Many processes, such as discrete event systems in engineering or population dynamics in biology, evolve in discrete space and continuous time. We consider the problem of optimal decision making in such discrete state and action space systems under partial observability. This places our work at the intersection of optimal filtering and optimal control. At the current state of research, a mathematical description for simultaneous decision making and filtering in continuous time with finite state and action spaces is still missing. In this paper, we give a mathematical description of a continuous-time partial observable Markov decision process (POMDP). By leveraging optimal filtering theory we derive a Hamilton-Jacobi-Bellman (HJB) type equation that characterizes the optimal solution. Using techniques from deep learning we approximately solve the resulting partial integro-differential equation. We present (i) an approach solving the decision problem offline by learning an approximation of the value function and (ii) an online algorithm which provides a solution in belief space using deep reinforcement learning. We show the applicability on a set of toy examples which pave the way for future methods providing solutions for high dimensional problems.

Agents trained via deep reinforcement learning (RL) routinely fail to generalize to unseen environments, even when these share the same underlying dynamics as the training levels. Understanding the generalization properties of RL is one of the challenges of modern machine learning. Towards this goal, we analyze policy learning in the context of Partially Observable Markov Decision Processes (POMDPs) and formalize the dynamics of training levels as instances. We prove that, independently of the exploration strategy, reusing instances introduces significant changes on the effective Markov dynamics the agent observes during training. Maximizing expected rewards impacts the learned belief state of the agent by inducing undesired instance-specific speed-running policies instead of generalizable ones, which are sub-optimal on the training set. We provide generalization bounds to the value gap in train and test environments based on the number of training instances, and use insights based on these to improve performance on unseen levels. We propose training a shared belief representation over an ensemble of specialized policies, from which we compute a consensus policy that is used for data collection, disallowing instance-specific exploitation. We experimentally validate our theory, observations, and the proposed computational solution over the CoinRun benchmark.

This work introduces macro-action discovery using value-of-information (VoI) for robust and efficient planning in partially observable Markov decision processes (POMDPs). POMDPs are a powerful framework for planning under uncertainty. Previous approaches have used high-level macro-actions within POMDP policies to reduce planning complexity. However, macro-action design is often heuristic and rarely comes with performance guarantees. Here, we present a method for extracting belief-dependent, variable-length macro-actions directly from a low-level POMDP model. We construct macro-actions by chaining sequences of open-loop actions together when the task-specific value of information (VoI) --- the change in expected task performance caused by observations in the current planning iteration --- is low. Importantly, we provide performance guarantees on the resulting VoI macro-action policies in the form of bounded regret relative to the optimal policy. In simulated tracking experiments, we achieve higher reward than both closed-loop and hand-coded macro-action baselines, selectively using VoI macro-actions to reduce planning complexity while maintaining near-optimal task performance.

Recently, model-based agents have achieved better performance than model-free ones using the same computational budget and training time in single-agent environments. However, due to the complexity of multi-agent systems, it is tough to learn the model of the environment. The significant compounding error may hinder the learning process when model-based methods are applied to multi-agent tasks. This paper proposes an implicit model-based multi-agent reinforcement learning method based on value decomposition methods. Under this method, agents can interact with the learned virtual environment and evaluate the current state value according to imagined future states in the latent space, making agents have the foresight. Our approach can be applied to any multi-agent value decomposition method. The experimental results show that our method improves the sample efficiency in different partially observable Markov decision process domains.

Reinforcement learning algorithms are widely used in domains where it is desirable to provide a personalized service. In these domains it is common that user data contains sensitive information that needs to be protected from third parties. Motivated by this, we study privacy in the context of finite-horizon Markov Decision Processes (MDPs) by requiring information to be obfuscated on the user side. We formulate this notion of privacy for RL by leveraging the local differential privacy (LDP) framework. We establish a lower bound for regret minimization in finite-horizon MDPs with LDP guarantees which shows that guaranteeing privacy has a multiplicative effect on the regret. This result shows that while LDP is an appealing notion of privacy, it makes the learning problem significantly more complex. Finally, we present an optimistic algorithm that simultaneously satisfies $\varepsilon$-LDP requirements, and achieves $\sqrt{K}/\varepsilon$ regret in any finite-horizon MDP after $K$ episodes, matching the lower bound dependency on the number of episodes $K$.

