Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed \emph{randomized weighted majority} (RWM), also known as "Hedge" or "Multiplicative Weights Update," which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribution over all possible actions. In typical settings of interest the action space is exponentially large, seemingly rendering RWM useless in practice. In this work, we refute this notion for a broad variety of \emph{structured} games, showing it is possible to efficiently (approximately) sample the action space in RWM in \emph{polylogarithmic} time. This gives the first efficient no-regret algorithms for problems such as the \emph{(discrete) Colonel Blotto game}, \emph{matroid congestion}, \emph{matroid security}, and basic \emph{dueling games}. As an immediate corollary, we give a polylogarithmic time meta-algorithm to compute approximate Nash Equilibria for these games that is exponentially faster than prior methods in several important settings. Further, our algorithm is the first to efficiently compute equilibria for more involved variants of these games with general sums, more than two players, and, for Colonel Blotto, multiple resource types.
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Self-stabilization is an excellent approach for adding fault tolerance to a distributed multi-agent system. However, two properties of self-stabilization theory, convergence and closure, may not be satisfied if agents are selfish. To guarantee convergence, we formulate the problem as a stochastic Bayesian game and introduce probabilistic self-stabilization to adjust the probabilities of rules with behavior strategies. This satisfies agents' self-interests such that no agent deviates the rules. To guarantee closure in the presence of selfish agents, we propose fault-containment as a method to constrain legitimate configurations of the self-stabilizing system to be Nash equilibria. We also assume selfish agents as capable of performing unauthorized actions at any time, which threatens both properties, and present a stepwise solution to handle it. As a case study, we consider the problem of distributed clustering and propose five self-stabilizing algorithms for forming clusters. Simulation results show that our algorithms react correctly to rule deviations and outperform comparable schemes in terms of fairness and stabilization time.
This paper focuses on robotic reinforcement learning with sparse rewards for natural language goal representations. An open problem is the sample-inefficiency that stems from the compositionality of natural language, and from the grounding of language in sensory data and actions. We address these issues with three contributions. We first present a mechanism for hindsight instruction replay utilizing expert feedback. Second, we propose a seq2seq model to generate linguistic hindsight instructions. Finally, we present a novel class of language-focused learning tasks. We show that hindsight instructions improve the learning performance, as expected. In addition, we also provide an unexpected result: We show that the learning performance of our agent can be improved by one third if, in a sense, the agent learns to talk to itself in a self-supervised manner. We achieve this by learning to generate linguistic instructions that would have been appropriate as a natural language goal for an originally unintended behavior. Our results indicate that the performance gain increases with the task-complexity.
This paper concerns the approximation of smooth, high-dimensional functions from limited samples using polynomials. This task lies at the heart of many applications in computational science and engineering -- notably, those arising from parametric modelling and uncertainty quantification. It is common to use Monte Carlo (MC) sampling in such applications, so as not to succumb to the curse of dimensionality. However, it is well known this strategy is theoretically suboptimal. There are many polynomial spaces of dimension $n$ for which the sample complexity scales log-quadratically in $n$. This well-documented phenomenon has led to a concerted effort to design improved, in fact, near-optimal strategies, whose sample complexities scale log-linearly, or even linearly in $n$. Paradoxically, in this work we show that MC is actually a perfectly good strategy in high dimensions. We first document this phenomenon via several numerical examples. Next, we present a theoretical analysis that resolves this paradox for holomorphic functions of infinitely-many variables. We show that there is a least-squares scheme based on $m$ MC samples whose error decays algebraically fast in $m/\log(m)$, with a rate that is the same as that of the best $n$-term polynomial approximation. This result is non-constructive, since it assumes knowledge of a suitable polynomial space in which to perform the approximation. We next present a compressed sensing-based scheme that achieves the same rate, except for a larger polylogarithmic factor. This scheme is practical, and numerically it performs as well as or better than well-known adaptive least-squares schemes. Overall, our findings demonstrate that MC sampling is eminently suitable for smooth function approximation when the dimension is sufficiently high. Hence the benefits of improved sampling strategies are generically limited to lower-dimensional settings.
