We propose a risk-averse statistical learning framework wherein the performance of a learning algorithm is evaluated by the conditional value-at-risk (CVaR) of losses rather than the expected loss. We devise algorithms based on stochastic gradient descent for this framework. While existing studies of CVaR optimization require direct access to the underlying distribution, our algorithms make a weaker assumption that only i.i.d.\ samples are given. For convex and Lipschitz loss functions, we show that our algorithm has $O(1/\sqrt{n})$-convergence to the optimal CVaR, where $n$ is the number of samples. For nonconvex and smooth loss functions, we show a generalization bound on CVaR. By conducting numerical experiments on various machine learning tasks, we demonstrate that our algorithms effectively minimize CVaR compared with other baseline algorithms.
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Microsoft Research（微软研究院）；Princeton University（普林斯顿大学）
This paper proves, in very general settings, that convex risk minimization is a procedure to select a unique conditional probability model determined by the classification problem. Unlike most previous work, we give results that are general enough to include cases in which no minimum exists, as occurs typically, for instance, with standard boosting algorithms. Concretely, we first show that any sequence of predictors minimizing convex risk over the source distribution will converge to this unique model when the class of predictors is linear (but potentially of infinite dimension). Secondly, we…
The Australian National University（澳大利亚国立大学）；Data61；Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
Conditional Value at Risk (CVaR) is a family of "coherent risk measures" which generalize the traditional mathematical expectation. Widely used in mathematical finance, it is garnering increasing interest in machine learning, e.g., as an alternate approach to regularization, and as a means for ensuring fairness. This paper presents a generalization bound for learning algorithms that minimize the CVaR of the empirical loss. The bound is of PAC-Bayesian type and is guaranteed to be small when the empirical CVaR is small. We achieve this by reducing the problem of estimating CVaR to that of merel…
We address imbalanced classification, the problem in which a label may have low marginal probability relative to other labels, by weighting losses according to the correct class. First, we examine the convergence rates of the expected excess weighted risk of plug-in classifiers where the weighting for the plug-in classifier and the risk may be different. This leads to irreducible errors that do not converge to the weighted Bayes risk, which motivates our consideration of robust risks. We define a robust risk that minimizes risk over a set of weightings and show excess risk bounds for this prob…
In-Context Learning as Nonparametric Conditional Probability Estimation: Risk Bounds and Optimality
Department of Statistics Beijing Normal University at Zhuhai（北京师范大学珠海分校统计学系）；Department of Statistics and Data Science Hunan University（湖南大学统计学与数据科学系）；School of Science Sun Yat-sen University（中山大学科学学院）
On the Generalization and Robustness in Conditional Value-at-Risk
Dept. of Mathematics IISER, Pune（数学系，印度科学研究所，普恩）；Dept. of Computer Science and Automation Indian Institute of Sciecne, Bangalore（计算机科学与自动化系，印度科学研究院，班加罗尔）；The Alan Turing Institute Greater London, England, UK（艾伦·图灵研究所，伦敦，英格兰，英国）
University of Toronto（多伦多大学）；Vector Institute（向量研究所）；Technion（以色列理工学院）；Google Research（谷歌研究院）；Mila（米拉研究所）；McGill（麦吉尔大学）
We study the mutual information between (certain summaries of) the output of a learning algorithm and its $n$ training data, conditional on a supersample of $n+1$ i.i.d. data from which the training data is chosen at random without replacement. These leave-one-out variants of the conditional mutual information (CMI) of an algorithm (Steinke and Zakynthinou, 2020) are also seen to control the mean generalization error of learning algorithms with bounded loss functions. For learning algorithms achieving zero empirical risk under 0-1 loss (i.e., interpolating algorithms), we provide an explicit c…
Distributionally Time-Varying Online Stochastic Optimization under Polyak-Łojasiewicz Condition with Application in Conditional Value-at-Risk Statistical Learning
The Australian National University（澳大利亚国立大学）；The University of Melbourne（墨尔本大学）
