Spring 2023

We want to minimize a __convex__, __continuous__ and __differentiable__ loss function \(\ell(w)\). In this section we discuss two of the most popular "hill-climbing" algorithms, gradient descent and Newton's method.

Initialize \(\mathbf{w}_0\)

Repeat until converge:

\(\mathbf{w}^{t+1}\) = \(\mathbf{w}^t\) + \(\mathbf{s}\)

If \(\|\mathbf{w}^{t+1}\) - \(\mathbf{w}^t\|_2\) < \(\epsilon\),

How can you minimize a function \(\ell\) if you don't know much about it? The trick is to assume it is much simpler than it really is. This can be done with Taylor's approximation. Provided that the norm \(\|\mathbf{s}\|_2\)is small (

\(\ell(\mathbf{w} + \mathbf{s})\) \(\approx\) \(\ell(\mathbf{w})\) + \(g(\mathbf{w})\) \(^\top\) \(\mathbf{s}\) + \(\frac{1}{2}\mathbf{s}^\top H(\mathbf{w})\mathbf{s}\)

Here, \(g(\mathbf{w})=\nabla\ell(\mathbf{w})\) is the gradient and \(H(\mathbf{w})=\nabla^{2}\ell(\mathbf{w})\) is the Hessian of \(\ell\). Both approximations are valid if \(\|\mathbf{s}\|_2\) is small, but the second one assumes that \(\ell\) is

In gradient descent we only use the gradient (first order). In other words, we assume that the function \(\ell\) around \(\mathbf{w}\) is linear and behaves like \(\ell(\mathbf{w}) + g(\mathbf{w})^\top\mathbf{s}\). Our goal is to find a vector \(\mathbf{s}\) that minimizes this function. In steepest descent we simply set

for some small \(\alpha\)>0. It is straight-forward to prove that in this case \(\ell(\mathbf{w}+\mathbf{s})<\ell(\mathbf{w})\).

Setting the

Initialize \(\mathbf{w}_0\) and \(\mathbf{z}\): \(\forall d\): \(w^0_d=0\) and \(z_d=0\)

Repeat until converge:

\(\mathbf{g}=\frac{\partial f(\mathbf{w})}{\partial \mathbf{w}}\) # Compute gradient

\(\forall d\): \(z_{d}\leftarrow z_{d}+g_{d}^2\)

\(\forall d\): \({w}_d^{t+1}\leftarrow {w}_d^t-\alpha \frac{{g}_d}{\sqrt{{z}_d+\epsilon}}\)

If \(\|\mathbf{w}^{t+1}\) - \(\mathbf{w}^t\|_2\) < \(\delta\),

Newton's method assumes that the loss \(\ell\) is __twice differentiable__ and uses the approximation with Hessian (2nd order Taylor approximation).
The **Hessian Matrix** contains all second order partial derivatives and is defined as

__Note__: A symmetric matrix \(\mathbf{M}\) is positive semi-definite if it has only non-negative eigenvalues or, equivalently, for any vector \(\mathbf{x}\) we must have \(\mathbf{x}^\top\mathbf{M}\mathbf{x}\geq 0\).

describes a convex parabola, and we can find its minimum by solving the following optimization problem:

To find the minimum of the objective, we take its first derivative with respect to \(\mathbf{s}\), equate it with zero, and solve for \(\mathbf{s}\): \begin{align} g(\mathbf{w}) + H(\mathbf{w})\mathbf{s}&=0\\ \Rightarrow \mathbf{s}&= -[H(\mathbf{w})]^{-1}g(\mathbf{w}). \end{align} This choice of \(\mathbf{s}\) converges extremely fast if the approximation is sufficiently accurate and the resulting step sufficiently small. Otherwise it can diverge. Divergence often happens if the function is flat or almost flat with respect to some dimension. In that case the second derivatives are close to zero, and their inverse becomes very large - resulting in gigantic steps. Different from gradient descent, here there is no step-size that guarantees that steps are all small and local. As the Taylor approximation is only accurate locally, large steps can move the current estimates far from regions where the Taylor approximation is accurate.

- The matrix \(H(\mathbf{w})\) scales \(d\times d\) and is expensive to compute. A good approximation can be to only compute its diagonal entries and multiply the update with a small step-size. Essentially you are then doing a hybrid between Newton's method and gradient descent, where you weigh the step-size for each dimension by the inverse Hessian.
- To avoid divergence of Newton's method, a good approach is to start with gradient descent (or even stochastic gradient descent) and then finish the optimization Newton's method. Typically, the second order approximation, used by Newton's Method, is more likely to be appropriate near the optimum.

(a) A starting point where Newton's Method converges in 8 iterations.

(b) A starting point where Newton's Method diverges.

(c) same starting point as in Figure 2, however Newton's method is only used after 6 gradient steps and converges in a few steps.

The three plots show a comparison of Newton's Method and Gradient Descent. Gradient Descent always converges after over 100 iterations from all initial starting points. If it converges (Figure 1), Newton's Method is much faster (convergence after 8 iterations) but it can diverge (Figure 2). Figure 3 shows the hybrid approach of taking 6 gradient descent steps and then switching to Newton's Method. It still converges in only 10 updates.