The shortest-path, commute time, and diffusion distances on undirected graphs have been widely employed in applications such as dimensionality reduction, link prediction, and trip planning. Increasingly, there is interest in using asymmetric structure of data derived from Markov chains and directed graphs, but few metrics are specifically adapted to this task. We introduce a metric on the state space of any ergodic, finite-state, time-homogeneous Markov chain and, in particular, on any Markov chain derived from a directed graph. Our construction is based on hitting probabilities, with nearness in the metric space related to the transfer of random walkers from one node to another at stationarity. Notably, our metric is insensitive to shortest and average path distances, thus giving new information compared to existing metrics. We use possible degeneracies in the metric to develop an interesting structural theory of directed graphs and explore a related quotienting procedure. Our metric can be computed in $O(n^3)$ time, where $n$ is the number of states, and in examples we scale up to $n=10,000$ nodes and $\approx 38M$ edges on a desktop computer. In several examples, we explore the nature of the metric, compare it to alternative methods, and demonstrate its utility for weak recovery of community structure in dense graphs, visualization, structure recovering, dynamics exploration, and multiscale cluster detection.

In audio applications, one of the most important representations of audio signals is the amplitude spectrogram. It is utilized in many machine-learning-based information processing methods including the ones using the restricted Boltzmann machines (RBM). However, the ordinary Gaussian-Bernoulli RBM (the most popular RBM among its variations) cannot directly handle amplitude spectra because the Gaussian distribution is a symmetric model allowing negative values which never appear in the amplitude. In this paper, after proposing a general gamma Boltzmann machine, we propose a practical model called the gamma-Bernoulli RBM that simultaneously handles both linear- and log-amplitude spectrograms. Its conditional distribution of the observable data is given by the gamma distribution, and thus the proposed RBM can naturally handle the data represented by positive numbers as the amplitude spectra. It can also treat amplitude in the logarithmic scale which is important for audio signals from the perceptual point of view. The advantage of the proposed model compared to the ordinary Gaussian-Bernoulli RBM was confirmed by PESQ and MSE in the experiment of representing the amplitude spectrograms of speech signals.

We consider un-discounted reinforcement learning (RL) in Markov decision processes (MDPs) under drifting non-stationarity, i.e., both the reward and state transition distributions are allowed to evolve over time, as long as their respective total variations, quantified by suitable metrics, do not exceed certain variation budgets. We first develop the Sliding Window Upper-Confidence bound for Reinforcement Learning with Confidence Widening (SWUCRL2-CW) algorithm, and establish its dynamic regret bound when the variation budgets are known. In addition, we propose the Bandit-over-Reinforcement Learning (BORL) algorithm to adaptively tune the SWUCRL2-CW algorithm to achieve the same dynamic regret bound, but in a parameter-free manner, i.e., without knowing the variation budgets. Notably, learning non-stationary MDPs via the conventional optimistic exploration technique presents a unique challenge absent in existing (non-stationary) bandit learning settings. We overcome the challenge by a novel confidence widening technique that incorporates additional optimism.

Compartmental models are used for predicting the scale and duration of epidemics, estimating epidemiological parameters such as reproduction numbers, and guiding outbreak control measures [Brauer, 2008, O'Neill, 2010, Kucharski et al., 2020]. They are increasingly important because they allow joint modelling of disease dynamics and multimodal data, such as medical test results, cell phone and transport flow data [Rubrichi et al., 2018, Wu et al., 2020], census and demographic information [Prem et al., 2020]. However, statistical inference in stochastic variants of compartmental models is a major computational challenge [Bretó, 2018]. The likelihood function for model parameters is usually intractable because it involves summation over a prohibitively large number of configurations of latent variables representing counts of subpopulations in disease states which cannot be observed directly. This has lead to the recent development of sophisticated computational methods for approximate inference involving various forms of stochastic simulation [Funk and King, 2020].

Motivated by broad applications in reinforcement learning and federated learning, we study local stochastic approximation over a network of agents, where their goal is to find the root of an operator composed of the local operators at the agents. Our focus is to characterize the finite-time performance of this method when the data at each agent are generated from Markov processes, and hence they are dependent. In particular, we provide the convergence rates of local stochastic approximation for both constant and time-varying step sizes. Our results show that these rates are within a logarithmic factor of the ones under independent data. We then illustrate the applications of these results to different interesting problems in multi-task reinforcement learning and federated learning.

We define an evolving in time Bayesian neural network called a Hidden Markov neural network. The weights of a feed-forward neural network are modelled with the hidden states of a Hidden Markov model, whose observed process is given by the available data. A filtering algorithm is used to learn a variational approximation to the evolving in time posterior over the weights. Training is pursued through a sequential version of Bayes by Backprop Blundell et al. 2015, which is enriched with a stronger regularization technique called variational DropConnect. The experiments test variational DropConnect on MNIST and display the performance of Hidden Markov neural networks on time series.

