optimal values $V^*(s)$. +Assume we have the optimal values $U^*(s)$. We need to do a mini-expectimax. -$$\pi^*(s)=\arg\max {}_a \Sigma_{s'} {T(s,a,s')} \Big [R(s,a,s') + {\gamma}V^{*}(s') \Big]$$ +$$\pi^*(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$$ This is called policy extraction, since it gets the policy implied by the values ->

Computing Actions from Q-Values

+>

Computing Actions from Q-Values

Assume we have the optimal Q-values. -This is kind of one-step expectimax with assumption that terminal values $V^\pi_k(s)$ are less or equal than result value from bellman equation. so we can conclude that new policy is better than the previous one. +This is kind of one-step expectimax with assumption that terminal values $U^\pi_k(s)$ are less or equal than result value from bellman equation. so we can conclude that new policy is better than the previous one. The action is completely trivial. -$$\pi^*(s)=\arg\max {}_a Q^*(s,a)$$ +$$\pi^*(s)=\underset{a}{\operatorname{argmax}} Q^*(s,a)$$ **Comparison:** actions are easier to select from q-values than values. **Efficiency:** -for each state we compute all the calcaulations and take max on different actions. so order is $O(S^2A)$ in one-step. +for each state we compute all the calcaulations and take max on different actions. so order is $O(\lvert S\rvert^{2} \lvert A\rvert)$ in one-step. -

Policy Iteration Summary

+

Policy Iteration Summary

- +
The policy iteration algorithm alternates the following two steps, beginning from some initial policy $\pi_0$ @@ -253,7 +265,7 @@ The policy iteration algorithm alternates the following two steps, beginning fro For fixed current policy $\pi$, find values with policy evaluation: -$$V_{K+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {T(s,\pi_{i}(s),s')} \Big [R(s,\pi_{i}(s),s') + {\gamma}V^{\pi_{i}}_{k}(s') \Big]$$ +$$U_{k+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi_{i}(s))} \Big [R(s,\pi_{i}(s),s') + {\gamma}U^{\pi_{i}}_{k}(s') \Big]$$ - **Policy improvement** @@ -264,7 +276,7 @@ $$V_{K+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {T(s,\pi_{i}(s),s')} \Big [R(s,\pi For fixed values, get a better policy using policy extraction: -$${\pi}_{i+1}(s)=max_{a}\Sigma_{s'} {T(s,a,s')} \Big [R(s,a,s') + {\gamma}V^{\pi_{i}}(s') \Big]$$ +$${\pi}_{i+1}(s)= \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ - **Repeat steps until policy converges** @@ -280,20 +292,45 @@ $${\pi}_{i+1}(s)=max_{a}\Sigma_{s'} {T(s,a,s')} \Big [R(s,a,s') + {\gamma}V^{\pi When algorithm stops changing utility, because we know that the utility function ${U_i}$ is a fixed point of the Bellman update, we conclude that it is a solution to the Bellman equations, and ${\pi_i}$ must be an optimal policy. -

Policy-Iteratoin pseudocode

+
+ +

Policy-Iteration pseudocode

+ +--- + +**function** Q-Value(MDP, $s, a, U$) **returns** a utility value
+  $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$ + +
+ +**function** Policy-Evaluation($\pi, U$, MDP) **returns** a utility function
+  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$, +          rewards $R(s,a,s^{\prime})$, discount $\gamma$ +          $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ +          $\delta$, the maximum relative change in the utility of any state +  **repeat** +      $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$ +     **for each** state s **in** S **do** +        $U^{\prime}(s) \leftarrow \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U(s^{\prime})]$ +        **if**   $\lvert U^{\prime}(s) - U(s)\lvert > \delta$   **then**   $\delta \leftarrow \lvert U^{\prime}(s) - U(s)\lvert$ +  **until** $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$ +  **return** $U$ -**function** Policy-Iteratoin(MDP) returns a utility function
-  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,
-  **local variables**: $U$ , a vector of utilities for states in $S$, initially zero
-                $\pi$, a policy vector indexed by state, initially random
-  **repeat**
-    $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP)
-    $unchanged? \leftarrow$ true
+
+ +**function** Policy-Iteration(MDP) **returns** a utility function +  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$, +        $U$ , a vector of utilities for states in $S$, initially zero +               $\pi$, a policy vector indexed by state, initially random +  **repeat** +    $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP) +    $unchanged? \leftarrow$ true     **for each** state s **in** S **do**
-      **if**   $\max\limits_{a \in A(s)} \Sigma_{s^{\prime}} P(s^{\prime}|s,a) U[s^{\prime}] > \Sigma_{s^{\prime}} P(s^{\prime}|s,\pi[s]) U[s^{\prime}]$   **then**   **do**
-        $\pi[s] \leftarrow \arg\max\limits_{a \in A(s)} \Sigma_{s^{\prime}} P(s^{\prime}|s,a) U[s^{\prime}]$
-        $unchanged? \leftarrow$ false
-  **until** $unchanged?$
+      $a^* \leftarrow \underset{a \in A(s)}{\operatorname{argmax}}$ Q-Value(MDP, $s, a, U$)
+      **if**   $\max\limits_{a \in A(s)}$ Q-Value(MDP, $s, a^*, U$) $>$ Q-Value(MDP, $s, \pi[s], U$)   **then**   **do** +        $\pi[s] \leftarrow a^*$ +        $unchanged? \leftarrow$ false +  **until** $unchanged?$   **return** $\pi$ @@ -301,12 +338,13 @@ When algorithm stops changing utility, because we know that the utility function Policy iteration is optimal **Convergence:** -Can converge mush faster under some codition +Can converge mush faster under some condition **Efficiency:** -$O(S^2A)=O(S^2)+O(S^2A)$ which are orders of policy evaluation and policy improvement. +$O(\lvert S\rvert^{2} \lvert A\rvert)=O(\lvert S\rvert^{2})+O(\lvert S\rvert^{2} \lvert A\rvert)$ which are orders of policy evaluation and policy improvement. ---- +
+

Conclusion

@@ -320,10 +358,13 @@ The value iteration algorithm is more straightforward to understand as you only Each of them has its cons and pros. Depending on the situation you can choose each of them, but policy iteration is more commonly used. -In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s, a, s') and T(s, a, s') and we have to do some actions to find or estimate them. +In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. + +
- +
+

References

From cefadb46e6dfcb13b4900e799ac2218a8ed7df2c Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 22:25:42 +0330 Subject: [PATCH 08/51] feat: update index.md --- .../index.md | 44 ++++++++++++++++--- 1 file changed, 39 insertions(+), 5 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index f1692173..a695b7eb 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -5,8 +5,14 @@ +
+ +
+

Table of Contents

+
+ - [Introduction](#introduction) - [Value Iteration](#value_iteration) - [Convergence of Value Iteration](#vi_convergence) @@ -28,17 +34,23 @@ --- +
+

Introduction

+
+ In the previous lecture, you became familiar with what MDP is. As you know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on his current state. It’s better to know MDP before Reinforcement Learning. In this lecture, we are going to talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us. ---- +

Value Iteration

+
+
The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. @@ -50,9 +62,13 @@ U_{i+1}(s) = \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\pr $$ where the update is assumed to be applied simultaneously to all the states at each iteration. If we apply the Bellman update infinitely often, we are guaranteed to reach an equilibrium, In fact, they are also the unique solutions, and the corresponding policy is optimal. +
+

Convergence of Value Iteration

+
+ Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{*} = B U^{*}$ that $U^{*}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$. >

Contraction

@@ -129,8 +145,12 @@ $$ $$ Thus, if $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} - U^* \Vert \leq \epsilon$. +
+

Value Iteration pseudocode

+
+ **function** Value-Iteration(MDP,$\epsilon$) returns a utility function   **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,           rewards $R(s,a,s^{\prime})$, discount $\gamma$ @@ -182,10 +202,12 @@ Figure shows how $N$ varies with $\gamma$, for different values of the ratio $c Time complexity of each iteration of value iteration is $O(\lvert S\rvert^{2} \lvert A\rvert)$. Thus, time complexity of value iteration is $O \Big (\lvert S\rvert^{2} \lvert A\rvert \Big \lceil \frac{\log{(\frac{R_{max}}{\epsilon (1-\gamma)})}}{\log{(\frac{1}{\gamma})}} \Big \rceil \Big )$ ---- +

Policy Iteration

+
+ In the previous section, we observed that it is possible to get an optimal policy even when the utility function estimate is inaccurate. If one action is clearly better than all others, then the exact magnitude of the utilities on the states involved need not be precise. This insight suggests an alternative way to find optimal policies. @@ -197,8 +219,12 @@ Once a policy, $\pi_0$ (could be initialized random), has been improved using $U Each policy is guaranteed to be a strict improvement over the previous one (unless it is already optimal). Because a finite MDP has only a finite number of policies, this process must converge to an optimal policy and optimal value function in a finite number of iterations. +
+

