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We study the model-based reward-free reinforcement learning with linear function approximation for episodic Markov decision processes (MDPs). In this setting, the agent works in two phases. In the exploration phase, the agent interacts with the environment and collects samples without the reward. In the planning phase, the agent is given a specific reward function and uses samples collected from the exploration phase to learn a good policy. We propose a new provably efficient algorithm, called UCRL-RFE under the Linear Mixture MDP assumption, where the transition probability kernel of the MDP can be parameterized by a linear function over certain feature mappings defined on the triplet of state, action, and next state. We show that to obtain an $\epsilon$-optimal policy for arbitrary reward function, UCRL-RFE needs to sample at most $\tilde O(H^5d^2\epsilon^{-2})$ episodes during the exploration phase. Here, $H$ is the length of the episode, $d$ is the dimension of the feature mapping. We also propose a variant of UCRL-RFE using Bernstein-type bonus and show that it needs to sample at most $\tilde O(H^4d(H + d)\epsilon^{-2})$ to achieve an $\epsilon$-optimal policy. By constructing a special class of linear Mixture MDPs, we also prove that for any reward-free algorithm, it needs to sample at least $\tilde \Omega(H^2d\epsilon^{-2})$ episodes to obtain an $\epsilon$-optimal policy. Our upper bound matches the lower bound in terms of the dependence on $\epsilon$ and the dependence on $d$ if $H \ge d$.

There is an increasing realization that algorithmic inductive biases are central in preventing overfitting; empirically, we often see a benign overfitting phenomenon in overparameterized settings for natural learning algorithms, such as stochastic gradient descent (SGD), where little to no explicit regularization has been employed. This work considers this issue in arguably the most basic setting: constant-stepsize SGD (with iterate averaging or tail averaging) for linear regression in the overparameterized regime. Our main result provides a sharp excess risk bound, stated in terms of the full eigenspectrum of the data covariance matrix, that reveals a bias-variance decomposition characterizing when generalization is possible: (i) the variance bound is characterized in terms of an effective dimension (specific for SGD) and (ii) the bias bound provides a sharp geometric characterization in terms of the location of the initial iterate (and how it aligns with the data covariance matrix). More specifically, for SGD with iterate averaging, we demonstrate the sharpness of the established excess risk bound by proving a matching lower bound (up to constant factors). For SGD with tail averaging, we show its advantage over SGD with iterate averaging by proving a better excess risk bound together with a nearly matching lower bound. Moreover, we reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares (minimum-norm interpolation) and ridge regression. Experimental results on synthetic data corroborate our theoretical findings.

Approximate linear programs (ALPs) are well-known models based on value function approximations (VFAs) to obtain policies and lower bounds on the optimal policy cost of discounted-cost Markov decision processes (MDPs). Formulating an ALP requires (i) basis functions, the linear combination of which defines the VFA, and (ii) a state-relevance distribution, which determines the relative importance of different states in the ALP objective for the purpose of minimizing VFA error. Both these choices are typically heuristic: basis function selection relies on domain knowledge while the state-relevance distribution is specified using the frequency of states visited by a heuristic policy. We propose a self-guided sequence of ALPs that embeds random basis functions obtained via inexpensive sampling and uses the known VFA from the previous iteration to guide VFA computation in the current iteration. Self-guided ALPs mitigate the need for domain knowledge during basis function selection as well as the impact of the initial choice of the state-relevance distribution, thus significantly reducing the ALP implementation burden. We establish high probability error bounds on the VFAs from this sequence and show that a worst-case measure of policy performance is improved. We find that these favorable implementation and theoretical properties translate to encouraging numerical results on perishable inventory control and options pricing applications, where self-guided ALP policies improve upon policies from problem-specific methods. More broadly, our research takes a meaningful step toward application-agnostic policies and bounds for MDPs.

We study the limiting behavior of the familywise error rate (FWER) of the Bonferroni procedure in a multiple testing problem. We establish that, in the equicorrelated normal setup, the FWER of Bonferroni's method tends to zero asymptotically (i.e for a sufficiently large number of hypotheses) for any positive equicorrelation. We extend this result for generalized familywise error rates.

