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``Relative-Continuity'' for Non-Lipschitz Non-Smooth Convex Optimization using Stochastic (or Deterministic) Mirror Descent

Haihao Lu(haihao***at***mit.edu)

Abstract: The usual approach to developing and analyzing first-order methods for non-smooth (stochastic or deterministic) convex optimization assumes that the objective function is uniformly Lipschitz continuous with parameter $M_f$. However, in many settings the non-differentiable convex function $f(\cdot)$ is not uniformly Lipschitz continuous -- for example (i) the classical support vector machine (SVM) problem, (ii) the problem of minimizing the maximum of convex quadratic functions, and even (iii) the univariate setting with $f(x) := \max\{0, x\} + x^2$. Herein we develop a notion of ``relative continuity'' that is determined relative to a user-specified ``reference function'' $h(\cdot)$ (that should be computationally tractable for algorithms), and we show that many non-differentiable convex functions are relatively continuous with respect to a correspondingly fairly-simple reference function $h(\cdot)$. We also similarly develop a notion of ``relative stochastic continuity'' for the stochastic setting. We analysis two standard algorithms -- the (deterministic) mirror descent algorithm and the stochastic mirror descent algorithm -- for solving optimization problems in these two new settings, and we develop for the first time computational guarantees for instances where the objective function is not uniformly Lipschitz continuous. This paper is a companion paper for non-differentiable convex optimization to the recent paper by Lu, Freund, and Nesterov, which developed similar sorts of results for differentiable convex optimization.

Keywords: large-scale optimization, stochastic first-order methods, non-Lipschitz non-smooth optimization, mirror descent

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: MIT Operations Research Center Working Paper

Download: [PDF]

Entry Submitted: 10/12/2017
Entry Accepted: 10/12/2017
Entry Last Modified: 10/12/2017

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