Policy Evaluation

+
+
>

Fixed Policy

@@ -220,8 +246,12 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( **Efficiency:** for each state we compute all the calculation which is the above equation. Thus order is $O(\lvert S\rvert^{2})$ per interation. +
+

Policy Extraction (Improvement)

+
+
>

Computing Actions from Values

@@ -252,8 +282,12 @@ actions are easier to select from q-values than values. **Efficiency:** for each state we compute all the calcaulations and take max on different actions. so order is $O(\lvert S\rvert^{2} \lvert A\rvert)$ in one-step. +
+

Policy Iteration Summary

+
+
The policy iteration algorithm alternates the following two steps, beginning from some initial policy $\pi_0$ @@ -296,7 +330,7 @@ When algorithm stops changing utility, because we know that the utility function

Policy-Iteration pseudocode

---- +
**function** Q-Value(MDP, $s, a, U$) **returns** a utility value
  $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$ @@ -343,11 +377,12 @@ Can converge mush faster under some condition **Efficiency:** $O(\lvert S\rvert^{2} \lvert A\rvert)=O(\lvert S\rvert^{2})+O(\lvert S\rvert^{2} \lvert A\rvert)$ which are orders of policy evaluation and policy improvement. -

Conclusion

+
+ These two methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. In policy iteration, you start with a random policy, then in several steps, you update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, you must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. You find a better policy using a one-step look-ahead. If the policy doesn’t change, it means you reach the optimal answer. @@ -363,7 +398,6 @@ In the next part, you will face reinforcement learning. The biggest change in RL
-

References

From aa54cc712909af9335d3bc672f980114b10f9ca8 Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 22:30:06 +0330 Subject: [PATCH 09/51] feat: update index.md --- .../index.md | 18 +++++++++++++----- 1 file changed, 13 insertions(+), 5 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index a695b7eb..d4f3ae78 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -11,8 +11,6 @@

Table of Contents

-
- - [Introduction](#introduction) - [Value Iteration](#value_iteration) - [Convergence of Value Iteration](#vi_convergence) @@ -32,8 +30,6 @@ - [Conclusion](#conclusion) - [References](#references) ---- -

Introduction

@@ -71,6 +67,8 @@ where the update is assumed to be applied simultaneously to all the states at ea Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{*} = B U^{*}$ that $U^{*}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$. +
+ >

Contraction

The basic concept used in showing that value iteration converges is the notion of a **contraction**. An important fact about the Bellman operator is that it's contraction. An operator $f:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is said to be a $\alpha$-contraction if $\alpha \in (0,1)$ and @@ -128,6 +126,8 @@ $$ That is, the Bellman operator is $\gamma -contraction$ on the space of utility functions. The fixed point of the Bellman operator is $U^*$. Hence, from the properties of contractions in general, it follows that value iteration always converges to $U^*$ whenever $\gamma < 1$. +
+ >

Error Bound

We can't have $\infty$ iterations to converges to $U^*$. If we view $\Vert U_{i+1} - U^* \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert \leq \delta$ we have @@ -165,6 +165,8 @@ Thus, if $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} -   **until** $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$   **return** $U$ +
+ >

Time Complexity

We want to calculate the number of iterations required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have @@ -213,7 +215,7 @@ If one action is clearly better than all others, then the exact magnitude of the This insight suggests an alternative way to find optimal policies. Policy iteration is a different approach to find the optimal policy for given states and actions. -

The idea of policy iteration

+

The idea of policy iteration

Once a policy, $\pi_0$ (could be initialized random), has been improved using $U^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $U^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. @@ -227,6 +229,8 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle
+
+ >

Fixed Policy

Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. @@ -254,6 +258,8 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(
+
+ >

Computing Actions from Values

Assume we have the optimal values $U^*(s)$. @@ -265,6 +271,8 @@ $$\pi^*(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \ This is called policy extraction, since it gets the policy implied by the values +
+ >

