Bilevel Optimization: Convergence Analysis and Enhanced Design

TL;DR

This paper provides a comprehensive convergence analysis of bilevel optimization algorithms, introduces a new stochastic method with improved efficiency, and validates these findings through experiments in meta-learning and hyperparameter tuning.

Contribution

It offers the first theoretical convergence rates for iterative differentiation, improves existing rates for implicit differentiation, and proposes a novel stochastic algorithm with superior complexity guarantees.

Findings

AID-based method's convergence rate is improved with practical parameter choices.

First theoretical convergence rate established for ITD-based methods.

stocBiO outperforms existing algorithms in efficiency for hyperparameter optimization.

Abstract

Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $\kappa$ and the target accuracy $\epsilon$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.