We study the task of learning from non-i.i.d. data. In particular, we aim at learning predictors that minimize the conditional risk for a stochastic process, i.e. the expected loss of the predictor on the next point conditioned on the set of training samples observed so far. For non-i.i.d. data, the training set contains information about the upcoming samples, so learning with respect to the conditional distribution can be expected to yield better predictors than one obtains from the classical setting of minimizing the marginal risk. Our main contribution is a practical estimator for the condi…
Imperial College Business School（帝国理工学院商学院）；Department of Statistics and Operations Research（统计与运筹学系）；Imperial College London（伦敦帝国理工学院）；UNC-Chapel Hill（北卡罗来纳大学教堂山分校）
We propose a general approach for supervised learning with structured output spaces, such as combinatorial and polyhedral sets, that is based on minimizing estimated conditional risk functions. Given a loss function defined over pairs of output labels, we first estimate the conditional risk function by solving a (possibly infinite) collection of regularized least squares problems. A prediction is made by solving an inference problem that minimizes the estimated conditional risk function over the output space. We show that this approach enables, in some cases, efficient training and inference w…
We study conditional risk minimization (CRM), i.e. the problem of learning a hypothesis of minimal risk for prediction at the next step of sequentially arriving dependent data. Despite it being a fundamental problem, successful learning in the CRM sense has so far only been demonstrated using theoretical algorithms that cannot be used for real problems as they would require storing all incoming data. In this work, we introduce MACRO, a meta-algorithm for CRM that does not suffer from this shortcoming, but nevertheless offers learning guarantees. Instead of storing all data it maintains and ite…
Indian Institute of Technology Madras（马德拉斯印度理工学院）；TCS Research（塔塔咨询服务研究院）
In several real-world applications involving decision making under uncertainty, the traditional expected value objective may not be suitable, as it may be necessary to control losses in the case of a rare but extreme event. Conditional Value-at-Risk (CVaR) is a popular risk measure for modeling the aforementioned objective. We consider the problem of estimating CVaR from i.i.d. samples of an unbounded random variable, which is either sub-Gaussian or sub-exponential. We derive a novel one-sided concentration bound for a natural sample-based CVaR estimator in this setting. Our bound relies on a …
Chongqing University（重庆大学）；Bank of China（中国银行）；University of Mississippi（密西西比大学）
Well-known for its simplicity and effectiveness in classification, AdaBoost, however, suffers from overfitting when class-conditional distributions have significant overlap. Moreover, it is very sensitive to noise that appears in the labels. This article tackles the above limitations simultaneously via optimizing a modified loss function (i.e., the conditional risk). The proposed approach has the following two advantages. (1) It is able to directly take into account label uncertainty with an associated label confidence. (2) It introduces a "trustworthiness" measure on training samples via the …
On the Global Convergence of Risk-Averse Natural Policy Gradient Methods with Expected Conditional Risk Measures
Department of Integrated Systems Engineering, The Ohio State University（俄亥俄州立大学整合系统工程系）；Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气工程与计算机科学系）
We show how to estimate a model's test error from unlabeled data, on distributions very different from the training distribution, while assuming only that certain conditional independencies are preserved between train and test. We do not need to assume that the optimal predictor is the same between train and test, or that the true distribution lies in any parametric family. We can also efficiently differentiate the error estimate to perform unsupervised discriminative learning. Our technical tool is the method of moments, which allows us to exploit conditional independencies in the absence of …
A Guide to Estimating Conditional Average Treatment Effects in Competing Risks Settings
Department of Statistics, TU Dortmund University（图恩-多特蒙德大学统计学系）；Research Center Trustworthy Data Science and Security, University Alliance Ruhr (UA Ruhr)（鲁尔大学联盟可信数据科学与安全研究中心）；Institute for Mathematics, University of Augsburg（艾希施泰特大学数学研究所）
