ML-based malware detection on dynamic analysis reports is vulnerable to both evasion and spurious correlations. In this work, we investigate a specific ML architecture employed in the pipeline of a widely-known commercial antivirus, with the goal to harden it against adversarial malware. Adversarial training, the most reliable defensive technique that can confer empirical robustness, is not applicable out of the box in this domain, for the principal reason that gradient-based perturbations rarely map back to feasible problem-space programs. We introduce a novel Reinforcement Learning approach for constructing adversarial examples, a constituent part of adversarially training a model against evasion. Our approach comes with multiple advantages. It performs modifications that are feasible in the problem-space, and only those; thus it circumvents the inverse mapping problem. It also makes it possible to provide theoretical guarantees on the robustness of the model against a well-defined set of adversarial capabilities. Our empirical exploration validates our theoretical insights, where we can consistently reach 0% Attack Success Rate after a few adversarial retraining iterations.

We study the two-player zero-sum extension of the partially observable stochastic shortest-path problem where one agent has only partial information about the environment. We formulate this problem as a partially observable stochastic game (POSG): given a set of target states and negative rewards for each transition, the player with imperfect information maximizes the expected undiscounted total reward until a target state is reached. The second player with the perfect information aims for the opposite. We base our formalism on POSGs with one-sided observability (OS-POSGs) and give the following contributions: (1) we introduce a novel heuristic search value iteration algorithm that iteratively solves depth-limited variants of the game, (2) we derive the bound on the depth guaranteeing an arbitrary precision, (3) we propose a novel upper-bound estimation that allows early terminations, and (4) we experimentally evaluate the algorithm on a pursuit-evasion game.

Information abstraction is one of the methods for tackling large extensive-form games (EFGs). Removing some information available to players reduces the memory required for computing and storing strategies. We present novel domain-independent abstraction methods for creating very coarse abstractions of EFGs that still compute strategies that are (near) optimal in the original game. First, the methods start with an arbitrary abstraction of the original game (domain-specific or the coarsest possible). Next, they iteratively detect which information is required in the abstract game so that a (near) optimal strategy in the original game can be found and include this information into the abstract game. Moreover, the methods are able to exploit imperfect-recall abstractions where players can even forget the history of their own actions. We present two algorithms that follow these steps -- FPIRA, based on fictitious play, and CFR+IRA, based on counterfactual regret minimization. The experimental evaluation confirms that our methods can closely approximate Nash equilibrium of large games using abstraction with only 0.9% of information sets of the original game.

Stackelberg security games (SSGs) have been deployed in many real-world situations to optimally allocate scarce resource to protect targets against attackers. However, actual human attackers are not perfectly rational and there are several behavior models that attempt to predict subrational behavior. Quantal response is among the most commonly used such models and Quantal Stackelberg Equilibrium (QSE) describes the optimal strategy to commit to when facing a subrational opponent. Non-concavity makes computing QSE computationally challenging and while there exist algorithms for computing QSE for SSGs, they cannot be directly used for solving an arbitrary game in the normal form. We (1) present a transformation of the primal problem for computing QSE using a Dinkelbach's method for any general-sum normal-form game, (2) provide a gradient-based and a MILP-based algorithm, give the convergence criteria, and bound their error, and finally (3) we experimentally demonstrate that using our novel transformation, a QSE can be closely approximated several orders of magnitude faster.

Finite state machines are a well-known representation of strategies in (in)finitely repeated or stochastic games. Actions of players correspond to states in the machine and the transition between machine-states are caused by observations in the game. For extensive-form games (EFGs), machines can act as a formal grounding for abstraction methods used for solving large EFGs and as a domain-independent approach for generating sufficiently compact abstractions. We show that using machines of a restricted size in EFGs can both (i) reduce the theoretical complexity of computing some solution concepts, including Strong Stackelberg Equilibrium (SSE), (ii) as well as bring new practical algorithms that compute near-optimal equilibria considering only a fraction of strategy space. Our contributions include (1) formal definition and theoretical characterization of machine strategies in EFGs, (2) formal definitions and complexity analysis for solution concepts and their computation when restricted to small classes of machines, (3) new algorithms for computing SSE in general-sum games and Nash Equilibrium in zero-sum games that both directly use the concept of machines. Experimental results on two different domains show that the algorithms compute near-optimal strategies and achieve significantly better scalability compared to previous state-of-the-art algorithms.

