Tighter Learning Guarantees on Digital Computers via Concentration of Measure on Finite Spaces

2024-02-08

AI Summary

In high-dimensional, small-sample regimes, conventional machine learning generalization bounds suffer from excessive looseness due to the “curse of dimensionality.” Method: This paper proposes an adaptive family of generalization bounds tailored to discretized Euclidean spaces. We first derive a non-asymptotic concentration inequality for finite metric spaces; introduce geometric representation dimension (m) as a pivotal parameter; and construct bounds with constant factor (c_m) scaling as (sqrt{m}). Tight analysis is achieved via metric embedding combined with discretization-based modeling. Contribution/Results: The proposed bounds yield significant tightening under practical sample sizes; retain the optimal (O(1/sqrt{N})) convergence rate; and break the exponential or polynomial dependence on ambient dimension inherent in classical bounds—achieving constant-factor improvement even in high-dimensional, low-precision settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Machine learning models with inputs in a Euclidean space $mathbb{R}^d$, when implemented on digital computers, generalize, and their generalization gap converges to $0$ at a rate of $c/N^{1/2}$ concerning the sample size $N$. However, the constant $c>0$ obtained through classical methods can be large in terms of the ambient dimension $d$ and machine precision, posing a challenge when $N$ is small to realistically large. In this paper, we derive a family of generalization bounds ${c_m/N^{1/(2vee m)}}_{m=1}^{infty}$ tailored for learning models on digital computers, which adapt to both the sample size $N$ and the so-called geometric representation dimension $m$ of the discrete learning problem. Adjusting the parameter $m$ according to $N$ results in significantly tighter generalization bounds for practical sample sizes $N$, while setting $m$ small maintains the optimal dimension-free worst-case rate of $mathcal{O}(1/N^{1/2})$. Notably, $c_{m}in mathcal{O}(m^{1/2})$ for learning models on discretized Euclidean domains. Furthermore, our adaptive generalization bounds are formulated based on our new non-asymptotic result for concentration of measure in finite metric spaces, established via leveraging metric embedding arguments.

Problem

• High-Dimensional Data
• Efficiency
• Reliability

Innovation

• Adaptive Adjustment
• Limited Data Space
• High-Dimensional Data Minimization