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In this work, we investigate stochastic approximation (SA) with Markovian data and nonlinear updates under constant stepsize. Existing work has primarily focused on either i.i.d. data or linear update rules. We take a new perspective and carefully examine the simultaneous presence of Markovian dependency of data and nonlinear update rules, delineating how the interplay between these two structures leads to complications that are not captured by prior techniques. By leveraging the smoothness and recurrence properties of the SA updates, we develop a fine-grained analysis of the correlation between the SA iterates and Markovian data. This enables us to overcome the obstacles in existing analysis and establish for the first time the weak convergence of the joint process. Furthermore, we present a precise characterization of the asymptotic bias of the SA iterates. As a by-product of our analysis, we derive finite-time bounds on higher moment and present non-asymptotic geometric convergence rates for the iterates, along with a Central Limit Theorem. 

Stochastic Approximation (SA) is a widely used algorithmic approach in various fields, including optimization and reinforcement learning (RL). Among RL algorithms, Q-learning is particularly popular due to its empirical success. In this paper, we study asynchronous Q-learning with constant stepsize, which is commonly used in practice for its fast convergence. By connecting the constant stepsize Q-learning to a time-homogeneous Markov chain, we show the distributional convergence of the iterates in Wasserstein distance and establish its exponential convergence rate. We also establish a Central Limit Theory for Q-learning iterates, demonstrating the asymptotic normality of the averaged iterates. Moreover, we provide an explicit expansion of the asymptotic bias of the averaged iterate in stepsize. Specifically, the bias is proportional to the stepsize up to higher-order terms and we provide an explicit expression for the linear coefficient. This precise characterization of the bias allows the application of Richardson-Romberg (RR) extrapolation technique to construct a new estimate that is provably closer to the optimal Q function. Numerical results corroborate our theoretical finding on the improvement of the RR extrapolation method. 

Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA. 

Motivated by Q-learning, we study nonsmooth contractive stochastic approximation (SA) with constant stepsize. We focus on two important classes of dynamics: 1) nonsmooth contractive SA with additive noise, and 2) synchronous and asynchronous Q-learning, which features both additive and multiplicative noise. For both dynamics, we establish weak convergence of the iterates to a stationary limit distribution in Wasserstein distance. Furthermore, we propose a prelimit coupling technique for establishing steady-state convergence and characterize the limit of the stationary distribution as the stepsize goes to zero. Using this result, we derive that the asymptotic bias of nonsmooth SA is proportional to the square root of the stepsize, which stands in sharp contrast to smooth SA. This bias characterization allows for the use of Richardson-Romberg extrapolation for bias reduction in nonsmooth SA. 

In response to COVID-19, a vast number of visualizations have been created to communicate information to the public. Information exposure in a public health crisis can impact people’s attitudes towards and responses to the crisis and risks, and ultimately the trajectory of a pandemic. As such, there is a need for work that documents, organizes, and investigates what COVID-19 visualizations have been presented to the public. We address this gap through an analysis of 668 COVID-19 visualizations. We present our findings through a conceptual framework derived from our analysis, that examines who, (uses) what data, (to communicate) what messages, in what form, under what circumstances in the context of COVID-19 crisis visualizations. We provide a set of factors to be considered within each component of the framework. We conclude with directions for future crisis visualization research. 

Temporal event sequence alignment has been used in many domains to visualize nuanced changes and interactions over time. Existing approaches align one or two sentinel events. Overview tasks require examining all alignments of interest using interaction and time or juxtaposition of many visualizations. Furthermore, any event attribute overviews are not closely tied to sequence visualizations. We present SEQUENCE BRAIDING, a novel overview visualization for temporal event sequences and attributes using a layered directed acyclic network. SEQUENCE BRAIDING visually aligns many temporal events and attribute groups simultaneously and supports arbitrary ordering, absence, and duplication of events. In a controlled experiment we compare SEQUENCE BRAIDING and IDMVis on user task completion time, correctness, error, and confidence. Our results provide good evidence that users of SEQUENCE BRAIDING can understand high-level patterns and trends faster and with similar error. A full version of this paper with all appendices; the evaluation stimuli, data, and analysis code; and source code are available at osf.io/mq2wt.