Hamiltonian Monte Carlo (HMC) is a Markov chain algorithm for sampling from a high-dimensional distribution with density $e^{-f(x)}$, given access to the gradient of $f$. A particular case of interest is that of a $d$-dimensional Gaussian distribution with covariance matrix $\Sigma$, in which case $f(x) = x^\top \Sigma^{-1} x$. We show that HMC can sample from a distribution that is $\varepsilon$-close in total variation distance using $\widetilde{O}(\sqrt{\kappa} d^{1/4} \log(1/\varepsilon))$ gradient queries, where $\kappa$ is the condition number of $\Sigma$. Our algorithm uses long and random integration times for the Hamiltonian dynamics. This contrasts with (and was motivated by) recent results that give an $\widetilde\Omega(\kappa d^{1/2})$ query lower bound for HMC with fixed integration times, even for the Gaussian case.

Communication helps agents to obtain information about others so that better coordinated behavior can be learned. Some existing work communicates predicted future trajectory with others, hoping to get clues about what others would do for better coordination. However, circular dependencies sometimes can occur when agents are treated synchronously so it is hard to coordinate decision-making. In this paper, we propose a novel communication scheme, Sequential Communication (SeqComm). SeqComm treats agents asynchronously (the upper-level agents make decisions before the lower-level ones) and has two communication phases. In negotiation phase, agents determine the priority of decision-making by communicating hidden states of observations and comparing the value of intention, which is obtained by modeling the environment dynamics. In launching phase, the upper-level agents take the lead in making decisions and communicate their actions with the lower-level agents. Theoretically, we prove the policies learned by SeqComm are guaranteed to improve monotonically and converge. Empirically, we show that SeqComm outperforms existing methods in various multi-agent cooperative tasks.

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies are often heuristic and rely on domain-specific knowledge or trial-and-error. We treat the process of adaptive mesh refinement as a local, sequential decision-making problem under incomplete information, formulating AMR as a partially observable Markov decision process. Using a deep reinforcement learning approach, we train policy networks for AMR strategy directly from numerical simulation. The training process does not require an exact solution or a high-fidelity ground truth to the partial differential equation at hand, nor does it require a pre-computed training dataset. The local nature of our reinforcement learning formulation allows the policy network to be trained inexpensively on much smaller problems than those on which they are deployed. The methodology is not specific to any particular partial differential equation, problem dimension, or numerical discretization, and can flexibly incorporate diverse problem physics. To that end, we apply the approach to a diverse set of partial differential equations, using a variety of high-order discontinuous Galerkin and hybridizable discontinuous Galerkin finite element discretizations. We show that the resultant deep reinforcement learning policies are competitive with common AMR heuristics, generalize well across problem classes, and strike a favorable balance between accuracy and cost such that they often lead to a higher accuracy per problem degree of freedom.

The Inverse Reinforcement Learning (\textit{IRL}) problem has seen rapid evolution in the past few years, with important applications in domains like robotics, cognition, and health. In this work, we explore the inefficacy of current IRL methods in learning an agent's reward function from expert trajectories depicting long-horizon, complex sequential tasks. We hypothesize that imbuing IRL models with structural motifs capturing underlying tasks can enable and enhance their performance. Subsequently, we propose a novel IRL method, SMIRL, that first learns the (approximate) structure of a task as a finite-state-automaton (FSA), then uses the structural motif to solve the IRL problem. We test our model on both discrete grid world and high-dimensional continuous domain environments. We empirically show that our proposed approach successfully learns all four complex tasks, where two foundational IRL baselines fail. Our model also outperforms the baselines in sample efficiency on a simpler toy task. We further show promising test results in a modified continuous domain on tasks with compositional reward functions.

Bandits with knapsacks (BwK) is an influential model of sequential decision-making under uncertainty that incorporates resource consumption constraints. In each round, the decision-maker observes an outcome consisting of a reward and a vector of nonnegative resource consumptions, and the budget of each resource is decremented by its consumption. In this paper we introduce a natural generalization of the stochastic BwK problem that allows non-monotonic resource utilization. In each round, the decision-maker observes an outcome consisting of a reward and a vector of resource drifts that can be positive, negative or zero, and the budget of each resource is incremented by its drift. Our main result is a Markov decision process (MDP) policy that has constant regret against a linear programming (LP) relaxation when the decision-maker knows the true outcome distributions. We build upon this to develop a learning algorithm that has logarithmic regret against the same LP relaxation when the decision-maker does not know the true outcome distributions. We also present a reduction from BwK to our model that shows our regret bound matches existing results.

