For the task of sampling from a density $\pi \propto \exp(-V)$ on $\mathbb{R}^d$, where $V$ is possibly non-convex but $L$-gradient Lipschitz, we prove that averaged Langevin Monte Carlo outputs a sample with $\varepsilon$-relative Fisher information after $O( L^2 d^2/\varepsilon^2)$ iterations. This is the sampling analogue of complexity bounds for finding an $\varepsilon$-approximate first-order stationary points in non-convex optimization and therefore constitutes a first step towards the general theory of non-log-concave sampling. We discuss numerous extensions and applications of our result; in particular, it yields a new state-of-the-art guarantee for sampling from distributions which satisfy a Poincar\'e inequality.

In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator and the necessary numerical elements are given to facilitate the implementation of the methodology in a parallel computing framework. In a second part, an application is presented to illustrate the proposed theory and is also, as such, a contribution to the three-dimensional stochastic homogenization of heterogeneous linear elastic media in the case of a non-separation of the microscale and macroscale. For the construction of the posterior probability measure by using the probabilistic learning inference, in addition to the constraints defined by given statistical moments of the random effective elasticity tensor, the second-order moment of the random normalized residue of the stochastic partial differential equation has been added as a constraint. This constraint guarantees that the algorithm seeks to bring the statistical moments closer to their targets while preserving a small residue.

We study a novel setting in offline reinforcement learning (RL) where a number of distributed machines jointly cooperate to solve the problem but only one single round of communication is allowed and there is a budget constraint on the total number of information (in terms of bits) that each machine can send out. For value function prediction in contextual bandits, and both episodic and non-episodic MDPs, we establish information-theoretic lower bounds on the minimax risk for distributed statistical estimators; this reveals the minimum amount of communication required by any offline RL algorithms. Specifically, for contextual bandits, we show that the number of bits must scale at least as $\Omega(AC)$ to match the centralised minimax optimal rate, where $A$ is the number of actions and $C$ is the context dimension; meanwhile, we reach similar results in the MDP settings. Furthermore, we develop learning algorithms based on least-squares estimates and Monte-Carlo return estimates and provide a sharp analysis showing that they can achieve optimal risk up to logarithmic factors. Additionally, we also show that temporal difference is unable to efficiently utilise information from all available devices under the single-round communication setting due to the initial bias of this method. To our best knowledge, this paper presents the first minimax lower bounds for distributed offline RL problems.

We consider stochastic optimization problems where data is drawn from a Markov chain. Existing methods for this setting crucially rely on knowing the mixing time of the chain, which in real-world applications is usually unknown. We propose the first optimization method that does not require the knowledge of the mixing time, yet obtains the optimal asymptotic convergence rate when applied to convex problems. We further show that our approach can be extended to: (i) finding stationary points in non-convex optimization with Markovian data, and (ii) obtaining better dependence on the mixing time in temporal difference (TD) learning; in both cases, our method is completely oblivious to the mixing time. Our method relies on a novel combination of multi-level Monte Carlo (MLMC) gradient estimation together with an adaptive learning method.

The ability of an agent to change its objectives in response to unexpected events is desirable in dynamic environments. In order to provide this capability to hierarchical task network (HTN) planning, we propose an extension of the paradigm called task modifiers, which are functions that receive a task list and a state and produce a new task list. We focus on a particular type of problems in which planning and execution are interleaved and the ability to handle exogenous events is crucial. To determine the efficacy of this approach, we evaluate the performance of our task modifier implementation in two environments, one of which is a simulation that differs substantially from traditional HTN domains.

Bayesian Neural Networks (BNNs) offer a mathematically grounded framework to quantify the uncertainty of model predictions but come with a prohibitive computation cost for both training and inference. In this work, we show a novel network architecture search (NAS) that optimizes BNNs for both accuracy and uncertainty while having a reduced inference latency. Different from canonical NAS that optimizes solely for in-distribution likelihood, the proposed scheme searches for the uncertainty performance using both in- and out-of-distribution data. Our method is able to search for the correct placement of Bayesian layer(s) in a network. In our experiments, the searched models show comparable uncertainty quantification ability and accuracy compared to the state-of-the-art (deep ensemble). In addition, the searched models use only a fraction of the runtime compared to many popular BNN baselines, reducing the inference runtime cost by $2.98 \times$ and $2.92 \times$ respectively on the CIFAR10 dataset when compared to MCDropout and deep ensemble.

In recent years, symbolic regression has been of wide interest to provide an interpretable symbolic representation of potentially large data relationships. Initially circled to genetic algorithms, symbolic regression methods now include a variety of Deep Learning based alternatives. However, these methods still do not generalize well to real-world data, mainly because they hardly include domain knowledge nor consider physical relationships between variables such as known equations and units. Regarding these issues, we propose a Reinforcement-Based Grammar-Guided Symbolic Regression (RBG2-SR) method that constrains the representational space with domain-knowledge using context-free grammar as reinforcement action space. We detail a Partially-Observable Markov Decision Process (POMDP) modeling of the problem and benchmark our approach against state-of-the-art methods. We also analyze the POMDP state definition and propose a physical equation search use case on which we compare our approach to grammar-based and non-grammarbased symbolic regression methods. The experiment results show that our method is competitive against other state-of-the-art methods on the benchmarks and offers the best error-complexity trade-off, highlighting the interest of using a grammar-based method in a real-world scenario.

Simulator-based models are models for which the likelihood is intractable but simulation of synthetic data is possible. They are often used to describe complex real-world phenomena, and as such can often be misspecified in practice. Unfortunately, existing Bayesian approaches for simulators are known to perform poorly in those cases. In this paper, we propose a novel algorithm based on the posterior bootstrap and maximum mean discrepancy estimators. This leads to a highly-parallelisable Bayesian inference algorithm with strong robustness properties. This is demonstrated through an in-depth theoretical study which includes generalisation bounds and proofs of frequentist consistency and robustness of our posterior. The approach is then assessed on a range of examples including a g-and-k distribution and a toggle-switch model.

This paper introduces a novel approach for abstraction selection in reinforcement learning problems modelled as factored Markov decision processes (MDPs), for which a state is described via a set of state components. In abstraction selection, an agent must choose an abstraction from a set of candidate abstractions, each build up from a different combination of state components.

Logistic regression is a commonly used representation for aggregators in Bayesian belief networks when a child has multiple parents. In this paper we consider extending logistic regression to relational models, where we want to model varying populations and interactions among parents. In this paper, we first examine the representational problems caused by population variation. We show how these problems arise even in simple cases with a single parametrized parent, and propose a linear relational logistic regression which we show can represent arbitrary linear (in population size) decision thresholds, whereas the traditional logistic regression cannot. Then we examine representing interactions among the parents of a child node, and representing non-linear dependency on population size.