The Nyström method for convex loss functions

• URL: http://arxiv.org/abs/2006.10016v4
• Date: Fri, 14 Mar 2025 17:16:59 GMT
• Title: The Nyström method for convex loss functions
• Authors: Andrea Della Vecchia, Ernesto De Vito, Jaouad Mourtada, Lorenzo Rosasco,
• Abstract summary: We investigate an extension of classical empirical risk computation, where the hypothesis space consists of a random subspace within a given Hilbert space.<n>Using random subspaces naturally leads to computational advantages, but a key question is whether it compromises the learning accuracy.
• Score: 15.389608666270817
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We investigate an extension of classical empirical risk minimization, where the hypothesis space consists of a random subspace within a given Hilbert space. Specifically, we examine the Nystr\"om method where the subspaces are defined by a random subset of the data. This approach recovers Nystr\"om approximations used in kernel methods as a specific case. Using random subspaces naturally leads to computational advantages, but a key question is whether it compromises the learning accuracy. Recently, the tradeoffs between statistics and computation have been explored for the square loss and self-concordant losses, such as the logistic loss. In this paper, we extend these analyses to general convex Lipschitz losses, which may lack smoothness, such as the hinge loss used in support vector machines. Our main results show the existence of various scenarios where computational gains can be achieved without sacrificing learning performance. When specialized to smooth loss functions, our analysis recovers most previous results. Moreover, it allows to consider classification problems and translate the surrogate risk bounds into classification error bounds. Indeed, this gives the opportunity to compare the effect of Nystr\"om approximations when combined with different loss functions such as the hinge or the square loss.

