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Register a new userA Near-Optimal Algorithm for Stochastic Bilevel Optimization via Double-Momentum
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Prashant Khanduri
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper proposes a new algorithm -- the underline{S}ingle-timescale Dounderline{u}ble-momentum underline{St}ochastic underline{A}pproxunderline{i}matiounderline{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 {aname}~requires $mathcal{O}(epsilon^{-3/2})$ iterations (each using ${cal O}(1)$ 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.
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Stochastic bilevel optimization generalizes the classic stochastic optimization from the minimization of a single objective to the minimization of an objective function that depends the solution of another optimization problem. Recently, stochastic bilevel optimization is regaining popularity in emerging machine learning applications such as hyper-parameter optimization and model-agnostic meta learning. To solve this class of stochastic optimization problems, existing methods require either double-loop or two-timescale updates, which are sometimes less efficient. This paper develops a new optimization method for a class of stochastic bilevel problems that we term Single-Timescale stochAstic BiLevEl optimization (STABLE) method. STABLE runs in a single loop fashion, and uses a single-timescale update with a fixed batch size. To achieve an $epsilon$-stationary point of the bilevel problem, STABLE requires ${cal O}(epsilon^{-2})$ samples in total; and to achieve an $epsilon$-optimal solution in the strongly convex case, STABLE requires ${cal O}(epsilon^{-1})$ samples. To the best of our knowledge, this is the first bilevel optimization algorithm achieving the same order of sample complexity as the stochastic gradient descent method for the single-level stochastic optimization.
In this paper, we consider non-convex stochastic bilevel optimization (SBO) problems that have many applications in machine learning. Although numerous studies have proposed stochastic algorithms for solving these problems, they are limited in two perspectives: (i) their sample complexities are high, which do not match the state-of-the-art result for non-convex stochastic optimization; (ii) their algorithms are tailored to problems with only one lower-level problem. When there are many lower-level problems, it could be prohibitive to process all these lower-level problems at each iteration. To address these limitations, this paper proposes fast randomized stochastic algorithms for non-convex SBO problems. First, we present a stochastic method for non-convex SBO with only one lower problem and establish its sample complexity of $O(1/epsilon^3)$ for finding an $epsilon$-stationary point under Lipschitz continuous conditions of stochastic oracles, matching the lower bound for stochastic smooth non-convex optimization. Second, we present a randomized stochastic method for non-convex SBO with $m>1$ lower level problems (multi-task SBO) by processing a constant number of lower problems at each iteration, and establish its sample complexity no worse than $O(m/epsilon^3)$, which could be a better complexity than that of simply processing all $m$ lower problems at each iteration. Lastly, we establish even faster convergence results for gradient-dominant functions. To the best of our knowledge, this is the first work considering multi-task SBO and developing state-of-the-art sample complexity results.
Bilevel optimization has been widely applied many machine learning problems such as hyperparameter optimization, policy optimization and meta learning. Although many bilevel optimization methods more recently have been proposed to solve the bilevel optimization problems, they still suffer from high computational complexities and do not consider the more general bilevel problems with nonsmooth regularization. In the paper, thus, we propose a class of efficient bilevel optimization methods based on Bregman distance. In our methods, we use the mirror decent iteration to solve the outer subproblem of the bilevel problem by using strongly-convex Bregman functions. Specifically, we propose a bilevel optimization method based on Bregman distance (BiO-BreD) for solving deterministic bilevel problems, which reaches the lower computational complexities than the best known results. We also propose a stochastic bilevel optimization method (SBiO-BreD) for solving stochastic bilevel problems based on the stochastic approximated gradients and Bregman distance. Further, we propose an accelerated version of SBiO-BreD method (ASBiO-BreD) by using the variance-reduced technique. Moreover, we prove that the ASBiO-BreD outperforms the best known computational complexities with respect to the condition number $kappa$ and the target accuracy $epsilon$ for finding an $epsilon$-stationary point of nonconvex-strongly-convex bilevel problems. In particular, our methods can solve the bilevel optimization problems with nonsmooth regularization with a lower computational complexity.
Stochastic gradient descent (SGD) is one of the most widely used optimization methods for parallel and distributed processing of large datasets. One of the key limitations of distributed SGD is the need to regularly communicate the gradients between different computation nodes. To reduce this communication bottleneck, recent work has considered a one-bit variant of SGD, where only the sign of each gradient element is used in optimization. In this paper, we extend this idea by proposing a stochastic variant of the proximal-gradient method that also uses one-bit per update element. We prove the theoretical convergence of the method for non-convex optimization under a set of explicit assumptions. Our results indicate that the compressed method can match the convergence rate of the uncompressed one, making the proposed method potentially appealing for distributed processing of large datasets.
Epoch gradient descent method (a.k.a. Epoch-GD) proposed by Hazan and Kale (2011) was deemed a breakthrough for stochastic strongly convex minimization, which achieves the optimal convergence rate of $O(1/T)$ with $T$ iterative updates for the {it objective gap}. However, its extension to solving stochastic min-max problems with strong convexity and strong concavity still remains open, and it is still unclear whether a fast rate of $O(1/T)$ for the {it duality gap} is achievable for stochastic min-max optimization under strong convexity and strong concavity. Although some recent studies have proposed stochastic algorithms with fast convergence rates for min-max problems, they require additional assumptions about the problem, e.g., smoothness, bi-linear structure, etc. In this paper, we bridge this gap by providing a sharp analysis of epoch-wise stochastic gradient descent ascent method (referred to as Epoch-GDA) for solving strongly convex strongly concave (SCSC) min-max problems, without imposing any additional assumption about smoothness or the functions structure. To the best of our knowledge, our result is the first one that shows Epoch-GDA can achieve the optimal rate of $O(1/T)$ for the duality gap of general SCSC min-max problems. We emphasize that such generalization of Epoch-GD for strongly convex minimization problems to Epoch-GDA for SCSC min-max problems is non-trivial and requires novel technical analysis. Moreover, we notice that the key lemma can also be used for proving the convergence of Epoch-GDA for weakly-convex strongly-concave min-max problems, leading to a nearly optimal complexity without resorting to smoothness or other structural conditions.
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