Computing Actions from Q-Values

Assume we have the optimal Q-values. From 3feffe94394275a4e74454b380645a3a91dfb8c1 Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 22:55:57 +0330 Subject: [PATCH 10/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index d4f3ae78..f3c8d4f5 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -235,10 +235,9 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. - -
- $U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big]$ -
+$U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big]$ + +
From 831d2de840cacc2c9f538327e6f0beab31d498af Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 23:01:11 +0330 Subject: [PATCH 11/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index f3c8d4f5..223de09a 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -235,7 +235,9 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. -$U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big]$ +$$ +U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big] +$$
From 57f3db25368599d93975a5a415803a2f9cf7353c Mon Sep 17 00:00:00 2001 From: aboots Date: Fri, 7 Jan 2022 23:18:12 +0330 Subject: [PATCH 12/51] feat: add notebook to index --- notebooks/index.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/index.yml b/notebooks/index.yml index 374032cc..43a61232 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -52,7 +52,7 @@ notebooks: kind: S2021, LN, Notebook - notebook: notebooks/16_deep_neural_networks/ kind: S2021, LN, Notebook - - md: notebooks/20_markov_decision_processes_part_2/ + - md: notebooks/20_markov_decision_processes_part_2/index.md kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ From 156b78193ebfa54d2bab9b6db42c31be5bb31113 Mon Sep 17 00:00:00 2001 From: Vahid Zehtab <33608325+vahidzee@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:31:27 +0330 Subject: [PATCH 13/51] Fix: 20_mdp_p2 address + metadata --- notebooks/20_markov_decision_processes_part_2/metadata.yml | 2 +- notebooks/index.yml | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/metadata.yml b/notebooks/20_markov_decision_processes_part_2/metadata.yml index d55c251f..9ad1e13a 100644 --- a/notebooks/20_markov_decision_processes_part_2/metadata.yml +++ b/notebooks/20_markov_decision_processes_part_2/metadata.yml @@ -32,5 +32,5 @@ authors: contact: - link: https://github.com/invisible0831 icon: fab fa-github - - link: mailto:amirsoli80@gmail.com + - link: mailto:amirsoli80@gmail.com icon: fas fa-envelope diff --git a/notebooks/index.yml b/notebooks/index.yml index 43a61232..05c80fbd 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -52,7 +52,7 @@ notebooks: kind: S2021, LN, Notebook - notebook: notebooks/16_deep_neural_networks/ kind: S2021, LN, Notebook - - md: notebooks/20_markov_decision_processes_part_2/index.md + - md: notebooks/20_markov_decision_processes_part_2 kind: S2021, LN, Notebook #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ From 28cb296b2656cf5c58bba21898d54f3da7a5ee5c Mon Sep 17 00:00:00 2001 From: Vahid Zehtab <33608325+vahidzee@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:49:45 +0330 Subject: [PATCH 14/51] Fix: publish_dir --- .github/workflows/main.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 7e8ef31a..4bf550f5 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true - publish_dir: . + publish_dir: webified exclude_assets: '.github' From eb8f4628ca3bf403c591a8f72acddbdf8076ee36 Mon Sep 17 00:00:00 2001 From: Vahid Zehtab <33608325+vahidzee@users.noreply.github.com> Date: Sat, 8 Jan 2022 00:55:05 +0330 Subject: [PATCH 15/51] Fix: metadata people --- notebooks/20_markov_decision_processes_part_2/metadata.yml | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/20_markov_decision_processes_part_2/metadata.yml b/notebooks/20_markov_decision_processes_part_2/metadata.yml index 9ad1e13a..059ac5d0 100644 --- a/notebooks/20_markov_decision_processes_part_2/metadata.yml +++ b/notebooks/20_markov_decision_processes_part_2/metadata.yml @@ -7,6 +7,7 @@ header: authors: label: position: top + kind: people content: - name: Mohammad Mahdi Abootorabi role: Author From f30c70f7b47ec66743cb8b0d71e1f5ef09f32678 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 01:23:14 +0330 Subject: [PATCH 16/51] feat: changes in yml files --- .github/workflows/main.yml | 2 +- .../authors_images/amirreza_soleimanbeigi.jpg | Bin 79621 -> 0 bytes .../mohammad_mahdi_abootorabi.jpg | Bin 53290 -> 0 bytes .../authors_images/yalda_shabanzadeh.jpg | Bin 25798 -> 0 bytes .../metadata.yml | 3 --- notebooks/index.yml | 2 +- 6 files changed, 2 insertions(+), 5 deletions(-) delete mode 100644 notebooks/20_markov_decision_processes_part_2/authors_images/amirreza_soleimanbeigi.jpg delete mode 100644 notebooks/20_markov_decision_processes_part_2/authors_images/mohammad_mahdi_abootorabi.jpg delete mode 100644 notebooks/20_markov_decision_processes_part_2/authors_images/yalda_shabanzadeh.jpg diff --git a/.github/workflows/main.yml b/.github/workflows/main.yml index 4bf550f5..70726084 100644 --- a/.github/workflows/main.yml +++ b/.github/workflows/main.yml @@ -24,5 +24,5 @@ jobs: github_token: ${{ secrets.GITHUB_TOKEN }} enable_jekyll: true allow_empty_commit: true - publish_dir: webified + publish_dir: './webified/' exclude_assets: '.github' diff --git a/notebooks/20_markov_decision_processes_part_2/authors_images/amirreza_soleimanbeigi.jpg b/notebooks/20_markov_decision_processes_part_2/authors_images/amirreza_soleimanbeigi.jpg deleted file mode 100644 index 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?u*%)4%EfVWRZpDqpW_buadY+8 z|H2;-MEU={3h@8nkBwgzs71;CUmv&>v7-Q_=kU*Szi3B1?VQEsk*3c2nZbU*zoEII-T%l*P##rZ#V37#U2Gco#Eze zt(oNpRWeIi`7+hANl4y;RV7ko&WzZGd=^hrjH0X^n@8FE=LK(!q+V6`09iK=2-KlS z<@x#}DR$zQSk)Zz*il5MK1WblyYcw}id$QoZbLU=VEu^nn6t*!k@;u)@Fzj4n%#%H zp`RzQci-hA*&h|7wWgmQKN+95+XG*I3nD&Y$4XNC398?|{Ie#byZG}IWdrdKnN9Lr z!$Vh4?-AwKI*A?iEh6%aC^RGSE55NTlg^iIq9h}!C zC-x@>O^K!xKM^eL$pNyDhZ?5;gYE@&wxhC*ojFQoXDfw)ug*99(r;5#3UV4Y3F)p) zOh%+HW6Not)H0G}?EKOuCMLOFICx&IqCZM=sE_dCWU(8fX~dN`Pb;Io^VAF^a1ZK7 zqa@~8tdG3#P3jJF22x2cT6vt3c3V*_!9q8`WAD=HyYys*v%U;ex)ZRQA2_)Bh=$m8 z=Kr-vyfO9VLy>eh2}y^JThM<{>0SZjbQ@F$Yn=g&5nnnaWb^UgUDCNHe?m#lPzBb` z_sLl@6%7)BeyBabo`oxR=jP0Ku1&-v35i z5GymktQHKRV+{sEL9Wh=vA+JOngs+8nu*6{4np92h29JE|x#yO2%`Mlh z2nuVZ$YhCpzLwWxS)6Bd;|p8oViob@#?xysUug#26 zzx7k@w|9<~z|7A?f3W`UexzQw|GOIB^%o$pvKjMwF?O;lsJo`if05VEp{9aPG18fv s&2;g~?-EU4)b9fm-{!CNO+6obp8a<2B$x!ezjaIQN1x`b=3nyv0pTtw&Hw-a diff --git a/notebooks/20_markov_decision_processes_part_2/metadata.yml b/notebooks/20_markov_decision_processes_part_2/metadata.yml index 059ac5d0..89372b46 100644 --- a/notebooks/20_markov_decision_processes_part_2/metadata.yml +++ b/notebooks/20_markov_decision_processes_part_2/metadata.yml @@ -11,7 +11,6 @@ authors: content: - name: Mohammad Mahdi Abootorabi role: Author - image: authors_images/mohammad_mahdi_abootorabi.jpg contact: - link: https://github.com/aboots icon: fab fa-github @@ -20,7 +19,6 @@ authors: - name: Yalda Shabanzadeh role: Author - image: authors_images/yalda_shabanzadeh.jpg contact: - link: https://github.com/yaldashbz icon: fab fa-github @@ -29,7 +27,6 @@ authors: - name: Amirreza Soleimanbeigi role: Author - image: authors_images/amirreza_soleimanbeigi.jpg contact: - link: https://github.com/invisible0831 icon: fab fa-github diff --git a/notebooks/index.yml b/notebooks/index.yml index 05c80fbd..fc771195 100644 --- a/notebooks/index.yml +++ b/notebooks/index.yml @@ -53,7 +53,7 @@ notebooks: - notebook: notebooks/16_deep_neural_networks/ kind: S2021, LN, Notebook - md: notebooks/20_markov_decision_processes_part_2 - kind: S2021, LN, Notebook + kind: F2021, LN #- notebook: notebooks/17_markov_decision_processes/ - notebook: notebooks/18_reinforcement_learning/ kind: S2021, LN, Notebook \ No newline at end of file From a5878bef331ec42c6eded12d269c31721359e34a Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 01:36:08 +0330 Subject: [PATCH 17/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 223de09a..6ae5a096 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -1,4 +1,3 @@ -

Markov Decision (from value iteration) @@ -65,7 +64,7 @@ where the update is assumed to be applied simultaneously to all the states at ea
-Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{*} = B U^{*}$ that $U^{*}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$. +Suppose we view the Bellman update as an operator $B:\mathbb{R}^{\lvert S\lvert}\rightarrow \mathbb{R}^{\lvert S\lvert}$ that maps functions to functions. Then as a result of Bellman equation $U^{\star} = B U^{\star}$ that $U^{\star}$ is the optimal utility function and the Bellman update equation can be written as $U_{i+1} = B U_{i}$.
From 03799e95a2675d2c4baccc9df14537d44c8fae5a Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 03:05:33 +0330 Subject: [PATCH 18/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 6ae5a096..5af0e9bf 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -76,9 +76,9 @@ $$ \forall x,y \in \mathbb{R}^{n}, \quad \Vert f(x)-f(y)\Vert \leq \alpha \Vert x-y \Vert $$ The **Banach fixed-point** theorem guarantees that if $f$ is a contraction, then $f$ has a unique -fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^* \in \mathbb{R}^{n}$ satisfying +fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^\star \in \mathbb{R}^{n}$ satisfying $$ -x^* = f(x^*) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R}^{n} +x^\star = f(x^\star) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R}^{n} $$ where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. From bd3dfcd74e39aaf662279d17aea36e028dc6d0e7 Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Sat, 8 Jan 2022 12:28:20 +0330 Subject: [PATCH 19/51] fix: add starts and enter --- notebooks/20_markov_decision_processes_part_2/index.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 5af0e9bf..b703be54 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -262,11 +262,12 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( >

