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We study methods based on reproducing kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process (MRP). We study a regularized form of the kernel least-squares temporal difference (LSTD) estimate; in the population limit of infinite data, it corresponds to the fixed point of a projected Bellman operator defined by the associated reproducing kernel Hilbert space. The estimator itself is obtained by computing the projected fixed point induced by a regularized version of the empirical operator; due to the underlying kernel structure, this reduces to solving a linear system involving kernel matrices. We analyze the error of this estimate in the $L^2(\mu)$-norm, where $\mu$ denotes the stationary distribution of the underlying Markov chain. Our analysis imposes no assumptions on the transition operator of the Markov chain, but rather only conditions on the reward function and population-level kernel LSTD solutions. We use empirical process theory techniques to derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator, as well as the instance-dependent variance of the Bellman residual error. In addition, we prove minimax lower bounds over sub-classes of MRPs, which shows that our rate is optimal in terms of the sample size $n$ and the effective horizon $H = (1 - \gamma)^{-1}$. Whereas existing worst-case theory predicts cubic scaling ($H^3$) in the effective horizon, our theory reveals that there is in fact a much wider range of scalings, depending on the kernel, the stationary distribution, and the variance of the Bellman residual error. Notably, it is only parametric and near-parametric problems that can ever achieve the worst-case cubic scaling.

Generalized Method of Moments (GMM) estimators in their various forms, including the popular Maximum Likelihood (ML) estimator, are frequently applied for the evaluation of complex econometric models with not analytically computable moment or likelihood functions. As the objective functions of GMM- and ML-estimators themselves constitute the approximation of an integral, more precisely of the expected value over the real world data space, the question arises whether the approximation of the moment function and the simulation of the entire objective function can be combined. Motivated by the popular Probit and Mixed Logit models, we consider double integrals with a linking function which stems from the considered estimator, e.g. the logarithm for Maximum Likelihood, and apply a sparse tensor product quadrature to reduce the computational effort for the approximation of the combined integral. Given H\"older continuity of the linking function, we prove that this approach can improve the order of the convergence rate of the classical GMM- and ML-estimator by a factor of two, even for integrands of low regularity or high dimensionality. This result is illustrated by numerical simulations of Mixed Logit and Multinomial Probit integrals which are estimated by ML- and GMM-estimators, respectively.

In this paper, we consider the multi-armed bandit problem with high-dimensional features. First, we prove a minimax lower bound, $\mathcal{O}\big((\log d)^{\frac{\alpha+1}{2}}T^{\frac{1-\alpha}{2}}+\log T\big)$, for the cumulative regret, in terms of horizon $T$, dimension $d$ and a margin parameter $\alpha\in[0,1]$, which controls the separation between the optimal and the sub-optimal arms. This new lower bound unifies existing regret bound results that have different dependencies on T due to the use of different values of margin parameter $\alpha$ explicitly implied by their assumptions. Second, we propose a simple and computationally efficient algorithm inspired by the general Upper Confidence Bound (UCB) strategy that achieves a regret upper bound matching the lower bound. The proposed algorithm uses a properly centered $\ell_1$-ball as the confidence set in contrast to the commonly used ellipsoid confidence set. In addition, the algorithm does not require any forced sampling step and is thereby adaptive to the practically unknown margin parameter. Simulations and a real data analysis are conducted to compare the proposed method with existing ones in the literature.

We explore the connection between outlier-robust high-dimensional statistics and non-convex optimization in the presence of sparsity constraints, with a focus on the fundamental tasks of robust sparse mean estimation and robust sparse PCA. We develop novel and simple optimization formulations for these problems such that any approximate stationary point of the associated optimization problem yields a near-optimal solution for the underlying robust estimation task. As a corollary, we obtain that any first-order method that efficiently converges to stationarity yields an efficient algorithm for these tasks. The obtained algorithms are simple, practical, and succeed under broader distributional assumptions compared to prior work.

In value-based deep reinforcement learning methods, approximation of value functions induces overestimation bias and leads to suboptimal policies. We show that in deep actor-critic methods that aim to overcome the overestimation bias, if the reinforcement signals received by the agent have a high variance, a significant underestimation bias arises. To minimize the underestimation, we introduce a parameter-free, novel deep Q-learning variant. Our Q-value update rule combines the notions behind Clipped Double Q-learning and Maxmin Q-learning by computing the critic objective through the nested combination of maximum and minimum operators to bound the approximate value estimates. We evaluate our modification on the suite of several OpenAI Gym continuous control tasks, improving the state-of-the-art in every environment tested.

Low-rank matrix approximation is one of the central concepts in machine learning, with applications in dimension reduction, de-noising, multivariate statistical methodology, and many more. A recent extension to LRMA is called low-rank matrix completion (LRMC). It solves the LRMA problem when some observations are missing and is especially useful for recommender systems. In this paper, we consider an element-wise weighted generalization of LRMA. The resulting weighted low-rank matrix approximation technique therefore covers LRMC as a special case with binary weights. WLRMA has many applications. For example, it is an essential component of GLM optimization algorithms, where an exponential family is used to model the entries of a matrix, and the matrix of natural parameters admits a low-rank structure. We propose an algorithm for solving the weighted problem, as well as two acceleration techniques. Further, we develop a non-SVD modification of the proposed algorithm that is able to handle extremely high-dimensional data. We compare the performance of all the methods on a small simulation example as well as a real-data application.

