A Note on Zeroth-Order Optimization on the Simplex

TL;DR

This paper introduces a new zeroth-order gradient estimator for functions on the probability simplex, enabling gradient-based algorithms to converge efficiently without explicit gradient information.

Contribution

It presents a novel simplex-only zeroth-order gradient estimator and demonstrates its effectiveness in projected gradient descent and exponential weights algorithms.

Findings

Converges at a rate of O(T^{-1/4}) using the proposed estimator.

Applicable to smooth functions on the probability simplex.

Enables gradient-based optimization without explicit gradient queries.

Abstract

We construct a zeroth-order gradient estimator for a smooth function defined on the probability simplex. The proposed estimator queries the simplex only. We prove that projected gradient descent and the exponential weights algorithm, when run with this estimator instead of exact gradients, converge at a $\mathcal O(T^{-1/4})$ rate.