We consider the problem of online learning in an episodic Markov decision process, where the reward function is allowed to change between episodes in an adversarial manner and the learner only observes the rewards associated with its actions. We assume that rewards and the transition function can be represented as linear functions in terms of a known low-dimensional feature map, which allows us to consider the setting where the state space is arbitrarily large. We also assume that the learner has a perfect knowledge of the MDP dynamics. Our main contribution is developing an algorithm whose expected regret after $T$ episodes is bounded by $\widetilde{\mathcal{O}}(\sqrt{dHT})$, where $H$ is the number of steps in each episode and $d$ is the dimensionality of the feature map.

A seminal result in game theory is von Neumann's minmax theorem, which states that zero-sum games admit an essentially unique equilibrium solution. Classical learning results build on this theorem to show that online no-regret dynamics converge to an equilibrium in a time-average sense in zero-sum games. In the past several years, a key research direction has focused on characterizing the transient behavior of such dynamics. General results in this direction show that broad classes of online learning dynamics are cyclic, and formally Poincar\'{e} recurrent, in zero-sum games. We analyze the robustness of these online learning behaviors in the case of periodic zero-sum games with a time-invariant equilibrium. This model generalizes the usual repeated game formulation while also being a realistic and natural model of a repeated competition between players that depends on exogenous environmental variations such as time-of-day effects, week-to-week trends, and seasonality. Interestingly, time-average convergence may fail even in the simplest such settings, in spite of the equilibrium being fixed. In contrast, using novel analysis methods, we show that Poincar\'{e} recurrence provably generalizes despite the complex, non-autonomous nature of these dynamical systems.

Learning to plan for long horizons is a central challenge in episodic reinforcement learning problems. A fundamental question is to understand how the difficulty of the problem scales as the horizon increases. Here the natural measure of sample complexity is a normalized one: we are interested in the \emph{number of episodes} it takes to provably discover a policy whose value is $\varepsilon$ near to that of the optimal value, where the value is measured by the \emph{normalized} cumulative reward in each episode. In a COLT 2018 open problem, Jiang and Agarwal conjectured that, for tabular, episodic reinforcement learning problems, there exists a sample complexity lower bound which exhibits a polynomial dependence on the horizon --- a conjecture which is consistent with all known sample complexity upper bounds. This work refutes this conjecture, proving that tabular, episodic reinforcement learning is possible with a sample complexity that scales only \emph{logarithmically} with the planning horizon. In other words, when the values are appropriately normalized (to lie in the unit interval), this results shows that long horizon RL is no more difficult than short horizon RL, at least in a minimax sense. Our analysis introduces two ideas: (i) the construction of an $\varepsilon$-net for near-optimal policies whose log-covering number scales only logarithmically with the planning horizon, and (ii) the Online Trajectory Synthesis algorithm, which adaptively evaluates all policies in a given policy class and enjoys a sample complexity that scales logarithmically with the cardinality of the given policy class. Both may be of independent interest.

We provide online convex optimization algorithms that guarantee improved full-matrix regret bounds. These algorithms extend prior work in several ways. First, we seamlessly allow for the incorporation of constraints without requiring unknown oracle-tuning for any learning rate parameters. Second, we improve the regret of the full-matrix AdaGrad algorithm by suggesting a better learning rate value and showing how to tune the learning rate to this value on-the-fly. Third, all our bounds are obtained via a general framework for constructing regret bounds that depend on an arbitrary sequence of norms.

We introduce an automatic curriculum algorithm, Variational Automatic Curriculum Learning (VACL), for solving challenging goal-conditioned cooperative multi-agent reinforcement learning problems. We motivate our curriculum learning paradigm through a variational perspective, where the learning objective can be decomposed into two terms: task learning on the current curriculum, and curriculum update to a new task distribution. Local optimization over the second term suggests that the curriculum should gradually expand the training tasks from easy to hard. Our VACL algorithm implements this variational paradigm with two practical components, task expansion and entity curriculum, which produces a series of training tasks over both the task configurations as well as the number of entities in the task. Experiment results show that VACL solves a collection of sparse-reward problems with a large number of agents. Particularly, using a single desktop machine, VACL achieves 98% coverage rate with 100 agents in the simple-spread benchmark and reproduces the ramp-use behavior originally shown in OpenAI's hide-and-seek project.

