Game-theoretic models of learning are a powerful set of models that optimize multi-objective architectures. Among these models are zero-sum architectures that have inspired adversarial learning frameworks. An important shortcoming of these zeros-sum architectures is that gradient-based training leads to weak convergence and cyclic dynamics.
We propose a novel follow the leader training algorithm for zeros-sum architectures that guarantees convergence to mixed Nash equilibrium without cyclic behaviors. It is a special type of follow the perturbed leader algorithm where perturbations are the result of a neural mediating agent.
We validate our theoretical results by applying this training algorithm to games with convex and non-convex loss as well as generative adversarial architectures. Moreover, we customize the implementation of this algorithm for adversarial imitation learning applications. At every step of the training, the mediator agent perturbs the observations with generated codes. As a result of these mediating codes, the proposed algorithm is also efficient for learning in environments with various factors of variations. We validate our assertion by using a procedurally generated game environment as well as synthetic data. Github implementation is available.
We study the problem of online learning with non-convex losses, where the learner has access to an offline optimization oracle. We show that the classical Follow the Perturbed Leader (FTPL) algorithm achieves optimal regret rate of $O(T^{-1/2})$ in this setting. This improves upon the previous best-known regret rate of $O(T^{-1/3})$ for FTPL. We further show that an optimistic variant of FTPL achieves better regret bounds when the sequence of losses encountered by the learner is `predictable'.
Institute for Information Transmission Problems, Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
In this paper the sequential prediction problem with expert advice is considered for the case where losses of experts suffered at each step cannot be bounded in advance. We present some modification of Kalai and Vempala algorithm of following the perturbed leader where weights depend on past losses of the experts. New notions of a volume and a scaled fluctuation of a game are introduced. We present a probabilistic algorithm protected from unrestrictedly large one-step losses. This algorithm has the optimal performance in the case when the scaled fluctuations of one-step losses of experts of th…
When applying aggregating strategies to Prediction with Expert Advice, the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative "Follow the Perturbed Leader" (FPL) algorithm from Kalai & Vempala (2003) (based on Hannan's algorithm) is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with…
When applying aggregating strategies to Prediction with Expert Advice, the learning rate must be adaptively tuned. The natural choice of sqrt(complexity/current loss) renders the analysis of Weighted Majority derivatives quite complicated. In particular, for arbitrary weights there have been no results proven so far. The analysis of the alternative "Follow the Perturbed Leader" (FPL) algorithm from Kalai (2003} (based on Hannan's algorithm) is easier. We derive loss bounds for adaptive learning rate and both finite expert classes with uniform weights and countable expert classes with arbitrary…
We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret…
Adaptive Learning Rates with Surrogate Probability for Follow-the-Perturbed-Leader
Korea Institute of Science and Technology（韩国科学技术院）；Kyoto University（京都大学）；RIKEN AIP（理化学研究所AIP）；The University of Tokyo（东京大学）；University of Science and Technology（科学技术大学）
Tsinghua University（清华大学）；University of Southern California（南加利福尼亚大学）
We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a c…
We present a new optimization-theoretic approach to analyzing Follow-the-Leader style algorithms, particularly in the setting where perturbations are used as a tool for regularization. We show that adding a strongly convex penalty function to the decision rule and adding stochastic perturbations to data correspond to deterministic and stochastic smoothing operations, respectively. We establish an equivalence between "Follow the Regularized Leader" and "Follow the Perturbed Leader" up to the smoothness properties. This intuition leads to a new generic analysis framework that recovers and improv…
In an online decision problem, one makes decisions often with a pool of decision sequence called experts but without knowledge of the future. After each step, one pays a cost based on the decision and observed rate. One reasonal goal would be to perform as well as the best expert in the pool. The modern and well-known way to attain this goal is the algorithm of exponential weighting. However, recently, another algorithm called follow the perturbed leader is developed and achieved about the same performance. In our work, we first show the properties shared in common by the two algorithms which …
University of Pennsylvania（宾夕法尼亚大学）；Microsoft Research（微软研究院）
We show a principled way of deriving online learning algorithms from a minimax analysis. Various upper bounds on the minimax value, previously thought to be non-constructive, are shown to yield algorithms. This allows us to seamlessly recover known methods and to derive new ones. Our framework also captures such "unorthodox" methods as Follow the Perturbed Leader and the R^2 forecaster. We emphasize that understanding the inherent complexity of the learning problem leads to the development of algorithms. We define local sequential Rademacher complexities and associated algorithms that allow us…
State University of New York at Buffalo（纽约州立大学布法罗分校）；Zhejiang University（浙江大学）
Recent work on adversarial attack has shown that Projected Gradient Descent (PGD) Adversary is a universal first-order adversary, and the classifier adversarially trained by PGD is robust against a wide range of first-order attacks. It is worth noting that the original objective of an attack/defense model relies on a data distribution $p(\mathbf{x})$, typically in the form of risk maximization/minimization, e.g., $\max/\min\mathbb{E}_{p(\mathbf(x))}\mathcal{L}(\mathbf{x})$ with $p(\mathbf{x})$ some unknown data distribution and $\mathcal{L}(\cdot)$ a loss function. However, since PGD generates…
Recent work on follow the perturbed leader (FTPL) algorithms for the adversarial multi-armed bandit problem has highlighted the role of the hazard rate of the distribution generating the perturbations. Assuming that the hazard rate is bounded, it is possible to provide regret analyses for a variety of FTPL algorithms for the multi-armed bandit problem. This paper pushes the inquiry into regret bounds for FTPL algorithms beyond the bounded hazard rate condition. There are good reasons to do so: natural distributions such as the uniform and Gaussian violate the condition. We give regret bounds f…
Institute for Information Transmission Problems, Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
The games of prediction with expert advice are considered in this paper. We present some modification of Kalai and Vempala algorithm of following the perturbed leader for the case of unrestrictedly large one-step gains. We show that in general case the cumulative gain of any probabilistic prediction algorithm can be much worse than the gain of some expert of the pool. Nevertheless, we give the lower bound for this cumulative gain in general case and construct a universal algorithm which has the optimal performance; we also prove that in case when one-step gains of experts of the pool have ``li…
We provide the first oracle efficient sublinear regret algorithms for adversarial versions of the contextual bandit problem. In this problem, the learner repeatedly makes an action on the basis of a context and receives reward for the chosen action, with the goal of achieving reward competitive with a large class of policies. We analyze two settings: i) in the transductive setting the learner knows the set of contexts a priori, ii) in the small separator setting, there exists a small set of contexts such that any two policies behave differently in one of the contexts in the set. Our algorithms…
Nash Convergence of Mean-Based Learning Algorithms in First-Price Auctions
Center on Frontiers of Computing Studies, Peking University（前沿计算研究中心，北京大学）；Department of Electrical Engineering and Computer Sciences, University of California, Berkeley（电气工程与计算机科学系，加州大学伯克利分校）；School of Engineering and Applied Sciences, Harvard University（工程与应用科学系，哈佛大学）；Department of Computer Science, Yale University（计算机科学系，耶鲁大学）
