This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program for MESP and show that its continuous relaxation yields a near-optimal solution. The results motivate us to study an efficient sampling algorithm and develop its approximation bound for MESP, which improves the best-known bound in literature. We then provide an efficient deterministic implementation of the sampling algorithm with the same approximation bound. By developing new mathematical tools for the singular matrices and analyzing the Lagrangian dual of the proposed convex integer program, we investigate the widely-used local search algorithm and prove its first-known approximation bound for MESP. The proof techniques further inspire us with an efficient implementation of the local search algorithm. Our numerical experiments demonstrate that these approximation algorithms can efficiently solve medium-sized and large-scale instances to near-optimality. Our proposed algorithms are coded and released as open-source software. Finally, we extend the analyses to the A-Optimal MESP (A-MESP), where the objective is to minimize the trace of the inverse of the selected principal submatrix.

Machine Learning Research Group and Oxford-Man Institute for Quantitative Finance, Department of Engineering Science, University of Oxford Abstract Graph spectral techniques for measuring graph similarity, or for learning the cluster number, require kernel smoothing. The choice of kernel function and bandwidth are typically chosen in an ad-hoc manner and heavily affect the resulting output. We prove that kernel smoothing biases the moments of the spectral density. We propose an information theoretically optimal approach to learn a smooth graph spectral density, which fully respects the moment information. Our method's computational cost is linear in the number of edges, and hence can be applied to large networks, with millions of nodes. We apply our method to the problems to graph similarity and cluster number learning, where we outperform comparable iterative spectral approaches on synthetic and real graphs. Keywords: Networks, Information Theory, Maximum Entropy, Graph Spectral Theory, Random matrix theory, iterative methods, kernel smoothing 1. Introduction: networks, their graph spectra and importance Many systems of interest can be naturally characterised by complex networks; examples include social networks (Mislove et al., 2007b; Flake et al., 2000; Leskovec et al., 2007), biological networks (Palla et al., 2005) and technological networks.

Maximum Entropy Models from Phase Harmonic Covariances Sixin Zhang 1, 4, St ephane Mallat 1, 2,3 1 ENS, PSL University, Paris, France 2 Coll ege de France, Paris, France 3 Flatiron Institute, New York, USA 4 Center for Data Science, Peking University, Beijing, China November 25, 2019 Abstract We define maximum entropy models of non-Gaussian stationary random vectors from covariances of nonlinear representations. These representations are calculated by multiplying the phase of Fourier or wavelet coefficients with harmonic integers, which amounts to compute a windowed Fourier transform along their phase. Rectifiers in neural networks compute such phase windowing. The covariance of these harmonic coefficients capture dependencies of Fourier and wavelet coefficients across frequencies, by canceling their random phase. We introduce maximum entropy models conditioned by such covariances over a graph of local interactions. These models are approximated by transporting an initial maximum ...

Deep networks have enabled reinforcement learning to scale to more complex and challenging domains, but these methods typically require large quantities of training data. An alternative is to use sample-efficient episodic control methods: neuro-inspired algorithms which use non-/semi-parametric models that predict values based on storing and retrieving previously experienced transitions. One way to further improve the sample efficiency of these approaches is to use more principled exploration strategies. In this work, we therefore propose maximum entropy mellowmax episodic control (MEMEC), which samples actions according to a Boltzmann policy with a state-dependent temperature. We demonstrate that MEMEC outperforms other uncertainty- and softmax-based exploration methods on classic reinforcement learning environments and Atari games, achieving both more rapid learning and higher final rewards.

Two hitherto disconnected threads of research, diverse exploration (DE) and maximum entropy RL have addressed a wide range of problems facing reinforcement learning algorithms via ostensibly distinct mechanisms. In this work, we identify a connection between these two approaches. First, a discriminator-based diversity objective is put forward and connected to commonly used divergence measures. We then extend this objective to the maximum entropy framework and propose an algorithm Maximum Entropy Diverse Exploration (MEDE) which provides a principled method to learn diverse behaviors. A theoretical investigation shows that the set of policies learned by MEDE capture the same modalities as the optimal maximum entropy policy. In effect, the proposed algorithm disentangles the maximum entropy policy into its diverse, constituent policies. Experiments show that MEDE is superior to the state of the art in learning high performing and diverse policies.

