We provide novel theoretical insights on structured prediction in the context of efficient convex surrogate loss minimization with consistency guarantees. For any task loss, we construct a convex surrogate that can be optimized via stochastic gradient descent and we prove tight bounds on the so-called "calibration function" relating the excess surrogate risk to the actual risk. In contrast to prior related work, we carefully monitor the effect of the exponential number of classes in the learning guarantees as well as on the optimization complexity. As an interesting consequence, we formalize the intuition that some task losses make learning harder than others, and that the classical 0-1 loss is ill-suited for general structured prediction.
Surrogate risk minimization is an ubiquitous paradigm in supervised machine learning, wherein a target problem is solved by minimizing a surrogate loss on a dataset. Surrogate regret bounds, also called excess risk bounds, are a common tool to prove generalization rates for surrogate risk minimization. While surrogate regret bounds have been developed for certain classes of loss functions, such as proper losses, general results are relatively sparse. We provide two general results. The first gives a linear surrogate regret bound for any polyhedral (piecewise-linear and convex) surrogate, meani…
Establishing Linear Surrogate Regret Bounds for Convex Smooth Losses via Convolutional Fenchel-Young Losses
College of Computing and Data Science, Nanyang Technological University（南洋理工大学计算与数据科学学院）；The Institute of Statistical Mathematics（统计数学研究所）；School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；Skywork AI
We use surrogate losses to obtain several new regret bounds and new algorithms for contextual bandit learning. Using the ramp loss, we derive new margin-based regret bounds in terms of standard sequential complexity measures of a benchmark class of real-valued regression functions. Using the hinge loss, we derive an efficient algorithm with a $\sqrt{dT}$-type mistake bound against benchmark policies induced by $d$-dimensional regressors. Under realizability assumptions, our results also yield classical regret bounds.
Surrogate Regret Bounds for Bipartite Ranking via Strongly Proper Losses
Indian Institute of Science（印度科学学院）
二分排名问题，其中实例被标记为正面或负面，目标是学习一个评分函数，该函数最大限度地减少对一对正面和负面实例的错误排名概率（或等效地，最大化ROC曲线下的面积），近年来已被广泛研究。一个占主导地位的理论和算法框架的问题已经减少二分排名成对分类，特别是，它是众所周知的，二分排名遗憾可以制定成对分类遗憾，这反过来又可以使用通常的遗憾界分类问题的上限。最近，Kotlowski et al.（2011）根据与标准（非成对）逻辑和指数损失的平衡版本相关的遗憾，显示了二分排序的遗憾界限。在本文中，我们表明，这种（非成对）代理遗憾界的二分排名可以得到一个广泛的类适当的（复合）损失，我们的术语作为强适当的。我们的证明技术比Kotlowski等人（2011）简单得多，并且依赖于Reid和Williamson（2010，2011）和其他人最近阐明的适当（复合）损失的性质。我们的结果产生明确的代理边界（没有隐藏的平衡条款）在各种强适当的损失，包括例如逻辑，指数，平方和平方铰链损失的特殊情况。我们还通过Clemencon和Robbiano（2011）的最新结果在某些低噪声条件下获得了更紧密的代理边界。
Radboud University（拉德堡德大学）；Delft University of Technology（代尔夫特理工大学）；University of Copenhagen（哥本哈根大学）
Consider a classification problem where we have both labeled and unlabeled data available. We show that for linear classifiers defined by convex margin-based surrogate losses that are decreasing, it is impossible to construct any semi-supervised approach that is able to guarantee an improvement over the supervised classifier measured by this surrogate loss on the labeled and unlabeled data. For convex margin-based loss functions that also increase, we demonstrate safe improvements are possible.
Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully hand-picked stepsize for fast convergence, which is notoriously tedious and time-consuming to tune. Over the last several years, a plethora of adaptive gradient-based algorithms have emerged to ameliorate this problem. They have proved efficient in reducing the labor of tuning in practice, but many of them lack theoretic guarantees even in the convex setting. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic opti…
Leiden University（莱顿大学）；Poznań University of Technology（波兹南理工大学）
A standard introduction to online learning might place Online Gradient Descent at its center and then proceed to develop generalizations and extensions like Online Mirror Descent and second-order methods. Here we explore the alternative approach of putting Exponential Weights (EW) first. We show that many standard methods and their regret bounds then follow as a special case by plugging in suitable surrogate losses and playing the EW posterior mean. For instance, we easily recover Online Gradient Descent by using EW with a Gaussian prior on linearized losses, and, more generally, all instances…
University of Michigan（密歇根大学）；Princeton University（普林斯顿大学）
We introduce UniLoss, a unified framework to generate surrogate losses for training deep networks with gradient descent, reducing the amount of manual design of task-specific surrogate losses. Our key observation is that in many cases, evaluating a model with a performance metric on a batch of examples can be refactored into four steps: from input to real-valued scores, from scores to comparisons of pairs of scores, from comparisons to binary variables, and from binary variables to the final performance metric. Using this refactoring we generate differentiable approximations for each non-diffe…
Non-Stationary Online Structured Prediction with Surrogate Losses
CyberAgent, Tokyo, Japan（日本东京CyberAgent公司）；National Institute of Informatics, Tokyo, Japan（日本东京信息机构）；Center for Advanced Intelligence Project, RIKEN, Tokyo, Japan（日本东京RIKEN高级智能项目）；The Institute of Statistical Mathematics, Tokyo, Japan（日本东京统计数学研究所）；Tohoku University, Miyagi, Japan（日本宫城县立东大学）；Nanyang Technological University, Singapore（新加坡南洋理工大学）
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…
Graph-Based Bayesian Optimization for Quantum Circuit Architecture Search with Uncertainty Calibrated Surrogates
Department of Physics, Indian Institute of Technology (BHU)（印度理工学院（BHU）物理系）；eBRAIN Lab, Division of Engineering, New York University Abu Dhabi (NYUAD)（纽约大学阿布扎赫分校（NYUAD）工程系eBRAIN实验室）；Center for Quantum and Topological Systems (CQTS), NYUAD Research Institute, NYUAD（纽约大学阿布扎赫分校（NYUAD）量子与拓扑系统中心（CQTS））
We formalize and study the natural approach of designing convex surrogate loss functions via embeddings, for problems such as classification, ranking, or structured prediction. In this approach, one embeds each of the finitely many predictions (e.g. rankings) as a point in $R^d$, assigns the original loss values to these points, and "convexifies" the loss in some way to obtain a surrogate. We establish a strong connection between this approach and polyhedral (piecewise-linear convex) surrogate losses: every discrete loss is embedded by some polyhedral loss, and every polyhedral loss embeds som…
Adversarial testing methods based on Projected Gradient Descent (PGD) are widely used for searching norm-bounded perturbations that cause the inputs of neural networks to be misclassified. This paper takes a deeper look at these methods and explains the effect of different hyperparameters (i.e., optimizer, step size and surrogate loss). We introduce the concept of MultiTargeted testing, which makes clever use of alternative surrogate losses, and explain when and how MultiTargeted is guaranteed to find optimal perturbations. Finally, we demonstrate that MultiTargeted outperforms more sophistica…
(Partial) ranking loss is a commonly used evaluation measure for multi-label classification, which is usually optimized with convex surrogates for computational efficiency. Prior theoretical work on multi-label ranking mainly focuses on (Fisher) consistency analyses. However, there is a gap between existing theory and practice -- some pairwise losses can lead to promising performance but lack consistency, while some univariate losses are consistent but usually have no clear superiority in practice. In this paper, we attempt to fill this gap through a systematic study from two complementary per…
When samples have internal structure, we often see a mismatch between the objective optimized during training and the model's goal during inference. For example, in sequence-to-sequence modeling we are interested in high-quality translated sentences, but training typically uses maximum likelihood at the word level. The natural training-time loss would involve a combinatorial problem -- dynamic programming-based global sequence alignment -- but solutions to combinatorial problems are not differentiable with respect to their input parameters, so surrogate, differentiable losses are used instead.…
RTL Nederland B.V.（RTL荷兰公司）；University of Amsterdam（阿姆斯特丹大学）
Multiclass multilabel classification is the task of attributing multiple labels to examples via predictions. Current models formulate a reduction of the multilabel setting into either multiple binary classifications or multiclass classification, allowing for the use of existing loss functions (sigmoid, cross-entropy, logistic, etc.). Multilabel classification reductions do not accommodate for the prediction of varying numbers of labels per example and the underlying losses are distant estimates of the performance metrics. We propose a loss function, sigmoidF1, which is an approximation of the …