Computing Actions from Values

-Assume we have the optimal values $U^*(s)$. +Assume we have the optimal values $U^\star(s)$. We need to do a mini-expectimax. -$$\pi^*(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$$ +$$\pi^\star(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\star}(s') \Big]$$ + This is called policy extraction, since it gets the policy implied by the values @@ -281,7 +282,7 @@ This is kind of one-step expectimax with assumption that terminal values $U^\pi_ The action is completely trivial. -$$\pi^*(s)=\underset{a}{\operatorname{argmax}} Q^*(s,a)$$ +$$\pi^\star(s)=\underset{a}{\operatorname{argmax}} Q^\star(s,a)$$ **Comparison:** @@ -320,6 +321,7 @@ $$U_{k+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi_{i}(s))} \Big $${\pi}_{i+1}(s)= \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ + - **Repeat steps until policy converges** The algorithm terminates when the policy improvement step yields no change in the utilities. From 4f5c0dc7acc9447000ce0219c46addfedf58ef4f Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Sat, 8 Jan 2022 12:48:20 +0330 Subject: [PATCH 20/51] fix: policy improvement underscore --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index b703be54..20855f7b 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -319,7 +319,7 @@ $$U_{k+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi_{i}(s))} \Big For fixed values, get a better policy using policy extraction: -$${\pi}_{i+1}(s)= \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ +$$\pi_{i+1}(s) = \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ - **Repeat steps until policy converges** From 17ebe6d69de720acbed3b5df838f8efe5b60ef18 Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Sat, 8 Jan 2022 12:59:02 +0330 Subject: [PATCH 21/51] fix: policy pseudocode --- .../index.md | 20 +++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 20855f7b..229126af 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -342,16 +342,16 @@ When algorithm stops changing utility, because we know that the utility function
-**function** Q-Value(MDP, $s, a, U$) **returns** a utility value
-  $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$ +**function** Q-Value($MDP, s, a, U$) **returns** a utility value
+  **return** $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$
**function** Policy-Evaluation($\pi, U$, MDP) **returns** a utility function
  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$, -          rewards $R(s,a,s^{\prime})$, discount $\gamma$ -          $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ -          $\delta$, the maximum relative change in the utility of any state +       rewards $R(s,a,s^{\prime})$, discount $\gamma$ +       $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ +       $\delta$, the maximum relative change in the utility of any state   **repeat**       $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$      **for each** state s **in** S **do** @@ -362,17 +362,17 @@ When algorithm stops changing utility, because we know that the utility function
-**function** Policy-Iteration(MDP) **returns** a utility function +**function** Policy-Iteration($MDP$) **returns** a utility function   **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,         $U$ , a vector of utilities for states in $S$, initially zero -               $\pi$, a policy vector indexed by state, initially random +       $\pi$, a policy vector indexed by state, initially random   **repeat**     $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP)     $unchanged? \leftarrow$ true     **for each** state s **in** S **do**
-      $a^* \leftarrow \underset{a \in A(s)}{\operatorname{argmax}}$ Q-Value(MDP, $s, a, U$)
-      **if**   $\max\limits_{a \in A(s)}$ Q-Value(MDP, $s, a^*, U$) $>$ Q-Value(MDP, $s, \pi[s], U$)   **then**   **do** -        $\pi[s] \leftarrow a^*$ +      $a^\star \leftarrow \underset{a \in A(s)}{\operatorname{argmax}}$ Q-Value($MDP, s, a, U$)
+      **if**   $\max\limits_{a \in A(s)}$ Q-Value($MDP, s, a^\star, U$) $>$ Q-Value($MDP, s, \pi[s], U$)   **then**   **do** +        $\pi[s] \leftarrow a^\star$         $unchanged? \leftarrow$ false   **until** $unchanged?$   **return** $\pi$ From 9413ce62ab2a70e24cc4bbaf9e2db8418adfd466 Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Sat, 8 Jan 2022 13:03:46 +0330 Subject: [PATCH 22/51] fix: policy pseudocode --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 229126af..c57496ae 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -347,7 +347,7 @@ When algorithm stops changing utility, because we know that the utility function
-**function** Policy-Evaluation($\pi, U$, MDP) **returns** a utility function
+**function** Policy-Evaluation($\pi, U, MDP$) **returns** a utility function
  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,        rewards $R(s,a,s^{\prime})$, discount $\gamma$        $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ From a8bf94a5380896aecf17143f47567af7f97bafd5 Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Sat, 8 Jan 2022 13:09:28 +0330 Subject: [PATCH 23/51] fix: beautify policy pseudocode --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index c57496ae..89eef07c 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -365,7 +365,7 @@ When algorithm stops changing utility, because we know that the utility function **function** Policy-Iteration($MDP$) **returns** a utility function   **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,         $U$ , a vector of utilities for states in $S$, initially zero -       $\pi$, a policy vector indexed by state, initially random +        $\pi$, a policy vector indexed by state, initially random   **repeat**     $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP)     $unchanged? \leftarrow$ true From c84cc1ea7f0588b1f8a28f55b0f4b2fbbfbefa4e Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:19:08 +0330 Subject: [PATCH 24/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 89eef07c..7a1c9e83 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -99,14 +99,14 @@ $$ $$ To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then -$$ +\begin{equation} \begin{aligned} \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\ & \leq f(b) - g(b) \\ & \leq \max_x (f(x) - g(x)) = \max_{x} \lvert f(x) - g(x) \lvert\\ \Rightarrow \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & \leq \max_{x} \lvert f(x) - g(x) \lvert \end{aligned} -$$ +\end{equation} Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have From 785952b7dface40d76e9dbe63b5d2c6150e0d2be Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:23:22 +0330 Subject: [PATCH 25/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 7a1c9e83..58aa4afd 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -99,6 +99,7 @@ $$ $$ To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then +$$ \begin{equation} \begin{aligned} \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\ @@ -107,6 +108,7 @@ To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetri \Rightarrow \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & \leq \max_{x} \lvert f(x) - g(x) \lvert \end{aligned} \end{equation} +$$ Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have From 84559a3b4a499b68cfb60bbc663ad5baad245173 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Sat, 8 Jan 2022 13:34:37 +0330 Subject: [PATCH 26/51] Update index.md --- .../index.md | 99 +++++++++---------- 1 file changed, 49 insertions(+), 50 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 58aa4afd..b80b91ef 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -76,7 +76,8 @@ $$ \forall x,y \in \mathbb{R}^{n}, \quad \Vert f(x)-f(y)\Vert \leq \alpha \Vert x-y \Vert $$ The **Banach fixed-point** theorem guarantees that if $f$ is a contraction, then $f$ has a unique -fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^\star \in \mathbb{R}^{n}$ satisfying +fixed point($x$ is a fixed point of the $f$ if $f(x) = x$) $x^\star \in \mathbb{R}^{n}$ satisfying + $$ x^\star = f(x^\star) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R}^{n} $$ @@ -100,47 +101,46 @@ $$ To see this, suppose $\max_{x} f(x) > \max_{x} g(x)$ (the other case is symmetric) and let $b = \underset{x}{\operatorname{argmax}} f(x)$. Then $$ -\begin{equation} \begin{aligned} -\lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\ -& \leq f(b) - g(b) \\ -& \leq \max_x (f(x) - g(x)) = \max_{x} \lvert f(x) - g(x) \lvert\\ +\lvert \max_{x} f(x) - \max_{x} g(x) \lvert & = f(b) - \max_{x} g(x) \\\\ +& \leq f(b) - g(b) \\\\ +& \leq \max_x (f(x) - g(x)) = \max_{x} \lvert f(x) - g(x) \lvert\\\\ \Rightarrow \lvert \max_{x} f(x) - \max_{x} g(x) \lvert & \leq \max_{x} \lvert f(x) - g(x) \lvert \end{aligned} -\end{equation} $$ Let $U_i$ and $U^{\prime}_{i}$ be any two utility functions. Then we have $$ \begin{aligned} -\Vert B U_i - B U^{\prime}_{i}\Vert & = \max_{s} \lvert B U_{i}(s) - B U^{\prime}_{i}(s) \lvert \\ -& = \max_{s} \Big \lvert \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U^{\prime}_{i}(s^{\prime})] \Big \lvert \\ -& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U^{\prime}_{i}(s^{\prime})] \Big \lvert & (Lemma)\\ -& = \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \sum_{s^\prime}^{} P(s^{\prime}|s,a)[U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime})] \Big \lvert \\ -& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \Big (\sum_{s^\prime}^{} (P(s^{\prime}|s,a)\Big ) \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big )\Big \lvert \\ -& = \Big \lvert \gamma \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big )\Big \lvert \\ -& \leq \gamma \max_{s^\prime} \Big \lvert U_{i}(s^{\prime}) - U^{\prime}_{i}(s^{\prime}) \Big \lvert \\ -\Rightarrow \Vert B U_i - B U^{\prime}_{i}\Vert & \leq \gamma \Big \Vert U_{i} - U^{\prime}_{i}\Big \Vert +\Vert B U_i - B U_{i}^{\prime}\Vert & = \max_{s} \lvert B U_{i}(s) - B U_{i}^{\prime}(s) \lvert \\\\ +& = \max_{s} \Big \lvert \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \max_{a \in A(s)} \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}^{\prime}(s^{\prime})] \Big \lvert \\\\ +& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}(s^{\prime})] - \sum_{s^\prime}^{} P(s^{\prime}|s,a)[R(s,a,s^{\prime})+\gamma U_{i}^{\prime}(s^{\prime})] \Big \lvert & (Lemma)\\\\ +& = \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \sum_{s^\prime}^{} P(s^{\prime}|s,a)[U_{i}(s^{\prime}) - U_{i}^{\prime}(s^{\prime})] \Big \lvert \\\\ +& \leq \max_{s} \max_{a \in A(s)} \Big \lvert \gamma \Big (\sum_{s^\prime}^{} (P(s^{\prime}|s,a)\Big ) \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U_{i}^{\prime}(s^{\prime}) \Big )\Big \lvert \\\\ +& = \Big \lvert \gamma \max_{s^\prime} \Big ( U_{i}(s^{\prime}) - U_{i}^{\prime}(s^{\prime}) \Big )\Big \lvert \\\\ +& \leq \gamma \max_{s^\prime} \Big \lvert U_{i}(s^{\prime}) - U_{i}^{\prime}(s^{\prime}) \Big \lvert \\\\ +\Rightarrow \Vert B U_i - B U_{i}^{\prime}\Vert & \leq \gamma \Big \Vert U_{i} - U_{i}^{\prime}\Big \Vert \end{aligned} $$ -That is, the Bellman operator is $\gamma -contraction$ on the space of utility functions. The fixed point of the Bellman operator is $U^*$. Hence, from the properties of contractions in general, it follows that value iteration always converges to $U^*$ whenever $\gamma < 1$. +That is, the Bellman operator is $\gamma -contraction$ on the space of utility functions. The fixed point of the Bellman operator is $U^{\star}$. Hence, from the properties of contractions in general, it follows that value iteration always converges to $U^{\star}$ whenever $\gamma < 1$.
>