Maximum entropy deep reinforcement learning (RL) methods have been demonstrated on a range of challenging continuous tasks. However, existing methods either suffer from severe instability when training on large off-policy data or cannot scale to tasks with very high state and action dimensionality such as 3D humanoid locomotion. Besides, the optimality of desired Boltzmann policy set for non-optimal soft value function is not persuasive enough. In this paper, we first derive soft policy gradient based on entropy regularized expected reward objective for RL with continuous actions. Then, we present an off-policy actor-critic, model-free maximum entropy deep RL algorithm called deep soft policy gradient (DSPG) by combining soft policy gradient with soft Bellman equation. To ensure stable learning while eliminating the need of two separate critics for soft value functions, we leverage double sampling approach to making the soft Bellman equation tractable. The experimental results demonstrate that our method outperforms in performance over off-policy prior methods.

Numerous learning methods for fuzzy cognitive maps (FCMs), such as the Hebbian-based and the population-based learning methods, have been developed for modeling and simulating dynamic systems. However, these methods are faced with several obvious limitations. Most of these models are extremely time consuming when learning the large-scale FCMs with hundreds of nodes. Furthermore, the FCMs learned by those algorithms lack robustness when the experimental data contain noise. In addition, reasonable distribution of the weights is rarely considered in these algorithms, which could result in the reduction of the performance of the resulting FCM. In this article, a straightforward, rapid, and robust learning method is proposed to learn FCMs from noisy data, especially, to learn large-scale FCMs. The crux of the proposed algorithm is to equivalently transform the learning problem of FCMs to a classic-constrained convex optimization problem in which the least-squares term ensures the robustness of the well-learned FCM and the maximum entropy term regularizes the distribution of the weights of the well-learned FCM. A series of experiments covering two frequently used activation functions (the sigmoid and hyperbolic tangent functions) are performed on both synthetic datasets with noise and real-world datasets. The experimental results show that the proposed method is rapid and robust against data containing noise and that the well-learned weights have better distribution. In addition, the FCMs learned by the proposed method also exhibit superior performance in comparison with the existing methods. Index Terms-Fuzzy cognitive maps (FCMs), maximum entropy, noisy data, rapid and robust learning.

Classical Principal Component Analysis (PCA) approximates data in terms of projections on a small number of orthogonal vectors. There are simple procedures to efficiently compute various functions of the data from the PCA approximation. The most important function is arguably the Euclidean distance between data items, This can be used, for example, to solve the approximate nearest neighbor problem. We use random variables to model the inherent uncertainty in such approximations, and apply the Maximum Entropy Method to infer the underlying probability distribution. We propose using the expected values of distances between these random variables as improved estimates of the distance. We show by analysis and experimentally that in most cases results obtained by our method are more accurate than what is obtained by the classical approach. This improves the accuracy of a classical technique that have been used with little change for over 100 years.

One reason for the emergence of bias in AI systems is biased data -- datasets that may not be true representations of the underlying distributions -- and may over or under-represent groups with respect to protected attributes such as gender or race. We consider the problem of correcting such biases and learning distributions that are "fair", with respect to measures such as proportional representation and statistical parity, from the given samples. Our approach is based on a novel formulation of the problem of learning a fair distribution as a maximum entropy optimization problem with a given expectation vector and a prior distribution. Technically, our main contributions are: (1) a new second-order method to compute the (dual of the) maximum entropy distribution over an exponentially-sized discrete domain that turns out to be faster than previous methods, and (2) methods to construct prior distributions and expectation vectors that provably guarantee that the learned distributions satisfy a wide class of fairness criteria. Our results also come with quantitative bounds on the total variation distance between the empirical distribution obtained from the samples and the learned fair distribution. Our experimental results include testing our approach on the COMPAS dataset and showing that the fair distributions not only improve disparate impact values but when used to train classifiers only incur a small loss of accuracy.

Making high quality inference on large, feature rich datasets under a constrained computational budget is arguably the primary goal of the learning community. This, however, comes with significant challenges. On the one hand, the exact computation of linear algebraic quantities may be prohibitively expensive, such as that of the log determinant. On the other hand, an analytic expression for the quantity of interest may not exist at all, such as the case for the entropy of a Gaussian mixture model, and approximate methods are often both inefficient and inaccurate.