Academy for Advanced Interdisciplinary Studies（前沿交叉学科研究院）；Peking University（北京大学）；MGH/BWH Center for Advanced Medical Computing and Analysis（麻省总医院/布莱根妇女医院高级医学计算与分析中心）；Massachusetts General Hospital and Harvard Medical School（麻省总医院和哈佛医学院）；Beijing International Center for Mathematical Research（北京国际数学研究中心）
We proved that a trained model in supervised deep learning minimizes the conditional risk for each input (Theorem 2.1). This property provided insights into the behavior of trained models and established a connection between supervised and unsupervised learning in some cases. In addition, when the labels are intractable but can be written as a conditional risk minimizer, we proved an equivalent form of the original supervised learning problem with accessible labels (Theorem 2.2). We demonstrated that many existing works, such as Noise2Score, Noise2Noise and score function estimation can be exp…
Generative Adversarial Net (GAN) has been proven to be a powerful machine learning tool in image data analysis and generation. In this paper, we propose to use Conditional Generative Adversarial Net (CGAN) to learn and simulate time series data. The conditions can be both categorical and continuous variables containing different kinds of auxiliary information. Our simulation studies show that CGAN is able to learn different kinds of normal and heavy tail distributions, as well as dependent structures of different time series and it can further generate conditional predictive distributions cons…
Institut Intelligence et Données (IID)（智能与数据研究所（IID））；Université Laval（拉瓦尔大学）；Quebec Artificial Intelligence Institute (Mila)（魁北克人工智能研究所（Mila））；Université de Montréal（蒙特利尔大学）
In the classical Reinforcement Learning (RL) setting, one aims to find a policy that maximizes its expected return. This objective may be inappropriate in safety-critical domains such as healthcare or autonomous driving, where intrinsic uncertainties due to stochastic policies and environment variability may lead to catastrophic failures. This can be addressed by using the Conditional-Value-at-Risk (CVaR) objective to instill risk-aversion in learned policies. In this paper, we propose Adversarial Cvar Reinforcement Learning (ACReL), a novel adversarial meta-algorithm to optimize the CVaR obje…
We propose a simple but effective modification of the discriminators, namely measure-conditional discriminators, as a plug-and-play module for different GANs. By taking the generated distributions as part of input so that the target optimum for the discriminator is stationary, the proposed discriminator is more robust than the vanilla one. A variant of the measure-conditional discriminator can also handle multiple target distributions, or act as a surrogate model of statistical distances such as KL divergence with applications to transfer learning.
The Ohio State University（俄亥俄州立大学）；University of Michigan（密歇根大学）
Traditional reinforcement learning (RL) aims to maximize the expected total reward, while the risk of uncertain outcomes needs to be controlled to ensure reliable performance in a risk-averse setting. In this paper, we consider the problem of maximizing dynamic risk of a sequence of rewards in infinite-horizon Markov Decision Processes (MDPs). We adapt the Expected Conditional Risk Measures (ECRMs) to the infinite-horizon risk-averse MDP and prove its time consistency. Using a convex combination of expectation and conditional value-at-risk (CVaR) as a special one-step conditional risk measure,…
Conditional Generative Modeling for Enhanced Credit Risk Management in Supply Chain Finance
School of Management, Fudan University（复旦大学管理学院）；Department of Decision Analytics and Operations, College of Business, City University of Hong Kong（城市大学商学院决策分析与运作部门）；Department of Industrial and Systems Engineering, University of Minnesota（明尼苏达大学工业与系统工程系）；The Laboratory for AI-Powered Financial Technologies（人工智能金融技术实验室）
RADE: Learning Risk-Adjustable Driving Environment via Multi-Agent Conditional Diffusion
Department of Civil and Environmental Engineering, University of Michigan（密歇根大学土木与环境工程系）；Department of Computer Science, University of Hong Kong（香港大学计算机科学系）；University of Michigan Transportation Research Institute（密歇根大学交通研究研究院）
Nanjing University of Science and Technology（南京理工大学）；PCA Lab, Key Lab of Intelligent Perception and Systems for High-Dimensional Information of Ministry of Education（PCA实验室，教育部高维信息智能感知与系统重点实验室）；School of Computer Science and Engineering（计算机科学与工程学院）
