Towards Fully Parameter-Free Stochastic Optimization: Grid Search with Self-Bounding Analysis

TL;DR

This paper introduces extsc{Grasp}, a grid search framework with self-bounding analysis for fully parameter-free stochastic optimization, applicable to convex and non-convex problems, achieving near-optimal convergence without prior parameter knowledge.

Contribution

The paper presents a novel grid search framework with self-bounding analysis that enables fully parameter-free stochastic optimization for both convex and non-convex problems.

Findings

Achieves near-optimal convergence rates in non-convex settings.

Demonstrates competitive performance in convex cases with acceleration.

Provides sharper guarantees for model ensemble under variance characterization.

Abstract

Parameter-free stochastic optimization aims to design algorithms that are agnostic to the underlying problem parameters while still achieving convergence rates competitive with optimally tuned methods. While some parameter-free methods do not require the specific values of the problem parameters, they still rely on prior knowledge, such as the lower or upper bounds of them. We refer to such methods as ``partially parameter-free''. In this work, we target achieving ``fully parameter-free'' methods, i.e., the algorithmic inputs do not need to satisfy any unverifiable condition related to the true problem parameters. We propose a powerful and general grid search framework, named \textsc{Grasp}, with a novel self-bounding analysis technique that effectively determines the search ranges of parameters, in contrast to previous work. Our method demonstrates generality in: (i) the non-convex case, where we propose a fully parameter-free method that achieves near-optimal convergence rate, up to logarithmic factors; (ii) the convex case, where our parameter-free methods are competitive with strong performance in terms of acceleration and universality. Finally, we contribute a sharper guarantee for the model ensemble, a final step of the grid search framework, under interpolated variance characterization.