We study the classical problem of community recovery in stochastic block models with a fixed number of communities, with a twist: We seek algorithms that are stable with respect to node-wise changes in the graph structure, formally defined as a differential privacy constraint. The algorithms we develop are based on spectral clustering, where we introduce privacy to the community recovery pipeline in the form of directly privatizing the adjacency matrix; private PCA; private convex optimization; private low-rank matrix estimation; and private approximate subspace estimation. Straightforward applications of existing private algorithms lead to a rapid increase in the privacy parameter $ε$ in order to ensure consistent estimation under node differential privacy, in contrast with the simpler setting of edge privacy. To alleviate these issues, we develop novel algorithms based on (1) sampling from an exponential mechanism with a Lipschitz extension and (2) a general framework for constructing smooth projections from the space of undirected graphs to the space of bounded-degree graphs, which can then be combined with various edge-private algorithms. Importantly, the methods we develop are all computable in polynomial-time as a function of the number of nodes in the graph. We also develop novel lower bounds on the growth rate of $ε$ required in order to achieve consistent community estimation under node privacy. On a technical note, our paper highlights the complications that arise when analyzing private algorithms under the non-standard scaling $ε\rightarrow \infty$ and proposes some solutions. We also provide a novel application of the HGR maximal correlation from information theory in the context of accuracy amplification in PAC learning, which may be of independent interest.

We quantify the efficiency of temporal difference (TD) learning over the direct, or Monte Carlo (MC), estimator for policy evaluation in reinforcement learning, with an emphasis on estimation of quantities related to rare events. Policy evaluation is complicated in the rare event setting by the long timescale of the event and by the need for \emph{relative accuracy} in estimates of very small values. Specifically, we focus on least-squares TD (LSTD) prediction for finite state Markov chains, and show that LSTD can achieve relative accuracy far more efficiently than MC. We prove a central limit theorem for the LSTD estimator and upper bound the \emph{relative asymptotic variance} by simple quantities characterizing the connectivity of states relative to the transition probabilities between them. Using this bound, we show that, even when both the timescale of the rare event and the relative accuracy of the MC estimator are exponentially large in the number of states, LSTD maintains a fixed level of relative accuracy with a total number of observed transitions of the Markov chain that is only \emph{polynomially} large in the number of states.

Directed exploration strategies for reinforcement learning are critical for learning an optimal policy in a minimal number of interactions with the environment. Many algorithms use optimism to direct exploration, either through visitation estimates or upper confidence bounds, as opposed to data-inefficient strategies like e-greedy that use random, undirected exploration. Most data-efficient exploration methods require significant computation, typically relying on a learned model to guide exploration. Least-squares methods have the potential to provide some of the data-efficiency benefits of model-based approaches--because they summarize past interactions--with the computation closer to that of model-free approaches. In this work, we provide a novel, computationally efficient, incremental exploration strategy, leveraging this property of least-squares temporal difference learning (LSTD). We derive upper confidence bounds on the action-values learned by LSTD, with context-dependent (or state-dependent) noise variance. Such context-dependent noise focuses exploration on a subset of variable states, and allows for reduced exploration in other states. We empirically demonstrate that our algorithm can converge more quickly than other incremental exploration strategies using confidence estimates on action-values.

Second, we t = rt+1+ t+1x>t+1w x>t w , TD-errorforw (see (2)). This t is tobelarger t.

Off-policy evaluation (OPE) is a fundamental task in reinforcement learning (RL). In the classic setting of linear OPE, finite-sample guarantees often take the form $$ \textrm{Evaluation error} \le \textrm{poly}(C^π, d, 1/n,\log(1/δ)), $$ where $d$ is the dimension of the features and $C^π$ is a coverage parameter that characterizes the degree to which the visited features lie in the span of the data distribution. While such guarantees are well-understood for several popular algorithms under stronger assumptions (e.g. Bellman completeness), the understanding is lacking and fragmented in the minimal setting where only the target value function is linearly realizable in the features. Despite recent interest in tight characterizations of the statistical rate in this setting, the right notion of coverage remains unclear, and candidate definitions from prior analyses have undesirable properties and are starkly disconnected from more standard definitions in the literature. We provide a novel finite-sample analysis of a canonical algorithm for this setting, LSTDQ. Inspired by an instrumental-variable view, we develop error bounds that depend on a novel coverage parameter, the feature-dynamics coverage, which can be interpreted as linear coverage in an induced dynamical system for feature evolution. With further assumptions -- such as Bellman-completeness -- our definition successfully recovers the coverage parameters specialized to those settings, finally yielding a unified understanding for coverage in linear OPE.

