Given a task of predicting $Y$ from $X$, a loss function $L$, and a set of probability distributions $Γ$ on $(X,Y)$, what is the optimal decision rule minimizing the worst-case expected loss over $Γ$? In this paper, we address this question by introducing a generalization of the principle of maximum entropy. Applying this principle to sets of distributions with marginal on $X$ constrained to be the empirical marginal from the data, we develop a general minimax approach for supervised learning problems. While for some loss functions such as squared-error and log loss, the minimax approach rederives well-knwon regression models, for the 0-1 loss it results in a new linear classifier which we call the maximum entropy machine. The maximum entropy machine minimizes the worst-case 0-1 loss over the structured set of distribution, and by our numerical experiments can outperform other well-known linear classifiers such as SVM. We also prove a bound on the generalization worst-case error in the minimax approach.
Institute of Telecommunications, Vienna University of Technology（维也纳工业大学电信研究所）；Technion---Israel Institute of Technology（以色列理工学院）
We consider the problem of learning a dictionary matrix from a number of observed signals, which are assumed to be generated via a linear model with a common underlying dictionary. In particular, we derive lower bounds on the minimum achievable worst case mean squared error (MSE), regardless of computational complexity of the dictionary learning (DL) schemes. By casting DL as a classical (or frequentist) estimation problem, the lower bounds on the worst case MSE are derived by following an established information-theoretic approach to minimax estimation. The main conceptual contribution of thi…
We study the stochastic Multi-Armed Bandit (MAB) problem under worst-case regret and heavy-tailed reward distribution. We modify the minimax policy MOSS for the sub-Gaussian reward distribution by using saturated empirical mean to design a new algorithm called Robust MOSS. We show that if the moment of order $1+ε$ for the reward distribution exists, then the refined strategy has a worst-case regret matching the lower bound while maintaining a distribution-dependent logarithm regret.
University of Oslo（奥斯陆大学）；University of Neuchatel（纳沙泰尔大学）；Zenseact；Chalmers University of Technology（查尔姆斯理工大学）
While the Bayesian decision-theoretic framework offers an elegant solution to the problem of decision making under uncertainty, one question is how to appropriately select the prior distribution. One idea is to employ a worst-case prior. However, this is not as easy to specify in sequential decision making as in simple statistical estimation problems. This paper studies (sometimes approximate) minimax-Bayes solutions for various reinforcement learning problems to gain insights into the properties of the corresponding priors and policies. We find that while the worst-case prior depends on the s…
Worst-case generation via minimax optimization in Wasserstein space
Department of Mathematics, Duke University（杜克大学数学系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
UC Berkeley（加州大学伯克利分校）；IBM Research – Almaden（IBM阿尔马登研究中心）；ICSI（国际计算机科学研究所）；Google Inc. Zürich（苏黎世谷歌公司）；UC Santa Cruz（加州大学圣克鲁兹分校）
In experimental design, we are given a large collection of vectors, each with a hidden response value that we assume derives from an underlying linear model, and we wish to pick a small subset of the vectors such that querying the corresponding responses will lead to a good estimator of the model. A classical approach in statistics is to assume the responses are linear, plus zero-mean i.i.d. Gaussian noise, in which case the goal is to provide an unbiased estimator with smallest mean squared error (A-optimal design). A related approach, more common in computer science, is to assume the respons…
Göteborgs Universitet（哥德堡大学）；Chalmers University of Technology（查尔姆斯理工大学）
Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.
