While multitask representation learning has become a popular approach in reinforcement learning (RL), theoretical understanding of why and when it works remains limited. This paper presents analyses for the statistical benefit of multitask representation learning in linear Markov Decision Process (MDP) under a generative model. In this paper, we consider an agent to learn a representation function $\phi$ out of a function class $\Phi$ from $T$ source tasks with $N$ data per task, and then use the learned $\hat{\phi}$ to reduce the required number of sample for a new task. We first discover a \emph{Least-Activated-Feature-Abundance} (LAFA) criterion, denoted as $\kappa$, with which we prove that a straightforward least-square algorithm learns a policy which is $\tilde{O}(H^2\sqrt{\frac{\mathcal{C}(\Phi)^2 \kappa d}{NT}+\frac{\kappa d}{n}})$ sub-optimal. Here $H$ is the planning horizon, $\mathcal{C}(\Phi)$ is $\Phi$'s complexity measure, $d$ is the dimension of the representation (usually $d\ll \mathcal{C}(\Phi)$) and $n$ is the number of samples for the new task. Thus the required $n$ is $O(\kappa d H^4)$ for the sub-optimality to be close to zero, which is much smaller than $O(\mathcal{C}(\Phi)^2\kappa d H^4)$ in the setting without multitask representation learning, whose sub-optimality gap is $\tilde{O}(H^2\sqrt{\frac{\kappa \mathcal{C}(\Phi)^2d}{n}})$. This theoretically explains the power of multitask representation learning in reducing sample complexity. Further, we note that to ensure high sample efficiency, the LAFA criterion $\kappa$ should be small. In fact, $\kappa$ varies widely in magnitude depending on the different sampling distribution for new task. This indicates adaptive sampling technique is important to make $\kappa$ solely depend on $d$. Finally, we provide empirical results of a noisy grid-world environment to corroborate our theoretical findings.

Online reinforcement learning (RL) has been widely applied in information processing scenarios, which usually exhibit much uncertainty due to the intrinsic randomness of channels and service demands. In this paper, we consider an un-discounted RL in general Markov decision processes (MDPs) with both endogeneous and exogeneous uncertainty, where both the rewards and state transition probability are unknown to the RL agent and evolve with the time as long as their respective variations do not exceed certain dynamic budget (i.e., upper bound). We first develop a variation-aware Bernstein-based upper confidence reinforcement learning (VB-UCRL), which we allow to restart according to a schedule dependent on the variations. We successfully overcome the challenges due to the exogeneous uncertainty and establish a regret bound of saving at most $\sqrt{S}$ or $S^{\frac{1}{6}}T^{\frac{1}{12}}$ compared with the latest results in the literature, where $S$ denotes the state size of the MDP and $T$ indicates the iteration index of learning steps.

A central goal of machine learning is to learn robust representations that capture the causal relationship between inputs features and output labels. However, minimizing empirical risk over finite or biased datasets often results in models latching on to spurious correlations between the training input/output pairs that are not fundamental to the problem at hand. In this paper, we define and analyze robust and spurious representations using the information-theoretic concept of minimal sufficient statistics. We prove that even when there is only bias of the input distribution (i.e. covariate shift), models can still pick up spurious features from their training data. Group distributionally robust optimization (DRO) provides an effective tool to alleviate covariate shift by minimizing the worst-case training loss over a set of pre-defined groups. Inspired by our analysis, we demonstrate that group DRO can fail when groups do not directly account for various spurious correlations that occur in the data. To address this, we further propose to minimize the worst-case losses over a more flexible set of distributions that are defined on the joint distribution of groups and instances, instead of treating each group as a whole at optimization time. Through extensive experiments on one image and two language tasks, we show that our model is significantly more robust than comparable baselines under various partitions. Our code is available at https://github.com/violet-zct/group-conditional-DRO.

Reinforcement learning (RL) is empirically successful in complex nonlinear Markov decision processes (MDPs) with continuous state spaces. By contrast, the majority of theoretical RL literature requires the MDP to satisfy some form of linear structure, in order to guarantee sample efficient RL. Such efforts typically assume the transition dynamics or value function of the MDP are described by linear functions of the state features. To resolve this discrepancy between theory and practice, we introduce the Effective Planning Window (EPW) condition, a structural condition on MDPs that makes no linearity assumptions. We demonstrate that the EPW condition permits sample efficient RL, by providing an algorithm which provably solves MDPs satisfying this condition. Our algorithm requires minimal assumptions on the policy class, which can include multi-layer neural networks with nonlinear activation functions. Notably, the EPW condition is directly motivated by popular gaming benchmarks, and we show that many classic Atari games satisfy this condition. We additionally show the necessity of conditions like EPW, by demonstrating that simple MDPs with slight nonlinearities cannot be solved sample efficiently.

