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Register a new userOptimistic mirror descent in saddle-point problems: Going the extra (gradient) mile
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Added by
Panayotis Mertikopoulos
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Owing to their connection with generative adversarial networks (GANs), saddle-point problems have recently attracted considerable interest in machine learning and beyond. By necessity, most theoretical guarantees revolve around convex-concave (or even linear) problems; however, making theoretical inroads towards efficient GAN training depends crucially on moving beyond this classic framework. To make piecemeal progress along these lines, we analyze the behavior of mirror descent (MD) in a class of non-monotone problems whose solutions coincide with those of a naturally associated variational inequality - a property which we call coherence. We first show that ordinary, vanilla MD converges under a strict version of this condition, but not otherwise; in particular, it may fail to converge even in bilinear models with a unique solution. We then show that this deficiency is mitigated by optimism: by taking an extra-gradient step, optimistic mirror descent (OMD) converges in all coherent problems. Our analysis generalizes and extends the results of Daskalakis et al. (2018) for optimistic gradient descent (OGD) in bilinear problems, and makes concrete headway for establishing convergence beyond convex-concave games. We also provide stochastic analogues of these results, and we validate our analysis by numerical experiments in a wide array of GAN models (including Gaussian mixture models, as well as the CelebA and CIFAR-10 datasets).
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Two of the most prominent algorithms for solving unconstrained smooth games are the classical stochastic gradient descent-ascent (SGDA) and the recently introduced stochastic consensus optimization (SCO) (Mescheder et al., 2017). SGDA is known to converge to a stationary point for specific classes of games, but current convergence analyses require a bounded variance assumption. SCO is used successfully for solving large-scale adversarial problems, but its convergence guarantees are limited to its deterministic variant. In this work, we introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO under this condition for solving a class of stochastic variational inequality problems that are potentially non-monotone. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size, and we propose insightful stepsize-switching rules to guarantee convergence to the exact solution. In addition, our convergence guarantees hold under the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching.
In this paper we consider online mirror descent (OMD) algorithms, a class of scalable online learning algorithms exploiting data geometric structures through mirror maps. Necessary and sufficient conditions are presented in terms of the step size sequence ${eta_t}_{t}$ for the convergence of an OMD algorithm with respect to the expected Bregman distance induced by the mirror map. The condition is $lim_{ttoinfty}eta_t=0, sum_{t=1}^{infty}eta_t=infty$ in the case of positive variances. It is reduced to $sum_{t=1}^{infty}eta_t=infty$ in the case of zero variances for which the linear convergence may be achieved by taking a constant step size sequence. A sufficient condition on the almost sure convergence is also given. We establish tight error bounds under mild conditions on the mirror map, the loss function, and the regularizer. Our results are achieved by some novel analysis on the one-step progress of the OMD algorithm using smoothness and strong convexity of the mirror map and the loss function.
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Non-convex optimization problems are challenging to solve; the success and computational expense of a gradient descent algorithm or variant depend heavily on the initialization strategy. Often, either random initialization is used or initialization rules are carefully designed by exploiting the nature of the problem class. As a simple alternative to hand-crafted initialization rules, we propose an approach for learning good initialization rules from previous solutions. We provide theoretical guarantees that establish conditions that are sufficient in all cases and also necessary in some under which our approach performs better than random initialization. We apply our methodology to various non-convex problems such as generating adversarial examples, generating post hoc explanations for black-box machine learning models, and allocating communication spectrum, and show consistent gains over other initialization techniques.
We propose Banker-OMD, a novel framework generalizing the classical Online Mirror Descent (OMD) technique in online learning algorithm design. Banker-OMD allows algorithms to robustly handle delayed feedback, and offers a general methodology for achieving $tilde{O}(sqrt{T} + sqrt{D})$-style regret bounds in various delayed-feedback online learning tasks, where $T$ is the time horizon length and $D$ is the total feedback delay. We demonstrate the power of Banker-OMD with applications to three important bandit scenarios with delayed feedback, including delayed adversarial Multi-armed bandits (MAB), delayed adversarial linear bandits, and a novel delayed best-of-both-worlds MAB setting. Banker-OMD achieves nearly-optimal performance in all the three settings. In particular, it leads to the first delayed adversarial linear bandit algorithm achieving $tilde{O}(text{poly}(n)(sqrt{T} + sqrt{D}))$ regret.
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