Optimization Exercise

During this exercise, you will implement gradient descent to solve optimization problems. For all activities, the task is to find the lowest point on a function surface or in mathematical language

$$ \min_{x}f(x) .$$

All modules in src require your attention.

• To get started, take a look at src/optimize_1d.py. Use the gradient to find the minumum of the parabola starting from five, or in other words

$$ \min_{x} x^2, \text{ with } x_0 = 5 .$$

The problem is illustrated below:

• Next we consider a paraboloid, $\cdot$ denotes the scalar product,

$$ \min_{\mathbf{x}} \mathbf{x} \cdot \mathbf{x}, \text{ with } \mathbf{x_0} = (2.9, -2.9) .$$

The paraboloid is already implemented in src/optimize_2d.py. Your task is to solve this problem using two-dimensional gradient descent. Once more the problem is illustrated below:

• Additionally we consider a bumpy paraboloid, $\cdot$ denotes the scalar product,

$$ \min_{\mathbf{x}} \mathbf{x} \cdot \mathbf{x} + \cos(2 \pi x_0) + \sin(2 \pi x_1 ), \text{ with } \mathbf{x_0} = (2.9, -2.9) .$$

The addtional sin and cosine terms will require momentum for convergence. The bumpy paraboloid is already implemented in src/optimize_2d_momentum_bumpy.py. Your task is to solve this problem using two-dimensional gradient descent with momentum. Once more the problem is illustrated below:

• Finally, to explore the automatic differentiation functionality we consider the problem,

$$ \min_{\mathbf{x}} \mathbf{x} \cdot \mathbf{x} + \cos(2 \pi x_0 ) + \sin(2 \pi x_1) + 0.5 \cdot \text{relu}(x_0) + 10 \cdot \tanh( |\mathbf{x} | ), \text{ with } \mathbf{x_0} = (2.9, -2.9) .$$

The function is already defined in src/optimize_2d_momentum_bumpy_jax.py. We don't have to find the gradient by hand! Use jax.grad (jax-documentation) to compute the gradient automatically. Use the result to find the minimum using momentum.

While coding use nox -s test, nox -s lint, and nox -s typing to check your code. Autoformatting help is available via nox -s format. Feel free to read mode about nox at https://nox.thea.codes/en/stable/ .