Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes

• URL: http://arxiv.org/abs/2504.04105v2
• Date: Fri, 18 Apr 2025 03:35:46 GMT
• Title: Minimax Optimal Convergence of Gradient Descent in Logistic Regression via Large and Adaptive Stepsizes
• Authors: Ruiqi Zhang, Jingfeng Wu, Licong Lin, Peter L. Bartlett,
• Abstract summary: We study $textitgradient descent$ (GD) for logistic regression on linearly separable data with stepsizes that adapt to the current risk.<n>We show that GD achieves a risk upper bounded by $exp(-Theta(eta))$, where $gamma$ is the margin of the dataset.<n> Notably, the classical $textitPerceptron$ (Novikoff, 1962), a first-order online method, also achieves a step complexity of $1/gamma2$, matching GD even in constants.
• Score: 32.64968378005787
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study $\textit{gradient descent}$ (GD) for logistic regression on linearly separable data with stepsizes that adapt to the current risk, scaled by a constant hyperparameter $\eta$. We show that after at most $1/\gamma^2$ burn-in steps, GD achieves a risk upper bounded by $\exp(-\Theta(\eta))$, where $\gamma$ is the margin of the dataset. As $\eta$ can be arbitrarily large, GD attains an arbitrarily small risk $\textit{immediately after the burn-in steps}$, though the risk evolution may be $\textit{non-monotonic}$. We further construct hard datasets with margin $\gamma$, where any batch (or online) first-order method requires $\Omega(1/\gamma^2)$ steps to find a linear separator. Thus, GD with large, adaptive stepsizes is $\textit{minimax optimal}$ among first-order batch methods. Notably, the classical $\textit{Perceptron}$ (Novikoff, 1962), a first-order online method, also achieves a step complexity of $1/\gamma^2$, matching GD even in constants. Finally, our GD analysis extends to a broad class of loss functions and certain two-layer networks.

Related papers

• Fast Last-Iterate Convergence of SGD in the Smooth Interpolation Regime [26.711510824243803]
  We study population convergence guarantees of gradient descent (SGD) for smooth convex objectives in the regime, where the noise at optimum is zero or near zero.<n>For a well-tuned stepsize we obtain a near optimal $widetildeO (1/T + sigma_star/sqrtT)$ rate for the last iterate.
  arXiv  Detail & Related papers  (2025-07-15T12:52:47Z)
• Large Stepsize Gradient Descent for Logistic Loss: Non-Monotonicity of the Loss Improves Optimization Efficiency [47.8739414267201]
  We consider gradient descent (GD) with a constant stepsize applied to logistic regression with linearly separable data. We show that GD exits this initial oscillatory phase rapidly -- in $mathcalO(eta)$ steps -- and subsequently achieves an $tildemathcalO (1 / (eta t) )$ convergence rate. Our results imply that, given a budget of $T$ steps, GD can achieve an accelerated loss of $tildemathcalO (1/T2)$ with an aggressive stepsize
  arXiv  Detail & Related papers  (2024-02-24T23:10:28Z)
• Scaling Up Differentially Private LASSO Regularized Logistic Regression via Faster Frank-Wolfe Iterations [51.14495595270775]
  We adapt the Frank-Wolfe algorithm for $L_1$ penalized linear regression to be aware of sparse inputs and to use them effectively. Our results demonstrate that this procedure can reduce runtime by a factor of up to $2,200times$, depending on the value of the privacy parameter $epsilon$ and the sparsity of the dataset.
  arXiv  Detail & Related papers  (2023-10-30T19:52:43Z)
• Differentially Private Stochastic Gradient Descent with Low-Noise [49.981789906200035]
  Modern machine learning algorithms aim to extract fine-grained information from data to provide accurate predictions, which often conflicts with the goal of privacy protection. This paper addresses the practical and theoretical importance of developing privacy-preserving machine learning algorithms that ensure good performance while preserving privacy.
  arXiv  Detail & Related papers  (2022-09-09T08:54:13Z)
• Best Policy Identification in Linear MDPs [70.57916977441262]
  We investigate the problem of best identification in discounted linear Markov+Delta Decision in the fixed confidence setting under a generative model. The lower bound as the solution of an intricate non- optimization program can be used as the starting point to devise such algorithms.
  arXiv  Detail & Related papers  (2022-08-11T04:12:50Z)
• High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation [89.21686761957383]
  We study the first gradient descent step on the first-layer parameters $boldsymbolW$ in a two-layer network. Our results demonstrate that even one step can lead to a considerable advantage over random features.
  arXiv  Detail & Related papers  (2022-05-03T12:09:59Z)
• A Variance-Reduced Stochastic Accelerated Primal Dual Algorithm [3.2958527541557525]
  Such problems arise frequently in machine learning in the context of robust empirical risk minimization. We consider the accelerated primal dual (SAPD) algorithm as a robust method against gradient noise. We show that our method improves upon SAPD both in practice and in theory.
  arXiv  Detail & Related papers  (2022-02-19T22:12:30Z)
• Continuous Submodular Maximization: Boosting via Non-oblivious Function [12.755674710719616]
  In this paper, we revisit the constrained and continuous submodular iterations in both offline and online settings. We use the factorrevealing optimization equation to derive an optimal auxiliary function $F$ for problem $max_boldsymbolxinmathCf(boldsymbolx)$. In the online setting, we propose boosting a gradient feedback algorithm, achieving a regret of $sqrtD$ (where $D$ is the sum of delays of gradient feedback against $(fracgamma2)$ adversarial.
  arXiv  Detail & Related papers  (2022-01-03T15:10:17Z)
• A first-order primal-dual method with adaptivity to local smoothness [64.62056765216386]
  We consider the problem of finding a saddle point for the convex-concave objective $min_x max_y f(x) + langle Ax, yrangle - g*(y)$, where $f$ is a convex function with locally Lipschitz gradient and $g$ is convex and possibly non-smooth. We propose an adaptive version of the Condat-Vu algorithm, which alternates between primal gradient steps and dual steps.
  arXiv  Detail & Related papers  (2021-10-28T14:19:30Z)
• High Dimensional Differentially Private Stochastic Optimization with Heavy-tailed Data [8.55881355051688]
  We provide the first study on the problem of DP-SCO with heavy-tailed data in the high dimensional space. We show that if the loss function is smooth and its gradient has bounded second order moment, it is possible to get a (high probability) error bound (excess population risk) of $tildeO(fraclog d(nepsilon)frac13)$ in the $epsilon$-DP model. In the second part of the paper, we study sparse learning with heavy-tailed data.
  arXiv  Detail & Related papers  (2021-07-23T11:03:21Z)
• Private Stochastic Convex Optimization: Optimal Rates in $\ell_1$ Geometry [69.24618367447101]
  Up to logarithmic factors the optimal excess population loss of any $(varepsilon,delta)$-differently private is $sqrtlog(d)/n + sqrtd/varepsilon n.$ We show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $sqrtlog(d)/n + (log(d)/varepsilon n)2/3.
  arXiv  Detail & Related papers  (2021-03-02T06:53:44Z)
• Gradient-Based Empirical Risk Minimization using Local Polynomial Regression [39.29885444997579]
  A major goal of this literature has been to compare different algorithms, such as gradient descent (GD) or gradient descent (SGD) We demonstrate that when the loss function is smooth in the data, we can learn the oracle at every iteration and beat the oracle complexities of both GD and SGD.
  arXiv  Detail & Related papers  (2020-11-04T20:10:31Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)

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.