Carnegie Mellon University（卡内基梅隆大学）；Conversant LLC（Conversant有限责任公司）；California Institute of Technology（加州理工学院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）；University of Pennsylvania（宾夕法尼亚大学）
We propose a robust adversarial prediction framework for general multiclass classification. Our method seeks predictive distributions that robustly optimize non-convex and non-continuous multiclass loss metrics against the worst-case conditional label distributions (the adversarial distributions) that (approximately) match the statistics of the training data. Although the optimized loss metrics are non-convex and non-continuous, the dual formulation of the framework is a convex optimization problem that can be recast as a risk minimization model with a prescribed convex surrogate loss we call …
We present Gaptron, a randomized first-order algorithm for online multiclass classification. In the full information setting we show expected mistake bounds with respect to the logistic loss, hinge loss, and the smooth hinge loss with constant regret, where the expectation is with respect to the learner's randomness. In the bandit classification setting we show that Gaptron is the first linear time algorithm with $O(K\sqrt{T})$ expected regret, where $K$ is the number of classes. Additionally, the expected mistake bound of Gaptron does not depend on the dimension of the feature vector, contrar…
The minimization of loss functions is the heart and soul of Machine Learning. In this paper, we propose an off-the-shelf optimization approach that can minimize virtually any non-differentiable and non-decomposable loss function (e.g. Miss-classification Rate, AUC, F1, Jaccard Index, Mathew Correlation Coefficient, etc.) seamlessly. Our strategy learns smooth relaxation versions of the true losses by approximating them through a surrogate neural network. The proposed loss networks are set-wise models which are invariant to the order of mini-batch instances. Ultimately, the surrogate losses are…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院，巴黎电信学院，LTCI实验室）
Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous…
Google Research（谷歌研究院）；Rutgers University（罗格斯大学）；Courant Institute of Mathematical Sciences（库朗数学科学研究所）
Adversarial robustness is an increasingly critical property of classifiers in applications. The design of robust algorithms relies on surrogate losses since the optimization of the adversarial loss with most hypothesis sets is NP-hard. But which surrogate losses should be used and when do they benefit from theoretical guarantees? We present an extensive study of this question, including a detailed analysis of the H-calibration and H-consistency of adversarial surrogate losses. We show that, under some general assumptions, convex loss functions, or the supremum-based convex losses often used in…
Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the underlying loss function is increasing in the number of incorrect predictions. However, gradient or cutting-plane computation for these functions is NP-hard for non-supermodular loss functions. We propose instead a novel surrogate loss function for submodular losses, the Lovász hinge…
The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所先进智能项目中心）
Complex classification performance metrics such as the F${}_β$-measure and Jaccard index are often used, in order to handle class-imbalanced cases such as information retrieval and image segmentation. These performance metrics are not decomposable, that is, they cannot be expressed in a per-example manner, which hinders a straightforward application of M-estimation widely used in supervised learning. In this paper, we consider linear-fractional metrics, which are a family of classification performance metrics that encompasses many standard ones such as the F${}_β$-measure and Jaccard index, an…
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；AIP（应用物理研究所）；University of Michigan（密歇根大学）
Adversarially robust classification seeks a classifier that is insensitive to adversarial perturbations of test patterns. This problem is often formulated via a minimax objective, where the target loss is the worst-case value of the 0-1 loss subject to a bound on the size of perturbation. Recent work has proposed convex surrogates for the adversarial 0-1 loss, in an effort to make optimization more tractable. A primary question is that of consistency, that is, whether minimization of the surrogate risk implies minimization of the adversarial 0-1 risk. In this work, we analyze this question thr…
We consider optimization of generalized performance metrics for binary classification by means of surrogate losses. We focus on a class of metrics, which are linear-fractional functions of the false positive and false negative rates (examples of which include $F_β$-measure, Jaccard similarity coefficient, AM measure, and many others). Our analysis concerns the following two-step procedure. First, a real-valued function $f$ is learned by minimizing a surrogate loss for binary classification on the training sample. It is assumed that the surrogate loss is a strongly proper composite loss functio…
University of Pennsylvania（宾夕法尼亚大学）；Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
