More data speeds up training time in learning halfspaces over sparse vectors

2013/11/10 by Amit Daniely, Nati Linial, Daniely, Amit +4 · 8 citations
Computer Science · Engineering · #Algorithms and Data Compression #Blind Source Separation Techniques #FOS: Computer and information sciences #Face and Expression Recognition #Imbalanced Data Classification Techniques #Machine Learning (cs.LG) #Machine Learning and Algorithms #Sparse and Compressive Sensing Techniques #cs.LG

paper · pdf · doi:10.48550/arxiv.1311.2271

13 pages

arxiv created 2013/11/10 · openalex publication_date 2013/11/10 · arxiv updated 2013/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The increased availability of data in recent years has led several authors to ask whether it is possible to use data as a \em computational resource. That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task? We give the first positive answer to this question for a \em natural supervised learning problem --- we consider agnostic PAC learning of halfspaces over 3-sparse vectors in \-1,1,0\ⁿ. This class is inefficiently learnable using O(n/ε²) examples. Our main contribution is a novel, non-cryptographic, methodology for establishing computational-statistical gaps, which allows us to show that, under a widely believed assumption that refuting random 3CNF formulas is hard, it is impossible to efficiently learn this class using only O(n/ε²) examples. We further show that under stronger hardness assumptions, even O(n^1.499/ε²) examples do not suffice. On the other hand, we show a new algorithm that learns this class efficiently using Ω(n²/ε²) examples. This formally establishes the tradeoff between sample and computational complexity for a natural supervised learning problem.

Cited by

• Strongly Refuting Random CSPs Below the Spectral Threshold
• Agnostic Learning by Refuting
• Sum-of-Squares Lower Bounds for Sparse PCA
• Improper Reinforcement Learning with Gradient-based Policy Optimization
• Learning Using 1-Local Membership Queries
• Computational-Statistical Tradeoffs from NP-hardness
• How to refute a random CSP
• From average case complexity to improper learning complexity

Related

• Complexity Theoretic Limitations on Learning Halfspaces
• Reliable Learning of Halfspaces under Gaussian Marginals
• Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise
• From average case complexity to improper learning complexity
• Sample-Optimal PAC Learning of Halfspaces with Malicious Noise