Optimal Implicit Bias in Linear Regression

• URL: http://arxiv.org/abs/2506.17187v1
• Date: Fri, 20 Jun 2025 17:41:39 GMT
• Title: Optimal Implicit Bias in Linear Regression
• Authors: Kanumuri Nithin Varma, Babak Hassibi,
• Abstract summary: We find the optimal implicit bias that leads to the best generalization performance.<n>In particular, we obtain a tight lower bound on the best generalization error possible among this class of interpolators.
• Score: 20.710343135282116
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Most modern learning problems are over-parameterized, where the number of learnable parameters is much greater than the number of training data points. In this over-parameterized regime, the training loss typically has infinitely many global optima that completely interpolate the data with varying generalization performance. The particular global optimum we converge to depends on the implicit bias of the optimization algorithm. The question we address in this paper is, ``What is the implicit bias that leads to the best generalization performance?". To find the optimal implicit bias, we provide a precise asymptotic analysis of the generalization performance of interpolators obtained from the minimization of convex functions/potentials for over-parameterized linear regression with non-isotropic Gaussian data. In particular, we obtain a tight lower bound on the best generalization error possible among this class of interpolators in terms of the over-parameterization ratio, the variance of the noise in the labels, the eigenspectrum of the data covariance, and the underlying distribution of the parameter to be estimated. Finally, we find the optimal convex implicit bias that achieves this lower bound under certain sufficient conditions involving the log-concavity of the distribution of a Gaussian convolved with the prior of the true underlying parameter.

Related papers

• Asymptotics of Linear Regression with Linearly Dependent Data [28.005935031887038]
  We study the computations of linear regression in settings with non-Gaussian covariates.<n>We show how dependencies influence estimation error and the choice of regularization parameters.
  arXiv  Detail & Related papers  (2024-12-04T20:31:47Z)
• Differentially Private Optimization with Sparse Gradients [60.853074897282625]
  We study differentially private (DP) optimization problems under sparsity of individual gradients. Building on this, we obtain pure- and approximate-DP algorithms with almost optimal rates for convex optimization with sparse gradients.
  arXiv  Detail & Related papers  (2024-04-16T20:01:10Z)
• Smoothing the Edges: Smooth Optimization for Sparse Regularization using Hadamard Overparametrization [10.009748368458409]
  We present a framework for smooth optimization of explicitly regularized objectives for (structured) sparsity. Our method enables fully differentiable approximation-free optimization and is thus compatible with the ubiquitous gradient descent paradigm in deep learning.
  arXiv  Detail & Related papers  (2023-07-07T13:06:12Z)
• Bayesian Analysis for Over-parameterized Linear Model via Effective Spectra [6.9060054915724]
  We introduce a data-adaptive Gaussian prior that targets the data's intrinsic complexity rather than its ambient dimension.<n>We establish contraction rates of the corresponding posterior distribution, which reveal how the mass in the spectrum affects the prediction error bounds.<n>Our findings demonstrate that Bayesian methods leveraging spectral information of the data are effective for estimation in non-sparse, high-dimensional settings.
  arXiv  Detail & Related papers  (2023-05-25T06:07:47Z)
• 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)
• Implicit differentiation for fast hyperparameter selection in non-smooth convex learning [87.60600646105696]
  We study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian.
  arXiv  Detail & Related papers  (2021-05-04T17:31:28Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Fundamental Limits of Ridge-Regularized Empirical Risk Minimization in High Dimensions [41.7567932118769]
  Empirical Risk Minimization algorithms are widely used in a variety of estimation and prediction tasks. In this paper, we characterize for the first time the fundamental limits on the statistical accuracy of convex ERM for inference.
  arXiv  Detail & Related papers  (2020-06-16T04:27:38Z)
• Asymptotics of Ridge (less) Regression under General Source Condition [26.618200633139256]
  We consider the role played by the structure of the true regression parameter. We show that (no regularisation) can be optimal even with bounded signal-to-noise ratio (SNR) This contrasts with previous work considering ridge regression with isotropic prior, in which case is only optimal in the limit of infinite SNR.
  arXiv  Detail & Related papers  (2020-06-11T13:00:21Z)
• Implicit differentiation of Lasso-type models for hyperparameter optimization [82.73138686390514]
  We introduce an efficient implicit differentiation algorithm, without matrix inversion, tailored for Lasso-type problems. Our approach scales to high-dimensional data by leveraging the sparsity of the solutions.
  arXiv  Detail & Related papers  (2020-02-20T18:43:42Z)
• Support recovery and sup-norm convergence rates for sparse pivotal estimation [79.13844065776928]
  In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. We show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators.
  arXiv  Detail & Related papers  (2020-01-15T16:11:04Z)

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.