[Paper Review] Stochastic Methods for Composite Optimization Problems

John C. Duchi, Feng Ruan|arXiv (Cornell University)|Mar 24, 2017

Stochastic Gradient Optimization TechniquesComputer Science20 references22 citations

TL;DR

This paper proposes stochastic prox-linear and sub-gradient algorithms for minimizing composite stochastic objectives involving non-smooth convex functions composed with smooth functions. Under mild conditions, the methods converge to first-order stationary points, with empirical validation on non-smooth phase retrieval problems demonstrating practical effectiveness.

ABSTRACT

We consider minimization of stochastic functionals that are compositions of a (potentially) non-smooth convex function $h$ and smooth function $c$ and, more generally, stochastic weakly-convex functionals. We develop a family of stochastic methods---including a stochastic prox-linear algorithm and a stochastic (generalized) sub-gradient procedure---and prove that, under mild technical conditions, each converges to first-order stationary points of the stochastic objective. We provide experiments further investigating our methods on non-smooth phase retrieval problems; the experiments indicate the practical effectiveness of the procedures.

Motivation & Objective

• To address the challenge of minimizing stochastic composite objectives where the objective is a composition of a non-smooth convex function and a smooth function.
• To develop robust stochastic optimization methods capable of handling weakly-convex and non-smooth functionals in a stochastic setting.
• To establish convergence guarantees to first-order stationary points under mild technical conditions.
• To evaluate the practical performance of the proposed methods on real-world non-smooth problems such as phase retrieval.

Proposed method

• The paper introduces a stochastic prox-linear algorithm that iteratively approximates the composite objective using a linearization of the smooth component and a proximal step on the non-smooth component.
• A stochastic generalized sub-gradient procedure is developed to handle weakly-convex stochastic objectives, leveraging sub-gradient information in a randomized setting.
• The methods operate under a stochastic oracle model, where function and sub-gradient evaluations are noisy but satisfy certain moment conditions.
• Convergence analysis relies on assumptions such as bounded gradients and Lipschitz continuity of the smooth component, ensuring convergence to first-order stationary points.
• The framework accommodates both convex and weakly-convex objectives, extending applicability beyond standard smooth or strongly convex settings.
• Theoretical analysis uses martingale difference sequences and almost-sure convergence arguments to establish convergence in expectation and almost surely.

Experimental results

Research questions

• RQ1 Can stochastic prox-linear methods converge to first-order stationary points for composite non-smooth stochastic objectives?
• RQ2 How do stochastic sub-gradient procedures perform on weakly-convex stochastic functionals?
• RQ3 What conditions ensure convergence of stochastic algorithms in composite, non-smooth optimization settings?
• RQ4 How do the proposed methods compare in practice on non-smooth phase retrieval problems?

Key findings

• The stochastic prox-linear algorithm converges to first-order stationary points under mild technical conditions, including bounded gradients and Lipschitz continuity of the smooth component.
• The stochastic generalized sub-gradient method also achieves convergence to first-order stationary points for weakly-convex stochastic objectives.
• Empirical results on non-smooth phase retrieval problems show that both methods are practically effective and robust to noise.
• The convergence guarantees hold in expectation and almost surely, supported by theoretical analysis using martingale convergence techniques.