Stochastic Bilevel Optimization with Heavy-Tailed Noise

• URL: http://arxiv.org/abs/2509.14952v1
• Date: Thu, 18 Sep 2025 13:37:40 GMT
• Title: Stochastic Bilevel Optimization with Heavy-Tailed Noise
• Authors: Zhuanghua Liu, Luo Luo,
• Abstract summary: This paper considers the smooth bilevel optimization in which the lower problem is convex strongly and the upper-level problem is possibly non-stationary noise level.
• Score: 27.792016944321627
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper considers the smooth bilevel optimization in which the lower-level problem is strongly convex and the upper-level problem is possibly nonconvex. We focus on the stochastic setting that the algorithm can access the unbiased stochastic gradient evaluation with heavy-tailed noise, which is prevalent in many machine learning applications such as training large language models and reinforcement learning. We propose a nested-loop normalized stochastic bilevel approximation (N$^2$SBA) for finding an $\epsilon$-stationary point with the stochastic first-order oracle (SFO) complexity of $\tilde{\mathcal{O}}\big(\kappa^{\frac{7p-3}{p-1}} \sigma^{\frac{p}{p-1}} \epsilon^{-\frac{4 p - 2}{p-1}}\big)$, where $\kappa$ is the condition number, $p\in(1,2]$ is the order of central moment for the noise, and $\sigma$ is the noise level. Furthermore, we specialize our idea to solve the nonconvex-strongly-concave minimax optimization problem, achieving an $\epsilon$-stationary point with the SFO complexity of $\tilde{\mathcal O}\big(\kappa^{\frac{2p-1}{p-1}} \sigma^{\frac{p}{p-1}} \epsilon^{-\frac{3p-2}{p-1}}\big)$. All above upper bounds match the best-known results under the special case of the bounded variance setting, i.e., $p=2$.

Related papers

• Faster Gradient Methods for Highly-smooth Stochastic Bilevel Optimization [27.377966916440432]
  We show the complexity of finding an $epsilon-gradient point for bilevel optimization when the upper-level problem is nonmath and the lower-level problem is convex.<n>Recent work proposed the first-order approximation, F$2$SA, achieving the $tildemathcalO(epsilon-4)$ lower bound for firstorder smooth problems.
  arXiv  Detail & Related papers  (2025-09-03T02:02:52Z)
• Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0, L_1)$-Smoothness [74.18546828528298]
  We show that SignSGD with Majority Voting can robustly work on the whole range of complexity with $kappakappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappakappakappa-1right, kappa
  arXiv  Detail & Related papers  (2025-02-11T19:54:11Z)
• Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem [16.9187409976238]
  We study a class of convex bilevel optimization problems, also known as simple bilevel optimization. We introduce novel bilevel optimization methods that approximate the solution set of the lower-level problem.
  arXiv  Detail & Related papers  (2023-08-15T02:37:11Z)
• Accelerating Inexact HyperGradient Descent for Bilevel Optimization [84.00488779515206]
  We present a method for solving general non-strongly-concave bilevel optimization problems. Our results also improve upon the existing complexity for finding second-order stationary points in non-strongly-concave problems.
  arXiv  Detail & Related papers  (2023-06-30T20:36:44Z)
• A Newton-CG based barrier-augmented Lagrangian method for general nonconvex conic optimization [53.044526424637866]
  In this paper we consider finding an approximate second-order stationary point (SOSP) that minimizes a twice different subject general non conic optimization. In particular, we propose a Newton-CG based-augmentedconjugate method for finding an approximate SOSP.
  arXiv  Detail & Related papers  (2023-01-10T20:43:29Z)
• An Efficient Stochastic Algorithm for Decentralized Nonconvex-Strongly-Concave Minimax Optimization [25.00475462213752]
  Decentralized Recursive desc. Method (DREAM) Concretely, it requires $mathcalO(minminappaappa3eps-3,kappa2 N)$ first-order oracle (SFO) calls and $tildemathcalO(kappa2 epsilon-2) communication rounds. Our numerical experiments validate the superiority of previous methods.
  arXiv  Detail & Related papers  (2022-12-05T16:09:39Z)
• 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)
• Decentralized Stochastic Variance Reduced Extragradient Method [25.21457349137344]
  This paper studies decentralized convex-concave minimax optimization problems of the form $min_xmax_y fx,y triqfrac1msumi=1m f_i triqfrac1msumi=1m f_i triqfrac1msumi=1m f_i triqfrac1msumi=1m f_i triqfrac1msum
  arXiv  Detail & Related papers  (2022-02-01T16:06:20Z)
• A Momentum-Assisted Single-Timescale Stochastic Approximation Algorithm for Bilevel Optimization [112.59170319105971]
  We propose a new algorithm -- the Momentum- Single-timescale Approximation (MSTSA) -- for tackling problems. MSTSA allows us to control the error in iterations due to inaccurate solution to the lower level subproblem.
  arXiv  Detail & Related papers  (2021-02-15T07:10:33Z)
• Hybrid Stochastic-Deterministic Minibatch Proximal Gradient: Less-Than-Single-Pass Optimization with Nearly Optimal Generalization [83.80460802169999]
  We show that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model. For loss factors, we prove that HSDMPG can attain an $mathcalObig (1/sttnbig)$ which is at the order of excess error on a learning model.
  arXiv  Detail & Related papers  (2020-09-18T02:18:44Z)
• Streaming Complexity of SVMs [110.63976030971106]
  We study the space complexity of solving the bias-regularized SVM problem in the streaming model. We show that for both problems, for dimensions of $frac1lambdaepsilon$, one can obtain streaming algorithms with spacely smaller than $frac1lambdaepsilon$.
  arXiv  Detail & Related papers  (2020-07-07T17:10:00Z)
• Second-Order Information in Non-Convex Stochastic Optimization: Power and Limitations [54.42518331209581]
  We find an algorithm which finds. epsilon$-approximate stationary point (with $|nabla F(x)|le epsilon$) using. $(epsilon,gamma)$surimate random random points. Our lower bounds here are novel even in the noiseless case.
  arXiv  Detail & Related papers  (2020-06-24T04:41:43Z)
• Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems [36.645753881826955]
  In this paper, we propose a novel method called RecurEnti Ascent (SREDA), which estimates more efficiently using variance. This method achieves the best known for this problem.
  arXiv  Detail & Related papers  (2020-01-11T09:05:03Z)

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.