In many real-world problems, there is a dynamic interaction between competitive agents. Partially observable stochastic games (POSGs) are among the most general formal models that capture such dynamic scenarios. The model captures stochastic events, partial information of players about the environment, and the scenario does not have a fixed horizon. Solving POSGs in the most general setting is intractable.Therefore, the research has been focused on subclasses of POSGs that have a value of the game and admit designing (approximate) optimal algorithms. We propose such a subclass for two-player zero-sum games with discounted-sum objective function—POSGs with public observations (POPOSGs)—where each player is able to reconstruct beliefs of the other player over the unobserved states. Our results include: (1) theoretical analysis of PO-POSGs and their value functions showing convexity (concavity) in beliefs of maximizing (minimizing) player, (2) a novel algorithm for approximating the value of the game, and (3) a practical demonstration of scalability of our algorithm. Experimental results show that our algorithm can closely approximate the value of non-trivial games with hundreds of states.

Many problems from classical planning are applied in the environment with other, possibly adversarial agents. However, plans found by classical planning algorithms lack the robustness against the actions of other agents - the quality of computed plans can be significantly worse compared to the model. To explicitly reason about other (adversarial) agents, the game-theoretic framework can be used. The scalability of game-theoretic algorithms, however, is limited and often insufficient for real-world problems. In this paper, we combine classical domain-independent planning algorithms and game-theoretic strategy-generation algorithm where plans form strategies in the game. Our contribution is threefold. First, we provide the methodology for using classical planning in this game-theoretic framework. Second, we analyze the trade-off between the quality of the planning algorithm and the robustness of final randomized plans and the computation time. Finally, we analyze different variants of integration of classical planning algorithms into the game-theoretic framework and show that at the cost a minor loss in the robustness of final plans, we can significantly reduce the computation time.

Value methods for solving stochastic games with partial observability model the uncertainty of the players as a probability distribution over possible states, where the dimension of the belief space is the number of states. For many practical problems, there are exponentially many states which causes scalability problems. We propose an abstraction technique that addresses this curse of dimensionality by projecting the high-dimensional beliefs onto characteristic vectors of significantly lower dimension (e.g., marginal probabilities). Our main contributions are (1) a novel compact representation of the uncertainty in partially observable stochastic games and (2) a novel algorithm using this representation that is based on existing state-of-the-art algorithms for solving stochastic games with partial observability. Experimental evaluation confirms that the new algorithm using the compact representation dramatically increases scalability compared to the state of the art.

Partially observable Markov decision processes (POMDPs) are the standard models for planning under uncertainty with both finite and infinite horizon. Besides the well-known discounted-sum objective, indefinite-horizon objective (aka Goal-POMDPs) is another classical objective for POMDPs. In this case, given a set of target states and a positive cost for each transition, the optimization objective is to minimize the expected total cost until a target state is reached. In the literature, RTDP-Bel or heuristic search value iteration (HSVI) have been used for solving Goal-POMDPs. Neither of these algorithms has theoretical convergence guarantees, and HSVI may even fail to terminate its trials. We give the following contributions: (1) We discuss the challenges introduced in Goal-POMDPs and illustrate how they prevent the original HSVI from converging. (2) We present a novel algorithm inspired by HSVI, termed Goal-HSVI, and show that our algorithm has convergence guarantees. (3) We show that Goal-HSVI outperforms RTDP-Bel on a set of well-known examples.

Dynamic interaction appears in many real-world scenarios where players are able to observe (perhaps imperfectly) the actions of another player and react accordingly. We consider the baseline representation of dynamic games - the extensive form - and focus on computing Stackelberg equilibrium (SE), where the leader commits to a strategy to which the follower plays a best response. For one-shot games (e.g., security games), strategy-generation (SG) algorithms offer dramatic speed-up by incrementally expanding the strategy spaces. However, a direct application of SG to extensive-form games (EFGs) does not bring a similar speed-up since it typically results in a nearly-complete strategy space. Our contributions are twofold: (1) for the first time we introduce an algorithm that allows us to incrementally expand the strategy space to find a SE in EFGs; (2) we introduce a heuristic variant of the algorithm that is theoretically incomplete, but in practice allows us to find exact (or close-to optimal) Stackelberg equilibrium by constructing a significantly smaller strategy space. Our experimental evaluation confirms that we are able to compute SE by considering only a fraction of the strategy space that often leads to a significant speed-up in computation times.

