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Bilevel optimization (BLO) is a popular approach with many applicationsincluding hyperparameter optimization, neural architecture search, adversarialrobustness and model-agnostic meta-learning. However, the approach suffers fromtime and memory complexity proportional to the length $r$ of its inneroptimization loop, which has led to several modifications being proposed. Onesuch modification is \textit{first-order} BLO (FO-BLO) which approximatesouter-level gradients by zeroing out second derivative terms, yieldingsignificant speed gains and requiring only constant memory as $r$ varies.Despite FO-BLO's popularity, there is a lack of theoretical understanding ofits convergence properties. We make progress by demonstrating a rich family ofexamples where FO-BLO-based stochastic optimization does not converge to astationary point of the BLO objective. We address this concern by proposing anew FO-BLO-based unbiased estimate of outer-level gradients, enabling us totheoretically guarantee this convergence, with no harm to memory and expectedtime complexity. Our findings are supported by experimental results on Omniglotand Mini-ImageNet, popular few-shot meta-learning benchmarks.

UFO-BLO: Unbiased First-Order Bilevel Optimization