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Debiasing a First-order Heuristic for Approximate Bi-level Optimization
Valerii Likhosherstov · Xingyou Song · Krzysztof Choromanski · Jared Quincy Davis · Adrian Weller

Wed Jul 21 05:20 AM -- 05:25 AM (PDT) @
Approximate bi-level optimization (ABLO) consists of (outer-level) optimization problems, involving numerical (inner-level) optimization loops. While ABLO has many applications across deep learning, it suffers from time and memory complexity proportional to the length $r$ of its inner optimization loop. To address this complexity, an earlier first-order method (FOM) was proposed as a heuristic which omits second derivative terms, yielding significant speed gains and requiring only constant memory. Despite FOM's popularity, there is a lack of theoretical understanding of its convergence properties. We contribute by theoretically characterizing FOM's gradient bias under mild assumptions. We further demonstrate a rich family of examples where FOM-based SGD does not converge to a stationary point of the ABLO objective. We address this concern by proposing an unbiased FOM (UFOM) enjoying constant memory complexity as a function of $r$. We characterize the introduced time-variance tradeoff, demonstrate convergence bounds, and find an optimal UFOM for a given ABLO problem. Finally, we propose an efficient adaptive UFOM scheme.

#### Author Information

##### Adrian Weller (University of Cambridge, Alan Turing Institute)

Adrian Weller is Programme Director for AI at The Alan Turing Institute, the UK national institute for data science and AI, and is a Turing AI Fellow leading work on trustworthy Machine Learning (ML). He is a Principal Research Fellow in ML at the University of Cambridge, and at the Leverhulme Centre for the Future of Intelligence where he is Programme Director for Trust and Society. His interests span AI, its commercial applications and helping to ensure beneficial outcomes for society. Previously, Adrian held senior roles in finance. He received a PhD in computer science from Columbia University, and an undergraduate degree in mathematics from Trinity College, Cambridge.