[Paper Review] Towards Optimal Problem Dependent Generalization Error Bounds in Statistical Learning Theory

Yunbei Xu, Assaf Zeevi|arXiv (Cornell University)|Nov 12, 2020

Sparse and Compressive Sensing Techniques80 references4 citations

TL;DR

This paper introduces a novel framework called 'uniform localized convergence' to derive sharp, problem-dependent generalization error bounds in statistical learning. It achieves optimal variance- and loss-dependent rates in both slow and fast rate regimes, enabling moment-penalized estimators and iterative algorithms like gradient descent to attain near-optimal finite-sample performance across non-convex learning, stochastic optimization, and missing data problems.

ABSTRACT

We study problem-dependent rates, i.e., generalization errors that scale near-optimally with the variance, the effective loss, or the gradient norms evaluated at the "best hypothesis." We introduce a principled framework dubbed "uniform localized convergence," and characterize sharp problem-dependent rates for central statistical learning problems. From a methodological viewpoint, our framework resolves several fundamental limitations of existing uniform convergence and localization analysis approaches. It also provides improvements and some level of unification in the study of localized complexities, one-sided uniform inequalities, and sample-based iterative algorithms. In the so-called "slow rate" regime, we provides the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general "rich" classes; we also establish improved loss-dependent rate for standard empirical risk minimization. In the "fast rate" regime, we establish finite-sample problem-dependent bounds that are comparable to precise asymptotics. In addition, we show that iterative algorithms like gradient descent and first-order Expectation-Maximization can achieve optimal generalization error in several representative problems across the areas of non-convex learning, stochastic optimization, and learning with missing data.

Motivation & Objective

• To address the lack of optimal problem-dependent generalization error bounds for 'rich' hypothesis classes in statistical learning theory.
• To overcome limitations of traditional uniform convergence and localization methods, such as local Rademacher complexity, which yield suboptimal sample size dependence.
• To unify and improve existing approaches to localized complexities, one-sided uniform inequalities, and regularization in iterative algorithms.
• To establish finite-sample, problem-dependent bounds in both slow and fast rate regimes that match asymptotic precision.
• To demonstrate that iterative algorithms like gradient descent and first-order EM can achieve optimal generalization error in non-convex and missing data settings.

Proposed method

• Proposes a new framework, 'uniform localized convergence,' which uses adaptive truncation levels and concentration over 'rings' to localize convergence around the optimal hypothesis.
• Employs a flexible surrogate function and concentrated function approach to derive uniform inequalities that depend on problem-specific parameters like variance and gradient norms.
• Introduces a moment-penalized estimator that achieves optimal variance-dependent rates in the slow rate regime for general rich classes.
• Applies one-sided uniform concentration inequalities to analyze the behavior of empirical processes near the optimal hypothesis, avoiding worst-case uniform bounds.
• Uses localized complexity measures and ring-based concentration to avoid geometric restrictions on the hypothesis class.
• Establishes finite-sample bounds in the parametric fast rate regime that match precise asymptotic behavior, even under zero curvature conditions (e.g., Huber loss).

Experimental results

Research questions

• RQ1 Can we derive generalization error bounds that are both optimal in sample size and sharp in problem-dependent parameters like variance and gradient norms?
• RQ2 How can we unify and improve upon existing localization techniques such as local Rademacher complexity and uniform convergence of gradients?
• RQ3 Can iterative algorithms like gradient descent and first-order EM achieve optimal problem-dependent generalization error in non-convex and missing data problems?
• RQ4 What is the role of adaptive truncation and ring-based concentration in achieving tighter, localized convergence bounds?
• RQ5 Can the proposed framework yield optimal rates for rich hypothesis classes where traditional methods fail?

Key findings

• The paper presents the first moment-penalized estimator that achieves optimal variance-dependent generalization error rates for general 'rich' classes in the slow rate regime.
• It establishes improved loss-dependent rates for standard empirical risk minimization by leveraging problem-specific loss structure.
• In the fast rate regime, the framework provides finite-sample bounds that are comparable in precision to asymptotic results, even under weak curvature conditions.
• Iterative algorithms such as gradient descent and first-order Expectation-Maximization are shown to achieve optimal problem-dependent generalization error in non-convex learning and missing data problems.
• The framework unifies localized complexities, one-sided uniform inequalities, and vector-based convergence results under a single principled approach.
• The method avoids geometric assumptions on the hypothesis class by using adaptive truncation and ring-based concentration, enabling broader applicability than prior localization methods.