To achieve such a goal, directed exploration strategies are key.

Directed exploration strategies for reinforcement learning are critical for learning an optimal policy in a minimal number of interactions with the environment. Many algorithms use optimism to direct exploration, either through visitation estimates or upper confidence bounds, as opposed to data-inefficient strategies like e-greedy that use random, undirected exploration. Most data-efficient exploration methods require significant computation, typically relying on a learned model to guide exploration. Least-squares methods have the potential to provide some of the data-efficiency benefits of model-based approaches--because they summarize past interactions--with the computation closer to that of model-free approaches. In this work, we provide a novel, computationally efficient, incremental exploration strategy, leveraging this property of least-squares temporal difference learning (LSTD). We derive upper confidence bounds on the action-values learned by LSTD, with context-dependent (or state-dependent) noise variance. Such context-dependent noise focuses exploration on a subset of variable states, and allows for reduced exploration in other states. We empirically demonstrate that our algorithm can converge more quickly than other incremental exploration strategies using confidence estimates on action-values.

We quantify the efficiency of temporal difference (TD) learning over the direct, or Monte Carlo (MC), estimator for policy evaluation in reinforcement learning, with an emphasis on estimation of quantities related to rare events. Policy evaluation is complicated in the rare event setting by the long timescale of the event and by the need for \emph{relative accuracy} in estimates of very small values. Specifically, we focus on least-squares TD (LSTD) prediction for finite state Markov chains, and show that LSTD can achieve relative accuracy far more efficiently than MC. We prove a central limit theorem for the LSTD estimator and upper bound the \emph{relative asymptotic variance} by simple quantities characterizing the connectivity of states relative to the transition probabilities between them. Using this bound, we show that, even when both the timescale of the rare event and the relative accuracy of the MC estimator are exponentially large in the number of states, LSTD maintains a fixed level of relative accuracy with a total number of observed transitions of the Markov chain that is only \emph{polynomially} large in the number of states.

This is a borderline paper. The paper is technically sound and addressing OPE in average-reward setting is an important problem. Despite that the work is an extension of Duan and Wang (for discounted setting) to the average-reward setting, the algorithm is somewhat different, as Duan and Wang uses FQE whereas the current paper performs stationary-distribution estimation. That said, there are a few weaknesses that the paper should try to address or at least discuss: 1. The entropy maximization is a novel algorithmic element which does not appear in previous approaches in the discounted setting.

Traditionally, TD and FQI are viewed as differing in the number of updates toward the target value function: TD makes one update, FQI makes an infinite number, and Partial Fitted Q-Iteration (PFQI) performs a finite number, such as the use of a target network in Deep Q-Networks (DQN) in the OPE setting. This perspective, however, fails to capture the convergence connections between these algorithms and may lead to incorrect conclusions, for example, that the convergence of TD implies the convergence of FQI. In this paper, we focus on linear value function approximation and offer a new perspective, unifying TD, FQI, and PFQI as the same iterative method for solving the Least Squares Temporal Difference (LSTD) system, but using different preconditioners and matrix splitting schemes. TD uses a constant preconditioner, FQI employs a data-feature adaptive preconditioner, and PFQI transitions between the two. Then, we reveal that in the context of linear function approximation, increasing the number of updates under the same target value function essentially represents a transition from using a constant preconditioner to data-feature adaptive preconditioner. This unifying perspective also simplifies the analyses of the convergence conditions for these algorithms and clarifies many issues. Consequently, we fully characterize the convergence of each algorithm without assuming specific properties of the chosen features (e.g., linear independence). We also examine how common assumptions about feature representations affect convergence, and discover new conditions on features that are important for convergence. These convergence conditions allow us to establish the convergence connections between these algorithms and to address important questions.