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1. Generalized Implicit Follow-The-Regularized-Leader

   Chen, Keyi; Orabona, Francesco (July 2023, International Conference on Machine Learning)

   We propose a new class of online learning algorithms, generalized implicit Follow-The-Regularized-Leader (FTRL), that expands the scope of FTRL framework. Generalized implicit FTRL can recover known algorithms, such as FTRL with linearized losses and implicit FTRL, and it allows the design of new update rules, as extensions of aProx and Mirror-Prox to FTRL. Our theory is constructive in the sense that it provides a simple unifying framework to design updates that directly improve the worst-case upper bound on the regret. The key idea is substituting the linearization of the losses with a Fenchel-Young inequality. We show the flexibility of the framework by proving that some known algorithms, like the Mirror-Prox updates, are instantiations of the generalized implicit FTRL. Finally, the new framework allows us to recover the temporal variation bound of implicit OMD, with the same computational complexity. 

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2. Solving Optimization Problems with Blackwell Approachability

   https://doi.org/10.1287/moor.2023.1376  

   Grand-Clément, Julien; Kroer, Christian (January 2023, Mathematics of Operations Research)

   In this paper, we propose a new algorithm for solving convex-concave saddle-point problems using regret minimization in the repeated game framework. To do so, we introduce the Conic Blackwell Algorithm + ([Formula: see text]), a new parameter- and scale-free regret minimizer for general convex compact sets. [Formula: see text] is based on Blackwell approachability and attains [Formula: see text] regret. We show how to efficiently instantiate [Formula: see text] for many decision sets of interest, including the simplex, [Formula: see text] norm balls, and ellipsoidal confidence regions in the simplex. Based on [Formula: see text], we introduce [Formula: see text], a new parameter-free algorithm for solving convex-concave saddle-point problems achieving a [Formula: see text] ergodic convergence rate. In our simulations, we demonstrate the wide applicability of [Formula: see text] on several standard saddle-point problems from the optimization and operations research literature, including matrix games, extensive-form games, distributionally robust logistic regression, and Markov decision processes. In each setting, [Formula: see text] achieves state-of-the-art numerical performance and outperforms classical methods, without the need for any choice of step sizes or other algorithmic parameters. Funding: J. Grand-Clément is supported by the Agence Nationale de la Recherche [Grant 11-LABX-0047] and by Hi! Paris. C. Kroer is supported by the Office of Naval Research [Grant N00014-22-1-2530] and by the National Science Foundation [Grant IIS-2147361]. 

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3. Stochastic regret minimization in extensive-form games

   Farina, G.; Kroer, C.; Sandholm, T. (January 2020, ICML)

   null (Ed.)

   Monte-Carlo counterfactual regret minimization (MCCFR) is the state-of-the-art algorithm for solving sequential games that are too large for full tree traversals. It works by using gradient es- timates that can be computed via sampling. How- ever, stochastic methods for sequential games have not been investigated extensively beyond MCCFR. In this paper we develop a new frame- work for developing stochastic regret minimiza- tion methods. This framework allows us to use any regret-minimization algorithm, coupled with any gradient estimator. The MCCFR algorithm can be analyzed as a special case of our frame- work, and this analysis leads to significantly stronger theoretical guarantees on convergence, while simultaneously yielding a simplified proof. Our framework allows us to instantiate several new stochastic methods for solving sequential games. We show extensive experiments on five games, where some variants of our methods out- perform MCCFR. 

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4. Model-Free Online Learning in Unknown Sequential Decision Making Problems and Games

   Farina, G.; Sandholm, T. (January 2021, Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   Regret minimization has proved to be a versatile tool for tree- form sequential decision making and extensive-form games. In large two-player zero-sum imperfect-information games, mod- ern extensions of counterfactual regret minimization (CFR) are currently the practical state of the art for computing a Nash equilibrium. Most regret-minimization algorithms for tree-form sequential decision making, including CFR, require (i) an exact model of the player’s decision nodes, observation nodes, and how they are linked, and (ii) full knowledge, at all times t, about the payoffs—even in parts of the decision space that are not encountered at time t. Recently, there has been growing interest towards relaxing some of those restric- tions and making regret minimization applicable to settings for which reinforcement learning methods have traditionally been used—for example, those in which only black-box access to the environment is available. We give the first, to our knowl- edge, regret-minimization algorithm that guarantees sublinear regret with high probability even when requirement (i)—and thus also (ii)—is dropped. We formalize an online learning setting in which the strategy space is not known to the agent and gets revealed incrementally whenever the agent encoun- ters new decision points. We give an efficient algorithm that achieves O(T 3/4) regret with high probability for that setting, even when the agent faces an adversarial environment. Our experiments show it significantly outperforms the prior algo- rithms for the problem, which do not have such guarantees. It can be used in any application for which regret minimization is useful: approximating Nash equilibrium or quantal response equilibrium, approximating coarse correlated equilibrium in multi-player games, learning a best response, learning safe opponent exploitation, and online play against an unknown opponent/environment. 

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5. Training GANs with Optimism

   Daskalakis, Constantinos; Ilyas, Andrew; Syrgkanis, Vasilis; Zeng, Haoyang (January 2018, International Conference on Learning Representations)

   We address the issue of limit cycling behavior in training Generative Adversarial Networks and propose the use of Optimistic Mirror Decent (OMD) for training Wasserstein GANs. Recent theoretical results have shown that optimistic mirror decent (OMD) can enjoy faster regret rates in the context of zero-sum games. WGANs is exactly a context of solving a zero-sum game with simultaneous no-regret dynamics. Moreover, we show that optimistic mirror decent addresses the limit cycling problem in training WGANs. We formally show that in the case of bi-linear zero-sum games the last iterate of OMD dynamics converges to an equilibrium, in contrast to GD dynamics which are bound to cycle. We also portray the huge qualitative difference between GD and OMD dynamics with toy examples, even when GD is modified with many adaptations proposed in the recent literature, such as gradient penalty or momentum. We apply OMD WGAN training to a bioinformatics problem of generating DNA sequences. We observe that models trained with OMD achieve consistently smaller KL divergence with respect to the true underlying distribution, than models trained with GD variants. Finally, we introduce a new algorithm, Optimistic Adam, which is an optimistic variant of Adam. We apply it to WGAN training on CIFAR10 and observe improved performance in terms of inception score as compared to Adam. 

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