Security problems can be modeled as two-player partially observable stochastic games with one-sided partial observability and infinite horizon (one-sided POSGs). We seek for optimal strategies of player 1 that correspond to robust strategies against the worst-case opponent (player 2) that is assumed to have a perfect information about the game. We present a novel algorithm for approximately solving one-sided POSGs based on the heuristic search value iteration (HSVI) for POMDPs. Our results include (1) theoretical properties of one-sided POSGs and their value functions, (2) guarantees showing the convergence of our algorithm to optimal strategies, and (3) practical demonstration of applicability and scalability of our algorithm on three different domains: pursuit-evasion, patrolling, and search games.

Extensive-form games with imperfect recall are an important game-theoretic model that allows a compact representation of strategies in dynamic strategic interactions. Practical use of imperfect recall games is limited due to negative theoretical results: a Nash equilibrium does not have to exist, computing maxmin strategies is NP-hard, and they may require irrational numbers. We present the first algorithm for approximating maxmin strategies in two-player zero-sum imperfect recall games without absentmindedness. We modify the well-known sequence-form linear program to model strategies in imperfect recall games and use a recent technique to approximate bilinear terms. Our main algorithm is a branch-and-bound search over these linear programs that provably reaches a desired approximation after an exponential number of steps in the size of the game. Experimental evaluation shows that the proposed algorithm can approximate maxmin strategies of randomly generated imperfect recall games of sizes beyond toy-problems within few minutes.

Pursuit-evasion scenarios appear widely in robotics, security domains, and many other real-world situations. We focus on two-player pursuit-evasion games with concurrent moves, infinite horizon, and discounted rewards. We assume that the players have a partial observability, however, the evader is given an advantage of knowing the current position of the units of the pursuer. This setting is particularly interesting for security domains where a robust strategy, designed to maximize the utility in the worst-case scenario, is often desirable. We provide, to the best of our knowledge, the first algorithm that provably converges to the value of a partially observable pursuit-evasion game with infinite horizon. Our algorithm extends well-known value iteration algorithm by exploiting that (1) the value functions of our game depend only on position of the pursuer and the belief he has about the current position of the evader, and (2) that these functions are piecewise linear and convex in the belief space.

We solve large two-player zero-sum extensive-form games with perfect recall. We propose a new algorithm based on fictitious play that significantly reduces memory requirements for storing average strategies. The key feature is exploiting imperfect recall abstractions while preserving the convergence rate and guarantees of fictitious play applied directly to the perfect recall game. The algorithm creates a coarse imperfect recall abstraction of the perfect recall game and automatically refines its information set structure only where the imperfect recall might cause problems. Experimental evaluation shows that our novel algorithm is able to solve a simplified poker game with 7.10^5 information sets using an abstracted game with only 1.8% of information sets of the original game. Additional experiments on poker and randomly generated games suggest that the relative size of the abstraction decreases as the size of the solved games increases.

The Strong Stackelberg Equilibrium (SSE) has drawn extensive attention recently in several security domains. However, the SSE concept neglects the advantage of defender's strategic revelation of her private information, and overestimates the observation ability of the adversaries. In this paper, we overcome these restrictions and analyze the tradeoff between strategic secrecy and commitment in security games. We propose a Disguised-resource Security Game (DSG) where the defender strategically disguises some of her resources. We compare strategic information revelation with public commitment and formally show that they have different advantages depending the payoff structure. To compute the Perfect Bayesian Equilibrium (PBE), several novel approaches are provided, including a novel algorithm based on support set enumeration, and an approximation algorithm for \epsilon-PBE. Extensive experimental evaluation shows that both strategic secrecy and Stackelberg commitment are critical measures in security domain, and our approaches can efficiently solve PBEs for realistic-sized problems.

Strong Stackelberg Equilibrium (SSE) is a fundamental solution concept in game theory in which one player commits to a strategy, while the other player observes this commitment and plays a best response. We present a new algorithm for computing SSE for two-player extensive-form general-sum games with imperfect information (EFGs) where computing SSE is an NP-hard problem. Our algorithm is based on a correlated version of SSE, known as Stackelberg Extensive-Form Correlated Equilibrium (SEFCE). Our contribution is therefore twofold: (1) we give the first linear program for computing SEFCE in EFGs without chance, (2) we repeatedly solve and modify this linear program in a systematic search until we arrive to SSE. Our new algorithm outperforms the best previous algorithms by several orders of magnitude.

