Learning Intersections of Halfspaces with Distribution Shift: Improved Algorithms and SQ Lower Bounds

• URL: http://arxiv.org/abs/2404.02364v2
• Date: Mon, 20 May 2024 18:25:35 GMT
• Title: Learning Intersections of Halfspaces with Distribution Shift: Improved Algorithms and SQ Lower Bounds
• Authors: Adam R. Klivans, Konstantinos Stavropoulos, Arsen Vasilyan,
• Abstract summary: Klivans, Stavropoulos, and Vasilyan initiated the study of testable learning with distribution shift. Their model deviates from all prior work in that no assumptions are made on $mathcalD'$.
• Score: 9.036777309376697
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent work of Klivans, Stavropoulos, and Vasilyan initiated the study of testable learning with distribution shift (TDS learning), where a learner is given labeled samples from training distribution $\mathcal{D}$, unlabeled samples from test distribution $\mathcal{D}'$, and the goal is to output a classifier with low error on $\mathcal{D}'$ whenever the training samples pass a corresponding test. Their model deviates from all prior work in that no assumptions are made on $\mathcal{D}'$. Instead, the test must accept (with high probability) when the marginals of the training and test distributions are equal. Here we focus on the fundamental case of intersections of halfspaces with respect to Gaussian training distributions and prove a variety of new upper bounds including a $2^{(k/\epsilon)^{O(1)}} \mathsf{poly}(d)$-time algorithm for TDS learning intersections of $k$ homogeneous halfspaces to accuracy $\epsilon$ (prior work achieved $d^{(k/\epsilon)^{O(1)}}$). We work under the mild assumption that the Gaussian training distribution contains at least an $\epsilon$ fraction of both positive and negative examples ($\epsilon$-balanced). We also prove the first set of SQ lower-bounds for any TDS learning problem and show (1) the $\epsilon$-balanced assumption is necessary for $\mathsf{poly}(d,1/\epsilon)$-time TDS learning for a single halfspace and (2) a $d^{\tilde{\Omega}(\log 1/\epsilon)}$ lower bound for the intersection of two general halfspaces, even with the $\epsilon$-balanced assumption. Our techniques significantly expand the toolkit for TDS learning. We use dimension reduction and coverings to give efficient algorithms for computing a localized version of discrepancy distance, a key metric from the domain adaptation literature.

Related papers

• Testable Learning with Distribution Shift [9.036777309376697]
  We define a new model called testable learning with distribution shift. We obtain provably efficient algorithms for certifying the performance of a classifier on a test distribution. We give several positive results for learning concept classes such as halfspaces, intersections of halfspaces, and decision trees.
  arXiv  Detail & Related papers  (2023-11-25T23:57:45Z)
• Near-Optimal Bounds for Learning Gaussian Halfspaces with Random Classification Noise [50.64137465792738]
  We show that any efficient SQ algorithm for the problem requires sample complexity at least $Omega(d1/2/(maxp, epsilon)2)$. Our lower bound suggests that this quadratic dependence on $1/epsilon$ is inherent for efficient algorithms.
  arXiv  Detail & Related papers  (2023-07-13T18:59:28Z)
• Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models [91.22420505636006]
  This paper deals with the problem of efficient sampling from a differential equation, given the drift function and the diffusion matrix. It is possible to obtain independent and identically distributed (i.i.d.) samples at precision $varepsilon$ with a cost that is $m2 d log (1/varepsilon)$ Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.
  arXiv  Detail & Related papers  (2023-03-30T02:50:49Z)
• Efficient Testable Learning of Halfspaces with Adversarial Label Noise [44.32410859776227]
  In the recently introduced testable learning model, one is required to produce a tester-learner such that if the data passes the tester, then one can trust the output of the robust learner on the data. Our tester-learner runs in time $poly(d/eps)$ and outputs a halfspace with misclassification error $O(opt)+eps$, where $opt$ is the 0-1 error of the best fitting halfspace.
  arXiv  Detail & Related papers  (2023-03-09T18:38:46Z)
• Near-Optimal Cryptographic Hardness of Agnostically Learning Halfspaces and ReLU Regression under Gaussian Marginals [43.0867217287089]
  We study the task of agnostically learning halfspaces under the Gaussian distribution. We prove a near-optimal computational hardness result for this task.
  arXiv  Detail & Related papers  (2023-02-13T16:46:23Z)
• Near Sample-Optimal Reduction-based Policy Learning for Average Reward MDP [58.13930707612128]
  This work considers the sample complexity of obtaining an $varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP) We prove an upper bound of $widetilde O(H varepsilon-3 ln frac1delta)$ samples per state-action pair, where $H := sp(h*)$ is the span of bias of any optimal policy, $varepsilon$ is the accuracy and $delta$ is the failure probability.
  arXiv  Detail & Related papers  (2022-12-01T15:57:58Z)
• Testing distributional assumptions of learning algorithms [5.204779946147061]
  We study the design of tester-learner pairs $(mathcalA,mathcalT)$. We show that if the distribution on examples in the data passes the tester $mathcalT$ then one can safely trust the output of the agnostic $mathcalA$ on the data.
  arXiv  Detail & Related papers  (2022-04-14T19:10:53Z)
• Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures [9.053430799456587]
  We study the problem of learning nonparametric distributions in a finite mixture. We establish tight bounds on the sample complexity for learning the component distributions in such models.
  arXiv  Detail & Related papers  (2022-03-28T23:53:48Z)
• Threshold Phenomena in Learning Halfspaces with Massart Noise [56.01192577666607]
  We study the problem of PAC learning halfspaces on $mathbbRd$ with Massart noise under Gaussian marginals. Our results qualitatively characterize the complexity of learning halfspaces in the Massart model.
  arXiv  Detail & Related papers  (2021-08-19T16:16:48Z)
• Hardness of Learning Halfspaces with Massart Noise [56.98280399449707]
  We study the complexity of PAC learning halfspaces in the presence of Massart (bounded) noise. We show that there an exponential gap between the information-theoretically optimal error and the best error that can be achieved by a SQ algorithm.
  arXiv  Detail & Related papers  (2020-12-17T16:43:11Z)
• Learning Halfspaces with Tsybakov Noise [50.659479930171585]
  We study the learnability of halfspaces in the presence of Tsybakov noise. We give an algorithm that achieves misclassification error $epsilon$ with respect to the true halfspace.
  arXiv  Detail & Related papers  (2020-06-11T14:25:02Z)

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.