Double-oracle sampling method for Stackelberg Equilibrium approximation in general-sum extensive-form games

The paper presents a new method for approximating Strong Stackelberg Equilibrium in general-sum sequential games with imperfect information and perfect recall. The proposed approach is generic as it does not rely on any specific properties of a particular game model. The method is based on iterative interleaving of the two following phases: (1) guided Monte Carlo Tree Search sampling of the Follower's strategy space and (2) building the Leader's behavior strategy tree for which the sampled Follower's strategy is an optimal response. The above solution scheme is evaluated with respect to expected Leader's utility and time requirements on three sets of interception games with variable characteristics, played on graphs. A comparison with three state-of-the-art MILP/LP-based methods shows that in vast majority of test cases proposed simulation-based approach leads to optimal Leader's strategies, while excelling the competitive methods in terms of better time scalability and lower memory requirements

Double-Oracle Deep Reinforcement Learning for Handling Exponential Action Space in Sequential Stackelberg Security Games

Standard Stackelberg Security Games (SSGs) assume attackers to be myopic players that select only a single target based on the defender's strategy. In this paper, we consider sequential SSGs, in which attackers launch multiple attacks sequentially. With a sequence of events, however, the defender's action space grows exponentially with the number of time steps, making the problem computationally intractable. To handle this issue, this paper presents the following contributions. First, we use the Double Oracle algorithm to iteratively derive player strategies. Second, we use Advantage Actor-Critic models to approximate best response policies for both players. Lastly, we represent the defender action space compactly with marginal probabilities instead of enumerating all possible actions. Overall, our experiments show that the Double Oracle algorithm not only allows us to search through defender strategies effectively and efficiently, but also provides optimal solutions that outperform other models at scalable settings

Function Approximation for Solving Stackelberg Equilibrium in Large Perfect Information Games

Function approximation (FA) has been a critical component in solving large zero-sum games. Yet, little attention has been given towards FA in solving \textit{general-sum} extensive-form games, despite them being widely regarded as being computationally more challenging than their fully competitive or cooperative counterparts. A key challenge is that for many equilibria in general-sum games, no simple analogue to the state value function used in Markov Decision Processes and zero-sum games exists. In this paper, we propose learning the \textit{Enforceable Payoff Frontier} (EPF) -- a generalization of the state value function for general-sum games. We approximate the optimal \textit{Stackelberg extensive-form correlated equilibrium} by representing EPFs with neural networks and training them by using appropriate backup operations and loss functions. This is the first method that applies FA to the Stackelberg setting, allowing us to scale to much larger games while still enjoying performance guarantees based on FA error. Additionally, our proposed method guarantees incentive compatibility and is easy to evaluate without having to depend on self-play or approximate best-response oracles.Comment: To appear in AAAI 202

Efficient Stackelberg Strategies for Finitely Repeated Games

We study the problem of efficiently computing optimal strategies in asymmetric leader-follower games repeated a finite number of times, which presents a different set of technical challenges than the infinite-horizon setting. More precisely, we give efficient algorithms for finding approximate Stackelberg equilibria in finite-horizon repeated two-player games, along with rates of convergence depending on the horizon $T$. We give two algorithms, one computing strategies with an optimal $\frac{1}{T}$ rate at the expense of an exponential dependence on the number of actions, and another (randomized) approach computing strategies with no dependence on the number of actions but a worse dependence on $T$ of $\frac{1}{T^{0.25}}$. Both algorithms build upon a linear program to produce simple automata leader strategies and induce corresponding automata best-responses for the follower. We complement these results by showing that approximating the Stackelberg value in three-player finite-horizon repeated games is a computationally hard problem via a reduction from the balanced vertex cover problem.Comment: An earlier version of this paper used incorrect asymptotic notation in the statement of the main hardness result as well as in the description of related hardness results in a table (in the related work section). The proofs and implication of the result remain unchanged, but a correction has been made to the statement of the resul

An Introduction to Bi-level Optimization: Foundations and Applications in Signal Processing and Machine Learning

Recently, bi-level optimization (BLO) has taken center stage in some very exciting developments in the area of signal processing (SP) and machine learning (ML). Roughly speaking, BLO is a classical optimization problem that involves two levels of hierarchy (i.e., upper and lower levels), wherein obtaining the solution to the upper-level problem requires solving the lower-level one. BLO has become popular largely because it is powerful in modeling problems in SP and ML, among others, that involve optimizing nested objective functions. Prominent applications of BLO range from resource allocation for wireless systems to adversarial machine learning. In this work, we focus on a class of tractable BLO problems that often appear in SP and ML applications. We provide an overview of some basic concepts of this class of BLO problems, such as their optimality conditions, standard algorithms (including their optimization principles and practical implementations), as well as how they can be leveraged to obtain state-of-the-art results for a number of key SP and ML applications. Further, we discuss some recent advances in BLO theory, its implications for applications, and point out some limitations of the state-of-the-art that require significant future research efforts. Overall, we hope that this article can serve to accelerate the adoption of BLO as a generic tool to model, analyze, and innovate on a wide array of emerging SP and ML applications

