Inference is typically intractable in high-treewidth undirected graphical models, making maximum likelihood learning a challenge. One way to overcome this is to restrict parameters to a tractable set, most typically the set of tree-structured parameters. This paper explores an alternative notion of a tractable set, namely a set of "fast-mixing parameters" where Markov chain Monte Carlo (MCMC) inference can be guaranteed to quickly converge to the stationary distribution. While it is common in practice to approximate the likelihood gradient using samples obtained from MCMC, such procedures lack theoretical guarantees. This paper proves that for any exponential family with bounded sufficient statistics, (not just graphical models) when parameters are constrained to a fast-mixing set, gradient descent with gradients approximated by sampling will approximate the maximum likelihood solution inside the set with high-probability.

We propose the infinite factorial dynamic model (iFDM), a general Bayesian nonparametric model for source separation. Our model builds on the Markov Indian buffet process to consider a potentially unbounded number of hidden Markov chains (sources) that evolve independently according to some dynamics, in which the state space can be either discrete or continuous. For posterior inference, we develop an algorithm based on particle Gibbs with ancestor sampling that can be efficiently applied to a wide range of source separation problems. We evaluate the performance of our iFDM on four well-known applications: multitarget tracking, cocktail party, power disaggregation, and multiuser detection. Our experimental results show that our approach for source separation does not only outperform previous approaches, but it can also handle problems that were computationally intractable for existing approaches.

Existing Graph Neural Network (GNN) methods that learn inductive unsupervised graph representations focus on learning node and edge representations by predicting observed edges in the graph. Although such approaches have shown advances in downstream node classification tasks, they are ineffective in jointly representing larger k-node sets, k{ }2. We propose MHM-GNN, an inductive unsupervised graph representation approach that combines joint k-node representations with energy-based models (hypergraph Markov networks) and GNNs. To address the intractability of the loss that arises from this combination, we endow our optimization with a loss upper bound using a finite-sample unbiased Markov Chain Monte Carlo estimator. Our experiments show that the unsupervised joint k-node representations of MHM-GNN produce better unsupervised representations than existing approaches from the literature.

Modeling the time evolution of discrete sets of items (e.g., genetic mutations) is a fundamental problem in many biomedical applications. We approach this problem through the lens of continuous-time Markov chains, and show that the resulting learning task is generally underspecified in the usual setting of cross-sectional data. We explore a perhaps surprising remedy: including a number of additional independent items can help determine time order, and hence resolve underspecification. This is in sharp contrast to the common practice of limiting the analysis to a small subset of relevant items, which is followed largely due to poor scaling of existing methods. To put our theoretical insight into practice, we develop an approximate likelihood maximization method for learning continuous-time Markov chains, which can scale to hundreds of items and is orders of magnitude faster than previous methods.

Combining discrete probability distributions and combinatorial optimization problems with neural network components has numerous applications but poses several challenges. We propose Implicit Maximum Likelihood Estimation (I-MLE), a framework for end-to-end learning of models combining discrete exponential family distributions and differentiable neural components. I-MLE is widely applicable as it only requires the ability to compute the most probable states and does not rely on smooth relaxations. The framework encompasses several approaches such as perturbation-based implicit differentiation and recent methods to differentiate through black-box combinatorial solvers. We introduce a novel class of noise distributions for approximating marginals via perturb-and-MAP.

Graph Neural Networks (GNNs) have emerged as the de facto standard for representation learning on graphs, owing to their ability to effectively integrate graph topology and node attributes. However, the inherent suboptimal nature of node connections, resulting from the complex and contingent formation process of graphs, presents significant challenges in modeling them effectively. To tackle this issue, Graph Structure Learning (GSL), a family of data-centric learning approaches, has garnered substantial attention in recent years. The core concept behind GSL is to jointly optimize the graph structure and the corresponding GNN models. Despite the proposal of numerous GSL methods, the progress in this field remains unclear due to inconsistent experimental protocols, including variations in datasets, data processing techniques, and splitting strategies.

We study the following independence testing problem: given access to samples from a distribution P over \{0,1\} n, decide whether P is a product distribution or whether it is \varepsilon -far in total variation distance from any product distribution. For arbitrary distributions, this problem requires \exp(n) samples. We show in this work that if P has a sparse structure, then in fact only linearly many samples are required.Specifically, if P is Markov with respect to a Bayesian network whose underlying DAG has in-degree bounded by d, then \tilde{\Theta}(2 {d/2}\cdot n/\varepsilon 2) samples are necessary and sufficient for independence testing.

We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size k and desired accuracy \varepsilon: \textbf{Distribution estimation} Under \ell_1 distance, PML yields optimal \Theta(k/(\varepsilon 2\log k)) sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual distribution; \textbf{Additive property estimation} For a broad class of additive properties, the PML plug-in estimator uses just four times the sample size required by the best estimator to achieve roughly twice its error, with exponentially higher confidence; \textbf{ \alpha -R\'enyi entropy estimation} For an integer \alpha 1, the PML plug-in estimator has optimal k {1-1/\alpha} sample complexity; for non-integer \alpha 3/4, the PML plug-in estimator has sample complexity lower than the state of the art; \textbf{Identity testing} In testing whether an unknown distribution is equal to or at least \varepsilon far from a given distribution in \ell_1 distance, a PML-based tester achieves the optimal sample complexity up to logarithmic factors of k . With minor modifications, most of these results also hold for a near-linear-time computable variant of PML.

Meticulously analysing the empirical strengths and weaknesses of reinforcement learning methods in hard (challenging) environments is essential to inspire innovations and assess progress in the field. In tabular reinforcement learning, there is no well-established standard selection of environments to conduct such analysis, which is partially due to the lack of a widespread understanding of the rich theory of hardness of environments. The goal of this paper is to unlock the practical usefulness of this theory through four main contributions. First, we present a systematic survey of the theory of hardness, which also identifies promising research directions. Second, we introduce \texttt{Colosseum}, a pioneering package that enables empirical hardness analysis and implements a principled benchmark composed of environments that are diverse with respect to different measures of hardness.

We propose a new complexity measure for Markov decision processes (MDPs), the maximum expected hitting cost (MEHC). This measure tightens the closely related notion of diameter [JOA10] by accounting for the reward structure. We show that this parameter replaces diameter in the upper bound on the optimal value span of an extended MDP, thus refining the associated upper bounds on the regret of several UCRL2-like algorithms. Furthermore, we show that potential-based reward shaping [NHR99] can induce equivalent reward functions with varying informativeness, as measured by MEHC. By analyzing the change in the maximum expected hitting cost, this work presents a formal understanding of the effect of potential-based reward shaping on regret (and sample complexity) in the undiscounted average reward setting.