Statistical Query Lower Bounds for Smoothed Agnostic Learning
Not on the map. Why?
Abstract (arXiv)
We study the complexity of smoothed agnostic learning, recently introduced by~citeCKKMS24, in which the learner competes with the best classifier in a target class under slight Gaussian perturbations of the inputs. Specifically, we focus on the prototypical task of agnostically learning halfspaces under subgaussian distributions in the smoothed model. The best known upper bound for this problem relies on L1-polynomial regression and has complexity d^tildeO(1/σ2) log(1/ε), where σ is the smoothing parameter and ε is the excess error. Our main result is a Statistical Query (SQ) lower bound providing formal evidence that this upper bound is close to best possible. In more detail, we show that (even for Gaussian marginals) any SQ algorithm for smoothed agnostic learning of halfspaces requires complexity d^Omega(1/σ2+log(1/ε)). This is the first non-trivial lower bound on the complexity of this task and nearly matches the known upper bound. Roughly speaking, we show that applying L1-polynomial regression to a smoothed version of the function is essentially best possible. Our techniques involve finding a moment-matching hard distribution by way of linear programming duality. This dual program corresponds exactly to finding a low-degree approximating polynomial to the smoothed version of the target function (which turns out to be the same condition required for the L1-polynomial regression to work). Our explicit SQ lower bound then comes from proving lower bounds on this approximation degree for the class of halfspaces.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 18 of its 41 references that are in the map. First posted on arXiv 24 Feb 2026.
Builds on
18 of 41 references.
References · 18
- 2016Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixturescited by 114
- 2020Near-Optimal SQ Lower Bounds for Agnostically Learning Halfspaces and ReLUs under Gaussian Marginalscited by 57
- 2021The Optimality of Polynomial Regression for Agnostic Learning under Gaussian Marginalscited by 52
- 2015Complexity Theoretic Limitations on Learning Halfspacescited by 50
- 2020Statistical-Query Lower Bounds via Functional Gradientscited by 50
- 2017Learning Geometric Concepts with Nasty Noisecited by 46
- 2015Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimizationcited by 41
- 2023Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginalscited by 31
- 2020Non-Convex SGD Learns Halfspaces with Adversarial Label Noisecited by 30
- 2016A General Characterization of the Statistical Query Complexitycited by 26
- 2022Cryptographic Hardness of Learning Halfspaces with Massart Noisecited by 25
- 2021Agnostic Proper Learning of Halfspaces under Gaussian Marginalscited by 21
6 more
- 2022Hardness of Agnostically Learning Halfspaces from Worst-Case Lattice Problemscited by 16
- 2024SQ Lower Bounds for Non-Gaussian Component Analysis with Weaker Assumptionscited by 13
- 2024Smoothed Analysis for Learning Concepts with Low Intrinsic Dimensioncited by 11
- 2025Algorithms and SQ Lower Bounds for Robustly Learning Real-valued Multi-index Modelscited by 5
- 2025Constructive Approximation under Carleman's Condition, with Applications to Smoothed Analysiscited by 3
- 2025Smoothed Agnostic Learning of Halfspaces over the Hypercubecited by 2
Built on by
Citing papers per year: 2026: 1.
1 paper.