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This paper resolves the open question of designing near-optimal algorithms for learning imperfect-information extensive-form games from bandit feedback. We present the first line of algorithms that require only $\widetilde{\mathcal{O}}((XA+YB)/\varepsilon^2)$ episodes of play to find an $\varepsilon$-approximate Nash equilibrium in two-player zero-sum games, where $X,Y$ are the number of information sets and $A,B$ are the number of actions for the two players. This improves upon the best known sample complexity of $\widetilde{\mathcal{O}}((X^2A+Y^2B)/\varepsilon^2)$ by a factor of $\widetilde{\mathcal{O}}(\max\{X, Y\})$, and matches the information-theoretic lower bound up to logarithmic factors. We achieve this sample complexity by two new algorithms: Balanced Online Mirror Descent, and Balanced Counterfactual Regret Minimization. Both algorithms rely on novel approaches of integrating \emph{balanced exploration policies} into their classical counterparts. We also extend our results to learning Coarse Correlated Equilibria in multi-player general-sum games.

For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.

Prediction models often fail if train and test data do not stem from the same distribution. Out-of-distribution (OOD) generalization to unseen, perturbed test data is a desirable but difficult-to-achieve property for prediction models and in general requires strong assumptions on the data generating process (DGP). In a causally inspired perspective on OOD generalization, the test data arise from a specific class of interventions on exogenous random variables of the DGP, called anchors. Anchor regression models, introduced by Rothenhaeusler et al. (2021), protect against distributional shifts in the test data by employing causal regularization. However, so far anchor regression has only been used with a squared-error loss which is inapplicable to common responses such as censored continuous or ordinal data. Here, we propose a distributional version of anchor regression which generalizes the method to potentially censored responses with at least an ordered sample space. To this end, we combine a flexible class of parametric transformation models for distributional regression with an appropriate causal regularizer under a more general notion of residuals. In an exemplary application and several simulation scenarios we demonstrate the extent to which OOD generalization is possible.

When are inferences (whether Direct-Likelihood, Bayesian, or Frequentist) obtained from partial data valid? This paper answers this question by offering a new asymptotic theory about inference with missing data that is more general than existing theories. By using more powerful tools from real analysis and probability theory than those used in previous research, it proves that as the sample size increases and the extent of missingness decreases, the average-loglikelihood function generated by partial data and that ignores the missingness mechanism will almost surely converge uniformly to that which would have been generated by complete data; and if the data are Missing at Random, this convergence depends only on sample size. Thus, inferences from partial data, such as posterior modes, uncertainty estimates, confidence intervals, likelihood ratios, test statistics, and indeed, all quantities or features derived from the partial-data loglikelihood function, will be consistently estimated. They will approximate their complete-data analogues. This adds to previous research which has only proved the consistency and asymptotic normality of the posterior mode, and developed separate theories for Direct-Likelihood, Bayesian, and Frequentist inference. Practical implications of this result are discussed, and the theory is verified using a previous study of International Human Rights Law.

We present a Bayesian nonparametric model for conditional distribution estimation using Bayesian additive regression trees (BART). The generative model we use is based on rejection sampling from a base model. Typical of BART models, our model is flexible, has a default prior specification, and is computationally convenient. To address the distinguished role of the response in the BART model we propose, we further introduce an approach to targeted smoothing which is possibly of independent interest for BART models. We study the proposed model theoretically and provide sufficient conditions for the posterior distribution to concentrate at close to the minimax optimal rate adaptively over smoothness classes in the high-dimensional regime in which many predictors are irrelevant. To fit our model we propose a data augmentation algorithm which allows for existing BART samplers to be extended with minimal effort. We illustrate the performance of our methodology on simulated data and use it to study the relationship between education and body mass index using data from the medical expenditure panel survey (MEPS).

