Offline reinforcement learning (RL) can be used to improve future performance by leveraging historical data. There exist many different algorithms for offline RL, and it is well recognized that these algorithms, and their hyperparameter settings, can lead to decision policies with substantially differing performance. This prompts the need for pipelines that allow practitioners to systematically perform algorithmhyperparameter selection for their setting. Critically, in most real-world settings, this pipeline must only involve the use of historical data. Inspired by statistical model selection methods for supervised learning, we introduce a task-and methodagnostic pipeline for automatically training, comparing, selecting, and deploying the best policy when the provided dataset is limited in size.

We consider the problem of estimating a structured multivariate density, subject to Markov conditions implied by an undirected graph. In the worst case, without Markovian assumptions, this problem suffers from the curse of dimensionality. Our main result shows how the curse of dimensionality can be avoided or greatly alleviated under the Markov property, and applies to arbitrary graphs. While existing results along these lines focus on sparsity or manifold assumptions, we introduce a new graphical quantity called "graph resilience" and show how it controls the sample complexity. Surprisingly, although one might expect the sample complexity of this problem to scale with local graph parameters such as the degree, this turns out not to be the case. Through explicit examples, we compute uniform deviation bounds and illustrate how the curse of dimensionality in density estimation can thus be circumvented. Notable examples where the rate improves substantially include sequential, hierarchical, and spatial data.

The most prevalent approach for estimating endogenous regression models with instruments is assuming low-dimensional linear relationships, i.e. h The coefficient in the final regression is taken to be the estimate of . Then a 2SLS estimation method is applied on these transformed feature spaces. The authors show asymptotic consistency of the resulting estimator, assuming that the approximation error goes to zero. Subsequently, they also estimate the function m(z) =E[y h(x) | z] based on another growing sieve. Though it may seem at first that the approach in that paper and ours are quite distinct, the population limit of our objective function coincides with theirs. To see this, consider the simplified version of our estimator presented in (6), where the function classes are already norm-constrained and no norm based regularization is imposed. Moreover, for a moment consider the population version of this estimator, i.e. min max (h, f) kfk Thus in the population limit and without norm regularization on the test function f, our criterion is equivalent to the minimum distance criterion analyzed in Chen and Pouzo [2012]. Another point of similarity is that we prove convergence of the estimator in terms of the pseudo-metric, the projected MSE defined in Section 4 of Chen and Pouzo [2012] - and like that paper we require additional conditions to relate the pseudo-metric to the true MSE. The present paper differs in a number of ways: (i) the finite sample criterion is different; (ii) we prove our results using localized Rademacher analysis which allows for weaker assumptions; (iii) we consider a broader range of estimation approaches than linear sieves, necessitating more of a focus on optimization. Digging into the second point, Chen and Pouzo [2012] take a more traditional parameter recovery approach which requires several minimum eigenvalue conditions and several regularity conditions to be satisfied for their estimation rate to hold (see e.g. This is analogous to a mean squared error proof in an exogenous linear regression setting, that requires the minimum eigenvalue of the feature co-variance to be bounded away from zero. Moreover, such parameter recovery methods seem limited to the growing sieve approach, since only then one has a clear finite dimensional parameter vector to work on for each fixed n.

In this paper, we propose a novel conceptual framework to detect outliers using optimal transport with a concave cost function. Conventional outlier detection approaches typically use a two-stage procedure: first, outliers are detected and removed, and then estimation is performed on the cleaned data. However, this approach does not inform outlier removal with the estimation task, leaving room for improvement. To address this limitation, we propose an automatic outlier rectification mechanism that integrates rectification and estimation within a joint optimization framework. We take the first step to utilize the optimal transport distance with a concave cost function to construct a rectification set in the space of probability distributions. Then, we select the best distribution within the rectification set to perform the estimation task. Notably, the concave cost function we introduced in this paper is the key to making our estimator effectively identify the outlier during the optimization process. We demonstrate the effectiveness of our approach over conventional approaches in simulations and empirical analyses for mean estimation, least absolute regression, and the fitting of option implied volatility surfaces.

We now see how the term "expert" can be misleading. In the revision, we will replace it with "oracle" We plan to include more results on harder domains. We will further clarify this in the revision. We will clarify and better motivate these equations. We will include an ablation study of UpdateInputWhitening in the revision.

We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean µ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates µ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-tailed distributions under mild moment assumptions. Our algorithm achieves the optimal asymptotic error using a number of samples scaling logarithmically with the ambient dimension. Importantly, the sample complexity of our method is optimal as a function of the failure probability, having an additive log(1/) dependence. Our algorithm leverages the stability-based approach from the algorithmic robust statistics literature, with crucial (and necessary) adaptations required in our setting. Our analysis may be of independent interest, involving the delicate design of a (non-spectral) decomposition for positive semi-definite matrices satisfying certain sparsity properties.

We consider the Hypothesis Transfer Learning (HTL) problem where one incorporates a hypothesis trained on the source domain into the learning procedure of the target domain. Existing theoretical analysis either only studies specific algorithms or only presents upper bounds on the generalization error but not on the excess risk. In this paper, we propose a unified algorithm-dependent framework for HTL through a novel notion of transformation function, which characterizes the relation between the source and the target domains. We conduct a general risk analysis of this framework and in particular, we show for the first time, if two domains are related, HTL enjoys faster convergence rates of excess risks for Kernel Smoothing and Kernel Ridge Regression than those of the classical non-transfer learning settings. Experiments on real world data demonstrate the effectiveness of our framework.

Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance estimators. Comparing functional connectivity of subjects in two populations calls for comparing these estimated GGMs. Our goal is to identify differences in GGMs known to have similar structure. We characterize the uncertainty of differences with confidence intervals obtained using a parametric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-sample settings. Characterizing the distributions of sparse models is inherently challenging as the penalties produce a biased estimator.

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes.

Transformer-based architectures have dominated various areas of machine learning in recent years. In this paper, we introduce a novel robust attention mechanism designed to enhance the resilience of transformer-based architectures. Crucially, this technique can be integrated into existing transformers as a plug-and-play layer, improving their robustness without the need for additional training or fine-tuning. Through comprehensive experiments and ablation studies, we demonstrate that our ProTransformer significantly enhances the robustness of transformer models across a variety of prediction tasks, attack mechanisms, backbone architectures, and data domains. Notably, without further fine-tuning, the ProTransformer consistently improves the performance of vanilla transformers by 19.5%, 28.3%, 16.1%, and 11.4% for BERT, ALBERT, DistilBERT, and RoBERTa, respectively, under the classical TextFooler attack. Furthermore, ProTransformer shows promising resilience in large language models (LLMs) against prompting-based attacks, improving the performance of T5 and LLaMA by 24.8% and 17.8%, respectively, and enhancing Vicuna by an average of 10.4% against the Jailbreaking attack. Beyond the language domain, ProTransformer also demonstrates outstanding robustness in both vision and graph domains.