A New Framework for High-Probability Regret Bounds in Empirical Risk Minimization
A new technical guide provides a comprehensive, modular framework for deriving high-probability regret bounds in Empirical Risk Minimization (ERM), a cornerstone of statistical machine learning. The work introduces a unifying three-step recipe for establishing these guarantees and extends the analysis to complex scenarios involving nuisance components, such as those found in causal inference and domain adaptation, offering novel results for the in-sample fitting regime.
The Three-Step Recipe for Standard ERM Guarantees
The core of the framework organizes many ERM rate derivations around a structured, three-step proof strategy. This modular approach begins with a basic inequality, proceeds to a uniform local concentration bound, and culminates in a fixed-point argument. This recipe yields regret bounds expressed in terms of a critical radius, a complexity measure defined via localized Rademacher complexity, under a mild Bernstein-type variance-risk condition.
To translate these abstract bounds into concrete, familiar rates, the guide details how to upper-bound the critical radius using tools like local maximal inequalities and metric-entropy integrals. This process recovers established statistical rates for well-known function classes, including VC-subgraph classes, Sobolev/Hölder classes, and bounded-variation classes, demonstrating the framework's broad applicability.
Extending to ERM with Nuisance Components
A significant portion of the analysis is dedicated to ERM problems involving nuisance parameters, which are common in modern statistical applications. These include methodologies like weighted ERM and Neyman-orthogonal losses, frequently employed in causal inference, missing data problems, and domain adaptation.
Following the orthogonal statistical learning framework, the guide highlights that these problems often admit regret-transfer bounds. These bounds effectively link the regret under an estimated loss function to the population regret under the target loss. The regret typically decomposes into two distinct components: (i) the statistical error under the estimated loss, and (ii) the approximation error stemming from nuisance estimation inaccuracies.
Novel Results for the In-Sample Regime
Under standard sample splitting or cross-fitting protocols, the first error term can be controlled using conventional fixed-loss ERM regret bounds, while the second depends solely on the nuisance-estimation accuracy. As a key novel contribution, the guide also treats the more challenging in-sample regime, where both the nuisances and the ERM predictor are fit on the same dataset without sample splitting.
For this setting, the authors derive new regret bounds and demonstrate that fast oracle rates—rates that match what would be achievable if the nuisance parameters were known—remain attainable. This is shown to hold under suitable smoothness conditions and Donsker-type conditions on the function class, providing important theoretical justification for practical single-sample procedures.
Why This Research Matters
- Unified Theoretical Framework: Provides a modular, "recipe-based" approach that unifies and simplifies the derivation of high-probability guarantees for ERM, a fundamental learning algorithm.
- Bridges Theory and Practice: Moves from abstract, high-level conditions to concrete rates for standard function classes, making advanced theoretical tools more accessible for applied researchers.
- Enables Complex Applications: The extended analysis for problems with nuisance components directly supports methodological advances in critical fields like causal machine learning and robust learning under distribution shift.
- Novel In-Sample Guarantees: Offers new theoretical insights for the common practice of fitting all models on the same data, showing that optimal rates can still be achieved under specific conditions, which has significant practical implications.