Comparison-Based Algorithms for One-Dimensional Stochastic Convex Optimization

TL;DR

This paper introduces a comparison-based algorithm for one-dimensional stochastic convex optimization that relies solely on comparative information, achieving optimal convergence rates without needing explicit sample or objective value data.

Contribution

The paper proposes a novel comparison-based algorithm (CBA) for 1D convex stochastic optimization that works with limited information and extends to multi-dimensional and non-convex problems.

Findings

CBA achieves the same convergence rate as optimal stochastic gradient methods.

Numerical experiments demonstrate the effectiveness of CBA in test problems.

The method extends to multi-dimensional quadratic and non-convex problems.

Abstract

Stochastic optimization finds a wide range of applications in operations research and management science. However, existing stochastic optimization techniques usually require the information of random samples (e.g., demands in the newsvendor problem) or the objective values at the sampled points (e.g., the lost sales cost), which might not be available in practice. In this paper, we consider a new setup for stochastic optimization, in which the decision maker has access to only comparative information between a random sample and two chosen decision points in each iteration. We propose a comparison-based algorithm (CBA) to solve such problems in one dimension with convex objective functions. Particularly, the CBA properly chooses the two points in each iteration and constructs an unbiased gradient estimate for the original problem. We show that the CBA achieves the same convergence rate as the optimal stochastic gradient methods (with the samples observed). We also consider extensions of our approach to multi-dimensional quadratic problems as well as problems with non-convex objective functions. Numerical experiments show that the CBA performs well in test problems.