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In this paper, we study safe data collection for the purpose of policy evaluation in tabular Markov decision processes (MDPs). In policy evaluation, we are given a target policy and asked to estimate the expected cumulative reward it will obtain. Policy evaluation requires data and we are interested in the question of what behavior policy should collect the data for the most accurate evaluation of the target policy. While prior work has considered behavior policy selection, in this paper, we additionally consider a safety constraint on the behavior policy. Namely, we assume there exists a known default policy that incurs a particular expected cost when run and we enforce that the cumulative cost of all behavior policies ran is better than a constant factor of the cost that would be incurred had we always run the default policy. We first show that there exists a class of intractable MDPs where no safe oracle algorithm with knowledge about problem parameters can efficiently collect data and satisfy the safety constraints. We then define the tractability condition for an MDP such that a safe oracle algorithm can efficiently collect data and using that we prove the first lower bound for this setting. We then introduce an algorithm SaVeR for this problem that approximates the safe oracle algorithm and bound the finite-sample mean squared error of the algorithm while ensuring it satisfies the safety constraint. Finally, we show in simulations that SaVeR produces low MSE policy evaluation while satisfying the safety constraint. 

In this paper, we study the problem of optimal data collection for policy evaluation in linear bandits. In policy evaluation, we are given a \textit{target} policy and asked to estimate the expected reward it will obtain when executed in a multi-armed bandit environment. Our work is the first work that focuses on such an optimal data collection strategy for policy evaluation involving heteroscedastic reward noise in the linear bandit setting. We first formulate an optimal design for weighted least squares estimates in the heteroscedastic linear bandit setting with the knowledge of noise variances. This design minimizes the mean squared error (MSE) of the estimated value of the target policy and is termed the oracle design. Since the noise variance is typically unknown, we then introduce a novel algorithm, SPEED (\textbf{S}tructured \textbf{P}olicy \textbf{E}valuation \textbf{E}xperimental \textbf{D}esign), that tracks the oracle design and derive its regret with respect to the oracle design. We show that regret scales as 𝑂˜(𝑑3𝑛−3/2) and prove a matching lower bound of Ω(𝑑2𝑛−3/2) . Finally, we evaluate SPEED on a set of policy evaluation tasks and demonstrate that it achieves MSE comparable to an optimal oracle and much lower than simply running the target policy. 

We study multi-task representation learning for the problem of pure exploration in bilinear bandits. In bilinear bandits, an action takes the form of a pair of arms from two different entity types and the reward is a bilinear function of the known feature vectors of the arms. In the \textit{multi-task bilinear bandit problem}, we aim to find optimal actions for multiple tasks that share a common low-dimensional linear representation. The objective is to leverage this characteristic to expedite the process of identifying the best pair of arms for all tasks. We propose the algorithm GOBLIN that uses an experimental design approach to optimize sample allocations for learning the global representation as well as minimize the number of samples needed to identify the optimal pair of arms in individual tasks. To the best of our knowledge, this is the first study to give sample complexity analysis for pure exploration in bilinear bandits with shared representation. Our results demonstrate that by learning the shared representation across tasks, we achieve significantly improved sample complexity compared to the traditional approach of solving tasks independently. 

This paper studies the problem of data collection for policy evaluation in Markov decision processes (MDPs). In policy evaluation, we are given a \textit{target} policy and asked to estimate the expected cumulative reward it will obtain in an environment formalized as an MDP. We develop theory for optimal data collection within the class of tree-structured MDPs by first deriving an oracle exploration strategy that uses knowledge of the variance of the reward distributions. We then introduce the \textbf{Re}duced \textbf{Var}iance Sampling (\rev\!) algorithm that approximates the oracle strategy when the reward variances are unknown a priori and bound its sub-optimality compared to the oracle strategy. Finally, we empirically validate that \rev leads to policy evaluation with mean squared error comparable to the oracle strategy and significantly lower than simply running the target policy. 

Active learning can reduce the number of samples needed to perform a hypothesis test and to estimate the parameters of a model. In this paper, we revisit the work of Chernoff that described an asymptotically optimal algorithm for performing a hypothesis test. We obtain a novel sample complexity bound for Chernoff’s algorithm, with a non-asymptotic term that characterizes its performance at a fixed confidence level. We also develop an extension of Chernoff sampling that can be used to estimate the parameters of a wide variety of models and we obtain a non-asymptotic bound on the estimation error. We apply our extension of Chernoff sampling to actively learn neural network models and to estimate parameters in real-data linear and non-linear regression problems, where our approach performs favorably to state-of-the-art methods. 

The level set estimation problem seeks to find all points in a domain  where the value of an unknown function 𝑓:→ℝ exceeds a threshold 𝛼 . The estimation is based on noisy function evaluations that may be acquired at sequentially and adaptively chosen locations in  . The threshold value 𝛼 can either be explicit and provided a priori, or implicit and defined relative to the optimal function value, i.e. 𝛼=(1−𝜖)𝑓(𝐱∗) for a given 𝜖>0 where 𝑓(𝐱∗) is the maximal function value and is unknown. In this work we provide a new approach to the level set estimation problem by relating it to recent adaptive experimental design methods for linear bandits in the Reproducing Kernel Hilbert Space (RKHS) setting. We assume that 𝑓 can be approximated by a function in the RKHS up to an unknown misspecification and provide novel algorithms for both the implicit and explicit cases in this setting with strong theoretical guarantees. Moreover, in the linear (kernel) setting, we show that our bounds are nearly optimal, namely, our upper bounds match existing lower bounds for threshold linear bandits. To our knowledge this work provides the first instance-dependent, non-asymptotic upper bounds on sample complexity of level-set estimation that match information theoretic lower bounds.