In this paper, we study the non-asymptotic and asymptotic performances of the optimal robust policy and value function of robust Markov Decision Processes(MDPs), where the optimal robust policy and value function are solved only from a generative model. While prior work focusing on non-asymptotic performances of robust MDPs is restricted in the setting of the KL uncertainty set and $(s,a)$-rectangular assumption, we improve their results and also consider other uncertainty sets, including $L_1$ and $\chi^2$ balls. Our results show that when we assume $(s,a)$-rectangular on uncertainty sets, the sample complexity is about $\widetilde{O}\left(\frac{|\mathcal{S}|^2|\mathcal{A}|}{\varepsilon^2\rho^2(1-\gamma)^4}\right)$. In addition, we extend our results from $(s,a)$-rectangular assumption to $s$-rectangular assumption. In this scenario, the sample complexity varies with the choice of uncertainty sets and is generally larger than the case under $(s,a)$-rectangular assumption. Moreover, we also show that the optimal robust value function is asymptotic normal with a typical rate $\sqrt{n}$ under $(s,a)$ and $s$-rectangular assumptions from both theoretical and empirical perspectives.
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Until recently, applications of neural networks in machine learning have almost exclusively relied on real-valued networks. It was recently observed, however, that complex-valued neural networks (CVNNs) exhibit superior performance in applications in which the input is naturally complex-valued, such as MRI fingerprinting. While the mathematical theory of real-valued networks has, by now, reached some level of maturity, this is far from true for complex-valued networks. In this paper, we analyze the expressivity of complex-valued networks by providing explicit quantitative error bounds for approximating $C^n$ functions on compact subsets of $\mathbb{C}^d$ by complex-valued neural networks that employ the modReLU activation function, given by $\sigma(z) = \mathrm{ReLU}(|z| - 1) \, \mathrm{sgn} (z)$, which is one of the most popular complex activation functions used in practice. We show that the derived approximation rates are optimal (up to log factors) in the class of modReLU networks with weights of moderate growth.
Unbiased and consistent variance estimators generally do not exist for design-based treatment effect estimators because experimenters never observe more than one potential outcome for any unit. The problem is exacerbated by interference and complex experimental designs. In this paper, we consider variance estimation for linear treatment effect estimators under interference and arbitrary experimental designs. Experimenters must accept conservative estimators in this setting, but they can strive to minimize the conservativeness. We show that this task can be interpreted as an optimization problem in which one aims to find the lowest estimable upper bound of the true variance given one's risk preference and knowledge of the potential outcomes. We characterize the set of admissible bounds in the class of quadratic forms, and we demonstrate that the optimization problem is a convex program for many natural objectives. This allows experimenters to construct less conservative variance estimators, making inferences about treatment effects more informative. The resulting estimators are guaranteed to be conservative regardless of whether the background knowledge used to construct the bound is correct, but the estimators are less conservative if the knowledge is reasonably accurate.
We consider reinforcement learning (RL) in Markov Decision Processes in which an agent repeatedly interacts with an environment that is modeled by a controlled Markov process. At each time step $t$, it earns a reward, and also incurs a cost-vector consisting of $M$ costs. We design learning algorithms that maximize the cumulative reward earned over a time horizon of $T$ time-steps, while simultaneously ensuring that the average values of the $M$ cost expenditures are bounded by agent-specified thresholds $c^{ub}_i,i=1,2,\ldots,M$. The considerations on the cumulative cost expenditures departs from the existing literature, in that the agent now additionally needs to balance the cost expenses in an online manner, while simultaneously performing the exploration-exploitation trade-off that is typically encountered in RL tasks. In order to measure the performance of a reinforcement learning algorithm that satisfies the average cost constraints, we define an $M+1$ dimensional regret vector that is composed of its reward regret, and $M$ cost regrets. The reward regret measures the sub-optimality in the cumulative reward, while the $i$-th component of the cost regret vector is the difference between its $i$-th cumulative cost expense and the expected cost expenditures $Tc^{ub}_i$. We prove that with a high probablity, the regret vector of UCRL-CMDP is upper-bounded as $O\left( S\sqrt{AT^{1.5}\log(T)}\right)$, where $S$ is the number of states, $A$ is the number of actions, and $T$ is the time horizon. We further show how to reduce the regret of a desired subset of the $M$ costs, at the expense of increasing the regrets of rewards and the remaining costs. To the best of our knowledge, ours is the only work that considers non-episodic RL under average cost constraints, and derive algorithms that can~\emph{tune the regret vector} according to the agent's requirements on its cost regrets.
The Robust Markov Decision Process (RMDP) framework focuses on designing control policies that are robust against the parameter uncertainties due to the mismatches between the simulator model and real-world settings. An RMDP problem is typically formulated as a max-min problem, where the objective is to find the policy that maximizes the value function for the worst possible model that lies in an uncertainty set around a nominal model. The standard robust dynamic programming approach requires the knowledge of the nominal model for computing the optimal robust policy. In this work, we propose a model-based reinforcement learning (RL) algorithm for learning an $\epsilon$-optimal robust policy when the nominal model is unknown. We consider three different forms of uncertainty sets, characterized by the total variation distance, chi-square divergence, and KL divergence. For each of these uncertainty sets, we give a precise characterization of the sample complexity of our proposed algorithm. In addition to the sample complexity results, we also present a formal analytical argument on the benefit of using robust policies. Finally, we demonstrate the performance of our algorithm on two benchmark problems.
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$.
Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.
Self-training algorithms, which train a model to fit pseudolabels predicted by another previously-learned model, have been very successful for learning with unlabeled data using neural networks. However, the current theoretical understanding of self-training only applies to linear models. This work provides a unified theoretical analysis of self-training with deep networks for semi-supervised learning, unsupervised domain adaptation, and unsupervised learning. At the core of our analysis is a simple but realistic ``expansion'' assumption, which states that a low-probability subset of the data must expand to a neighborhood with large probability relative to the subset. We also assume that neighborhoods of examples in different classes have minimal overlap. We prove that under these assumptions, the minimizers of population objectives based on self-training and input-consistency regularization will achieve high accuracy with respect to ground-truth labels. By using off-the-shelf generalization bounds, we immediately convert this result to sample complexity guarantees for neural nets that are polynomial in the margin and Lipschitzness. Our results help explain the empirical successes of recently proposed self-training algorithms which use input consistency regularization.
Importance sampling is one of the most widely used variance reduction strategies in Monte Carlo rendering. In this paper, we propose a novel importance sampling technique that uses a neural network to learn how to sample from a desired density represented by a set of samples. Our approach considers an existing Monte Carlo rendering algorithm as a black box. During a scene-dependent training phase, we learn to generate samples with a desired density in the primary sample space of the rendering algorithm using maximum likelihood estimation. We leverage a recent neural network architecture that was designed to represent real-valued non-volume preserving ('Real NVP') transformations in high dimensional spaces. We use Real NVP to non-linearly warp primary sample space and obtain desired densities. In addition, Real NVP efficiently computes the determinant of the Jacobian of the warp, which is required to implement the change of integration variables implied by the warp. A main advantage of our approach is that it is agnostic of underlying light transport effects, and can be combined with many existing rendering techniques by treating them as a black box. We show that our approach leads to effective variance reduction in several practical scenarios.
This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.
We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.
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