Many-agent Reinforcement Learning

Multi-agent reinforcement learning (RL) solves the problem of how each agent should behave optimally in a stochastic environment in which multiple agents are learning simultaneously. It is an interdisciplinary domain with a long history that lies in the joint area of psychology, control theory, game theory, reinforcement learning, and deep learning. Following the remarkable success of the AlphaGO series in single-agent RL, 2019 was a booming year that witnessed significant advances in multi-agent RL techniques; impressive breakthroughs have been made on developing AIs that outperform humans on many challenging tasks, especially multi-player video games. Nonetheless, one of the key challenges of multi-agent RL techniques is the scalability; it is still non-trivial to design efficient learning algorithms that can solve tasks including far more than two agents ($N \gg 2$), which I name by \emph{many-agent reinforcement learning} (MARL\footnote{I use the world of ``MARL" to denote multi-agent reinforcement learning with a particular focus on the cases of many agents; otherwise, it is denoted as ``Multi-Agent RL" by default.}) problems. In this thesis, I contribute to tackling MARL problems from four aspects. Firstly, I offer a self-contained overview of multi-agent RL techniques from a game-theoretical perspective. This overview fills the research gap that most of the existing work either fails to cover the recent advances since 2010 or does not pay adequate attention to game theory, which I believe is the cornerstone to solving many-agent learning problems. Secondly, I develop a tractable policy evaluation algorithm -- $\alpha^\alpha$-Rank -- in many-agent systems. The critical advantage of $\alpha^\alpha$-Rank is that it can compute the solution concept of $\alpha$-Rank tractably in multi-player general-sum games with no need to store the entire pay-off matrix. This is in contrast to classic solution concepts such as Nash equilibrium which is known to be $PPAD$-hard in even two-player cases. $\alpha^\alpha$-Rank allows us, for the first time, to practically conduct large-scale multi-agent evaluations. Thirdly, I introduce a scalable policy learning algorithm -- mean-field MARL -- in many-agent systems. The mean-field MARL method takes advantage of the mean-field approximation from physics, and it is the first provably convergent algorithm that tries to break the curse of dimensionality for MARL tasks. With the proposed algorithm, I report the first result of solving the Ising model and multi-agent battle games through a MARL approach. Fourthly, I investigate the many-agent learning problem in open-ended meta-games (i.e., the game of a game in the policy space). Specifically, I focus on modelling the behavioural diversity in meta-games, and developing algorithms that guarantee to enlarge diversity during training. The proposed metric based on determinantal point processes serves as the first mathematically rigorous definition for diversity. Importantly, the diversity-aware learning algorithms beat the existing state-of-the-art game solvers in terms of exploitability by a large margin. On top of the algorithmic developments, I also contribute two real-world applications of MARL techniques. Specifically, I demonstrate the great potential of applying MARL to study the emergent population dynamics in nature, and model diverse and realistic interactions in autonomous driving. Both applications embody the prospect that MARL techniques could achieve huge impacts in the real physical world, outside of purely video games

Special Topics in Information Technology

This open access book presents thirteen outstanding doctoral dissertations in Information Technology from the Department of Electronics, Information and Bioengineering, Politecnico di Milano, Italy. Information Technology has always been highly interdisciplinary, as many aspects have to be considered in IT systems. The doctoral studies program in IT at Politecnico di Milano emphasizes this interdisciplinary nature, which is becoming more and more important in recent technological advances, in collaborative projects, and in the education of young researchers. Accordingly, the focus of advanced research is on pursuing a rigorous approach to specific research topics starting from a broad background in various areas of Information Technology, especially Computer Science and Engineering, Electronics, Systems and Control, and Telecommunications. Each year, more than 50 PhDs graduate from the program. This book gathers the outcomes of the thirteen best theses defended in 2019-20 and selected for the IT PhD Award. Each of the authors provides a chapter summarizing his/her findings, including an introduction, description of methods, main achievements and future work on the topic. Hence, the book provides a cutting-edge overview of the latest research trends in Information Technology at Politecnico di Milano, presented in an easy-to-read format that will also appeal to non-specialists

PARL: A Unified Framework for Policy Alignment in Reinforcement Learning

We present a novel unified bilevel optimization-based framework, \textsf{PARL}, formulated to address the recently highlighted critical issue of policy alignment in reinforcement learning using utility or preference-based feedback. We identify a major gap within current algorithmic designs for solving policy alignment due to a lack of precise characterization of the dependence of the alignment objective on the data generated by policy trajectories. This shortfall contributes to the sub-optimal performance observed in contemporary algorithms. Our framework addressed these concerns by explicitly parameterizing the distribution of the upper alignment objective (reward design) by the lower optimal variable (optimal policy for the designed reward). Interestingly, from an optimization perspective, our formulation leads to a new class of stochastic bilevel problems where the stochasticity at the upper objective depends upon the lower-level variable. To demonstrate the efficacy of our formulation in resolving alignment issues in RL, we devised an algorithm named \textsf{A-PARL} to solve PARL problem, establishing sample complexity bounds of order $\mathcal{O}(1/T)$. Our empirical results substantiate that the proposed \textsf{PARL} can address the alignment concerns in RL by showing significant improvements (up to 63\% in terms of required samples) for policy alignment in large-scale environments of the Deepmind control suite and Meta world tasks

Computing Optimal Equilibria and Mechanisms via Learning in Zero-Sum Extensive-Form Games

We introduce a new approach for computing optimal equilibria via learning in games. It applies to extensive-form settings with any number of players, including mechanism design, information design, and solution concepts such as correlated, communication, and certification equilibria. We observe that optimal equilibria are minimax equilibrium strategies of a player in an extensive-form zero-sum game. This reformulation allows to apply techniques for learning in zero-sum games, yielding the first learning dynamics that converge to optimal equilibria, not only in empirical averages, but also in iterates. We demonstrate the practical scalability and flexibility of our approach by attaining state-of-the-art performance in benchmark tabular games, and by computing an optimal mechanism for a sequential auction design problem using deep reinforcement learning