Conditional independence (CI) testing is frequently used in data analysis and machine learning for various scientific fields and it forms the basis of constraint-based causal discovery. Oftentimes, CI testing relies on strong, rather unrealistic assumptions. One of these assumptions is homoskedasticity, in other words, a constant conditional variance is assumed. We frame heteroskedasticity in a structural causal model framework and present an adaptation of the partial correlation CI test that works well in the presence of heteroskedastic noise, given that expert knowledge about the heteroskedastic relationships is available. Further, we provide theoretical consistency results for the proposed CI test which carry over to causal discovery under certain assumptions.

In this paper, we present a new practical method for Bayesian learning that can rapidly draw representative samples from complex posterior distributions with multiple isolated modes in the presence of mini-batch noise. This is achieved by simulating a collection of replicas in parallel with different temperatures and periodically swapping them. When evolving the replicas' states, the Nos\'e-Hoover dynamics is applied, which adaptively neutralizes the mini-batch noise. To perform proper exchanges, a new protocol is developed with a noise-aware test of acceptance, by which the detailed balance is reserved in an asymptotic way. While its efficacy on complex multimodal posteriors has been illustrated by testing over synthetic distributions, experiments with deep Bayesian neural networks on large-scale datasets have shown its significant improvements over strong baselines.

Submodular and supermodular functions have found wide applicability in machine learning, capturing notions such as diversity and regularity, respectively. These notions have deep consequences for optimization, and the problem of (approximately) optimizing submodular functions has received much attention. However, beyond optimization, these notions allow specifying expressive probabilistic models that can be used to quantify predictive uncertainty via marginal inference. Prominent, well-studied special cases include Ising models and determinantal point processes, but the general class of log-submodular and log-supermodular models is much richer and little studied. In this paper, we investigate the use of Markov chain Monte Carlo sampling to perform approximate inference in general log-submodular and log-supermodular models. In particular, we consider a simple Gibbs sampling procedure, and establish two sufficient conditions, the first guaranteeing polynomial-time, and the second fast (O(nlogn)) mixing.

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.

Reward-free reinforcement learning (RL) is a framework which is suitable for both the batch RL setting and the setting where there are many reward functions of interest. During the exploration phase, an agent collects samples without using a pre-specified reward function. After the exploration phase, a reward function is given, and the agent uses samples collected during the exploration phase to compute a near-optimal policy. Jin et al. [2020] showed that in the tabular setting, the agent only needs to collect polynomial number of samples (in terms of the number states, the number of actions, and the planning horizon) for reward-free RL. However, in practice, the number of states and actions can be large, and thus function approximation schemes are required for generalization.

We examine an adaptive learning framework for nonatomic congestion games where the players' cost functions may be subject to exogenous fluctuations (e.g., due to disturbances in the network, variations in the traffic going through a link). In this setting, the popular multiplicative/ exponential weights algorithm enjoys an \mathcal{O}(1/\sqrt{T}) equilibrium convergence rate; however, this rate is suboptimal in static environments---i.e., when the network is not subject to randomness. In this static regime, accelerated algorithms achieve an \mathcal{O}(1/T {2}) convergence speed, but they fail to converge altogether in stochastic problems. To fill this gap, we propose a novel, adaptive exponential weights method---dubbed AdaWeight---that seamlessly interpolates between the \mathcal{O}(1/T {2}) and \mathcal{O}(1/\sqrt{T}) rates in the static and stochastic regimes respectively. Importantly, this "best-of-both-worlds" guarantee does not require any prior knowledge of the problem's parameters or tuning by the optimizer; in addition, the method's convergence speed depends subquadratically on the size of the network (number of vertices and edges), so it scales gracefully to large, real-life urban networks.

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.

We study the problem of estimating a nonparametric probability distribution under a family of losses called Besov IPMs. This family is quite large, including, for example, L p distances, total variation distance, and generalizations of both Wasserstein (earthmover's) and Kolmogorov-Smirnov distances. For a wide variety of settings, we provide both lower and upper bounds, identifying precisely how the choice of loss function and assumptions on the data distribution interact to determine the mini-max optimal convergence rate. We also show that, in many cases, linear distribution estimates, such as the empirical distribution or kernel density estimator, cannot converge at the optimal rate. These bounds generalize, unify, or improve on several recent and classical results.

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.

Zap Q-learning is a recent class of reinforcement learning algorithms, motivated primarily as a means to accelerate convergence. Stability theory has been absent outside of two restrictive classes: the tabular setting, and optimal stopping. This paper introduces a new framework for analysis of a more general class of recursive algorithms known as stochastic approximation. Based on this general theory, it is shown that Zap Q-learning is consistent under a non-degeneracy assumption, even when the function approximation architecture is nonlinear. Zap Q-learning with neural network function approximation emerges as a special case, and is tested on examples from OpenAI Gym.