Significant advances in maximum flow algorithms have changed the relative performance of various approaches to isotonic regression. If the transitive closure is given then the standard approach used for $L_0$ (Hamming distance) isotonic regression (finding anti-chains in the transitive closure of the violator graph), combined with new flow algorithms, gives an $L_1$ algorithm taking $\tilde{\Theta}(n^2+n^\frac{3}{2} \log U )$ time, where $U$ is the maximum vertex weight. The previous fastest was $\Theta(n^3)$. Similar results are obtained for $L_2$ and for $L_p$ approximations, $1 < p < \infty$. For weighted points in $d$-dimensional space with coordinate-wise ordering, $d \geq 3$, $L_0, L_1$ and $L_2$ regressions can be found in only $o(n^\frac{3}{2} \log^d n \log U)$ time, improving on the previous best of $\tilde{\Theta}(n^2 \log^d n)$.

We introduce a novel rule-based approach for handling regression problems. The new methodology carries elements from two frameworks: (i) it provides information about the uncertainty of the parameters of interest using Bayesian inference, and (ii) it allows the incorporation of expert knowledge through rule-based systems. The blending of those two different frameworks can be particularly beneficial for various domains (e.g. engineering), where, even though the significance of uncertainty quantification motivates a Bayesian approach, there is no simple way to incorporate researcher intuition into the model. We validate our models by applying them to synthetic applications: a simple linear regression problem and two more complex structures based on partial differential equations. Finally, we review the advantages of our methodology, which include the simplicity of the implementation, the uncertainty reduction due to the added information and, in some occasions, the derivation of better point predictions, and we address limitations, mainly from the computational complexity perspective, such as the difficulty in choosing an appropriate algorithm and the added computational burden.

We study query and computationally efficient planning algorithms with linear function approximation and a simulator. We assume that the agent only has local access to the simulator, meaning that the agent can only query the simulator at states that have been visited before. This setting is more practical than many prior works on reinforcement learning with a generative model. We propose an algorithm named confident Monte Carlo least square policy iteration (Confident MC-LSPI) for this setting. Under the assumption that the Q-functions of all deterministic policies are linear in known features of the state-action pairs, we show that our algorithm has polynomial query and computational complexities in the dimension of the features, the effective planning horizon and the targeted sub-optimality, while these complexities are independent of the size of the state space. One technical contribution of our work is the introduction of a novel proof technique that makes use of a virtual policy iteration algorithm. We use this method to leverage existing results on $\ell_\infty$-bounded approximate policy iteration to show that our algorithm can learn the optimal policy for the given initial state even only with local access to the simulator. We believe that this technique can be extended to broader settings beyond this work.

We develop a post-selective Bayesian framework to jointly and consistently estimate parameters in group-sparse linear regression models. After selection with the Group LASSO (or generalized variants such as the overlapping, sparse, or standardized Group LASSO), uncertainty estimates for the selected parameters are unreliable in the absence of adjustments for selection bias. Existing post-selective approaches are limited to uncertainty estimation for (i) real-valued projections onto very specific selected subspaces for the group-sparse problem, (ii) selection events categorized broadly as polyhedral events that are expressible as linear inequalities in the data variables. Our Bayesian methods address these gaps by deriving a likelihood adjustment factor, and an approximation thereof, that eliminates bias from selection. Paying a very nominal price for this adjustment, experiments on simulated data, and data from the Human Connectome Project demonstrate the efficacy of our methods for a joint estimation of group-sparse parameters and their uncertainties post selection.

We present a means of formulating and solving the well known structure-and-motion problem in computer vision with probabilistic graphical models. We model the unknown camera poses and 3D feature coordinates as well as the observed 2D projections as Gaussian random variables, using sigma point parameterizations to effectively linearize the nonlinear relationships between these variables. Those variables involved in every projection are grouped into a cluster, and we connect the clusters in a cluster graph. Loopy belief propagation is performed over this graph, in an iterative re-initialization and estimation procedure, and we find that our approach shows promise in both simulation and on real-world data. The PGM is easily extendable to include additional parameters or constraints.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.