BCAM-Basque Center of Applied Mathematics（BCAM巴斯克应用数学中心）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Conventional techniques for supervised classification constrain the classification rules considered and use surrogate losses for classification 0-1 loss. Favored families of classification rules are those that enjoy parametric representations suitable for surrogate loss minimization, and low complexity properties suitable for overfitting control. This paper presents classification techniques based on robust risk minimization (RRM) that we call linear probabilistic classifiers (LPCs). The proposed techniques consider unconstrained classification rules, optimize the classification 0-1 loss, and …
We consider semi-supervised regression when the predictor variables are drawn from an unknown manifold. A simple two step approach to this problem is to: (i) estimate the manifold geodesic distance between any pair of points using both the labeled and unlabeled instances; and (ii) apply a k nearest neighbor regressor based on these distance estimates. We prove that given sufficiently many unlabeled points, this simple method of geodesic kNN regression achieves the optimal finite-sample minimax bound on the mean squared error, as if the manifold were known. Furthermore, we show how this approac…
As opposed to standard empirical risk minimization (ERM), distributionally robust optimization aims to minimize the worst-case risk over a larger ambiguity set containing the original empirical distribution of the training data. In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. In particular, we prove generalization bounds that involve the covering number properties of the original ERM problem. As an illustrative example, we provide generalization guarantees for transport-based domain adaptation problems where the Wa…
We study the supervised clustering problem under the two-component anisotropic Gaussian mixture model in high dimensions and in the non-asymptotic setting. We first derive a lower and a matching upper bound for the minimax risk of clustering in this framework. We also show that in the high-dimensional regime, the linear discriminant analysis (LDA) classifier turns out to be sub-optimal in the minimax sense. Next, we characterize precisely the risk of $\ell_2$-regularized supervised least squares classifiers. We deduce the fact that the interpolating solution may outperform the regularized clas…
One of the key drivers of complexity in the classical (stochastic) multi-armed bandit (MAB) problem is the difference between mean rewards in the top two arms, also known as the instance gap. The celebrated Upper Confidence Bound (UCB) policy is among the simplest optimism-based MAB algorithms that naturally adapts to this gap: for a horizon of play n, it achieves optimal O(log n) regret in instances with "large" gaps, and a near-optimal O(\sqrt{n log n}) minimax regret when the gap can be arbitrarily "small." This paper provides new results on the arm-sampling behavior of UCB, leading to seve…
Georgia Institute of Technology（佐治亚理工学院）；GE Global Research（通用电气全球研究中心）
Since its inception, the modus operandi of multi-task learning (MTL) has been to minimize the task-wise mean of the empirical risks. We introduce a generalized loss-compositional paradigm for MTL that includes a spectrum of formulations as a subfamily. One endpoint of this spectrum is minimax MTL: a new MTL formulation that minimizes the maximum of the tasks' empirical risks. Via a certain relaxation of minimax MTL, we obtain a continuum of MTL formulations spanning minimax MTL and classical MTL. The full paradigm itself is loss-compositional, operating on the vector of empirical risks. It inc…
We study the kernelized bandit problem, that involves designing an adaptive strategy for querying a noisy zeroth-order-oracle to efficiently learn about the optimizer of an unknown function $f$ with a norm bounded by $M<\infty$ in a Reproducing Kernel Hilbert Space~(RKHS) associated with a positive definite kernel $K$. Prior results, working in a \emph{minimax framework}, have characterized the worst-case~(over all functions in the problem class) limits on regret achievable by \emph{any} algorithm, and have constructed algorithms with matching~(modulo polylogarithmic factors) worst-case perfor…
Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）
We consider the problem of global optimization of an unknown non-convex smooth function with zeroth-order feedback. In this setup, an algorithm is allowed to adaptively query the underlying function at different locations and receives noisy evaluations of function values at the queried points (i.e. the algorithm has access to zeroth-order information). Optimization performance is evaluated by the expected difference of function values at the estimated optimum and the true optimum. In contrast to the classical optimization setup, first-order information like gradients are not directly accessibl…
We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can …
University of California Santa Barbara（加州大学圣巴巴拉分校）
Machine learning models are vulnerable to adversarial attacks that can often cause misclassification by introducing small but well designed perturbations. In this paper, we explore, in the setting of classical composite hypothesis testing, a defense strategy based on the generalized likelihood ratio test (GLRT), which jointly estimates the class of interest and the adversarial perturbation. We evaluate the GLRT approach for the special case of binary hypothesis testing in white Gaussian noise under $\ell_{\infty}$ norm-bounded adversarial perturbations, a setting for which a minimax strategy o…
BCAM-Basque Center of Applied Mathematics（巴斯克应用数学中心）；IKERBASQUE-Basque Foundation for Science（巴斯克科学基金会）；École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Supervised classification techniques use training samples to find classification rules with small expected 0-1 loss. Conventional methods achieve efficient learning and out-of-sample generalization by minimizing surrogate losses over specific families of rules. This paper presents minimax risk classifiers (MRCs) that do not rely on a choice of surrogate loss and family of rules. MRCs achieve efficient learning and out-of-sample generalization by minimizing worst-case expected 0-1 loss w.r.t. uncertainty sets that are defined by linear constraints and include the true underlying distribution. I…