This paper considers two fundamental sequential decision-making problems: the problem of prediction with expert advice and the multi-armed bandit problem. We focus on stochastic regimes in which an adversary may corrupt losses, and we investigate what level of robustness can be achieved against adversarial corruptions. The main contribution of this paper is to show that optimal robustness can be expressed by a square-root dependency on the amount of corruption. More precisely, we show that two classes of algorithms, anytime Hedge with decreasing learning rate and algorithms with second-order regret bounds, achieve $O( \frac{\log N}{\Delta} + \sqrt{ \frac{C \log N }{\Delta} } )$-regret, where $N, \Delta$, and $C$ represent the number of experts, the gap parameter, and the corruption level, respectively. We further provide a matching lower bound, which means that this regret bound is tight up to a constant factor. For the multi-armed bandit problem, we also provide a nearly tight lower bound up to a logarithmic factor.

We consider the power of local algorithms for approximately solving Max $k$XOR, a generalization of two constraint satisfaction problems previously studied with classical and quantum algorithms (MaxCut and Max E3LIN2). On instances with either random signs or no overlapping clauses and $D+1$ clauses per variable, we calculate the average satisfying fraction of the depth-1 QAOA and compare with a generalization of the local threshold algorithm. Notably, the quantum algorithm outperforms the threshold algorithm for $k > 4$. On the other hand, we highlight potential difficulties for the QAOA to achieve computational quantum advantage on this problem. We first compute a tight upper bound on the maximum satisfying fraction of nearly all large random regular Max $k$XOR instances by numerically calculating the ground state energy density $P(k)$ of a mean-field $k$-spin glass. The upper bound grows with $k$ much faster than the performance of both one-local algorithms. We also identify a new obstruction result for low-depth quantum circuits (including the QAOA) when $k=3$, generalizing a result of Bravyi et al [arXiv:1910.08980] when $k=2$. We conjecture that a similar obstruction exists for all $k$.

Contextual multi-armed bandit (MAB) achieves cutting-edge performance on a variety of problems. When it comes to real-world scenarios such as recommendation system and online advertising, however, it is essential to consider the resource consumption of exploration. In practice, there is typically non-zero cost associated with executing a recommendation (arm) in the environment, and hence, the policy should be learned with a fixed exploration cost constraint. It is challenging to learn a global optimal policy directly, since it is a NP-hard problem and significantly complicates the exploration and exploitation trade-off of bandit algorithms. Existing approaches focus on solving the problems by adopting the greedy policy which estimates the expected rewards and costs and uses a greedy selection based on each arm's expected reward/cost ratio using historical observation until the exploration resource is exhausted. However, existing methods are hard to extend to infinite time horizon, since the learning process will be terminated when there is no more resource. In this paper, we propose a hierarchical adaptive contextual bandit method (HATCH) to conduct the policy learning of contextual bandits with a budget constraint. HATCH adopts an adaptive method to allocate the exploration resource based on the remaining resource/time and the estimation of reward distribution among different user contexts. In addition, we utilize full of contextual feature information to find the best personalized recommendation. Finally, in order to prove the theoretical guarantee, we present a regret bound analysis and prove that HATCH achieves a regret bound as low as $O(\sqrt{T})$. The experimental results demonstrate the effectiveness and efficiency of the proposed method on both synthetic data sets and the real-world applications.

This work considers the problem of provably optimal reinforcement learning for episodic finite horizon MDPs, i.e. how an agent learns to maximize his/her long term reward in an uncertain environment. The main contribution is in providing a novel algorithm --- Variance-reduced Upper Confidence Q-learning (vUCQ) --- which enjoys a regret bound of $\widetilde{O}(\sqrt{HSAT} + H^5SA)$, where the $T$ is the number of time steps the agent acts in the MDP, $S$ is the number of states, $A$ is the number of actions, and $H$ is the (episodic) horizon time. This is the first regret bound that is both sub-linear in the model size and asymptotically optimal. The algorithm is sub-linear in that the time to achieve $\epsilon$-average regret for any constant $\epsilon$ is $O(SA)$, which is a number of samples that is far less than that required to learn any non-trivial estimate of the transition model (the transition model is specified by $O(S^2A)$ parameters). The importance of sub-linear algorithms is largely the motivation for algorithms such as $Q$-learning and other "model free" approaches. vUCQ algorithm also enjoys minimax optimal regret in the long run, matching the $\Omega(\sqrt{HSAT})$ lower bound. Variance-reduced Upper Confidence Q-learning (vUCQ) is a successive refinement method in which the algorithm reduces the variance in $Q$-value estimates and couples this estimation scheme with an upper confidence based algorithm. Technically, the coupling of both of these techniques is what leads to the algorithm enjoying both the sub-linear regret property and the asymptotically optimal regret.