Neural Information Processing Systems http://nips.cc/

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximum entropy distribution is intractable [BGS14]. We provide computationally efficient versions of this principle when the mean parameters are pairwise moments: we design distributions that approximately match given pairwise moments, while having entropy which is comparable to the maximum entropy distribution matching those moments. We additionally provide surprising applications of the approximate maximum entropy principle to designing provable variational methods for partition function calculations for Ising models without any assumptions on the potentials of the model. More precisely, we show that we can get approximation guarantees for the log-partition function comparable to those in the low-temperature limit, which is the setting of optimization of quadratic forms over the hypercube.

Maximum entropy principle (MEP) offers an effective and unbiased approach to inferring unknown probability distributions when faced with incomplete information, while neural networks provide the flexibility to learn complex distributions from data. This paper proposes a novel neural network architecture, the MEP-Net, which combines the MEP with neural networks to generate probability distributions from moment constraints. We also provide a comprehensive overview of the fundamentals of the maximum entropy principle, its mathematical formulations, and a rigorous justification for its applicability for non-equilibrium systems based on the large deviations principle. Through fruitful numerical experiments, we demonstrate that the MEP-Net can be particularly useful in modeling the evolution of probability distributions in biochemical reaction networks and in generating complex distributions from data.

The well known maximum-entropy principle due to Jaynes, which states that given mean parameters, the maximum entropy distribution matching them is in an exponential family has been very popular in machine learning due to its "Occam's razor" interpretation. Unfortunately, calculating the potentials in the maximumentropy distribution is intractable [BGS14]. We provide computationally efficient versions of this principle when the mean parameters are pairwise moments: we design distributions that approximately match given pairwise moments, while having entropy which is comparable to the maximum entropy distribution matching those moments. We additionally provide surprising applications of the approximate maximum entropy principle to designing provable variational methods for partition function calculations for Ising models without any assumptions on the potentials of the model. More precisely, we show that we can get approximation guarantees for the log-partition function comparable to those in the low-temperature limit, which is the setting of optimization of quadratic forms over the hypercube.

In this paper, we introduce the maximum casual entropy Inverse Reinforcement Learning (IRL) problem for discrete-time mean-field games (MFGs) under an infinite-horizon discounted-reward optimality criterion. The state space of a typical agent is finite. Our approach begins with a comprehensive review of the maximum entropy IRL problem concerning deterministic and stochastic Markov decision processes (MDPs) in both finite and infinite-horizon scenarios. Subsequently, we formulate the maximum casual entropy IRL problem for MFGs--a non-convex optimization problem with respect to policies. Leveraging the linear programming formulation of MDPs, we restructure this IRL problem into a convex optimization problem and establish a gradient descent algorithm to compute the optimal solution with a rate of convergence. Finally, we present a new algorithm by formulating the MFG problem as a generalized Nash equilibrium problem (GNEP), which is capable of computing the mean-field equilibrium (MFE) for the forward RL problem. This method is employed to produce data for a numerical example. We note that this novel algorithm is also applicable to general MFE computations. Keywords: Mean-field games, inverse reinforcement learning, maximum causal entropy, discounted reward.

Herding is a deterministic algorithm used to generate data points that can be regarded as random samples satisfying input moment conditions. The algorithm is based on the complex behavior of a high-dimensional dynamical system and is inspired by the maximum entropy principle of statistical inference. In this paper, we propose an extension of the herding algorithm, called entropic herding, which generates a sequence of distributions instead of points. Entropic herding is derived as the optimization of the target function obtained from the maximum entropy principle. Using the proposed entropic herding algorithm as a framework, we discuss a closer connection between herding and the maximum entropy principle. Specifically, we interpret the original herding algorithm as a tractable version of entropic herding, the ideal output distribution of which is mathematically represented. We further discuss how the complex behavior of the herding algorithm contributes to optimization. We argue that the proposed entropic herding algorithm extends the application of herding to probabilistic modeling. In contrast to original herding, entropic herding can generate a smooth distribution such that both efficient probability density calculation and sample generation become possible. To demonstrate the viability of these arguments in this study, numerical experiments were conducted, including a comparison with other conventional methods, on both synthetic and real data.

While the utility of well-chosen abstractions for understanding and predicting the behaviour of complex systems is well appreciated, precisely what an abstraction $\textit{is}$ has so far has largely eluded mathematical formalization. In this paper, we aim to set out a mathematical theory of abstraction. We provide a precise characterisation of what an abstraction is and, perhaps more importantly, suggest how abstractions can be learnt directly from data both for static datasets and for dynamical systems. We define an abstraction to be a small set of `summaries' of a system which can be used to answer a set of queries about the system or its behaviour. The difference between the ground truth behaviour of the system on the queries and the behaviour of the system predicted only by the abstraction provides a measure of the `leakiness' of the abstraction which can be used as a loss function to directly learn abstractions from data. Our approach can be considered a generalization of classical statistics where we are not interested in reconstructing `the data' in full, but are instead only concerned with answering a set of arbitrary queries about the data. While highly theoretical, our results have deep implications for statistical inference and machine learning and could be used to develop explicit methods for learning precise kinds of abstractions directly from data.

In this article, I will explain what the maximum entropy principle is, how to apply it and why it's useful in the context of Bayesian inference. The code to reproduce the results and figures can be found in this notebook. The maximum entropy principle is a method to create probability distributions that is most consistent with a given set of assumptions and nothing more. The rest of the article will explain what this means. First, we need to a way to measure the uncertainty in a probability distribution.