Generalization and Robustness of the Tilted Empirical Risk (2409.19431v3)

Abstract: The generalization error (risk) of a supervised statistical learning algorithm quantifies its prediction ability on previously unseen data. Inspired by exponential tilting, \citet{li2020tilted} proposed the {\it tilted empirical risk} (TER) as a non-linear risk metric for machine learning applications such as classification and regression problems. In this work, we examine the generalization error of the tilted empirical risk in the robustness regime under \textit{negative tilt}. Our first contribution is to provide uniform and information-theoretic bounds on the {\it tilted generalization error}, defined as the difference between the population risk and the tilted empirical risk, under negative tilt for unbounded loss function under bounded $(1+\epsilon)$-th moment of loss function for some $\epsilon\in(0,1]$ with a convergence rate of $O(n^{-\epsilon/(1+\epsilon)})$ where $n$ is the number of training samples, revealing a novel application for TER under no distribution shift. Secondly, we study the robustness of the tilted empirical risk with respect to noisy outliers at training time and provide theoretical guarantees under distribution shift for the tilted empirical risk. We empirically corroborate our findings in simple experimental setups where we evaluate our bounds to select the value of tilt in a data-driven manner.

• The paper presents uniform and information-theoretic bounds on tilted generalization error, achieving convergence rates of O(1/√n) for bounded loss functions.
• It demonstrates TERM’s practical robustness by analyzing negative tilt settings that handle unbounded loss functions and mitigate distribution shifts.
• The study extends to KL-regularized TERM by deriving tighter bounds for the tilted Gibbs posterior, achieving a convergence rate of O(1/n) and improved model complexity control.

An Analysis of the Generalization Error of Tilted Empirical Risk

The paper "Generalization Error of the Tilted Empirical Risk" provides a comprehensive analysis of the generalization error, or risk, associated with the Tilted Empirical Risk Minimization (TERM) framework. This work is particularly motivated by the successes demonstrated by Li et al. (2020) in handling class imbalance, mitigating the effects of outliers, and enhancing fairness in supervised machine learning tasks through the application of TERM. The primary contribution of the paper lies in providing uniform and information-theoretic bounds on the generalization error of the tilted empirical risk (TER), alongside exploring its practical robustness and examining the KL-regularized TERM problem.

Key Contributions and Theoretical Framework

The analysis in this paper is framed around the concept of tilted generalization error, $\mathrm{gen}_{\gamma}(h,S)$, defined as the difference between the population risk and the TER. Several critical contributions are facilitated through this lens, including:

1. Uniform and Information-Theoretic Bounds:
   • For Bounded Loss Functions: The paper presents upper bounds on the tilted generalization error, achieving convergence rates of $O(1/\sqrt{n})$. This is accomplished through both uniform and information-theoretic approaches for bounded loss functions. Results are showcased using the theoretical frameworks of VC-dimension and Rademacher complexity.
   • For Unbounded Loss Functions: When handling unbounded loss functions, the analysis is confined to negative tilt ($\gamma < 0$) scenarios with bounded second-moment assumptions. The authors derive similar convergence rates of $O(1/\sqrt{n})$ under these relaxed conditions.
2. Robustness Under Distribution Shift: The authors investigate the robustness of TERM under conditions where the training dataset distribution is shifted due to noise or outliers. They demonstrate how TER, particularly with negative tilt, can be robust to such adverse scenarios, reflecting practical implications for real-world noisy and imbalanced datasets.
3. KL-Regularized TERM: The paper also extends its analysis to KL-regularized TERM, deriving an upper bound on the expected tilted generalization error of the tilted Gibbs posterior, achieving an impressive convergence rate of $O(1/n)$. This part of the paper asserts the benefit of regularizing the TER through KL divergence for better control over the learning algorithm's complexity.

Detailed Insights into Results and Implications

Uniform Bounds for Bounded Loss Functions

For bounded loss functions, the TER is analyzed using uniform bounds based on the variance of the exponential of the loss (Proposition 3.1). The main theoretical result provides that the tilted generalization error converges at a rate of $O(1/\sqrt{n})$. The theorems address both the uniform upper and lower bounds, illustrating the impact of the tilt parameter $\gamma$ and its influence on the generalization performance.

Information-Theoretic Bounds

Deploying information-theoretic tools, the authors establish bounds leveraging mutual information between the hypothesis and the dataset. These bounds emphasize the role of information measures in characterizing the expected tilted generalization error. The theorems provide nuanced comparisons between different tilt parameters and indicate that the convergence rate remains $O(1/\sqrt{n})$ for bounded loss scenarios.

Analysis Under Unbounded Loss Functions

For scenarios involving unbounded loss functions, results are constrained to negative tilts. The analysis shows that even under less stringent conditions, such as bounded second moments, the TER maintains a convergence rate of $O(1/\sqrt{n})$. This portion of the paper provides a critical extension of TERM's applicability in broader and more practical contexts where the assumption of bounded loss may not hold.

Robustness to Distribution Shift

This section examines TER's robustness by modeling the distributional shift due to noise or outliers. The results showcase that TER, particularly with negative tilt, offers robustness against such adverse conditions, providing theoretical justification for TERM’s empirical successes reported by Li et al. (2020).

KL-Regularization and Tilted Gibbs Posterior

The exploration of KL-regularized TERM reveals that the tilted Gibbs posterior serves as an optimal solution in minimizing a regularized form of TER. The regularization helps in controlling the complexity of the hypothesis space, ensuring tighter bounds on the generalization error. This analysis further highlights potential paths for integrating TERM with Bayesian inference frameworks to achieve enhanced learning performance with rigorous theoretical backing.

Future Directions and Practical Implications

The implications of this research are multifaceted, bridging theory with practice. Future directions include extending the analysis to positive tilts under unbounded loss settings and exploring tighter boundaries when $\gamma \to \infty$. The theoretical groundwork laid by this paper facilitates the development of robust and generalizable machine learning models, particularly in domains beset with imbalanced and noisy data distributions. Further investigation into TERM within federated learning environments and its intersections with other regularization techniques presents compelling avenues for extending the practical utility of this framework.

Overall, the paper contributes significant theoretical insights into the generalization properties of TERM, cementing its potential as a robust alternative in supervised learning tasks that demand resilience against data irregularities and imbalances.