We consider a natural model of online preference aggregation, where sets of preferred items R 2, ..., R t items in each R t, k t aiming that at least k t appear high in \pi_t. This is a fundamental problem in preference aggregation with applications to e.g., ordering product or news items in web pages based on user scrolling and click patterns. The widely studied Generalized Min-Sum-Set-Cover (GMSSC) problem serves as a formal model for the setting above. GMSSC is NP-hard and the standard application of no-regret online learning algorithms is computationally inefficient, because they operate in the space of rankings. In this work, we show how to achieve low regret for GMSSC in polynomial-time. We employ dimensionality reduction from rankings to the space of doubly stochastic matrices, where we apply Online Gradient Descent. A key step is to show how subgradients can be computed efficiently, by solving the dual of a configuration LP. Using deterministic and randomized rounding schemes, we map doubly stochastic matrices back to rankings with a small loss in the GMSSC objective.

Progress in Reinforcement Learning (RL) algorithms goes hand-in-hand with the development of challenging environments that test the limits of current methods. While existing RL environments are either sufficiently complex or based on fast simulation, they are rarely both. Here, we present the NetHack Learning Environment (NLE), a scalable, procedurally generated, stochastic, rich, and challenging environment for RL research based on the popular single-player terminal-based roguelike game, NetHack. We argue that NetHack is sufficiently complex to drive long-term research on problems such as exploration, planning, skill acquisition, and language-conditioned RL, while dramatically reducing the computational resources required to gather a large amount of experience. We compare NLE and its task suite to existing alternatives, and discuss why it is an ideal medium for testing the robustness and systematic generalization of RL agents. We demonstrate empirical success for early stages of the game using a distributed Deep RL baseline and Random Network Distillation exploration, alongside qualitative analysis of various agents trained in the environment.

Raven's Progressive Matrices are multiple-choice intelligence tests, where one tries to complete the missing location in a 3x3 grid of abstract images. Previous attempts to address this test have focused solely on selecting the right answer out of the multiple choices. In this work, we focus, instead, on generating a correct answer given the grid, which is a harder task, by definition. The proposed neural model combines multiple advances in generative models, including employing multiple pathways through the same network, using the reparameterization trick along two pathways to make their encoding compatible, a selective application of variational losses, and a complex perceptual loss that is coupled with a selective backpropagation procedure. Our algorithm is able not only to generate a set of plausible answers but also to be competitive to the state of the art methods in multiple-choice tests.

Due to the drastic gap in complexity between sequential and batch statistical learning, recent work has studied a smoothed sequential learning setting, where Nature is constrained to select contexts with density bounded by $1/\sigma$ with respect to a known measure $\mu$. Unfortunately, for some function classes, there is an exponential gap between the statistically optimal regret and that which can be achieved efficiently. In this paper, we give a computationally efficient algorithm that is the first to enjoy the statistically optimal $\log(T/\sigma)$ regret for realizable $K$-wise linear classification. We extend our results to settings where the true classifier is linear in an over-parameterized polynomial featurization of the contexts, as well as to a realizable piecewise-regression setting assuming access to an appropriate ERM oracle. Somewhat surprisingly, standard disagreement-based analyses are insufficient to achieve regret logarithmic in $1/\sigma$. Instead, we develop a novel characterization of the geometry of the disagreement region induced by generalized linear classifiers. Along the way, we develop numerous technical tools of independent interest, including a general anti-concentration bound for the determinant of certain matrix averages.

Unmeasured confounding can render identification strategies based on adjustment functionals invalid. We study the "Napkin graph", a causal structure that encapsulates patterns of M-bias, instrumental variables, and the classical back-door and front-door models within a single graphical framework, yet requires a nonstandard identification strategy: the average treatment effect is expressed as a ratio of two g-formulas. We develop novel estimators for this functional, including doubly robust one-step and targeted minimum loss-based estimators that remain asymptotically linear when nuisance functions are estimated at slower-than-parametric rates using machine learning. We also show how a generalized independence restriction encoded by the Napkin graph, known as a Verma constraint, can be exploited to improve efficiency, illustrating more generally how such constraints in hidden variable DAGs can inform semiparametric inference. The proposed methods are validated through simulations and applied to the Finnish Life Course study to estimate the effect of educational attainment on income. An accompanying R package, napkincausal, implements all proposed procedures.