Error Bound

-We can't have $\infty$ iterations to converges to $U^*$. If we view $\Vert U_{i+1} - U^* \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert \leq \delta$ we have +We can't have $\infty$ iterations to converges to $U^{\star}$. If we view $\Vert U_{i+1} - U^{\star} \Vert$ as the error, we want to relate a bound for error to a bound for $\Vert U_{i+1} - U_{i} \Vert$. If $\Vert U_{i+1} - U_{i} \Vert \leq \delta$ we have + $$ \begin{aligned} -\Vert U_{i+1} - U^* \Vert & = \max_{s} \lvert U_{i+1}(s) - U^{*}(s) \lvert \\ -& = \max_{s} \Big \lvert (U_{i+1}(s) - U_{i+2}(s)) + (U_{i+2}(s) - U_{i+3}(s)) + ...\Big \lvert \\ -& \leq \max_{s} \Big ( \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \Big ) \\ -& \leq \max_{s} \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \max_{s} \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \\ -& = \Vert B U_{i} - B U_{i+1}\Vert + \Vert B U_{i+1} - B U_{i+2}\Vert + ... \\ -& \leq \gamma \Vert U_{i} - U_{i+1}\Vert + \gamma^{2} \Vert U_{i} - U_{i+1}\Vert + ... \\ -& = \frac{\gamma}{1 - \gamma} \Vert U_{i} - U_{i+1}\Vert & \text{for } \gamma < 1\\ +\Vert U_{i+1} - U^{\star} \Vert & = \max_{s} \lvert U_{i+1}(s) - U^{\star}(s) \lvert \\\\ +& = \max_{s} \Big \lvert (U_{i+1}(s) - U_{i+2}(s)) + (U_{i+2}(s) - U_{i+3}(s)) + ...\Big \lvert \\\\ +& \leq \max_{s} \Big ( \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \Big ) \\\\ +& \leq \max_{s} \lvert U_{i+1}(s) - U_{i+2}(s)\lvert + \max_{s} \lvert U_{i+2}(s) - U_{i+3}(s)\lvert + ... \\\\ +& = \Vert B U_{i} - B U_{i+1}\Vert + \Vert B U_{i+1} - B U_{i+2}\Vert + ... \\\\ +& \leq \gamma \Vert U_{i} - U_{i+1}\Vert + \gamma^{2} \Vert U_{i} - U_{i+1}\Vert + ... \\\\ +& = \frac{\gamma}{1 - \gamma} \Vert U_{i} - U_{i+1}\Vert & \text{for } \gamma < 1\\\\ & \leq \frac{\gamma}{1 - \gamma} \delta \end{aligned} $$ @@ -170,11 +170,12 @@ Thus, if $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} - >

Time Complexity

-We want to calculate the number of iterations required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have +We want to calculate the number of iterations required to reach an error of at most $\epsilon$. Suppose that the rewards are bounded by $\pm R_{max}$. if $\gamma < 1$ we have + $$ \begin{aligned} -U(s0,a0,s1,...) & = \sum_{t=0}^{\infty} \gamma^{t} R(s_{t},a_{t},s_{t+1}) \\ -& \leq \sum_{t=0}^{\infty} \gamma^{t} R_{max} \\ +U(s0,a0,s1,...) & = \sum_{t=0}^{\infty} \gamma^{t} R(s_{t},a_{t},s_{t+1}) \\\\ +& \leq \sum_{t=0}^{\infty} \gamma^{t} R_{max} \\\\ U(s) & \leq \frac{R_{max}}{1-\gamma} \end{aligned} $$ @@ -182,17 +183,17 @@ $$ If we run for $N$ iterations, we have $$ \begin{aligned} -\Vert U_{N} - U^* \Vert & = \Vert B U_{N-1} - B U^* \Vert \\ -& \leq \gamma \Vert U_{N-1} - U^* \Vert \\ -& = \gamma \Vert B U_{N-2} - B U^* \Vert \\ -& \leq \gamma^{2} \Vert U_{N-2} - U^* \Vert \\ -& \leq \gamma^{N} \Vert U_{0} - U^* \Vert \\ -& = \gamma^{N} \Vert U^* \Vert & \forall s \text{ } U_{0}(s) = 0 \\ +\Vert U_{N} - U^* \Vert & = \Vert B U_{N-1} - B U^{\star} \Vert \\\\ +& \leq \gamma \Vert U_{N-1} - U^{\star} \Vert \\\\ +& = \gamma \Vert B U_{N-2} - B U^{\star} \Vert \\\\ +& \leq \gamma^{2} \Vert U_{N-2} - U^{\star} \Vert \\\\ +& \leq \gamma^{N} \Vert U_{0} - U^{\star} \Vert \\\\ +& = \gamma^{N} \Vert U^{\star} \Vert & \forall s \text{ } U_{0}(s) = 0 \\\\ & \leq \gamma^{N} \frac{R_{max}}{1-\gamma} \end{aligned} $$ -if $\gamma^{N} \frac{R_{max}}{1-\gamma} \leq \epsilon$, then $\Vert U_{N} - U^* \Vert \leq \epsilon$. Thus +if $\gamma^{N} \frac{R_{max}}{1-\gamma} \leq \epsilon$, then $\Vert U_{N} - U^{\star} \Vert \leq \epsilon$. Thus $$ N = \Big \lceil \frac{\log{(\frac{R_{max}}{\epsilon (1-\gamma)})}}{\log{(\frac{1}{\gamma})}} \Big \rceil $$ @@ -264,12 +265,11 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( >

Computing Actions from Values

-Assume we have the optimal values $U^\star(s)$. +Assume we have the optimal values $U^{\star}(s)$. We need to do a mini-expectimax. -$$\pi^\star(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\star}(s') \Big]$$ - +$$\pi^{\star}(s)=\underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\star}(s') \Big]$$ This is called policy extraction, since it gets the policy implied by the values @@ -284,7 +284,7 @@ This is kind of one-step expectimax with assumption that terminal values $U^\pi_ The action is completely trivial. -$$\pi^\star(s)=\underset{a}{\operatorname{argmax}} Q^\star(s,a)$$ +$$\pi^{\star}(s)=\underset{a}{\operatorname{argmax}} Q^{\star}(s,a)$$ **Comparison:** @@ -321,8 +321,7 @@ $$U_{k+1}^{\pi_{i}}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi_{i}(s))} \Big For fixed values, get a better policy using policy extraction: -$$\pi_{i+1}(s) = \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ - +$$\pi_{i+1}(s)= \underset{a}{\operatorname{argmax}} \Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\pi_{i}}(s') \Big]$$ - **Repeat steps until policy converges** @@ -344,16 +343,16 @@ When algorithm stops changing utility, because we know that the utility function
-**function** Q-Value($MDP, s, a, U$) **returns** a utility value
-  **return** $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{*}(s') \Big]$ +**function** Q-Value(MDP, $s, a, U$) **returns** a utility value
+  $\Sigma_{s'} {P(s^{\prime}|s,a)} \Big [R(s,a,s') + {\gamma}U^{\star}(s') \Big]$
-**function** Policy-Evaluation($\pi, U, MDP$) **returns** a utility function
+**function** Policy-Evaluation($\pi, U$, MDP) **returns** a utility function
  **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$, -       rewards $R(s,a,s^{\prime})$, discount $\gamma$ -       $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ -       $\delta$, the maximum relative change in the utility of any state +          rewards $R(s,a,s^{\prime})$, discount $\gamma$ +          $U$, $U^{\prime}$, utility functions for states in S with policy $\pi$ +          $\delta$, the maximum relative change in the utility of any state   **repeat**       $U \leftarrow U^{\prime}$;$\delta \leftarrow 0$      **for each** state s **in** S **do** @@ -364,17 +363,17 @@ When algorithm stops changing utility, because we know that the utility function
-**function** Policy-Iteration($MDP$) **returns** a utility function +**function** Policy-Iteration(MDP) **returns** a utility function   **inputs**: MDP, an MDP with states $S$ , actions $A(s)$, transition model $P(s^{\prime}|s,a)$,         $U$ , a vector of utilities for states in $S$, initially zero -        $\pi$, a policy vector indexed by state, initially random +               $\pi$, a policy vector indexed by state, initially random   **repeat**     $U \leftarrow $ Policy-Evaluation($\pi$, $U$, MDP)     $unchanged? \leftarrow$ true     **for each** state s **in** S **do**
-      $a^\star \leftarrow \underset{a \in A(s)}{\operatorname{argmax}}$ Q-Value($MDP, s, a, U$)
-      **if**   $\max\limits_{a \in A(s)}$ Q-Value($MDP, s, a^\star, U$) $>$ Q-Value($MDP, s, \pi[s], U$)   **then**   **do** -        $\pi[s] \leftarrow a^\star$ +      $a^{\star} \leftarrow \underset{a \in A(s)}{\operatorname{argmax}}$ Q-Value(MDP, $s, a, U$)
+      **if**   $\max\limits_{a \in A(s)}$ Q-Value(MDP, $s, a^{\star}, U$) $>$ Q-Value(MDP, $s, \pi[s], U$)   **then**   **do** +        $\pi[s] \leftarrow a^{\star}$         $unchanged? \leftarrow$ false   **until** $unchanged?$   **return** $\pi$ From f0c753f462c3ccca47c94fd6499389ffcfa593ae Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:42:24 +0330 Subject: [PATCH 27/51] feat: update index.md references --- notebooks/20_markov_decision_processes_part_2/index.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index b80b91ef..642a71b7 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -414,8 +414,8 @@ In the next part, you will face reinforcement learning. The biggest change in RL

