UFO-BLO: Unbiased First-Order Bilevel Optimization

TL;DR

This paper introduces UFO-BLO, an unbiased first-order bilevel optimization method that guarantees convergence without increasing memory or time complexity, supported by theoretical analysis and experiments on meta-learning benchmarks.

Contribution

We propose a new unbiased FO-BLO gradient estimator that ensures convergence, addressing limitations of existing FO-BLO methods.

Findings

Unbiased FO-BLO guarantees convergence to stationary points.

Experimental results validate the effectiveness on few-shot learning benchmarks.

Existing FO-BLO can fail to converge, which our method overcomes.

Abstract

Bilevel optimization (BLO) is a popular approach with many applications including hyperparameter optimization, neural architecture search, adversarial robustness and model-agnostic meta-learning. However, the approach suffers from time and memory complexity proportional to the length $r$ of its inner optimization loop, which has led to several modifications being proposed. One such modification is \textit{first-order} BLO (FO-BLO) which approximates outer-level gradients by zeroing out second derivative terms, yielding significant speed gains and requiring only constant memory as $r$ varies. Despite FO-BLO's popularity, there is a lack of theoretical understanding of its convergence properties. We make progress by demonstrating a rich family of examples where FO-BLO-based stochastic optimization does not converge to a stationary point of the BLO objective. We address this concern by proposing a new FO-BLO-based unbiased estimate of outer-level gradients, enabling us to theoretically guarantee this convergence, with no harm to memory and expected time complexity. Our findings are supported by experimental results on Omniglot and Mini-ImageNet, popular few-shot meta-learning benchmarks.