A New Framework for High-Probability Regret Bounds in Empirical Risk Minimization
A new technical guide on arXiv establishes a unified, 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 methodology offers researchers a powerful toolkit for verifying performance guarantees across a wide range of function classes and loss functions.
The Three-Step Recipe for Standard ERM Analysis
The core of the framework organizes standard ERM rate derivations around a three-step proof strategy. First, a basic inequality relates the empirical and population risk. 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 variance-risk condition, making it broadly applicable. To translate these abstract bounds into concrete rates, 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 rates for classical 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 modern statistical settings where the loss function depends on nuisance parameters. These arise in pivotal areas like causal inference, missing data problems, and domain adaptation, often modeled via weighted ERM or Neyman-orthogonal losses.
Following the orthogonal statistical learning framework, the analysis provides regret-transfer bounds. These bounds decompose the total regret when using an estimated loss into two components: the statistical error under the estimated loss and the approximation error from nuisance estimation. Under sample splitting or cross-fitting, the statistical error term can be controlled using the standard ERM bounds, isolating the impact of nuisance estimation accuracy.
A Novel Analysis for the In-Sample Regime
As a novel contribution, the guide also treats the more challenging in-sample regime, where the nuisance components and the ERM predictor are fit on the same dataset without sample splitting. It derives new regret bounds for this setting, demonstrating that fast oracle rates—rates achievable if the nuisances were known—can remain attainable. This result holds under suitable smoothness conditions and Donsker-type conditions on the nuisance function class, providing crucial theoretical justification for practical, single-sample algorithms.
Why This Research Matters
- Unified Theoretical Toolkit: Provides a modular, three-step blueprint for deriving regret bounds, simplifying and standardizing analysis for a wide spectrum of ERM problems.
- Bridges Theory and Practice: Directly addresses the complexity of modern machine learning applications in causal and robust AI by providing guarantees for problems with nuisance parameters.
- Enables Efficient Algorithm Design: The analysis of the in-sample regime shows that fast rates are possible without sample splitting, guiding the development of more data-efficient methods.
- Connects Complexity and Rate: Explicitly links final statistical rates to fundamental complexity measures like the critical radius and metric entropy, offering deep insight into model behavior.