References

+ AI course teached by Dr. Rohban at Sharif University of Technology, Fall 2021 -+ Russell, S. J., Norvig, P., & Davis, E. (4rd. Ed). Artificial Intelligence: A modern approach. Pearson Educación. -+ https://www.geeksforgeeks.org -+ http://ai.berkeley.edu -+ https://towardsdatascience.com -+ https://ai.stanford.edu/~gwthomas/notes/mdps.pdf ++ Russell, S. J., Norvig, P., & Davis, E. (4th. Ed). Artificial Intelligence: A modern approach. Pearson Educación. ++ [Geeks For Geeks](https://www.geeksforgeeks.org) ++ [Towards Data Science](https://towardsdatascience.com) ++ [ai.stanford.edu](https://ai.stanford.edu/~gwthomas/notes/mdps.pdf) ++ [ai.berkeley.edu](http://ai.berkeley.edu) From e5e839fd9cdabdee36eac96480a93fa8988e26d3 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:47:09 +0330 Subject: [PATCH 28/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 642a71b7..7fd10239 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -217,7 +217,9 @@ If one action is clearly better than all others, then the exact magnitude of the This insight suggests an alternative way to find optimal policies. Policy iteration is a different approach to find the optimal policy for given states and actions. +

The idea of policy iteration

+
Once a policy, $\pi_0$ (could be initialized random), has been improved using $U^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $U^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. From 6b15f331dede9bcfc13c93abd7b05e99cca9aab3 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:50:14 +0330 Subject: [PATCH 29/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 7fd10239..cbf5965c 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -209,7 +209,7 @@ Time complexity of each iteration of value iteration is $O(\lvert S\rvert^{2} \l

Policy Iteration

- +---
In the previous section, we observed that it is possible to get an optimal policy even when the utility function estimate is inaccurate. From 4fd483823a4b4912cfe160a7aa317fc8c2b8bba3 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 13:53:58 +0330 Subject: [PATCH 30/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 8 ++++++-- 1 file changed, 6 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index cbf5965c..b3b9c802 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -9,6 +9,7 @@

Table of Contents

+--- - [Introduction](#introduction) - [Value Iteration](#value_iteration) @@ -32,6 +33,7 @@

Introduction

+---
@@ -43,6 +45,7 @@ In this lecture, we are going to talk about two different methods for solving MD

Value Iteration

+---
@@ -393,7 +396,7 @@ $O(\lvert S\rvert^{2} \lvert A\rvert)=O(\lvert S\rvert^{2})+O(\lvert S\rvert^{2}

Conclusion

- +---
These two methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. @@ -414,7 +417,8 @@ In the next part, you will face reinforcement learning. The biggest change in RL

References

- +--- +
+ AI course teached by Dr. Rohban at Sharif University of Technology, Fall 2021 + Russell, S. J., Norvig, P., & Davis, E. (4th. Ed). Artificial Intelligence: A modern approach. Pearson Educación. + [Geeks For Geeks](https://www.geeksforgeeks.org) From a0e320a55d863ac6af6d5728e32a87efa1930437 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 14:00:31 +0330 Subject: [PATCH 31/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index b3b9c802..9ff2342e 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -412,7 +412,7 @@ Each of them has its cons and pros. Depending on the situation you can choose ea In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. -
+

From d9391ba36c837e3fc8f1ec5d7d675867eb26d0be Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 14:07:36 +0330 Subject: [PATCH 32/51] feat: update index.md --- .../index.md | 25 ++++++++++++++++++- 1 file changed, 24 insertions(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 9ff2342e..007c7fe2 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -412,7 +412,30 @@ Each of them has its cons and pros. Depending on the situation you can choose ea In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. -
+ + + + + +
+ +
+ +
From ff4e6a8509a3eeae5412036a0ff6ab5195894c00 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 14:18:02 +0330 Subject: [PATCH 33/51] feat: update index.md --- .../index.md | 153 +++++++++++++++++- 1 file changed, 145 insertions(+), 8 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 007c7fe2..482f9cca 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -49,7 +49,30 @@ In this lecture, we are going to talk about two different methods for solving MD
-
+ + + + + +
+ +
+ + The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. @@ -203,7 +226,30 @@ $$ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value iteration converges exponentially fast but $N$ grows rapidly as $\gamma$ becomes close to 1. We can get fast convergence if we make $\gamma$ small. -
+ + + + + +
+ +
+ + Figure shows how $N$ varies with $\gamma$, for different values of the ratio $c = \frac{2\epsilon}{R_{max}}$. @@ -234,7 +280,30 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle
-
+ + + + + +
+ +
+ +
@@ -248,7 +317,30 @@ $$
-
+ + + + + +
+ +
+ + **Idea:** Calculate values for the fixed policy: @@ -264,7 +356,30 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(
-
+ + + + + +
+ +
+ +
@@ -304,7 +419,30 @@ for each state we compute all the calcaulations and take max on different action
-
+ + + + + +
+ +
+ + The policy iteration algorithm alternates the following two steps, beginning from some initial policy $\pi_0$ @@ -411,7 +549,6 @@ Each of them has its cons and pros. Depending on the situation you can choose ea In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. - From f26b1f233d0620533910b81a079168e2f97fce94 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 15:28:15 +0330 Subject: [PATCH 35/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 6 ++---- 1 file changed, 2 insertions(+), 4 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 91d7abb6..fb3c3d47 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -558,12 +558,12 @@ In the next part, you will face reinforcement learning. The biggest change in RL } @media (min-width: 450px) { .center-fit { - max-width: 300px; + max-width: 800px; max-height: 100vh; margin: auto; } } - @media (max-width: 450px) and min-width(0px) { + @media (max-width: 450px) and (min-width: 0px) { .center-fit { max-width: 100%; max-height: 100vh; @@ -583,8 +583,6 @@ In the next part, you will face reinforcement learning. The biggest change in RL -
-

References

---
From c351facba2d48e3a4160bdf26f76d1ed017b33a3 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 15:51:04 +0330 Subject: [PATCH 36/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index fb3c3d47..3b9bf83c 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -551,6 +551,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL + -
- -
+
+ +
From a8aaf4945a4ee9f5d648ba922572febf2936603e Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 16:26:19 +0330 Subject: [PATCH 38/51] feat: update index.md --- .../index.md | 268 +++++++++++------- 1 file changed, 170 insertions(+), 98 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index e29de624..88ef4c61 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -48,29 +48,40 @@ In this lecture, we are going to talk about two different methods for solving MD ---
- + -
- -
+
+ +
@@ -228,26 +239,38 @@ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value i + -
- -
+
+ +
@@ -279,29 +302,40 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle

Policy Evaluation


- + -
- -
+
+ +
@@ -319,29 +353,42 @@ $$ + -
- -
+
+ +
+ **Idea:** Calculate values for the fixed policy: $U_0^\pi=0$ @@ -358,29 +405,42 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( + -
- -
+
+ +
+
>