The F-measure is a widely used performance measure for multi-label classification, where multiple labels can be active in an instance simultaneously (e.g. in image tagging, multiple tags can be active in any image). In particular, the F-measure explicitly balances recall (fraction of active labels predicted to be active) and precision (fraction of labels predicted to be active that are actually so), both of which are important in evaluating the overall performance of a multi-label classifier. As with most discrete prediction problems, however, directly optimizing the F-measure is computational…
University of Amsterdam（阿姆斯特丹大学）；Universidad Carlos III de Madrid（马德里卡洛斯三世大学）
We study the statistical properties of learning to defer (L2D) to multiple experts. In particular, we address the open problems of deriving a consistent surrogate loss, confidence calibration, and principled ensembling of experts. Firstly, we derive two consistent surrogates -- one based on a softmax parameterization, the other on a one-vs-all (OvA) parameterization -- that are analogous to the single expert losses proposed by Mozannar and Sontag (2020) and Verma and Nalisnick (2022), respectively. We then study the frameworks' ability to estimate P( m_j = y | x ), the probability that the jth…
University of California, Berkeley（加州大学伯克利分校）；Google Research Accra（谷歌阿克拉研究院）；University of Illinois at Urbana Champaign（伊利诺伊大学厄巴纳-香槟分校）
The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously understood when taking the $k$-argmax of a vector is guaranteed to return the $k$-argmax of another vector, though doing so is crucial to describe Bayes optimality; we do both tasks. Then, we define top-$k$ calibration and show it is necessary and sufficient for consistency. Based o…
We study consistency properties of surrogate loss functions for general multiclass learning problems, defined by a general multiclass loss matrix. We extend the notion of classification calibration, which has been studied for binary and multiclass 0-1 classification problems (and for certain other specific learning problems), to the general multiclass setting, and derive necessary and sufficient conditions for a surrogate loss to be calibrated with respect to a loss matrix in this setting. We then introduce the notion of convex calibration dimension of a multiclass loss matrix, which measures …
Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）
Active learning is a type of sequential design for supervised machine learning, in which the learning algorithm sequentially requests the labels of selected instances from a large pool of unlabeled data points. The objective is to produce a classifier of relatively low risk, as measured under the 0-1 loss, ideally using fewer label requests than the number of random labeled data points sufficient to achieve the same. This work investigates the potential uses of surrogate loss functions in the context of active learning. Specifically, it presents an active learning algorithm based on an arbitra…
Toyota Technological Institute of Technology（丰田工业大学）
In this dissertation, we focus on several important problems in structured prediction. In structured prediction, the label has a rich intrinsic substructure, and the loss varies with respect to the predicted label and the true label pair. Structured SVM is an extension of binary SVM to adapt to such structured tasks. In the first part of the dissertation, we study the surrogate losses and its efficient methods. To minimize the empirical risk, a surrogate loss which upper bounds the loss, is used as a proxy to minimize the actual loss. Since the objective function is written in terms of the sur…
In this paper we refine the process of computing calibration functions for a number of multiclass classification surrogate losses. Calibration functions are a powerful tool for easily converting bounds for the surrogate risk (which can be computed through well-known methods) into bounds for the true risk, the probability of making a mistake. They are particularly suitable in non-parametric settings, where the approximation error can be controlled, and provide tighter bounds than the common technique of upper-bounding the 0-1 loss by the surrogate loss. The abstract nature of the more sophistic…
We formalize and study the natural approach of designing convex surrogate loss functions via embeddings, for problems such as classification, ranking, or structured prediction. In this approach, one embeds each of the finitely many predictions (e.g.\ rankings) as a point in $\mathbb{R}^d$, assigns the original loss values to these points, and "convexifies" the loss in some way to obtain a surrogate. We establish a strong connection between this approach and polyhedral (piecewise-linear convex) surrogate losses. Given any polyhedral loss $L$, we give a construction of a link function through wh…
The University of Tokyo（东京大学）；RIKEN AIP（理化学研究所先进智能项目中心）
Contrastive representation learning encourages data representation to make semantically similar pairs closer than randomly drawn negative samples, which has been successful in various domains such as vision, language, and graphs. Recent theoretical studies have attempted to explain the benefit of the large negative sample size by upper-bounding the downstream classification loss with the contrastive loss. However, the previous surrogate bounds have two drawbacks: they are only legitimate for a limited range of negative sample sizes and prohibitively large even within that range. Due to these d…