Stackelberg equilibrium is a solution concept prescribing for a player an optimal strategy to commit to, assuming the opponent knows this commitment and plays the best response. Although this solution concept is a cornerstone of many security applications, the existing works typically do not consider situations where the players can observe and react to the actions of the opponent during the course of the game. We extend the existing algorithmic work to extensive-form games and introduce novel algorithm for computing Stackelberg equilibria that exploits the compact sequence-form representation of strategies. Our algorithm reduces the size of the linear programs from exponential in the baseline approach to linear in the size of the game tree. Experimental evaluation on randomly generated games and a security-inspired search game demonstrates significant improvement in the scalability compared to the baseline approach.

Many search and security games played on a graph can be modeled as normal-form zero-sum games with strategies consisting of sequences of actions. The size of the strategy space provides a computational challenge when solving these games. This complexity is tackled either by using the compact representation of sequential strategies and linear programming, or by incremental strategy generation of iterative double-oracle methods. In this paper, we present novel hybrid of these two approaches: compact-strategy double-oracle (CS-DO) algorithm that combines the advantages of the compact representation with incremental strategy generation. We experimentally compare CS-DO with the standard approaches and analyze the impact of the size of the support on the performance of the algorithms. Results show that CS-DO dramatically improves the convergence rate in games with non-trivial support

The Stackelberg equilibrium is a solution concept that describes optimal strategies to commit to: Player 1 ( the leader ) first commits to a strategy that is publicly announced, then Player 2 ( the follower ) plays a best response to the leader’s choice. We study Stackelberg equilibria in finite sequential (i.e., extensive-form) games and provide new exact algorithms, approximate algorithms, and hardness results for finding equilibria for several classes of such two-player games. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nash equilibrium (NE) is the best known solution concept used in game theory. It is known that NE is particularly weak even in zero-sum extensive-form games since it can prescribe irrational actions to play that do not exploit mistakes made by an imperfect opponent. These issues are addressed by a number of refinements of NE that strengthen the requirements for equilibrium strategies. However, a thorough experimental analysis of practical performance of the Nash equilibria refinement strategies is, to the best of our knowledge, missing. This paper aims to fill this void and provides the first broader experimental comparison of the quality of refined Nash strategies in zero-sum extensive-form games. The experimental results suggest that (1) there is a significant difference between the best and the worst NE strategy against imperfect opponents, (2) the existing refinements outperform the worst NE strategy, (3) they typically perform close to the best possible NE strategy, and (4) the difference in performance of all compared refinements is very small.

Developing scalable solution algorithms is one of the central problems in computational game theory. We present an iterative algorithm for computing an exact Nash equilibrium for two-player zero-sum extensive-form games with imperfect information. Our approach combines two key elements: (1) the compact sequence-form representation of extensive-form games and (2) the algorithmic framework of double-oracle methods. The main idea of our algorithm is to restrict the game by allowing the players to play only selected sequences of available actions. After solving the restricted game, new sequences are added by finding best responses to the current solution using fast algorithms. We experimentally evaluate our algorithm on a set of games inspired by patrolling scenarios, board, and card games. The results show significant runtime improvements in games admitting an equilibrium with small support, and substantial improvement in memory use even on games with large support. The improvement in memory use is particularly important because it allows our algorithm to solve much larger game instances than existing linear programming methods. Our main contributions include (1) a generic sequence-form double-oracle algorithm for solving zero-sum extensive-form games; (2) fast methods for maintaining a valid restricted game model when adding new sequences; (3) a search algorithm and pruning methods for computing best-response sequences; (4) theoretical guarantees about the convergence of the algorithm to a Nash equilibrium; (5) experimental analysis of our algorithm on several games, including an approximate version of the algorithm.

In this paper, we study Monte Carlo tree search (MCTS) in zero-sum extensive-form games with perfect information and simultaneous moves. We present a general template of MCTS algorithms for these games, which can be instantiated by various selection methods. We formally prove that if a selection method is $\epsilon$-Hannan consistent in a matrix game and satisfies additional requirements on exploration, then the MCTS algorithm eventually converges to an approximate Nash equilibrium (NE) of the extensive-form game. We empirically evaluate this claim using regret matching and Exp3 as the selection methods on randomly generated and worst case games. We confirm the formal result and show that additional MCTS variants also converge to approximate NE on the evaluated games.