Although the field of multi-agent reinforcement learning (MARL) has made considerable progress in the last years, solving systems with a large number of agents remains a hard challenge. Graphon mean field games (GMFGs) enable the scalable analysis of MARL problems that are otherwise intractable. By the mathematical structure of graphons, this approach is limited to dense graphs which are insufficient to describe many real-world networks such as power law graphs. Our paper introduces a novel formulation of GMFGs, called LPGMFGs, which leverages the graph theoretical concept of $L^p$ graphons and provides a machine learning tool to efficiently and accurately approximate solutions for sparse network problems. This especially includes power law networks which are empirically observed in various application areas and cannot be captured by standard graphons. We derive theoretical existence and convergence guarantees and give empirical examples that demonstrate the accuracy of our learning approach for systems with many agents. Furthermore, we rigorously extend the Online Mirror Descent (OMD) learning algorithm to our setup to accelerate learning speed, allow for agent interaction through the mean field in the transition kernel, and empirically show its capabilities. In general, we provide a scalable, mathematically well-founded machine learning approach to a large class of otherwise intractable problems of great relevance in numerous research fields.
In this paper we derive a Toeplitz-structured closed form of the unique positive semi-definite stabilizing solution for the discrete-time algebraic Riccati equations, especially for the case that the state matrix is not stable. Based on the found form and fast Fourier transform, we propose a new algorithm for solving both discrete-time and continuous-time large-scale algebraic Riccati equations with low-rank structure. It works without unnecessary assumptions, complicated shift selection strategies, or matrix calculations of the cubic order with respect to the problem scale. Numerical examples are given to illustrate its features. Besides, we show that it is theoretically equivalent to several algorithms existing in the literature in the sense that they all produce the same sequence under the same parameter setting.
We study a posterior sampling approach to efficient exploration in constrained reinforcement learning. Alternatively to existing algorithms, we propose two simple algorithms that are more efficient statistically, simpler to implement and computationally cheaper. The first algorithm is based on a linear formulation of CMDP, and the second algorithm leverages the saddle-point formulation of CMDP. Our empirical results demonstrate that, despite its simplicity, posterior sampling achieves state-of-the-art performance and, in some cases, significantly outperforms optimistic algorithms.
In this paper, we consider a discrete-time Stackelberg graphon mean field game with a finite number of leaders, a finite number of major followers and an infinite number of minor followers. The leaders and the followers each observe types privately that evolve as conditionally independent controlled Markov processes. The leaders and the followers sequentially make strategic decisions where each follower's actions affect her neighbors, which is captured in a graph generated by a known graphon, however, the leaders' actions affect everyone. The leaders are of "Stackelberg" kind which means each of them commits to a dynamic policy and all the followers(both major and minor) best respond to that policy and each other. Knowing that the minor followers would best respond (in the sense of a mean-field game) while the major followers will best respond (in the sense of NAsh) based on their policies, each leader chooses a policy that maximizes her reward knowing that other leader's are doing the same. We refer to the resulting outcome as a Stackelberg Graphon Mean Field Equilibrium with multiple leaders (SGMFE-ML). In this paper, we provide a master equation of this game that allows one to compute all SGMFE-ML. We further extend this notion to the case when there are an infinite number of leaders.
We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.
Multi-agent influence diagrams (MAIDs) are a popular form of graphical model that, for certain classes of games, have been shown to offer key complexity and explainability advantages over traditional extensive form game (EFG) representations. In this paper, we extend previous work on MAIDs by introducing the concept of a MAID subgame, as well as subgame perfect and trembling hand perfect equilibrium refinements. We then prove several equivalence results between MAIDs and EFGs. Finally, we describe an open source implementation for reasoning about MAIDs and computing their equilibria.
For deploying a deep learning model into production, it needs to be both accurate and compact to meet the latency and memory constraints. This usually results in a network that is deep (to ensure performance) and yet thin (to improve computational efficiency). In this paper, we propose an efficient method to train a deep thin network with a theoretic guarantee. Our method is motivated by model compression. It consists of three stages. In the first stage, we sufficiently widen the deep thin network and train it until convergence. In the second stage, we use this well-trained deep wide network to warm up (or initialize) the original deep thin network. This is achieved by letting the thin network imitate the immediate outputs of the wide network from layer to layer. In the last stage, we further fine tune this well initialized deep thin network. The theoretical guarantee is established by using mean field analysis, which shows the advantage of layerwise imitation over traditional training deep thin networks from scratch by backpropagation. We also conduct large-scale empirical experiments to validate our approach. By training with our method, ResNet50 can outperform ResNet101, and BERT_BASE can be comparable with BERT_LARGE, where both the latter models are trained via the standard training procedures as in the literature.
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