We consider the problem of forming prediction sets in an online setting where the distribution generating the data is allowed to vary over time. Previous approaches to this problem suffer from over-weighting historical data and thus may fail to quickly react to the underlying dynamics. Here we correct this issue and develop a novel procedure with provably small regret over all local time intervals of a given width. We achieve this by modifying the adaptive conformal inference (ACI) algorithm of Gibbs and Cand\`{e}s (2021) to contain an additional step in which the step-size parameter of ACI's gradient descent update is tuned over time. Crucially, this means that unlike ACI, which requires knowledge of the rate of change of the data-generating mechanism, our new procedure is adaptive to both the size and type of the distribution shift. Our methods are highly flexible and can be used in combination with any baseline predictive algorithm that produces point estimates or estimated quantiles of the target without the need for distributional assumptions. We test our techniques on two real-world datasets aimed at predicting stock market volatility and COVID-19 case counts and find that they are robust and adaptive to real-world distribution shifts.

We present the problem of reinforcement learning with exogenous termination. We define the Termination Markov Decision Process (TerMDP), an extension of the MDP framework, in which episodes may be interrupted by an external non-Markovian observer. This formulation accounts for numerous real-world situations, such as a human interrupting an autonomous driving agent for reasons of discomfort. We learn the parameters of the TerMDP and leverage the structure of the estimation problem to provide state-wise confidence bounds. We use these to construct a provably-efficient algorithm, which accounts for termination, and bound its regret. Motivated by our theoretical analysis, we design and implement a scalable approach, which combines optimism (w.r.t. termination) and a dynamic discount factor, incorporating the termination probability. We deploy our method on high-dimensional driving and MinAtar benchmarks. Additionally, we test our approach on human data in a driving setting. Our results demonstrate fast convergence and significant improvement over various baseline approaches.

We study the global convergence of a Fisher-Rao policy gradient flow for infinite-horizon entropy-regularised Markov decision processes with Polish state and action space. The flow is a continuous-time analogue of a policy mirror descent method. We establish the global well-posedness of the gradient flow and demonstrate its exponential convergence to the optimal policy. Moreover, we prove the flow is stable with respect to gradient evaluation, offering insights into the performance of a natural policy gradient flow with log-linear policy parameterisation. To overcome challenges stemming from the lack of the convexity of the objective function and the discontinuity arising from the entropy regulariser, we leverage the performance difference lemma and the duality relationship between the gradient and mirror descent flows.

Autonomous agents operating in real-world scenarios frequently encounter uncertainty and make decisions based on incomplete information. Planning under uncertainty can be mathematically formalized using partially observable Markov decision processes (POMDPs). However, finding an optimal plan for POMDPs can be computationally expensive and is feasible only for small tasks. In recent years, approximate algorithms, such as tree search and sample-based methodologies, have emerged as state-of-the-art POMDP solvers for larger problems. Despite their effectiveness, these algorithms offer only probabilistic and often asymptotic guarantees toward the optimal solution due to their dependence on sampling. To address these limitations, we derive a deterministic relationship between a simplified solution that is easier to obtain and the theoretically optimal one. First, we derive bounds for selecting a subset of the observations to branch from while computing a complete belief at each posterior node. Then, since a complete belief update may be computationally demanding, we extend the bounds to support reduction of both the state and the observation spaces. We demonstrate how our guarantees can be integrated with existing state-of-the-art solvers that sample a subset of states and observations. As a result, the returned solution holds deterministic bounds relative to the optimal policy. Lastly, we substantiate our findings with supporting experimental results.

This paper studies Hoeffding's inequality for Markov chains under the generalized concentrability condition defined via integral probability metric (IPM). The generalized concentrability condition establishes a framework that interpolates and extends the existing hypotheses of Markov chain Hoeffding-type inequalities. The flexibility of our framework allows Hoeffding's inequality to be applied beyond the ergodic Markov chains in the traditional sense. We demonstrate the utility by applying our framework to several non-asymptotic analyses arising from the field of machine learning, including (i) a generalization bound for empirical risk minimization with Markovian samples, (ii) a finite sample guarantee for Ployak-Ruppert averaging of SGD, and (iii) a new regret bound for rested Markovian bandits with general state space. Keywords: Hoeffding's inequality, Markov chains, Dobrushin coefficient, integral probability metric, concentration of measures, ergodicity, empirical risk minimization, stochastic gradient descent, rested bandit

Markov Decision Processes (MDPs) are a formal framework for modeling and solving sequential decision-making problems. In finite-time horizons such problems are relevant for instance for optimal stopping or specific supply chain problems, but also in the training of large language models. In contrast to infinite horizon MDPs optimal policies are not stationary, policies must be learned for every single epoch. In practice all parameters are often trained simultaneously, ignoring the inherent structure suggested by dynamic programming. This paper introduces a combination of dynamic programming and policy gradient called dynamic policy gradient, where the parameters are trained backwards in time. For the tabular softmax parametrisation we carry out the convergence analysis for simultaneous and dynamic policy gradient towards global optima, both in the exact and sampled gradient settings without regularisation. It turns out that the use of dynamic policy gradient training much better exploits the structure of finite-time problems which is reflected in improved convergence bounds.

