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We study the benign overfitting theory in the prediction of the conditional average treatment effect (CATE), with linear regression models. As the development of machine learning for causal inference, a wide range of large-scale models for causality are gaining attention. One problem is that suspicions have been raised that the large-scale models are prone to overfitting to observations with sample selection, hence the large models may not be suitable for causal prediction. In this study, to resolve the suspicious, we investigate on the validity of causal inference methods for overparameterized models, by applying the recent theory of benign overfitting (Bartlett et al., 2020). Specifically, we consider samples whose distribution switches depending on an assignment rule, and study the prediction of CATE with linear models whose dimension diverges to infinity. We focus on two methods: the T-learner, which based on a difference between separately constructed estimators with each treatment group, and the inverse probability weight (IPW)-learner, which solves another regression problem approximated by a propensity score. In both methods, the estimator consists of interpolators that fit the samples perfectly. As a result, we show that the T-learner fails to achieve the consistency except the random assignment, while the IPW-learner converges the risk to zero if the propensity score is known. This difference stems from that the T-learner is unable to preserve eigenspaces of the covariances, which is necessary for benign overfitting in the overparameterized setting. Our result provides new insights into the usage of causal inference methods in the overparameterizated setting, in particular, doubly robust estimators.

In this paper, we analyze the convergence %semi-convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the $k$th iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.

Recent works have investigated the sample complexity necessary for fair machine learning. The most advanced of such sample complexity bounds are developed by analyzing multicalibration uniform convergence for a given predictor class. We present a framework which yields multicalibration error uniform convergence bounds by reparametrizing sample complexities for Empirical Risk Minimization (ERM) learning. From this framework, we demonstrate that multicalibration error exhibits dependence on the classifier architecture as well as the underlying data distribution. We perform an experimental evaluation to investigate the behavior of multicalibration error for different families of classifiers. We compare the results of this evaluation to multicalibration error concentration bounds. Our investigation provides additional perspective on both algorithmic fairness and multicalibration error convergence bounds. Given the prevalence of ERM sample complexity bounds, our proposed framework enables machine learning practitioners to easily understand the convergence behavior of multicalibration error for a myriad of classifier architectures.

This paper considers adaptive, minimax estimation of a quadratic functional in a nonparametric instrumental variables (NPIV) model, which is an important problem in optimal estimation of a nonlinear functional of an ill-posed inverse regression with an unknown operator. We first show that a leave-one-out, sieve NPIV estimator of the quadratic functional can attain a convergence rate that coincides with the lower bound previously derived in Chen and Christensen [2018]. The minimax rate is achieved by the optimal choice of the sieve dimension (a key tuning parameter) that depends on the smoothness of the NPIV function and the degree of ill-posedness, both are unknown in practice. We next propose a Lepski-type data-driven choice of the key sieve dimension adaptive to the unknown NPIV model features. The adaptive estimator of the quadratic functional is shown to attain the minimax optimal rate in the severely ill-posed case and in the regular mildly ill-posed case, but up to a multiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.

We examine global non-asymptotic convergence properties of policy gradient methods for multi-agent reinforcement learning (RL) problems in Markov potential games (MPG). To learn a Nash equilibrium of an MPG in which the size of state space and/or the number of players can be very large, we propose new independent policy gradient algorithms that are run by all players in tandem. When there is no uncertainty in the gradient evaluation, we show that our algorithm finds an $\epsilon$-Nash equilibrium with $O(1/\epsilon^2)$ iteration complexity which does not explicitly depend on the state space size. When the exact gradient is not available, we establish $O(1/\epsilon^5)$ sample complexity bound in a potentially infinitely large state space for a sample-based algorithm that utilizes function approximation. Moreover, we identify a class of independent policy gradient algorithms that enjoys convergence for both zero-sum Markov games and Markov cooperative games with the players that are oblivious to the types of games being played. Finally, we provide computational experiments to corroborate the merits and the effectiveness of our theoretical developments.

The \emph{turnpike property} in contemporary macroeconomics asserts that if an economic planner seeks to move an economy from one level of capital to another, then the most efficient path, as long as the planner has enough time, is to rapidly move stock to a level close to the optimal stationary or constant path, then allow for capital to develop along that path until the desired term is nearly reached, at which point the stock ought to be moved to the final target. Motivated in part by its nature as a resource allocation strategy, over the past decade, the turnpike property has also been shown to hold for several classes of partial differential equations arising in mechanics. When formalized mathematically, the turnpike theory corroborates the insights from economics: for an optimal control problem set in a finite-time horizon, optimal controls and corresponding states, are close (often exponentially), during most of the time, except near the initial and final time, to the optimal control and corresponding state for the associated stationary optimal control problem. In particular, the former are mostly constant over time. This fact provides a rigorous meaning to the asymptotic simplification that some optimal control problems appear to enjoy over long time intervals, allowing the consideration of the corresponding stationary problem for computing and applications. We review a slice of the theory developed over the past decade --the controllability of the underlying system is an important ingredient, and can even be used to devise simple turnpike-like strategies which are nearly optimal--, and present several novel applications, including, among many others, the characterization of Hamilton-Jacobi-Bellman asymptotics, and stability estimates in deep learning via residual neural networks.

Artificial intelligence (AI) has become a part of everyday conversation and our lives. It is considered as the new electricity that is revolutionizing the world. AI is heavily invested in both industry and academy. However, there is also a lot of hype in the current AI debate. AI based on so-called deep learning has achieved impressive results in many problems, but its limits are already visible. AI has been under research since the 1940s, and the industry has seen many ups and downs due to over-expectations and related disappointments that have followed. The purpose of this book is to give a realistic picture of AI, its history, its potential and limitations. We believe that AI is a helper, not a ruler of humans. We begin by describing what AI is and how it has evolved over the decades. After fundamentals, we explain the importance of massive data for the current mainstream of artificial intelligence. The most common representations for AI, methods, and machine learning are covered. In addition, the main application areas are introduced. Computer vision has been central to the development of AI. The book provides a general introduction to computer vision, and includes an exposure to the results and applications of our own research. Emotions are central to human intelligence, but little use has been made in AI. We present the basics of emotional intelligence and our own research on the topic. We discuss super-intelligence that transcends human understanding, explaining why such achievement seems impossible on the basis of present knowledge,and how AI could be improved. Finally, a summary is made of the current state of AI and what to do in the future. In the appendix, we look at the development of AI education, especially from the perspective of contents at our own university.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.

Estimating post-click conversion rate (CVR) accurately is crucial for ranking systems in industrial applications such as recommendation and advertising. Conventional CVR modeling applies popular deep learning methods and achieves state-of-the-art performance. However it encounters several task-specific problems in practice, making CVR modeling challenging. For example, conventional CVR models are trained with samples of clicked impressions while utilized to make inference on the entire space with samples of all impressions. This causes a sample selection bias problem. Besides, there exists an extreme data sparsity problem, making the model fitting rather difficult. In this paper, we model CVR in a brand-new perspective by making good use of sequential pattern of user actions, i.e., impression -> click -> conversion. The proposed Entire Space Multi-task Model (ESMM) can eliminate the two problems simultaneously by i) modeling CVR directly over the entire space, ii) employing a feature representation transfer learning strategy. Experiments on dataset gathered from Taobao's recommender system demonstrate that ESMM significantly outperforms competitive methods. We also release a sampling version of this dataset to enable future research. To the best of our knowledge, this is the first public dataset which contains samples with sequential dependence of click and conversion labels for CVR modeling.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.