Computing Actions from Values

@@ -421,26 +481,38 @@ for each state we compute all the calcaulations and take max on different action + -
- -
+
+ +
From 336995f27628cf469f56eb567df7e7c8305b002d Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 16:39:42 +0330 Subject: [PATCH 39/51] feat: update index.md --- .../index.md | 32 +++++++++---------- 1 file changed, 16 insertions(+), 16 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 88ef4c61..29463ce5 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -57,12 +57,12 @@ In this lecture, we are going to talk about two different methods for solving MD padding: 0; } .center-fit { - max-width: 300px; + max-width: 400px; max-height: 100vh; margin: auto; } .imgbox { - height: 70%; + height: 40%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -72,7 +72,7 @@ In this lecture, we are going to talk about two different methods for solving MD margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -251,7 +251,7 @@ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value i margin: auto; } .imgbox { - height: 70%; + height: 40%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -261,7 +261,7 @@ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value i margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -316,7 +316,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle margin: auto; } .imgbox { - height: 70%; + height: 40%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -326,7 +326,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -360,12 +360,12 @@ $$ padding: 0; } .center-fit { - max-width: 300px; + max-width: 600px; max-height: 100vh; margin: auto; } .imgbox { - height: 70%; + height: 40%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -375,7 +375,7 @@ $$ margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -417,7 +417,7 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( margin: auto; } .imgbox { - height: 70%; + height: 50%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -427,7 +427,7 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -493,7 +493,7 @@ for each state we compute all the calcaulations and take max on different action margin: auto; } .imgbox { - height: 70%; + height: 50%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -503,7 +503,7 @@ for each state we compute all the calcaulations and take max on different action margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } @@ -635,7 +635,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL margin: auto; } .imgbox { - height: 70%; + height: 50%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -645,7 +645,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL margin: auto; } .imgbox { - height: 30%; + height: 40%; display: grid; } } From 7ed1fe733122f144e3ec9b2df1d60e581749c71c Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 16:44:00 +0330 Subject: [PATCH 40/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 29463ce5..ec52547c 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -62,7 +62,7 @@ In this lecture, we are going to talk about two different methods for solving MD margin: auto; } .imgbox { - height: 40%; + height: 50%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -72,7 +72,7 @@ In this lecture, we are going to talk about two different methods for solving MD margin: auto; } .imgbox { - height: 40%; + height: 50%; display: grid; } } From d91be80d6ed7b40fdca30698ad2950afbf1b29a8 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 16:49:11 +0330 Subject: [PATCH 41/51] feat: update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 ++ 1 file changed, 2 insertions(+) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index ec52547c..049d0e51 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -84,6 +84,7 @@ In this lecture, we are going to talk about two different methods for solving MD
+
The Bellman equation is the basis of the **value iteration** algorithm for solving MDPs. We would like to solve these simultaneous equations to find the utilities. One thing to try is an iterative approach. We start with arbitrary initial values for the utilities, calculate the right-hand side of the equation, and plug it into the left-hand side. Thereby updating the utility of each state from the utilities of its neighbors. We repeat this until we reach an equilibrium. @@ -657,6 +658,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL +

References

--- From 0e0f83bb99d720f3e26558fdf640977703571385 Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 17:21:40 +0330 Subject: [PATCH 42/51] feat: update index.md --- .../20_markov_decision_processes_part_2/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 049d0e51..7e4b6bf6 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -62,7 +62,7 @@ In this lecture, we are going to talk about two different methods for solving MD margin: auto; } .imgbox { - height: 50%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -252,7 +252,7 @@ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value i margin: auto; } .imgbox { - height: 40%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -317,7 +317,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle margin: auto; } .imgbox { - height: 40%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -366,7 +366,7 @@ $$ margin: auto; } .imgbox { - height: 40%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -418,7 +418,7 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( margin: auto; } .imgbox { - height: 50%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -494,7 +494,7 @@ for each state we compute all the calcaulations and take max on different action margin: auto; } .imgbox { - height: 50%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -636,7 +636,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL margin: auto; } .imgbox { - height: 50%; + height: 80%; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { From b157050d38521182969523974ce6a4747b7974ef Mon Sep 17 00:00:00 2001 From: aboots Date: Sat, 8 Jan 2022 17:27:58 +0330 Subject: [PATCH 43/51] feat: update index.md --- .../20_markov_decision_processes_part_2/index.md | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 7e4b6bf6..971ebe49 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -62,7 +62,7 @@ In this lecture, we are going to talk about two different methods for solving MD margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -252,7 +252,7 @@ $N$ does not depend much on the ratio $\frac{\epsilon}{R_{max}}$ because value i margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -317,7 +317,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -366,7 +366,7 @@ $$ margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -418,7 +418,7 @@ $U_{k+1}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi( margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -494,7 +494,7 @@ for each state we compute all the calcaulations and take max on different action margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { @@ -636,7 +636,7 @@ In the next part, you will face reinforcement learning. The biggest change in RL margin: auto; } .imgbox { - height: 80%; + height: 400px; display: grid; } @media only screen and (max-width: 450px) and (min-width: 0px) { From f0079bd1c37f967b222e5bd9831622beb5d34989 Mon Sep 17 00:00:00 2001 From: aboots Date: Wed, 19 Jan 2022 21:45:49 +0330 Subject: [PATCH 44/51] fix: english improvement in index.md --- .../index.md | 29 +++++++++---------- 1 file changed, 14 insertions(+), 15 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 971ebe49..83f9e39f 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -15,18 +15,18 @@ - [Value Iteration](#value_iteration) - [Convergence of Value Iteration](#vi_convergence) - [Contraction](#vi_cont) - - [Error Bound ](#vi_error) - - [Value Iteration pseudocode](#vi_code) + - [Error Bound](#vi_error) + - [Value Iteration Pseudocode](#vi_code) - [Time Complexity](#time_vi) - [Policy Iteration](#pi) - - [The idea of policy iteration](#pi_id) + - [The Idea of Policy Iteration](#pi_id) - [Policy Evaluation](#pe) - [Fixed Policy](#fp) - [Policy Extraction (Improvement)](#pex) - [Computing Actions from Values](#pi_cafv) - [Computing Actions from Q-Values](#pi_cafq) - [Policy Iteration Summary](#pis) - - [Policy Iteration pseudocode](#pi_code) + - [Policy Iteration Pseudocode](#pi_code) - [Conclusion](#conclusion) - [References](#references) @@ -37,7 +37,7 @@
-In the previous lecture, you became familiar with what MDP is. As you know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on his current state. It’s better to know MDP before Reinforcement Learning. +In the previous lecture, you became familiar with what MDP is. As you are aware, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on its current state. It is better to know MDP before Reinforcement Learning. In this lecture, we are going to talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us. @@ -186,7 +186,7 @@ Thus, if $\delta \leq \frac{1 - \gamma}{\gamma} \epsilon$, then $\Vert U_{i+1} -
-

Value Iteration pseudocode

+

Value Iteration Pseudocode


@@ -291,7 +291,7 @@ This insight suggests an alternative way to find optimal policies. Policy iteration is a different approach to find the optimal policy for given states and actions.
-

The idea of policy iteration

+

The Idea of Policy Iteration


Once a policy, $\pi_0$ (could be initialized random), has been improved using $U^{\pi_0}$ to yield a better policy, $\pi_1$, we can then compute $U^{\pi_1}$ and improve it again to yield an even better $\pi_2$. We can thus obtain a sequence of monotonically improving policies and value functions. @@ -344,8 +344,7 @@ Each policy is guaranteed to be a strict improvement over the previous one (unle >

Fixed Policy

-Back to expectimax tree, there was different actions for node s, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node s. This cause an important change in Bellman equation: there will be no need to take maximum in the equation. - +Back to the expectimax tree, there were different actions for node $s$, so we had different choices; but if the policy is fixed ($\pi(s)$), there is only one fixed action for the node $s$. This causes an important change in the Bellman equation: there will be no need to take maximum in the equation. $$ U_{k}^{\pi}(s) \leftarrow \Sigma_{s'} {P(s^{\prime}|s,\pi(s))} \Big [R(s,\pi(s),s') + {\gamma}U^{\pi}_{k}(s') \Big] $$ @@ -555,7 +554,7 @@ When algorithm stops changing utility, because we know that the utility function
-

