Dimensional Peeking for Low-Variance Gradients in Zeroth-Order Discrete Optimization via Simulation

TL;DR

This paper introduces dimensional peeking, a variance reduction technique for gradient estimation in high-dimensional discrete optimization, significantly improving convergence speed and competitiveness of zeroth-order methods.

Contribution

It proposes a novel variance reduction method called dimensional peeking that lifts sampling granularity, enhancing gradient estimates without bias in simulation-based discrete optimization.

Findings

Variance reductions up to 7.9 times observed.

Improved optimization progress over meta-heuristics.

Enhanced competitiveness of zeroth-order methods in high-dimensional problems.

Abstract

Gradient-based optimization methods are commonly used to identify local optima in high-dimensional spaces. When derivatives cannot be evaluated directly, stochastic estimators can provide approximate gradients. However, these estimators' perturbation-based sampling of the objective function introduces variance that can lead to slow convergence. In this paper, we present dimensional peeking, a variance reduction method for gradient estimation in discrete optimization via simulation. By lifting the sampling granularity from scalar values to classes of values that follow the same control flow path, we increase the information gathered per simulation evaluation. Our derivation from an established smoothed gradient estimator shows that the method does not introduce any bias. We present an implementation via a custom numerical data type to transparently carry out dimensional peeking over C++ programs. Variance reductions by factors of up to 7.9 are observed for three simulation-based optimization problems with high-dimensional input. The optimization progress compared to three meta-heuristics shows that dimensional peeking increases the competitiveness of zeroth-order optimization for discrete and non-convex simulations.