Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our loss results in unconstrained convex objective functions, supports infinite (or very large) class spaces, and naturally defines a geometric generalization of the softmax operator. The geometric properties of this loss make it suitable for predicting sparse and singular distributions, for instance supported on curves or hyper-surfaces. We study the theoretical properties of our loss and show-case its effectiveness on two applications: ordinal regression and drawing generation.
Distributionally robust optimization (DRO) is a widely-used approach to learn models that are robust against distribution shift. Compared with the standard optimization setting, the objective function in DRO is more difficult to optimize, and most of the existing theoretical results make strong assumptions on the loss function. In this work we bridge the gap by studying DRO algorithms for general smooth non-convex losses. By carefully exploiting the specific form of the DRO objective, we are able to provide non-asymptotic convergence guarantees even though the objective function is possibly no…
Online Learning on Hidden-Convex Losses via Algorithmic Equivalence: Optimal Regret, Geometric Barrier, and Bandit Feedback
Singapore University of Technology and Design（新加坡科技设计大学）；National Technical University of Athens（雅典国家技术大学）；National and Kapodistrian University of Athens（雅典国家与卡多斯大学）；University of California, Irvine（加州大学 Irvine 分校）；Archimedes, Athena Research Center, Greece（希腊阿提卡研究中心 Archimedes）；National University of Singapore, Centre for Quantum Technologies（新加坡国立大学 量子技术中心）
A Geometric Approach to Constrained Online Learning
Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur（计算机科学与工程系，印度理工学院Kharagpur分校）；School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India（技术与计算机科学学院，印度基础研究科学研究院，孟买，印度）
There is growing evidence that converting targets to soft targets in supervised learning can provide considerable gains in performance. Much of this work has considered classification, converting hard zero-one values to soft labels---such as by adding label noise, incorporating label ambiguity or using distillation. In parallel, there is some evidence from a regression setting in reinforcement learning that learning distributions can improve performance. In this work, we investigate the reasons for this improvement, in a regression setting. We introduce a novel distributional regression loss, …
Image hashing is a principled approximate nearest neighbor approach to find similar items to a query in a large collection of images. Hashing aims to learn a binary-output function that maps an image to a binary vector. For optimal retrieval performance, producing balanced hash codes with low-quantization error to bridge the gap between the learning stage's continuous relaxation and the inference stage's discrete quantization is important. However, in the existing deep supervised hashing methods, coding balance and low-quantization error are difficult to achieve and involve several losses. We …
This thesis presents some geometric insights into three different types of two player prediction games -- namely general learning task, prediction with expert advice, and online convex optimization. These games differ in the nature of the opponent (stochastic, adversarial, or intermediate), the order of the players' move, and the utility function. The insights shed some light on the understanding of the intrinsic barriers of the prediction problems and the design of computationally efficient learning algorithms with strong theoretical guarantees (such as generalizability, statistical consisten…
Relative Wasserstein Angle and the Problem of the $W_2$-Nearest Gaussian Distribution
Department of Computer Science, George Washington University, Washington D.C., USA（计算机科学系，乔治·华盛顿大学，华盛顿特区，美国）；MAE, George Washington University, Washington D.C., USA（机械工程与应用科学系，乔治·华盛顿大学，华盛顿特区，美国）
Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization
Graduate School of Information Science and Technology, Hokkaido University, Japan（信息科学与技术研究生学校，北海道大学，日本）；Bioinformatics Center, Kyoto University, Japan（生物信息学中心，京都大学，日本）；Institute of Information Technology, Vietnam Academy of Science and Technology, Vietnam（信息技术研究所，越南科学与技术学院，越南）
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…
In this work we establish the equivalence of algorithmic regularization and explicit convex penalization for generic convex losses. We introduce a geometric condition for the optimization path of a convex function, and show that if such a condition is satisfied, the optimization path of an iterative algorithm on the unregularized optimization problem can be represented as the solution path of a corresponding penalized problem.
The critical locus of the loss function of a neural network is determined by the geometry of the functional space and by the parameterization of this space by the network's weights. We introduce a natural distinction between pure critical points, which only depend on the functional space, and spurious critical points, which arise from the parameterization. We apply this perspective to revisit and extend the literature on the loss function of linear neural networks. For this type of network, the functional space is either the set of all linear maps from input to output space, or a determinantal…
