Nonsmooth Optimization with Zeroth Order Comparison Feedback

TL;DR

This paper introduces a novel zeroth-order optimization method for nonsmooth, nonconvex functions using noisy pairwise comparisons, achieving explicit complexity bounds with unbiased estimators.

Contribution

It develops an unbiased estimator for directional differences from comparison data and provides complexity bounds for nonconvex optimization with various link functions.

Findings

Constructs an unbiased estimator with finite expected cost.

Provides explicit comparison-complexity bounds for common link functions.

Achieves convergence to a Goldstein stationary point under mild conditions.

Abstract

We study unconstrained optimization problems of nonsmooth, nonconvex Lipschitz functions, using only noisy pairwise comparisons governed by a known link function. Our goal is to compute a $(\delta,\varepsilon)$-Goldstein stationary point. We combine randomized smoothing with a novel unbiased reduction from comparisons to local value differences. By leveraging a Russian-roulette truncation on the Bernoulli-product expansion of the inverse link, we construct an exactly unbiased estimator for directional differences. This estimator has finite expected cost and variance scaling quadratically with the function gap, $\mathcal{O}(B^2)$, under mild conditions. Plugging this into the smoothed gradient identity enables a standard nonconvex SGD analysis, yielding explicit comparison-complexity bounds for common symmetric links such as logistic, probit, and cauchit.