Related papers

• Refined Risk Bounds for Unbounded Losses via Transductive Priors [58.967816314671296]
  We revisit the sequential variants of linear regression with the squared loss, classification problems with hinge loss, and logistic regression. Our key tools are based on the exponential weights algorithm with carefully chosen transductive priors.
  arXiv  Detail & Related papers  (2024-10-29T00:01:04Z)
• EnsLoss: Stochastic Calibrated Loss Ensembles for Preventing Overfitting in Classification [1.3778851745408134]
  We propose a novel ensemble method, namely EnsLoss, to combine loss functions within the Empirical risk minimization framework. We first transform the CC conditions of losses into loss-derivatives, thereby bypassing the need for explicit loss functions. We theoretically establish the statistical consistency of our approach and provide insights into its benefits.
  arXiv  Detail & Related papers  (2024-09-02T02:40:42Z)
• On the Performance of Empirical Risk Minimization with Smoothed Data [59.3428024282545]
  Empirical Risk Minimization (ERM) is able to achieve sublinear error whenever a class is learnable with iid data. We show that ERM is able to achieve sublinear error whenever a class is learnable with iid data.
  arXiv  Detail & Related papers  (2024-02-22T21:55:41Z)
• On the Dynamics Under the Unhinged Loss and Beyond [104.49565602940699]
  We introduce the unhinged loss, a concise loss function, that offers more mathematical opportunities to analyze closed-form dynamics. The unhinged loss allows for considering more practical techniques, such as time-vary learning rates and feature normalization.
  arXiv  Detail & Related papers  (2023-12-13T02:11:07Z)
• Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels [57.46832672991433]
  We propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS) We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We develop an expectation-propagation expectation-maximization algorithm for efficient posterior inference and function estimation.
  arXiv  Detail & Related papers  (2023-10-09T03:55:09Z)
• Random Smoothing Regularization in Kernel Gradient Descent Learning [24.383121157277007]
  We present a framework for random smoothing regularization that can adaptively learn a wide range of ground truth functions belonging to the classical Sobolev spaces. Our estimator can adapt to the structural assumptions of the underlying data and avoid the curse of dimensionality.
  arXiv  Detail & Related papers  (2023-05-05T13:37:34Z)
• Cross-Entropy Loss Functions: Theoretical Analysis and Applications [27.3569897539488]
  We present a theoretical analysis of a broad family of loss functions, that includes cross-entropy (or logistic loss), generalized cross-entropy, the mean absolute error and other cross-entropy-like loss functions. We show that these loss functions are beneficial in the adversarial setting by proving that they admit $H$-consistency bounds. This leads to new adversarial robustness algorithms that consist of minimizing a regularized smooth adversarial comp-sum loss.
  arXiv  Detail & Related papers  (2023-04-14T17:58:23Z)
• Kernel-based off-policy estimation without overlap: Instance optimality beyond semiparametric efficiency [53.90687548731265]
  We study optimal procedures for estimating a linear functional based on observational data. For any convex and symmetric function class $mathcalF$, we derive a non-asymptotic local minimax bound on the mean-squared error.
  arXiv  Detail & Related papers  (2023-01-16T02:57:37Z)
• Regularized ERM on random subspaces [17.927376388967144]
  We consider possibly data dependent subspaces spanned by a random subset of the data, recovering as a special case Nystrom approaches for kernel methods. Considering random subspaces naturally leads to computational savings, but the question is whether the corresponding learning accuracy is degraded.
  arXiv  Detail & Related papers  (2022-12-04T16:12:11Z)
• On the Benefits of Large Learning Rates for Kernel Methods [110.03020563291788]
  We show that a phenomenon can be precisely characterized in the context of kernel methods. We consider the minimization of a quadratic objective in a separable Hilbert space, and show that with early stopping, the choice of learning rate influences the spectral decomposition of the obtained solution.
  arXiv  Detail & Related papers  (2022-02-28T13:01:04Z)
• Smoothed Embeddings for Certified Few-Shot Learning [63.68667303948808]
  We extend randomized smoothing to few-shot learning models that map inputs to normalized embeddings. Our results are confirmed by experiments on different datasets.
  arXiv  Detail & Related papers  (2022-02-02T18:19:04Z)
• More is Less: Inducing Sparsity via Overparameterization [2.885175627590247]
  In deep learning it is common to over parameterize neural networks, that is, to use more parameters than training samples. Quite surprisingly, generalize the neural network via (stochastic) gradient descent leads to that very well. Our proof relies on analyzing a certain Bregman divergence of the flow.
  arXiv  Detail & Related papers  (2021-12-21T07:55:55Z)
• Rethinking preventing class-collapsing in metric learning with margin-based losses [81.22825616879936]
  Metric learning seeks embeddings where visually similar instances are close and dissimilar instances are apart. margin-based losses tend to project all samples of a class onto a single point in the embedding space. We propose a simple modification to the embedding losses such that each sample selects its nearest same-class counterpart in a batch.
  arXiv  Detail & Related papers  (2020-06-09T09:59:25Z)
• Learning from Non-Random Data in Hilbert Spaces: An Optimal Recovery Perspective [12.674428374982547]
  We consider the regression problem from an Optimal Recovery perspective. We first develop a semidefinite program for calculating the worst-case error of any recovery map in finite-dimensional Hilbert spaces. We show that Optimal Recovery provides a formula which is user-friendly from an algorithmic point-of-view.
  arXiv  Detail & Related papers  (2020-06-05T21:49:07Z)
• Classification vs regression in overparameterized regimes: Does the loss function matter? [21.75115239010008]
  We show that solutions obtained by least-squares minimum-norm, typically used for regression, are identical to those produced by the hard-margin support vector machine (SVM) Our results demonstrate the very different roles and properties of loss functions used at the training phase (optimization) and the testing phase (generalization)
  arXiv  Detail & Related papers  (2020-05-16T17:58:25Z)
• Nonlinear Functional Output Regression: a Dictionary Approach [1.8160945635344528]
  We introduce projection learning (PL), a novel dictionary-based approach that learns to predict a function that is expanded on a dictionary. PL minimizes an empirical risk based on a functional loss. PL enjoys a particularily attractive trade-off between computational cost and performances.
  arXiv  Detail & Related papers  (2020-03-03T10:31:17Z)
• Supervised Learning: No Loss No Cry [51.07683542418145]
  Supervised learning requires the specification of a loss function to minimise. This paper revisits the sc SLIsotron algorithm of Kakade et al. (2011) through a novel lens. We show how it provides a principled procedure for learning the loss.
  arXiv  Detail & Related papers  (2020-02-10T05:30:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.