Convergence Properties of Stochastic Hypergradients

TL;DR

This paper analyzes stochastic approximation methods for hypergradients in bilevel optimization, providing theoretical bounds and demonstrating practical benefits for large-scale machine learning tasks.

Contribution

It introduces a stochastic variant of implicit differentiation for hypergradients, with error bounds independent of solver choices, and validates the approach through experiments.

Findings

The proposed method achieves bounded mean square error in hypergradient approximation.

Stochastic hypergradients outperform deterministic methods in large-scale settings.

Numerical results confirm theoretical error bounds and practical efficiency.

Abstract

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of the gradient of the upper-level objective (hypergradient). In this work, we study stochastic approximation schemes for the hypergradient, which are important when the lower-level problem is empirical risk minimization on a large dataset. The method that we propose is a stochastic variant of the approximate implicit differentiation approach in (Pedregosa, 2016). We provide bounds for the mean square error of the hypergradient approximation, under the assumption that the lower-level problem is accessible only through a stochastic mapping which is a contraction in expectation. In particular, our main bound is agnostic to the choice of the two stochastic solvers employed by the procedure. We provide numerical experiments to support our theoretical analysis and to show the advantage of using stochastic hypergradients in practice.