Spectral Projected Subgradient Method for Nonsmooth Convex Optimization Problems

TL;DR

This paper introduces a spectral projected subgradient method for nonsmooth convex optimization with stochastic objectives, combining sample average approximation and adaptive step sizes to improve convergence and computational efficiency.

Contribution

It proposes a novel algorithm integrating spectral coefficients with SAA subgradients and an Armijo-like line search for better performance in stochastic nonsmooth optimization.

Findings

Line search significantly improves convergence.

Variable sample strategy outperforms full sample approach.

Method demonstrates strong empirical performance on machine learning problems.

Abstract

We consider constrained optimization problems with a nonsmooth objective function in the form of mathematical expectation. The Sample Average Approximation (SAA) is used to estimate the objective function and variable sample size strategy is employed. The proposed algorithm combines an SAA subgradient with the spectral coefficient in order to provide a suitable direction which improves the performance of the first order method as shown by numerical results. The step sizes are chosen from the predefined interval and the almost sure convergence of the method is proved under the standard assumptions in stochastic environment. To enhance the performance of the proposed algorithm, we further specify the choice of the step size by introducing an Armijo-like procedure adapted to this framework. Considering the computational cost on machine learning problems, we conclude that the line search improves the performance significantly. Numerical experiments conducted on finite sums problems also reveal that the variable sample strategy outperforms the full sample approach.