We propose in this paper a general framework for deriving loss functions for structured prediction. Inourframework,theuserchooses aconvexsetincluding the output space and provides an oracle forprojectingonto that set.

Fenchel-Young losses are a family of loss functions, encompassing the squared,logistic and sparsemax losses, among others. They are convex w.r.t. the modeloutput and the target, separately. Each Fenchel-Young loss is implicitly associatedwith a link function, that maps model outputs to predictions. For instance, thelogistic loss is associated with the soft argmax link function. Can we build newloss functions associated with the same link function as Fenchel-Young losses?In this paper, we introduce Fitzpatrick losses, a new family of separately convexloss functions based on the Fitzpatrick function. A well-known theoretical tool inmaximal monotone operator theory, the Fitzpatrick function naturally leads to arefined Fenchel-Young inequality, making Fitzpatrick losses tighter than Fenchel-Young losses, while maintaining the same link function for prediction. As anexample, we introduce the Fitzpatrick logistic loss and the Fitzpatrick sparsemaxloss, counterparts of the logistic and the sparsemax losses. This yields two newtighter losses associated with the soft argmax and the sparse argmax, two of themost ubiquitous output layers used in machine learning. We study in details theproperties of Fitzpatrick losses and, in particular, we show that they can be seen asFenchel-Young losses using a modified, target-dependent generating function.

Online structured prediction, including online classification as a special case, is the task of sequentially predicting labels from input features. Therein the surrogate regret -- the cumulative excess of the target loss (e.g., 0-1 loss) over the surrogate loss (e.g., logistic loss) of the fixed best estimator -- has gained attention, particularly because it often admits a finite bound independent of the time horizon $T$. However, such guarantees break down in non-stationary environments, where every fixed estimator may incur the surrogate loss growing linearly with $T$. We address this by proving a bound of the form $F_T + C(1 + P_T)$ on the cumulative target loss, where $F_T$ is the cumulative surrogate loss of any comparator sequence, $P_T$ is its path length, and $C > 0$ is some constant. This bound depends on $T$ only through $F_T$ and $P_T$, often yielding much stronger guarantees in non-stationary environments. Our core idea is to synthesize the dynamic regret bound of the online gradient descent (OGD) with the technique of exploiting the surrogate gap. Our analysis also sheds light on a new Polyak-style learning rate for OGD, which systematically offers target-loss guarantees and exhibits promising empirical performance. We further extend our approach to a broader class of problems via the convolutional Fenchel--Young loss. Finally, we prove a lower bound showing that the dependence on $F_T$ and $P_T$ is tight.

Fenchel-Young losses are a family of loss functions, encompassing the squared,logistic and sparsemax losses, among others. They are convex w.r.t. the modeloutput and the target, separately. Each Fenchel-Young loss is implicitly associatedwith a link function, that maps model outputs to predictions. For instance, thelogistic loss is associated with the soft argmax link function. Can we build newloss functions associated with the same link function as Fenchel-Young losses?In this paper, we introduce Fitzpatrick losses, a new family of separately convexloss functions based on the Fitzpatrick function.

Estimating parameters from samples of an optimal probability distribution is essential in applications ranging from socio-economic modeling to biological system analysis. In these settings, the probability distribution arises as the solution to an optimization problem that captures either static interactions among agents or the dynamic evolution of a system over time. Our approach relies on minimizing a new class of loss functions, called sharpened Fenchel-Young losses, which measure the sub-optimality gap of the optimization problem over the space of measures. We study the stability of this estimation method when only a finite number of sample is available. The parameters to be estimated typically correspond to a cost function in static problems and to a potential function in dynamic problems. To analyze stability, we introduce a general methodology that leverages the strong convexity of the loss function together with the sample complexity of the forward optimization problem. Our analysis emphasizes two specific settings in the context of optimal transport, where our method provides explicit stability guarantees: The first is inverse unbalanced optimal transport (iUOT) with entropic regularization, where the parameters to estimate are cost functions that govern transport computations; this method has applications such as link prediction in machine learning. The second is inverse gradient flow (iJKO), where the objective is to recover a potential function that drives the evolution of a probability distribution via the Jordan-Kinderlehrer-Otto (JKO) time-discretization scheme; this is particularly relevant for understanding cell population dynamics in single-cell genomics. Finally, we validate our approach through numerical experiments on Gaussian distributions, where closed-form solutions are available, to demonstrate the practical performance of our methods

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical risk, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel-Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.