Computational level explanations based on optimal feedback control with signal-dependent noise have been able to account for a vast array of phenomena in human sensorimotor behavior. However, commonly a cost function needs to be assumed for a task and the optimality of human behavior is evaluated by comparing observed and predicted trajectories. Here, we introduce inverse optimal control with signal-dependent noise, which allows inferring the cost function from observed behavior. To do so, we formalize the problem as a partially observable Markov decision process and distinguish between the agent's and the experimenter's inference problems. Specifically, we derive a probabilistic formulation of the evolution of states and belief states and an approximation to the propagation equation in the linear-quadratic Gaussian problem with signal-dependent noise. We extend the model to the case of partial observability of state variables from the point of view of the experimenter. We show the feasibility of the approach through validation on synthetic data and application to experimental data. Our approach enables recovering the costs and benefits implicit in human sequential sensorimotor behavior, thereby reconciling normative and descriptive approaches in a computational framework.

A fundamental question in neuroscience is how the brain creates an internal model of the world to guide actions using sequences of ambiguous sensory information. This is naturally formulated as a reinforcement learning problem under partial observations, where an agent must estimate relevant latent variables in the world from its evidence, anticipate possible future states, and choose actions that optimize total expected reward. This problem can be solved by control theory, which allows us to find the optimal actions for a given system dynamics and objective function. However, animals often appear to behave suboptimally.

In probably approximately correct (PAC) reinforcement learning (RL), an agent is required to identify an $\epsilon$-optimal policy with probability $1-\delta$. While minimax optimal algorithms exist for this problem, its instance-dependent complexity remains elusive in episodic Markov decision processes (MDPs). In this paper, we propose the first nearly matching (up to a horizon squared factor and logarithmic terms) upper and lower bounds on the sample complexity of PAC RL in deterministic episodic MDPs with finite state and action spaces. In particular, our bounds feature a new notion of sub-optimality gap for state-action pairs that we call the deterministic return gap. While our instance-dependent lower bound is written as a linear program, our algorithms are very simple and do not require solving such an optimization problem during learning. Their design and analyses employ novel ideas, including graph-theoretical concepts (minimum flows) and a new maximum-coverage exploration strategy.

Continuous hidden Markov models (HMMs) assume that observations are generated from a mixture of Gaussian densities, limiting their ability to model more complex distributions. In this work, we address this shortcoming and propose novel continuous HMM models, dubbed FlowHMMs, that enable learning general continuous observation densities without constraining them to follow a Gaussian distribution or their mixtures. To that end, we leverage deep flow-based architectures that model complex, non-Gaussian functions and propose two variants of training a FlowHMM model. The first one, based on gradient-based technique, can be applied directly to continuous multidimensional data, yet its application to larger data sequences remains computationally expensive. Therefore, we also present a second approach to training our FlowHMM that relies on the co-occurrence matrix of discretized observations and considers the joint distribution of pairs of co-observed values, hence rendering the training time independent of the training sequence length. As a result, we obtain a model that can be flexibly adapted to the characteristics and dimensionality of the data. We perform a variety of experiments in which we compare both training strategies with a baseline of Gaussian mixture models. We show, that in terms of quality of the recovered probability distribution, accuracy of prediction of hidden states, and likelihood of unseen data, our approach outperforms the standard Gaussian methods.

We study a constructive procedure that approximates Gateaux derivatives for statistical functionals by finite-differencing, with attention to causal inference functionals. We focus on the case where probability distributions are not known a priori but need also to be estimated from data, leading to empirical Gateaux derivatives, and study relationships between empirical, numerical, and analytical Gateaux derivatives. Starting with a case study of counterfactual mean estimation, we verify the exact relationship between finite-differences and the analytical Gateaux derivative. We then derive requirements on the rates of numerical approximation in perturbation and smoothing that preserve statistical benefits. We study more complicated functionals such as dynamic treatment regimes and the linear-programming formulation for policy optimization infinite-horizon Markov decision processes. In the case of the latter, this approach can be used to approximate bias adjustments in the presence of arbitrary constraints, illustrating the usefulness of constructive approaches for Gateaux derivatives. We find that, omitting unfavorable dimension dependence of smoothing, although rate-double robustness permits for coarser rates of perturbation size than implied by generic approximation analysis of finite-differences for the case of the counterfactual mean, this is not the case for the infinite-horizon MDP policy value.