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, meaning that surrogate generalization rates translate directly to target rates. The second shows that for sufficiently non-polyhedral surrogates, the regret bound is a square root, meaning fast surrogate generalization rates translate to slow rates for the target. Together, these results suggest polyhedral surrogates are optimal in many cases.
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 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…
The problem of bipartite ranking, where instances are labeled positive or negative and the goal is to learn a scoring function that minimizes the probability of mis-ranking a pair of positive and negative instances (or equivalently, that maximizes the area under the ROC curve), has been widely studied in recent years. A dominant theoretical and algorithmic framework for the problem has been to reduce bipartite ranking to pairwise classification; in particular, it is well known that the bipartite ranking regret can be formulated as a pairwise classification regret, which in turn can be upper bo…
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…
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…
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…
Technion - Israel Institute of Technology（以色列理工学院）
Hoffman's classical result gives a bound on the distance of a point from a convex and compact polytope in terms of the magnitude of violation of the constraints. Recently, several results showed that Hoffman's bound can be used to derive strongly-convex-like rates for first-order methods for \textit{offline} convex optimization of curved, though not strongly convex, functions, over polyhedral sets. In this work, we use this classical result for the first time to obtain faster rates for \textit{online convex optimization} over polyhedral sets with curved convex, though not strongly convex, loss…
Bayesian optimization (BO) is a popular method for black-box optimization, which relies on uncertainty as part of its decision-making process when deciding which experiment to perform next. However, not much work has addressed the effect of uncertainty on the performance of the BO algorithm and to what extent calibrated uncertainties improve the ability to find the global optimum. In this work, we provide an extensive study of the relationship between the BO performance (regret) and uncertainty calibration for popular surrogate models and compare them across both synthetic and real-world exper…
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加利福尼亚大学）；Columbia University（哥伦比亚大学）
While extensive-form games (EFGs) can be converted into normal-form games (NFGs), doing so comes at the cost of an exponential blowup of the strategy space. So, progress on NFGs and EFGs has historically followed separate tracks, with the EFG community often having to catch up with advances (e.g., last-iterate convergence and predictive regret bounds) from the larger NFG community. In this paper we show that the Optimistic Multiplicative Weights Update (OMWU) algorithm -- the premier learning algorithm for NFGs -- can be simulated on the normal-form equivalent of an EFG in linear time per iter…
Universal Online Convex Optimization with $1$ Projection per Round
National Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术国家重点实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）
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 …
Mathematics and Computer Science, Marburg University（马尔堡大学数学与计算机科学学院）
We consider the problem of rank loss minimization in the setting of multilabel classification, which is usually tackled by means of convex surrogate losses defined on pairs of labels. Very recently, this approach was put into question by a negative result showing that commonly used pairwise surrogate losses, such as exponential and logistic losses, are inconsistent. In this paper, we show a positive result which is arguably surprising in light of the previous one: the simpler univariate variants of exponential and logistic surrogates (i.e., defined on single labels) are consistent for rank los…
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）
National Key Laboratory for Novel Software Technology（新型软件技术国家重点实验室）；Nanjing University（南京大学）
In this paper, we study online convex optimization in dynamic environments, and aim to bound the dynamic regret with respect to any sequence of comparators. Existing work have shown that online gradient descent enjoys an $O(\sqrt{T}(1+P_T))$ dynamic regret, where $T$ is the number of iterations and $P_T$ is the path-length of the comparator sequence. However, this result is unsatisfactory, as there exists a large gap from the $Ω(\sqrt{T(1+P_T)})$ lower bound established in our paper. To address this limitation, we develop a novel online method, namely adaptive learning for dynamic environment …
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…
Top-$k$ classification is a generalization of multiclass classification used widely in information retrieval, image classification, and other extreme classification settings. Several hinge-like (piecewise-linear) surrogates have been proposed for the problem, yet all are either non-convex or inconsistent. For the proposed hinge-like surrogates that are convex (i.e., polyhedral), we apply the recent embedding framework of Finocchiaro et al. (2019; 2022) to determine the prediction problem for which the surrogate is consistent. These problems can all be interpreted as variants of top-$k$ classif…