Key insights from the studies Arjevani and Field (2022); Chizat, Oyallon, and Bach (2018); Du, Zhai, P oczos, and Singh (2018); Yun, Sra, and Jadbabaie (2018) emphasize the pivotal role of over-parameterization in finding the global optimum and enhancing the generalization ability of deep neural networks (DNNs). Recent work has shown that the evolution of the trainable parameters in continuous-width DNNs during training can be captured by the neural tangent kernel (NTK) Arora, Du, Hu, Li, and Wang (2019); Du, Lee, Li, Wang, and Zhai (2018); Jacot, Gabriel, and Hongler (2018); Mohamadi, Bae, and Sutherland (2023); Wang, Li, and Sun (2023); Zou, Cao, Zhou, and Gu (2018). An alternative research direction attempts to examine the infinite-width neural network from a mean-field perspective (Chizat & Bach, 2018; Mei, Montanari, & Nguyen, 2018; Nguyen & Pham, 2023; Sirignano & Spiliopoulos, 2018). However, in practical applications, neural networks are of finite width, and under this condition, it remains unclear whether NTK theory and mean-field theory can adequately characterize the convergence properties of neural networks Seleznova and Kutyniok (2021). Therefore, the mechanisms of non-convex optimization in deep learning, and the impact of over-parameterization on model training, remain incompletely resolved.

Data-driven inverse optimization seeks to estimate unknown parameters in an optimization model from observations of optimization solutions. Many existing methods are ineffective in handling noisy and suboptimal solution observations and also suffer from computational challenges. In this paper, we build a connection between inverse optimization and the Fenchel-Young (FY) loss originally designed for structured prediction, proposing a FY loss approach to data-driven inverse optimization. This new approach is amenable to efficient gradient-based optimization, hence much more efficient than existing methods. We provide theoretical guarantees for the proposed method and use extensive simulation and real-data experiments to demonstrate its significant advantage in parameter estimation accuracy, decision error and computational speed.

From a variational perspective, many statistical learning criteria involve seeking a distribution that balances empirical risk and regularization. In this paper, we broaden this perspective by introducing a new general class of variational methods based on Fenchel-Young (FY) losses, treated as divergences that generalize (and encompass) the familiar Kullback-Leibler divergence at the core of classical variational learning. Our proposed formulation -- FY variational learning -- includes as key ingredients new notions of FY free energy, FY evidence, FY evidence lower bound, and FY posterior. We derive alternating minimization and gradient backpropagation algorithms to compute (or lower bound) the FY evidence, which enables learning a wider class of models than previous variational formulations. This leads to generalized FY variants of classical algorithms, such as an FY expectation-maximization (FYEM) algorithm, and latent-variable models, such as an FY variational autoencoder (FYVAE). Our new methods are shown to be empirically competitive, often outperforming their classical counterparts, and most importantly, to have qualitatively novel features. For example, FYEM has an adaptively sparse E-step, while the FYVAE can support models with sparse observations and sparse posteriors.

The gradient descent (GD) has been one of the most common optimizer in machine learning. In particular, the loss landscape of a neural network is typically sharpened during the initial phase of training, making the training dynamics hover on the edge of stability. This is beyond our standard understanding of GD convergence in the stable regime where arbitrarily chosen stepsize is sufficiently smaller than the edge of stability. Recently, Wu et al. (COLT2024) have showed that GD converges with arbitrary stepsize under linearly separable logistic regression. Although their analysis hinges on the self-bounding property of the logistic loss, which seems to be a cornerstone to establish a modified descent lemma, our pilot study shows that other loss functions without the self-bounding property can make GD converge with arbitrary stepsize. To further understand what property of a loss function matters in GD, we aim to show arbitrary-stepsize GD convergence for a general loss function based on the framework of \emph{Fenchel--Young losses}. We essentially leverage the classical perceptron argument to derive the convergence rate for achieving $\epsilon$-optimal loss, which is possible for a majority of Fenchel--Young losses. Among typical loss functions, the Tsallis entropy achieves the GD convergence rate $T=\Omega(\epsilon^{-1/2})$, and the R{\'e}nyi entropy achieves the far better rate $T=\Omega(\epsilon^{-1/3})$. We argue that these better rate is possible because of \emph{separation margin} of loss functions, instead of the self-bounding property.