A Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-MomentumDownload PDF

Published: 09 Nov 2021, Last Modified: 05 May 2023NeurIPS 2021 PosterReaders: Everyone
Keywords: stochastic bilevel optimization, non-convex optimization, single time-scale stochastic approximation
Abstract: This paper proposes a new algorithm -- the \underline{S}ingle-timescale Do\underline{u}ble-momentum \underline{St}ochastic \underline{A}pprox\underline{i}matio\underline{n} (SUSTAIN) -- for tackling stochastic unconstrained bilevel optimization problems. We focus on bilevel problems where the lower level subproblem is strongly-convex and the upper level objective function is smooth. Unlike prior works which rely on \emph{two-timescale} or \emph{double loop} techniques, we design a stochastic momentum-assisted gradient estimator for both the upper and lower level updates. The latter allows us to control the error in the stochastic gradient updates due to inaccurate solution to both subproblems. If the upper objective function is smooth but possibly non-convex, we show that {SUSTAIN}~requires $O(\epsilon^{-3/2})$ iterations (each using $O(1)$ samples) to find an $\epsilon$-stationary solution. The $\epsilon$-stationary solution is defined as the point whose squared norm of the gradient of the outer function is less than or equal to $\epsilon$. The total number of stochastic gradient samples required for the upper and lower level objective functions matches the best-known complexity for single-level stochastic gradient algorithms. We also analyze the case when the upper level objective function is strongly-convex.
Code Of Conduct: I certify that all co-authors of this work have read and commit to adhering to the NeurIPS Statement on Ethics, Fairness, Inclusivity, and Code of Conduct.
Supplementary Material: pdf
14 Replies