Institute of High Performance Computing（高性能计算研究所）；California Institute of Technology（加州理工学院）
We study Smoothed Online Convex Optimization, a version of online convex optimization where the learner incurs a penalty for changing her actions between rounds. Given a $Ω(\sqrt{d})$ lower bound on the competitive ratio of any online algorithm, where $d$ is the dimension of the action space, we ask under what conditions this bound can be beaten. We introduce a novel algorithmic framework for this problem, Online Balanced Descent (OBD), which works by iteratively projecting the previous point onto a carefully chosen level set of the current cost function so as to balance the switching costs an…
To deal with complicated constraints via locally light computations in distributed online learning, a recent study has presented a projection-free algorithm called distributed online conditional gradient (D-OCG), and achieved an $O(T^{3/4})$ regret bound for convex losses, where $T$ is the number of total rounds. However, it requires $T$ communication rounds, and cannot utilize the strong convexity of losses. In this paper, we propose an improved variant of D-OCG, namely D-BOCG, which can attain the same $O(T^{3/4})$ regret bound with only $O(\sqrt{T})$ communication rounds for convex losses, …
Department of Computer Science, The University of Manchester（曼彻斯特大学计算机科学系）；Department of ISEM, National University of Singapore（新加坡国立大学ISEM系）；Alliance Manchester Business School, University of Manchester（曼彻斯特大学阿联酋商业学院）
From Switching to Dynamic Regret: A Simple Reduction via Unbiased Random Sequences
State Key Laboratory for Novel Software Technology, Nanjing University（南京大学计算机软件新技术全国重点实验室）；School of Artificial Intelligence, Nanjing University（南京大学人工智能学院）；School of Software Technology, Zhejiang University（浙江大学软件学院）
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.
Nanjing University（南京大学）；the University of Iowa（爱荷华大学）
In this paper, we revisit the problem of smoothed online learning, in which the online learner suffers both a hitting cost and a switching cost, and target two performance metrics: competitive ratio and dynamic regret with switching cost. To bound the competitive ratio, we assume the hitting cost is known to the learner in each round, and investigate the simple idea of balancing the two costs by an optimization problem. Surprisingly, we find that minimizing the hitting cost alone is $\max(1, \frac{2}α)$-competitive for $α$-polyhedral functions and $1 + \frac{4}λ$-competitive for $λ$-quadratic …
The acquisition function, a critical component in Bayesian optimization (BO), can often be written as the expectation of a utility function under a surrogate model. However, to ensure that acquisition functions are tractable to optimize, restrictions must be placed on the surrogate model and utility function. To extend BO to a broader class of models and utilities, we propose likelihood-free BO (LFBO), an approach based on likelihood-free inference. LFBO directly models the acquisition function without having to separately perform inference with a probabilistic surrogate model. We show that co…
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…
Efficient Black-box Adversarial Attacks via Bayesian Optimization Guided by a Function Prior
Tsinghua University（清华大学）；JQ Investments（JQ投资）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；RealAI；Kaiyuan International Mathematical Sciences Institute（开源国际数学科学研究院）
Robust Bayesian Optimization via Tempered Posteriors
Booth School of Business, University of Chicago（芝加哥大学商学院）；Rice University（里士满大学）；Computational Research Division, Lawrence Berkeley National Laboratory（伯克利国家实验室计算研究部）
We study stochastic linear optimization problem with bandit feedback. The set of arms take values in an $N$-dimensional space and belong to a bounded polyhedron described by finitely many linear inequalities. We provide a lower bound for the expected regret that scales as $Ω(N\log T)$. We then provide a nearly optimal algorithm and show that its expected regret scales as $O(N\log^{1+ε}(T))$ for an arbitrary small $ε>0$. The algorithm alternates between exploration and exploitation intervals sequentially where deterministic set of arms are played in the exploration intervals and greedily select…
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…
Bayesian Optimization of CVaR with Unknown Hyperparameters
LMU Munich（慕尼黑大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）；RPTU Kaiserslautern-Landau（凯撒斯劳滕-兰道工业大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）；L3S Research Center（L3S研究中心）
We study an online contextual decision-making problem with resource constraints. At each time period, the decision-maker first predicts a reward vector and resource consumption matrix based on a given context vector and then solves a downstream optimization problem to make a decision. The final goal of the decision-maker is to maximize the summation of the reward and the utility from resource consumption, while satisfying the resource constraints. We propose an algorithm that mixes a prediction step based on the "Smart Predict-then-Optimize (SPO)" method with a dual update step based on mirror…
