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Register a new userEnhanced Bilevel Optimization via Bregman Distance
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Feihu Huang
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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.
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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.
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.
Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.
Bilevel optimization has arisen as a powerful tool for many machine learning problems such as meta-learning, hyperparameter optimization, and reinforcement learning. In this paper, we investigate the nonconvex-strongly-convex bilevel optimization problem. For deterministic bilevel optimization, we provide a comprehensive convergence rate analysis for two popular algorithms respectively based on approximate implicit differentiation (AID) and iterative differentiation (ITD). For the AID-based method, we orderwisely improve the previous convergence rate analysis due to a more practical parameter selection as well as a warm start strategy, and for the ITD-based method we establish the first theoretical convergence rate. Our analysis also provides a quantitative comparison between ITD and AID based approaches. For stochastic bilevel optimization, we propose a novel algorithm named stocBiO, which features a sample-efficient hypergradient estimator using efficient Jacobian- and Hessian-vector product computations. We provide the convergence rate guarantee for stocBiO, and show that stocBiO outperforms the best known computational complexities orderwisely with respect to the condition number $kappa$ and the target accuracy $epsilon$. We further validate our theoretical results and demonstrate the efficiency of bilevel optimization algorithms by the experiments on meta-learning and hyperparameter optimization.
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