Debiasing a First-order Heuristic for Approximate Bi-level Optimization

Valerii Likhosherstov, Xingyou Song, Krzysztof Choromanski, Jared Q Davis, Adrian Weller
Proceedings of the 38th International Conference on Machine Learning, PMLR 139:6621-6630, 2021.

Abstract

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v139-likhosherstov21a, title = {Debiasing a First-order Heuristic for Approximate Bi-level Optimization}, author = {Likhosherstov, Valerii and Song, Xingyou and Choromanski, Krzysztof and Davis, Jared Q and Weller, Adrian}, booktitle = {Proceedings of the 38th International Conference on Machine Learning}, pages = {6621--6630}, year = {2021}, editor = {Meila, Marina and Zhang, Tong}, volume = {139}, series = {Proceedings of Machine Learning Research}, month = {18--24 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v139/likhosherstov21a/likhosherstov21a.pdf}, url = {https://proceedings.mlr.press/v139/likhosherstov21a.html}, abstract = {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.} }
Endnote
%0 Conference Paper %T Debiasing a First-order Heuristic for Approximate Bi-level Optimization %A Valerii Likhosherstov %A Xingyou Song %A Krzysztof Choromanski %A Jared Q Davis %A Adrian Weller %B Proceedings of the 38th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2021 %E Marina Meila %E Tong Zhang %F pmlr-v139-likhosherstov21a %I PMLR %P 6621--6630 %U https://proceedings.mlr.press/v139/likhosherstov21a.html %V 139 %X 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.
APA
Likhosherstov, V., Song, X., Choromanski, K., Davis, J.Q. & Weller, A.. (2021). Debiasing a First-order Heuristic for Approximate Bi-level Optimization. Proceedings of the 38th International Conference on Machine Learning, in Proceedings of Machine Learning Research 139:6621-6630 Available from https://proceedings.mlr.press/v139/likhosherstov21a.html.

Related Material