Super fast rates in structured prediction

• URL: http://arxiv.org/abs/2102.00760v1
• Date: Mon, 1 Feb 2021 10:50:04 GMT
• Title: Super fast rates in structured prediction
• Authors: Vivien Cabannes and Alessandro Rudi and Francis Bach
• Abstract summary: We show how we can leverage the fact that discrete problems are essentially predicting a discrete output when continuous problems are predicting a continuous value. We first illustrate it for predictors based on nearest neighbors, generalizing rates known for binary classification to any discrete problem within the framework of structured prediction. We then consider kernel ridge regression where we improve known rates in $n-1/4$ to arbitrarily fast rates, depending on a parameter characterizing the hardness of the problem.
• Score: 88.99819200562784
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Discrete supervised learning problems such as classification are often tackled by introducing a continuous surrogate problem akin to regression. Bounding the original error, between estimate and solution, by the surrogate error endows discrete problems with convergence rates already shown for continuous instances. Yet, current approaches do not leverage the fact that discrete problems are essentially predicting a discrete output when continuous problems are predicting a continuous value. In this paper, we tackle this issue for general structured prediction problems, opening the way to "super fast" rates, that is, convergence rates for the excess risk faster than $n^{-1}$, where $n$ is the number of observations, with even exponential rates with the strongest assumptions. We first illustrate it for predictors based on nearest neighbors, generalizing rates known for binary classification to any discrete problem within the framework of structured prediction. We then consider kernel ridge regression where we improve known rates in $n^{-1/4}$ to arbitrarily fast rates, depending on a parameter characterizing the hardness of the problem, thus allowing, under smoothness assumptions, to bypass the curse of dimensionality.

Related papers

• A Statistical Theory of Regularization-Based Continual Learning [10.899175512941053]
  We provide a statistical analysis of regularization-based continual learning on a sequence of linear regression tasks. We first derive the convergence rate for the oracle estimator obtained as if all data were available simultaneously. A byproduct of our theoretical analysis is the equivalence between early stopping and generalized $ell$-regularization.
  arXiv  Detail & Related papers  (2024-06-10T12:25:13Z)
• Faster Convergence of Stochastic Accelerated Gradient Descent under Interpolation [51.248784084461334]
  We prove new convergence rates for a generalized version of Nesterov acceleration underrho conditions. Our analysis reduces the dependence on the strong growth constant from $$ to $sqrt$ as compared to prior work.
  arXiv  Detail & Related papers  (2024-04-03T00:41:19Z)
• Breaking the Heavy-Tailed Noise Barrier in Stochastic Optimization Problems [56.86067111855056]
  We consider clipped optimization problems with heavy-tailed noise with structured density. We show that it is possible to get faster rates of convergence than $mathcalO(K-(alpha - 1)/alpha)$, when the gradients have finite moments of order. We prove that the resulting estimates have negligible bias and controllable variance.
  arXiv  Detail & Related papers  (2023-11-07T17:39:17Z)
• Distributionally Robust Optimization with Bias and Variance Reduction [9.341215359733601]
  We show that Prospect, a gradient-based algorithm, enjoys linear convergence for smooth regularized losses. We also show that Prospect can converge 2-3$times$ faster than baselines such as gradient-based methods.
  arXiv  Detail & Related papers  (2023-10-21T00:03:54Z)
• Empirical Risk Minimization with Shuffled SGD: A Primal-Dual Perspective and Improved Bounds [12.699376765058137]
  gradient descent (SGD) is perhaps the most prevalent optimization method in modern machine learning. It is only very recently that SGD with sampling without replacement -- shuffled SGD -- has been analyzed. We prove fine-grained complexity bounds that depend on the data matrix and are never worse than what is predicted by the existing bounds.
  arXiv  Detail & Related papers  (2023-06-21T18:14:44Z)
• Instance-Dependent Generalization Bounds via Optimal Transport [51.71650746285469]
  Existing generalization bounds fail to explain crucial factors that drive the generalization of modern neural networks. We derive instance-dependent generalization bounds that depend on the local Lipschitz regularity of the learned prediction function in the data space. We empirically analyze our generalization bounds for neural networks, showing that the bound values are meaningful and capture the effect of popular regularization methods during training.
  arXiv  Detail & Related papers  (2022-11-02T16:39:42Z)
• A Primal-Dual Approach to Solving Variational Inequalities with General Constraints [54.62996442406718]
  Yang et al. (2023) recently showed how to use first-order gradient methods to solve general variational inequalities. We prove the convergence of this method and show that the gap function of the last iterate of the method decreases at a rate of $O(frac1sqrtK)$ when the operator is $L$-Lipschitz and monotone.
  arXiv  Detail & Related papers  (2022-10-27T17:59:09Z)
• Robust Linear Predictions: Analyses of Uniform Concentration, Fast Rates and Model Misspecification [16.0817847880416]
  We offer a unified framework that includes a broad variety of linear prediction problems on a Hilbert space. We show that for misspecification level $epsilon$, these estimators achieve an error rate of $O(maxleft|mathcalO|1/2n-1/2, |mathcalI|1/2n-1 right+epsilon)$, matching the best-known rates in literature.
  arXiv  Detail & Related papers  (2022-01-06T08:51:08Z)
• Fast Rates for the Regret of Offline Reinforcement Learning [69.23654172273085]
  We study the regret of reinforcement learning from offline data generated by a fixed behavior policy in an infinitehorizon discounted decision process (MDP) We show that given any estimate for the optimal quality function $Q*$, the regret of the policy it defines converges at a rate given by the exponentiation of the $Q*$-estimate's pointwise convergence rate.
  arXiv  Detail & Related papers  (2021-01-31T16:17:56Z)

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.