A New Framework for High-Probability Regret Bounds in Empirical Risk Minimization
A new technical guide, published on arXiv, establishes a unified and modular framework for deriving high-probability regret bounds in Empirical Risk Minimization (ERM). The work provides a structured "recipe" for analyzing statistical learning algorithms, extending its principles to complex scenarios involving nuisance components common in causal inference and domain adaptation. This reference aims to streamline the often-complex process of proving generalization guarantees for a wide array of machine learning models.
The Three-Step Recipe for Standard ERM Regret
The core of the framework organizes rate derivations around a three-step proof strategy applicable under mild conditions. First, a basic inequality relates the empirical and population risks. Second, a uniform local concentration bound controls the fluctuations of the empirical process. Finally, a fixed-point argument yields the final regret bound expressed in terms of a critical radius, a complexity measure defined via localized Rademacher complexity. This approach requires only a mild Bernstein-type condition relating variance and risk.
To make these abstract bounds concrete, the guide demonstrates how to upper-bound the critical radius using tools like local maximal inequalities and metric-entropy integrals. This process recovers well-known statistical learning rates for classical function classes, including VC-subgraph classes, Sobolev/Hölder classes, and bounded-variation classes, thereby validating the framework's generality and power.
Extending the Framework to Problems with Nuisance Parameters
A significant portion of the guide is dedicated to ERM in the presence of nuisance components, a common challenge in modern data science. This includes settings like weighted ERM and the use of Neyman-orthogonal losses, which are pivotal in robust causal inference, missing data problems, and domain adaptation.
Following the orthogonal statistical learning framework, the authors present regret-transfer bounds. These bounds cleverly link the regret under an estimated, data-dependent loss to the desired population regret under the target loss. The regret typically decomposes into two interpretable parts: (i) the statistical error under the estimated loss, and (ii) an approximation error stemming solely from the accuracy of the nuisance parameter estimation.
Novel Analysis for the In-Sample Regime
While sample splitting or cross-fitting can simplify analysis by isolating the two error terms, the guide makes a novel contribution by also treating the more challenging in-sample regime. Here, the nuisance parameters and the ERM predictor are fit on the same dataset, creating statistical dependencies. The authors derive new regret bounds for this setting, showing that fast oracle rates—rates as if the nuisance were known—remain attainable. This is possible under suitable smoothness conditions and Donsker-type conditions on the function class, providing crucial theoretical justification for practical single-sample algorithms.
Why This Framework Matters for Machine Learning
- Unified Proof Strategy: It provides a clear, three-step "recipe" (basic inequality, uniform concentration, fixed-point argument) that demystifies the derivation of generalization bounds for ERM, making advanced theoretical tools more accessible.
- Bridges Theory and Practice: By explicitly handling nuisance parameters via regret-transfer bounds, the framework directly addresses the complexity of real-world problems in causal inference and robust ML, connecting theoretical guarantees to practical algorithm design.
- Enables New Analyses: The novel treatment of the in-sample regime proves that efficient learning is possible without sample splitting under certain conditions, offering greater flexibility and data efficiency for practitioners.