Many problems in computational science and engineering can be described in terms of approximating a smooth function of $d$ variables, defined over an unknown domain of interest $\Omega\subset \mathbb{R}^d$, from sample data. Here both the curse of dimensionality ($d\gg 1$) and the lack of domain knowledge with $\Omega$ potentially irregular and/or disconnected are confounding factors for sampling-based methods. Na\"{i}ve approaches often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20\% wasted samples in some problems. In surrogate model construction in computational uncertainty quantification (UQ), the high cost of computing samples needs a more efficient sampling procedure. In the last years, methods for computing such approximations from sample data have been studied in the case of irregular domains. The advantages of computing sampling measures depending on an approximation space $P$ of $\dim(P)=N$ have been shown. In particular, such methods confer advantages such as stability and well-conditioning, with $\mathcal{O}(N\log(N))$ as sample complexity. The recently-proposed adaptive sampling for general domains (ASGD) strategy is one method to construct these sampling measures. The main contribution of this paper is to improve ASGD by adaptively updating the sampling measures over unknown domains. We achieve this by first introducing a general domain adaptivity strategy (GDAS), which approximates the function and domain of interest from sample points. Second, we propose adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance. Then, we derive least squares techniques for polynomial approximation on unknown domains. Numerical results show that the ASUD approach can reduce the computational cost by as 50\% when compared with uniform sampling.

Approximate Bayesian Computation (ABC) enables statistical inference in complex models whose likelihoods are difficult to calculate but easy to simulate from. ABC constructs a kernel-type approximation to the posterior distribution through an accept/reject mechanism which compares summary statistics of real and simulated data. To obviate the need for summary statistics, we directly compare empirical distributions with a Kullback-Leibler (KL) divergence estimator obtained via classification. In particular, we blend flexible machine learning classifiers within ABC to automate fake/real data comparisons. We consider the traditional accept/reject kernel as well as an exponential weighting scheme which does not require the ABC acceptance threshold. Our theoretical results show that the rate at which our ABC posterior distributions concentrate around the true parameter depends on the estimation error of the classifier. We derive limiting posterior shape results and find that, with a properly scaled exponential kernel, asymptotic normality holds. We demonstrate the usefulness of our approach on simulated examples as well as real data in the context of stock volatility estimation.

Reinforcement Learning (RL) has the promise of providing data-driven support for decision-making in a wide range of problems in healthcare, education, business, and other domains. Classical RL methods focus on the mean of the total return and, thus, may provide misleading results in the setting of the heterogeneous populations that commonly underlie large-scale datasets. We introduce the K-Heterogeneous Markov Decision Process (K-Hetero MDP) to address sequential decision problems with population heterogeneity. We propose the Auto-Clustered Policy Evaluation (ACPE) for estimating the value of a given policy, and the Auto-Clustered Policy Iteration (ACPI) for estimating the optimal policy in a given policy class. Our auto-clustered algorithms can automatically detect and identify homogeneous sub-populations, while estimating the Q function and the optimal policy for each sub-population. We establish convergence rates and construct confidence intervals for the estimators obtained by the ACPE and ACPI. We present simulations to support our theoretical findings, and we conduct an empirical study on the standard MIMIC-III dataset. The latter analysis shows evidence of value heterogeneity and confirms the advantages of our new method.

Bayesian bandit algorithms with approximate inference have been widely used in practice with superior performance. Yet, few studies regarding the fundamental understanding of their performances are available. In this paper, we propose a Bayesian bandit algorithm, which we call Generalized Bayesian Upper Confidence Bound (GBUCB), for bandit problems in the presence of approximate inference. Our theoretical analysis demonstrates that in Bernoulli multi-armed bandit, GBUCB can achieve $O(\sqrt{T}(\log T)^c)$ frequentist regret if the inference error measured by symmetrized Kullback-Leibler divergence is controllable. This analysis relies on a novel sensitivity analysis for quantile shifts with respect to inference errors. To our best knowledge, our work provides the first theoretical regret bound that is better than $o(T)$ in the setting of approximate inference. Our experimental evaluations on multiple approximate inference settings corroborate our theory, showing that our GBUCB is consistently superior to BUCB and Thompson sampling.

With the capability of modeling bidirectional contexts, denoising autoencoding based pretraining like BERT achieves better performance than pretraining approaches based on autoregressive language modeling. However, relying on corrupting the input with masks, BERT neglects dependency between the masked positions and suffers from a pretrain-finetune discrepancy. In light of these pros and cons, we propose XLNet, a generalized autoregressive pretraining method that (1) enables learning bidirectional contexts by maximizing the expected likelihood over all permutations of the factorization order and (2) overcomes the limitations of BERT thanks to its autoregressive formulation. Furthermore, XLNet integrates ideas from Transformer-XL, the state-of-the-art autoregressive model, into pretraining. Empirically, XLNet outperforms BERT on 20 tasks, often by a large margin, and achieves state-of-the-art results on 18 tasks including question answering, natural language inference, sentiment analysis, and document ranking.