A graph generative model defines a distribution over graphs. One type of generative model is constructed by autoregressive neural networks, which sequentially add nodes and edges to generate a graph. However, the likelihood of a graph under the autoregressive model is intractable, as there are numerous sequences leading to the given graph; this makes maximum likelihood estimation challenging. Instead, in this work we derive the exact joint probability over the graph and the node ordering of the sequential process. From the joint, we approximately marginalize out the node orderings and compute a lower bound on the log-likelihood using variational inference. We train graph generative models by maximizing this bound, without using the ad-hoc node orderings of previous methods. Our experiments show that the log-likelihood bound is significantly tighter than the bound of previous schemes. Moreover, the models fitted with the proposed algorithm can generate high-quality graphs that match the structures of target graphs not seen during training. We have made our code publicly available at \hyperref[https://github.com/tufts-ml/graph-generation-vi]{https://github.com/tufts-ml/graph-generation-vi}.

We study automated intrusion prevention using reinforcement learning. In a novel approach, we formulate the problem of intrusion prevention as an optimal stopping problem. This formulation allows us insight into the structure of the optimal policies, which turn out to be threshold based. Since the computation of the optimal defender policy using dynamic programming is not feasible for practical cases, we approximate the optimal policy through reinforcement learning in a simulation environment. To define the dynamics of the simulation, we emulate the target infrastructure and collect measurements. Our evaluations show that the learned policies are close to optimal and that they indeed can be expressed using thresholds.

We focus on the problem of efficient sampling and learning of probability densities by incorporating symmetries in probabilistic models. We first introduce Equivariant Stein Variational Gradient Descent algorithm -- an equivariant sampling method based on Stein's identity for sampling from densities with symmetries. Equivariant SVGD explicitly incorporates symmetry information in a density through equivariant kernels which makes the resultant sampler efficient both in terms of sample complexity and the quality of generated samples. Subsequently, we define equivariant energy based models to model invariant densities that are learned using contrastive divergence. By utilizing our equivariant SVGD for training equivariant EBMs, we propose new ways of improving and scaling up training of energy based models. We apply these equivariant energy models for modelling joint densities in regression and classification tasks for image datasets, many-body particle systems and molecular structure generation.

Learning the causal structure that underlies data is a crucial step towards robust real-world decision making. The majority of existing work in causal inference focuses on determining a single directed acyclic graph (DAG) or a Markov equivalence class thereof. However, a crucial aspect to acting intelligently upon the knowledge about causal structure which has been inferred from finite data demands reasoning about its uncertainty. For instance, planning interventions to find out more about the causal mechanisms that govern our data requires quantifying epistemic uncertainty over DAGs. While Bayesian causal inference allows to do so, the posterior over DAGs becomes intractable even for a small number of variables. Aiming to overcome this issue, we propose a form of variational inference over the graphs of Structural Causal Models (SCMs). To this end, we introduce a parametric variational family modelled by an autoregressive distribution over the space of discrete DAGs. Its number of parameters does not grow exponentially with the number of variables and can be tractably learned by maximising an Evidence Lower Bound (ELBO). In our experiments, we demonstrate that the proposed variational posterior is able to provide a good approximation of the true posterior.

Actor-critic methods integrating target networks have exhibited a stupendous empirical success in deep reinforcement learning. However, a theoretical understanding of the use of target networks in actor-critic methods is largely missing in the literature. In this paper, we bridge this gap between theory and practice by proposing the first theoretical analysis of an online target-based actor-critic algorithm with linear function approximation in the discounted reward setting. Our algorithm uses three different timescales: one for the actor and two for the critic. Instead of using the standard single timescale temporal difference (TD) learning algorithm as a critic, we use a two timescales target-based version of TD learning closely inspired from practical actor-critic algorithms implementing target networks. First, we establish asymptotic convergence results for both the critic and the actor under Markovian sampling. Then, we provide a finite-time analysis showing the impact of incorporating a target network into actor-critic methods.

Marginalising over families of Gaussian Process kernels produces flexible model classes with well-calibrated uncertainty estimates. Existing approaches require likelihood evaluations of many kernels, rendering them prohibitively expensive for larger datasets. We propose a Bayesian Quadrature scheme to make this marginalisation more efficient and thereby more practical. Through use of the maximum mean discrepancies between distributions, we define a kernel over kernels that captures invariances between Spectral Mixture (SM) Kernels. Kernel samples are selected by generalising an information-theoretic acquisition function for warped Bayesian Quadrature. We show that our framework achieves more accurate predictions with better calibrated uncertainty than state-of-the-art baselines, especially when given limited (wall-clock) time budgets.