A New Framework for High-Probability Regret Bounds in Empirical Risk Minimization
A new technical guide, published on arXiv, establishes a comprehensive and modular framework for deriving high-probability regret bounds in Empirical Risk Minimization (ERM). The work provides a unified "recipe" for analyzing statistical learning algorithms, extending its reach to complex scenarios involving nuisance components common in causal inference and domain adaptation. This systematic approach promises to streamline theoretical analysis across a wide spectrum of machine learning problems.
The Three-Step Recipe for Standard ERM Analysis
The authors organize the derivation of ERM convergence rates around a core three-step strategy. This modular process begins with establishing a basic inequality, followed by applying a uniform local concentration bound, and concludes with a fixed-point argument. This methodology yields regret bounds expressed in terms of a critical radius, a complexity measure defined via localized Rademacher complexity, under a mild Bernstein-type condition linking variance and risk.
To translate these abstract bounds into concrete, familiar rates, the guide demonstrates how to upper-bound the critical radius using tools like local maximal inequalities and metric-entropy integrals. This process recovers established statistical rates for several fundamental function classes, including VC-subgraph classes, Sobolev/Hölder classes, and bounded-variation classes.
Extending the Framework to Problems with Nuisance Parameters
A significant portion of the guide is dedicated to ERM in settings with nuisance components, such as weighted ERM and losses designed with Neyman orthogonality. These problems are ubiquitous in applied fields like causal inference, missing data, and domain adaptation. The analysis is framed within the orthogonal statistical learning paradigm.
The key insight is that these problems often admit regret-transfer bounds. These bounds decompose the overall regret into two distinct components: the statistical error under the estimated loss function and the approximation error stemming from inaccuracies in nuisance parameter estimation. When using sample splitting or cross-fitting techniques, the first component can be controlled using standard ERM bounds, while the second depends solely on the convergence rate of the nuisance estimator.
Novel Analysis for the In-Sample Regime
As a novel contribution, the guide also tackles the more challenging in-sample regime, where both the nuisance parameters and the primary ERM model are trained on the same dataset. The authors derive new regret bounds for this setting, demonstrating that fast oracle rates—rates that match the performance if the nuisance components were known—remain attainable. This result holds under suitable smoothness conditions and Donsker-type conditions on the function class, providing crucial theoretical justification for practical single-sample algorithms.
Why This Research Matters
- Unified Theoretical Framework: Provides a standardized, modular "recipe" for deriving high-probability guarantees, simplifying and unifying analysis across diverse ERM problems.
- Bridges Theory and Practice: Extends rigorous regret bounds to modern, complex learning scenarios involving nuisance parameters, which are central to causal machine learning and robust AI.
- Enables Efficient Single-Sample Algorithms: The novel in-sample analysis shows that fast convergence is possible without sample splitting, offering theoretical support for more data-efficient methodologies.
- Connects Abstract and Concrete Results: Systematically shows how abstract complexity measures translate into concrete, well-known statistical rates for standard function classes.