A conceptually appealing approach for learning Extensive-Form Games (EFGs) is to convert them to Normal-Form Games (NFGs). This approach enables us to directly translate state-of-the-art techniques and analyses in NFGs to learning EFGs, but typically suffers from computational intractability due to the exponential blow-up of the game size introduced by the conversion. In this paper, we address this problem in natural and important setups for the \emph{$\Phi$-Hedge} algorithm---A generic algorithm capable of learning a large class of equilibria for NFGs. We show that $\Phi$-Hedge can be directly used to learn Nash Equilibria (zero-sum settings), Normal-Form Coarse Correlated Equilibria (NFCCE), and Extensive-Form Correlated Equilibria (EFCE) in EFGs. We prove that, in those settings, the \emph{$\Phi$-Hedge} algorithms are equivalent to standard Online Mirror Descent (OMD) algorithms for EFGs with suitable dilated regularizers, and run in polynomial time. This new connection further allows us to design and analyze a new class of OMD algorithms based on modifying its log-partition function. In particular, we design an improved algorithm with balancing techniques that achieves a sharp $\widetilde{\mathcal{O}}(\sqrt{XAT})$ EFCE-regret under bandit-feedback in an EFG with $X$ information sets, $A$ actions, and $T$ episodes. To our best knowledge, this is the first such rate and matches the information-theoretic lower bound.

A conceptually appealing approach for learning Extensive-Form Games (EFGs) is to convert them to Normal-Form Games (NFGs). This approach enables us to directly translate state-of-the-art techniques and analyses in NFGs to learning EFGs, but typically suffers from computational intractability due to the exponential blow-up of the game size introduced by the conversion. In this paper, we address this problem in natural and important setups for the \emph{ \Phi -Hedge} algorithm---A generic algorithm capable of learning a large class of equilibria for NFGs. We show that \Phi -Hedge can be directly used to learn Nash Equilibria (zero-sum settings), Normal-Form Coarse Correlated Equilibria (NFCCE), and Extensive-Form Correlated Equilibria (EFCE) in EFGs. We prove that, in those settings, the \emph{ \Phi -Hedge} algorithms are equivalent to standard Online Mirror Descent (OMD) algorithms for EFGs with suitable dilated regularizers, and run in polynomial time.

We present local mappings that relate the marginal probabilities of a global probability mass function represented by its primal normal factor graph to the corresponding marginal probabilities in its dual normal factor graph. The mapping is based on the Fourier transform of the local factors of the models. Details of the mapping are provided for the Ising model, where it is proved that the local extrema of the fixed points are attained at the phase transition of the two-dimensional nearest-neighbor Ising model. The results are further extended to the Potts model, to the clock model, and to Gaussian Markov random fields. By employing the mapping, we can transform simultaneously all the estimated marginal probabilities from the dual domain to the primal domain (and vice versa), which is advantageous if estimating the marginals can be carried out more efficiently in the dual domain. An example of particular significance is the ferromagnetic Ising model in a positive external magnetic field. For this model, there exists a rapidly mixing Markov chain (called the subgraphs--world process) to generate configurations in the dual normal factor graph of the model. Our numerical experiments illustrate that the proposed procedure can provide more accurate estimates of marginal probabilities of a global probability mass function in various settings.

The dual normal factor graph and the factor graph duality theorem have been considered for discrete graphical models. In this paper, we show an application of the factor graph duality theorem to continuous graphical models. Specifically, we propose a method to solve exactly the Gaussian graphical models defined on the ladder graph if certain conditions on the local covariance matrices are satisfied. Unlike the conventional approaches, the efficiency of the method depends on the position of the zeros in the local covariance matrices. The method and details of the dualization are illustrated on two toy examples.

Solution concepts of traditional game theory assume entirely rational players; therefore, their ability to exploit subrational opponents is limited. One type of subrationality that describes human behavior well is the quantal response. While there exist algorithms for computing solutions against quantal opponents, they either do not scale or may provide strategies that are even worse than the entirely-rational Nash strategies. This paper aims to analyze and propose scalable algorithms for computing effective and robust strategies against a quantal opponent in normal-form and extensive-form games. Our contributions are: (1) we define two different solution concepts related to exploiting quantal opponents and analyze their properties; (2) we prove that computing these solutions is computationally hard; (3) therefore, we evaluate several heuristic approximations based on scalable counterfactual regret minimization (CFR); and (4) we identify a CFR variant that exploits the bounded opponents better than the previously used variants while being less exploitable by the worst-case perfectly-rational opponent.

We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph. Analogously, we give a combinatorial characterization of the Bethe partition function, whose original definition was also of an analytical nature. As we point out, our approach has similarities to the replica method, but also stark differences. The above findings are a natural backdrop for introducing a decoder for graph-based codes that we will call symbolwise graph-cover decoding, a decoder that extends our earlier work on blockwise graph-cover decoding. Both graph-cover decoders are theoretical tools that help towards a better understanding of message-passing iterative decoding, namely blockwise graph-cover decoding links max-product (min-sum) algorithm decoding with linear programming decoding, and symbolwise graph-cover decoding links sum-product algorithm decoding with Bethe free energy function minimization at temperature one. In contrast to the Gibbs entropy function, which is a concave function, the Bethe entropy function is in general not concave everywhere. In particular, we show that every code picked from an ensemble of regular low-density parity-check codes with minimum Hamming distance growing (with high probability) linearly with the block length has a Bethe entropy function that is convex in certain regions of its domain.