Our implementation of recurrent model-free RL outperforms the on-policy version (PPO/A2C-GRU), and a recent model-based POMDP algorithm (VRM) on most tasks of a POMDP benchmark where VRM was evaluated in their paper. While algorithms for decision-making typically focus on relatively easy problems where everything is known, most realistic problems involve noise and incomplete information. Complex algorithms have been proposed to tackle these complex problems, but there's a simple approach that (in theory) works on both the easy and the complex problems. We show how to make this simple approach work in practice. Decision-making tasks in the real world are messy, with noise, occlusions, and uncertainty that are typically missing from their canonical problem formulation as a Markov decision process (MDP; Bellman, 1957).

One of the most complex tasks of decision making and planning is to gather information. This task becomes even more complex when the state is high-dimensional and its belief cannot be expressed with a parametric distribution. Although the state is high-dimensional, in many problems only a small fraction of it might be involved in transitioning the state and generating observations. We exploit this fact to calculate an information-theoretic expected reward, mutual information (MI), over a much lower-dimensional subset of the state, to improve efficiency and without sacrificing accuracy. A similar approach was used in previous works, yet specifically for Gaussian distributions, and we here extend it for general distributions. Moreover, we apply the dimensionality reduction for cases in which the new states are augmented to the previous, yet again without sacrificing accuracy. We then continue by developing an estimator for the MI which works in a Sequential Monte Carlo (SMC) manner, and avoids the reconstruction of future belief's surfaces. Finally, we show how this work is applied to the informative planning optimization problem. This work is then evaluated in a simulation of an active SLAM problem, where the improvement in both accuracy and timing is demonstrated.

In this paper, we analyze the impact of data freshness on real-time supervised learning, where a neural network is trained to infer a time-varying target (e.g., the position of the vehicle in front) based on features (e.g., video frames) observed at a sensing node (e.g., camera or lidar). One might expect that the performance of real-time supervised learning degrades monotonically as the feature becomes stale. Using an information-theoretic analysis, we show that this is true if the feature and target data sequence can be closely approximated as a Markov chain; it is not true if the data sequence is far from Markovian. Hence, the prediction error of real-time supervised learning is a function of the Age of Information (AoI), where the function could be non-monotonic. Several experiments are conducted to illustrate the monotonic and non-monotonic behaviors of the prediction error. To minimize the inference error in real-time, we propose a new "selection-from-buffer" model for sending the features, which is more general than the "generate-at-will" model used in earlier studies. By using Gittins and Whittle indices, low-complexity scheduling strategies are developed to minimize the inference error, where a new connection between the Gittins index theory and Age of Information (AoI) minimization is discovered. These scheduling results hold (i) for minimizing general AoI functions (monotonic or non-monotonic) and (ii) for general feature transmission time distributions. Data-driven evaluations are presented to illustrate the benefits of the proposed scheduling algorithms.

We consider a discounted cost constrained Markov decision process (CMDP) policy optimization problem, in which an agent seeks to maximize a discounted cumulative reward subject to a number of constraints on discounted cumulative utilities. To solve this constrained optimization program, we study an online actor-critic variant of a classic primal-dual method where the gradients of both the primal and dual functions are estimated using samples from a single trajectory generated by the underlying time-varying Markov processes. This online primal-dual natural actor-critic algorithm maintains and iteratively updates three variables: a dual variable (or Lagrangian multiplier), a primal variable (or actor), and a critic variable used to estimate the gradients of both primal and dual variables. These variables are updated simultaneously but on different time scales (using different step sizes) and they are all intertwined with each other. Our main contribution is to derive a finite-time analysis for the convergence of this algorithm to the global optimum of a CMDP problem. Specifically, we show that with a proper choice of step sizes the optimality gap and constraint violation converge to zero in expectation at a rate $\mathcal{O}(1/K^{1/6})$, where K is the number of iterations. To our knowledge, this paper is the first to study the finite-time complexity of an online primal-dual actor-critic method for solving a CMDP problem. We also validate the effectiveness of this algorithm through numerical simulations.

Restricted Boltzmann Machines (RBMs) offer a versatile architecture for unsupervised machine learning that can in principle approximate any target probability distribution with arbitrary accuracy. However, the RBM model is usually not directly accessible due to its computational complexity, and Markov-chain sampling is invoked to analyze the learned probability distribution. For training and eventual applications, it is thus desirable to have a sampler that is both accurate and efficient. We highlight that these two goals generally compete with each other and cannot be achieved simultaneously. More specifically, we identify and quantitatively characterize three regimes of RBM learning: independent learning, where the accuracy improves without losing efficiency; correlation learning, where higher accuracy entails lower efficiency; and degradation, where both accuracy and efficiency no longer improve or even deteriorate. These findings are based on numerical experiments and heuristic arguments.