BCAM-Basque Center for Applied Mathematics（巴斯克应用数学中心）；IKERBASQUE-Basque Foundation for Science（巴斯克科学基金会）；Department of Electronic Engineering, Tsinghua University（清华大学电子工程系）
The maximum entropy principle advocates to evaluate events' probabilities using a distribution that maximizes entropy among those that satisfy certain expectations' constraints. Such principle can be generalized for arbitrary decision problems where it corresponds to minimax approaches. This paper establishes a framework for supervised classification based on the generalized maximum entropy principle that leads to minimax risk classifiers (MRCs). We develop learning techniques that determine MRCs for general entropy functions and provide performance guarantees by means of convex optimization. …
Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心（BCAM））；IKERBASQUE-Basque Foundation for Science（IKERBASQUE-巴斯克科学基金会）；Federal University of Bahia（巴伊亚联邦大学）；National Research Institute for Mathematics and Computer Science (CWI)（国家数学与计算机科学研究院（CWI））
Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Alibaba Group（阿里巴巴集团）；Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；Sun Yat-sen University（中山大学）；Peng Cheng Laboratory（鹏城实验室）
In recent years, great progress has been made to incorporate unlabeled data to overcome the inefficiently supervised problem via semi-supervised learning (SSL). Most state-of-the-art models are based on the idea of pursuing consistent model predictions over unlabeled data toward the input noise, which is called consistency regularization. Nonetheless, there is a lack of theoretical insights into the reason behind its success. To bridge the gap between theoretical and practical results, we propose a worst-case consistency regularization technique for SSL in this paper. Specifically, we first pr…
In this paper, we consider learning scenarios where the learned model is evaluated under an unknown test distribution which potentially differs from the training distribution (i.e. distribution shift). The learner has access to a family of weight functions such that the test distribution is a reweighting of the training distribution under one of these functions, a setting typically studied under the name of Distributionally Robust Optimization (DRO). We consider the problem of deriving regret bounds in the classical learning theory setting, and require that the resulting regret bounds hold uni…
We study high-dimensional asymptotic performance limits of binary supervised classification problems where the class conditional densities are Gaussian with unknown means and covariances and the number of signal dimensions scales faster than the number of labeled training samples. We show that the Bayes error, namely the minimum attainable error probability with complete distributional knowledge and equally likely classes, can be arbitrarily close to zero and yet the limiting minimax error probability of every supervised learning algorithm is no better than a random coin toss. In contrast to r…
We consider the problem of learning from training data obtained in different contexts, where the underlying context distribution is unknown and is estimated empirically. We develop a robust method that takes into account the uncertainty of the context distribution. Unlike the conventional and overly conservative minimax approach, we focus on excess risks and construct distribution sets with statistical coverage to achieve an appropriate trade-off between performance and robustness. The proposed method is computationally scalable and shown to interpolate between empirical risk minimization and …
We present a new machine learning approach to estimate personalized treatment effects in the classical potential outcomes framework with binary outcomes. To overcome the problem that both treatment and control outcomes for the same unit are required for supervised learning, we propose surrogate loss functions that incorporate both treatment and control data. The new surrogates yield tighter bounds than the sum of losses for treatment and control groups. A specific choice of loss function, namely a type of hinge loss, yields a minimax support vector machine formulation. The resulting optimizati…
We address the problem of aggregating an ensemble of predictors with known loss bounds in a semi-supervised binary classification setting, to minimize prediction loss incurred on the unlabeled data. We find the minimax optimal predictions for a very general class of loss functions including all convex and many non-convex losses, extending a recent analysis of the problem for misclassification error. The result is a family of semi-supervised ensemble aggregation algorithms which are as efficient as linear learning by convex optimization, but are minimax optimal without any relaxations. Their de…
Université de Toulouse（图卢兹大学）；ONERA - The French Aerospace Lab（法国航空航天实验室（ONERA））；ISAE-SUPAERO（法国高等航空航天学院）
This work tackles the problem of robust zero-shot planning in non-stationary stochastic environments. We study Markov Decision Processes (MDPs) evolving over time and consider Model-Based Reinforcement Learning algorithms in this setting. We make two hypotheses: 1) the environment evolves continuously with a bounded evolution rate; 2) a current model is known at each decision epoch but not its evolution. Our contribution can be presented in four points. 1) we define a specific class of MDPs that we call Non-Stationary MDPs (NSMDPs). We introduce the notion of regular evolution by making an hyp…
We develop a worst-case analysis of aggregation of classifier ensembles for binary classification. The task of predicting to minimize error is formulated as a game played over a given set of unlabeled data (a transductive setting), where prior label information is encoded as constraints on the game. The minimax solution of this game identifies cases where a weighted combination of the classifiers can perform significantly better than any single classifier.