We study minimax optimal reinforcement learning in episodic factored Markov decision processes (FMDPs), which are MDPs with conditionally independent transition components. Assuming the factorization is known, we propose two model-based algorithms. The first one achieves minimax optimal regret guarantees for a rich class of factored structures, while the second one enjoys better computational complexity with a slightly worse regret. A key new ingredient of our algorithms is the design of a bonus term to guide exploration. We complement our algorithms by presenting several structure-dependent lower bounds on regret for FMDPs that reveal the difficulty hiding in the intricacy of the structures.

In many sequential decision making settings, the agent lacks complete information about the underlying state of the system, a phenomenon known as partial observability. Partial observability significantly complicates the tasks of reinforcement learning and planning, because the non-Markovian nature of the observations forces the agent to maintain memory and reason about beliefs of the system state, all while exploring to collect information about the environment. For example, a robot may not be able to perceive all objects in the environment due to occlusions, and it must reason about how these objects may move to avoid collisions [Cassandra et al., 1996]. Similar reasoning problems arise in imperfect information games [Brown and Sandholm, 2018], medical diagnosis [Hauskrecht and Fraser, 2000], and elsewhere [Rafferty et al., 2011]. Furthermore, from a theoretical perspective, well-known complexity-theoretic results show that learning and planning in partially observable environments is statistically and computationally intractable in general [Papadimitriou and Tsitsiklis, 1987, Mundhenk et al., 2000, Vlassis et al., 2012, Mossel and Roch, 2005]. The standard formulation for reinforcement learning with partial observability is the Partially Observable Markov Decision Process (POMDP), in which an agent operating on noisy observations makes decisions that influence the evolution of a latent state. The complexity barriers apply for this model, but they are of a worst case nature, and they do not preclude efficient algorithms for interesting subclasses of POMDPs. Thus we ask: Can we develop efficient algorithms for reinforcement learning in large classes of POMDPs?

We study risk-sensitive reinforcement learning in episodic Markov decision processes with unknown transition kernels, where the goal is to optimize the total reward under the risk measure of exponential utility. We propose two provably efficient model-free algorithms, Risk-Sensitive Value Iteration (RSVI) and Risk-Sensitive Q-learning (RSQ). These algorithms implement a form of risk-sensitive optimism in the face of uncertainty, which adapts to both risk-seeking and risk-averse modes of exploration. We prove that RSVI attains an $\tilde{O}\big(\lambda(|\beta| H^2) \cdot \sqrt{H^{3} S^{2}AT} \big)$ regret, while RSQ attains an $\tilde{O}\big(\lambda(|\beta| H^2) \cdot \sqrt{H^{4} SAT} \big)$ regret, where $\lambda(u) = (e^{3u}-1)/u$ for $u>0$. In the above, $\beta$ is the risk parameter of the exponential utility function, $S$ the number of states, $A$ the number of actions, $T$ the total number of timesteps, and $H$ the episode length. On the flip side, we establish a regret lower bound showing that the exponential dependence on $|\beta|$ and $H$ is unavoidable for any algorithm with an $\tilde{O}(\sqrt{T})$ regret (even when the risk objective is on the same scale as the original reward), thus certifying the near-optimality of the proposed algorithms. Our results demonstrate that incorporating risk awareness into reinforcement learning necessitates an exponential cost in $|\beta|$ and $H$, which quantifies the fundamental tradeoff between risk sensitivity (related to aleatoric uncertainty) and sample efficiency (related to epistemic uncertainty). To the best of our knowledge, this is the first regret analysis of risk-sensitive reinforcement learning with the exponential utility.

Many modern data analyses benefit from explicitly modeling dependence structure in data - such as measurements across time or space, ordered words in a sentence, or genes in a genome. A gold standard evaluation technique is structured cross-validation (CV), which leaves out some data subset (such as data within a time interval or data in a geographic region) in each fold. But CV here can be prohibitively slow due to the need to rerun already-expensive learning algorithms many times. Previous work has shown approximate cross-validation (ACV) methods provide a fast and provably accurate alternative in the setting of empirical risk minimization. But this existing ACV work is restricted to simpler models by the assumptions that (i) data across CV folds are independent and (ii) an exact initial model fit is available. In structured data analyses, both these assumptions are often untrue. In the present work, we address (i) by extending ACV to CV schemes with dependence structure between the folds. To address (ii), we verify - both theoretically and empirically - that ACV quality deteriorates smoothly with noise in the initial fit. We demonstrate the accuracy and computational benefits of our proposed methods on a diverse set of real-world applications.