Partially observable Markov decision processes (POMDP) are a useful model for decision-making under partial observability and stochastic actions. Partially Observable Monte-Carlo Planning is an online algorithm for deciding on the next action to perform, using a Monte-Carlo tree search approach, based on the UCT (UCB applied to trees) algorithm for fully observable Markov-decision processes. POMCP develops an action-observation tree, and at the leaves, uses a rollout policy to provide a value estimate for the leaf. As such, POMCP is highly dependent on the rollout policy to compute good estimates, and hence identify good actions. Thus, many practitioners who use POMCP are required to create strong, domain-specific heuristics. In this paper, we model POMDPs as stochastic contingent planning problems. This allows us to leverage domain-independent heuristics that were developed in the planning community. We suggest two heuristics, the first is based on the well-known h_add heuristic from classical planning, and the second is computed in belief space, taking the value of information into account.

In recent years, the integration of Automated Planning (AP) and Reinforcement Learning (RL) has seen a surge of interest. To perform this integration, a general framework for Sequential Decision Making (SDM) would prove immensely useful, as it would help us understand how AP and RL fit together. In this preliminary work, we attempt to provide such a framework, suitable for any method ranging from Classical Planning to Deep RL, by drawing on concepts from Probability Theory and Bayesian inference. We formulate an SDM task as a set of training and test Markov Decision Processes (MDPs), to account for generalization. We provide a general algorithm for SDM which we hypothesize every SDM method is based on. According to it, every SDM algorithm can be seen as a procedure that iteratively improves its solution estimate by leveraging the task knowledge available. Finally, we derive a set of formulas and algorithms for calculating interesting properties of SDM tasks and methods, which make possible their empirical evaluation and comparison.

Whittle index policy is a heuristic to the intractable restless multi-armed bandits (RMAB) problem. Although it is provably asymptotically optimal, finding Whittle indices remains difficult. In this paper, we present Neural-Q-Whittle, a Whittle index based Q-learning algorithm for RMAB with neural network function approximation, which is an example of nonlinear two-timescale stochastic approximation with Q-function values updated on a faster timescale and Whittle indices on a slower timescale. Despite the empirical success of deep Q-learning, the non-asymptotic convergence rate of Neural-Q-Whittle, which couples neural networks with two-timescale Q-learning largely remains unclear. This paper provides a finite-time analysis of Neural-Q-Whittle, where data are generated from a Markov chain, and Q-function is approximated by a ReLU neural network. Our analysis leverages a Lyapunov drift approach to capture the evolution of two coupled parameters, and the nonlinearity in value function approximation further requires us to characterize the approximation error. Combing these provide Neural-Q-Whittle with $\mathcal{O}(1/k^{2/3})$ convergence rate, where $k$ is the number of iterations.

Model-based reinforcement learning seeks to simultaneously learn the dynamics of an unknown stochastic environment and synthesise an optimal policy for acting in it. Ensuring the safety and robustness of sequential decisions made through a policy in such an environment is a key challenge for policies intended for safety-critical scenarios. In this work, we investigate two complementary problems: first, computing reach-avoid probabilities for iterative predictions made with dynamical models, with dynamics described by Bayesian neural network (BNN); second, synthesising control policies that are optimal with respect to a given reach-avoid specification (reaching a "target" state, while avoiding a set of "unsafe" states) and a learned BNN model. Our solution leverages interval propagation and backward recursion techniques to compute lower bounds for the probability that a policy's sequence of actions leads to satisfying the reach-avoid specification. Such computed lower bounds provide safety certification for the given policy and BNN model. We then introduce control synthesis algorithms to derive policies maximizing said lower bounds on the safety probability. We demonstrate the effectiveness of our method on a series of control benchmarks characterized by learned BNN dynamics models. On our most challenging benchmark, compared to purely data-driven policies the optimal synthesis algorithm is able to provide more than a four-fold increase in the number of certifiable states and more than a three-fold increase in the average guaranteed reach-avoid probability.