Article Contents
Article Contents

# A regularization interpretation of the proximal point method for weakly convex functions

• * Corresponding author: tim.hoheisel@mcgill.ca

The third author is supported by AFOSR FA9550-18-1-0167

• Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.

Mathematics Subject Classification: Primary: 26B25, 65K10; Secondary: 90C25.

 Citation:

• Figure 1.  Illustration of Example 1

Figure 2.  Top: Iterations of gradient descent and the proximal point method. The proximal point method gets closer to the global mininum with the same number of total gradient evaluations. Bottom: function values at each iteration of gradient descent and at each outer proximal point iteration for the Rosenbrock function (a fair comparison is used by counting total gradient evaluations, using 10 or 20 for the approximate proximal point). After 2000 gradient evaluations the function value for gradient descent is still order 1, while the proximal point method is order $10^{-2}$, moreover the function values appear to decrease at a first order rate

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