Policy-Iteration pseudocode

+

Policy Iteration Pseudocode


@@ -610,15 +609,15 @@ $O(\lvert S\rvert^{2} \lvert A\rvert)=O(\lvert S\rvert^{2})+O(\lvert S\rvert^{2} ---
-These two methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. +Policy iteration and value iteration methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. -In policy iteration, you start with a random policy, then in several steps, you update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, you must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. You find a better policy using a one-step look-ahead. If the policy doesn’t change, it means you reach the optimal answer. +In policy iteration, we start with a random policy, then in several steps, we update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, we must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. we find a better policy using a one-step look-ahead. If the policy does not change, it means we reach the optimal answer. -In value iteration, you start with a random value function. In each step, you improve the values and (implicitly) the policy together. We don't care much about policy apparently, but taking maximum action will improve it as well. +In value iteration, we start with a random value function. In each step, we improve the values and (implicitly) the policy together. We don't care much about policy apparently, but taking maximum action will improve it as well. -The value iteration algorithm is more straightforward to understand as you only have to do one thing in each step. On the other hand, in practical terms, often policy iteration algorithm converges within fewer iterations and is much faster. Maximum selection in the value iteration is not here and it has a great impact on this fact. In theory, however, policy iteration must takes the same number of iterations as value iteration in the worst case. They differ only in whether we plug in a fixed policy or max over action +The value iteration algorithm is more straightforward to understand as we only have to do one thing in each step. On the other hand, in practical terms, often policy iteration algorithm converges within fewer iterations and is much faster. Maximum selection in the value iteration is not here and it has a great impact on this fact. In theory, however, policy iteration must takes the same number of iterations as value iteration in the worst case. They differ only in whether we plug in a fixed policy or max over action -Each of them has its cons and pros. Depending on the situation you can choose each of them, but policy iteration is more commonly used. +Each of them has its cons and pros. Depending on the situation, we can choose each of them, but policy iteration is more commonly used. In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. From f17bfe65d3e1a6166cd9174d2466389bf18caeeb Mon Sep 17 00:00:00 2001 From: aboots Date: Wed, 19 Jan 2022 21:52:43 +0330 Subject: [PATCH 45/51] fix: english improvement in index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 83f9e39f..33890b72 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -286,9 +286,9 @@ Time complexity of each iteration of value iteration is $O(\lvert S\rvert^{2} \l
In the previous section, we observed that it is possible to get an optimal policy even when the utility function estimate is inaccurate. -If one action is clearly better than all others, then the exact magnitude of the utilities on the states involved need not be precise. +If one action is clearly better than all others, then the exact magnitude of the utilities on the states involved does not need to be precise. This insight suggests an alternative way to find optimal policies. -Policy iteration is a different approach to find the optimal policy for given states and actions. +Policy iteration is a different approach to finding the optimal policy for given states and actions.

The Idea of Policy Iteration

From cac25e16a5f2ffaa76467126b688a7325d7aaa27 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Sun, 23 Jan 2022 19:31:00 +0330 Subject: [PATCH 46/51] Update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 33890b72..d5ffc057 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -121,7 +121,7 @@ x^\star = f(x^\star) = lim_{k \to \infty} f^{k}(x) \quad \forall x \in \mathbb{R $$ where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. -Hence, two important properties of contractions are +Hence, two important properties of contractions are - A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. - When the operator is applied to any argument, the output must get closer to the fixed point (because the fixed point does not move), so repeated application of a contraction always reaches the fixed point in the limit. From e5a5ca6efe07a671cec1828aec47f73f61d90339 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Sun, 23 Jan 2022 19:40:56 +0330 Subject: [PATCH 47/51] Update index.md --- notebooks/20_markov_decision_processes_part_2/index.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index d5ffc057..13ded080 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -122,7 +122,7 @@ $$ where $f^{k+1}(x) = f(f^{k}(x))$ for $k > 0$ and $f^{1}(x) = f(x)$. Hence, two important properties of contractions are -- A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. +- A contraction has only one fixed point; if there were two fixed points, they would not get closer together when the function was applied, so it would not be a contraction. - When the operator is applied to any argument, the output must get closer to the fixed point (because the fixed point does not move), so repeated application of a contraction always reaches the fixed point in the limit. We will use the **max norm** to measure distances between utility functions: From 65068476770f9142c56aa5fd339d0ffe088b6df8 Mon Sep 17 00:00:00 2001 From: yaldashbz Date: Thu, 27 Jan 2022 09:55:51 +0330 Subject: [PATCH 48/51] fix: english bug --- notebooks/20_markov_decision_processes_part_2/index.md | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 13ded080..ace6673d 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -286,9 +286,8 @@ Time complexity of each iteration of value iteration is $O(\lvert S\rvert^{2} \l
In the previous section, we observed that it is possible to get an optimal policy even when the utility function estimate is inaccurate. -If one action is clearly better than all others, then the exact magnitude of the utilities on the states involved does not need to be precise. -This insight suggests an alternative way to find optimal policies. -Policy iteration is a different approach to finding the optimal policy for given states and actions. +If one action is better than all others, then the exact magnitude of the utilities on the states involved does not need to be precise. +This insight suggests an alternative way to find optimal policies. Policy iteration is a different approach to finding the optimal policy for given states and actions.

The Idea of Policy Iteration

From 9fdb5a19940681a1c7a6587998d7d5df8d0a09e5 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Fri, 28 Jan 2022 23:34:22 +0330 Subject: [PATCH 49/51] change --- notebooks/20_markov_decision_processes_part_2/index.md | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index ace6673d..807e8664 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -37,9 +37,8 @@
-In the previous lecture, you became familiar with what MDP is. As you are aware, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on its current state. It is better to know MDP before Reinforcement Learning. - -In this lecture, we are going to talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us. +In the previous lecture, we became familiar with what MDP is. As we know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on its current state. It is better to know MDP before Reinforcement Learning. +In this lecture, we will talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us.
From df9406016223f7489982122ffb2bf9239682d775 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Fri, 28 Jan 2022 23:52:11 +0330 Subject: [PATCH 50/51] change conclusion --- .../20_markov_decision_processes_part_2/index.md | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 807e8664..35203c35 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -607,17 +607,17 @@ $O(\lvert S\rvert^{2} \lvert A\rvert)=O(\lvert S\rvert^{2})+O(\lvert S\rvert^{2} ---
-Policy iteration and value iteration methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As you see, in both you must use Bellman equations. +Policy iteration and value iteration methods are compared and summarized in this section. In general, both are kinds of dynamic programming algorithms and guarantee convergence. As we saw, in both, we must use Bellman equations. -In policy iteration, we start with a random policy, then in several steps, we update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, we must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. we find a better policy using a one-step look-ahead. If the policy does not change, it means we reach the optimal answer. +In policy iteration, we start with a random policy, and then in several steps, we update utilities with a fixed policy. We called this part policy evaluation. $O(|S|^ 2)$ time required per iteration. Next, we must go to the policy improvement phase. it takes $O(|S|^ 2|A|)$ time. We find a better policy using a one-step look-ahead. If the policy does not change, it means we reach the optimal answer. -In value iteration, we start with a random value function. In each step, we improve the values and (implicitly) the policy together. We don't care much about policy apparently, but taking maximum action will improve it as well. +In value iteration, we start with a random value function. In each step, we improve the values and (implicitly) the policy together. We do not care much about policy, apparently, but taking maximum action will improve it as well. -The value iteration algorithm is more straightforward to understand as we only have to do one thing in each step. On the other hand, in practical terms, often policy iteration algorithm converges within fewer iterations and is much faster. Maximum selection in the value iteration is not here and it has a great impact on this fact. In theory, however, policy iteration must takes the same number of iterations as value iteration in the worst case. They differ only in whether we plug in a fixed policy or max over action +The value iteration algorithm is more straightforward to understand as we only have to do one thing in each step. On the other hand, in practical terms, often, policy iteration algorithm converges within fewer iterations and is much faster. Maximum selection in the value iteration is not here, and it has a significant impact on this fact. In theory, however, policy iteration must take the same number of iterations as value iteration in the worst case. They differ only in whether we plug in a fixed policy or max over action. -Each of them has its cons and pros. Depending on the situation, we can choose each of them, but policy iteration is more commonly used. +Each of them has its cons and pros. We can choose each of them depending on the situation, but policy iteration is more commonly used. -In the next part, you will face reinforcement learning. The biggest change in RL is that we are unaware of R(s,a,s') and P(s'|s,a) and we have to do some actions to find or estimate them. +In the next part, we will face reinforcement learning. The most significant change in RL is that we are unaware of R(s,a,s') and P(s'|s,a), and we have to do some actions to find or estimate them. From 0ad1eab9e52cd4fb3ef423a30c05e24cdd4813d2 Mon Sep 17 00:00:00 2001 From: Amirreza Soleimanbeigi <67834187+invisible0831@users.noreply.github.com> Date: Fri, 28 Jan 2022 23:57:32 +0330 Subject: [PATCH 51/51] chage intro --- notebooks/20_markov_decision_processes_part_2/index.md | 1 + 1 file changed, 1 insertion(+) diff --git a/notebooks/20_markov_decision_processes_part_2/index.md b/notebooks/20_markov_decision_processes_part_2/index.md index 35203c35..5ce1c986 100644 --- a/notebooks/20_markov_decision_processes_part_2/index.md +++ b/notebooks/20_markov_decision_processes_part_2/index.md @@ -38,6 +38,7 @@
In the previous lecture, we became familiar with what MDP is. As we know, MDP can be remembered as a mathematical framework used for modeling decision-making problems. Usually, an agent can do some actions, but the outcomes are stochastic and not entirely controllable. The goal is to decide the best action to select based on its current state. It is better to know MDP before Reinforcement Learning. + In this lecture, we will talk about two different methods for solving MDPs. These two methods are value iteration and policy iteration. Stay with us.