Core Concepts

The paper provides uniform and information-theoretic bounds on the tilted generalization error of the tilted empirical risk minimization, with convergence rates of O(1/√n) for bounded loss functions and unbounded loss functions with bounded second moments.

Abstract

The paper studies the generalization error of the tilted empirical risk minimization (TERM) framework proposed by Li et al. (2021). The key contributions are:

- Uniform and information-theoretic bounds on the tilted generalization error for bounded loss functions, showing a convergence rate of O(1/√n).
- Uniform and information-theoretic bounds on the tilted generalization error for unbounded loss functions with bounded second moments, also showing a convergence rate of O(1/√n).
- Analysis of the robustness of TERM under distribution shift induced by noise or outliers in the training data for unbounded loss functions with bounded second moments.
- Study of the KL-regularized TERM problem, deriving an upper bound on the expected tilted generalization error with a convergence rate of O(1/n).

The paper provides theoretical justification for the empirical success of the TERM framework in handling class imbalance, mitigating the effect of outliers, and enabling fairness between subgroups.

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arxiv.org

Stats

The loss function ℓ(h, z) is bounded between 0 and M for all h in the hypothesis space H and z in the instance space Z.
The expected population risk under the true distribution μ is bounded by Ru_T.
The second moment of the loss function under the true distribution μ is bounded by Mu_2.
The expected population risk under the distribution shift ̃μ is bounded by Rs_T.
The second moment of the loss function under the distribution shift ̃μ is bounded by Ms_2.

Quotes

"The generalization error (risk) of a supervised statistical learning algorithm quantifies its prediction ability on previously unseen data."
"Inspired by exponential tilting, Li et al. (2021) proposed the tilted empirical risk as a non-linear risk metric for machine learning applications such as classification and regression problems."
"We provide uniform and information-theoretic bounds on the tilted generalization error, defined as the difference between the population risk and the tilted empirical risk, with a convergence rate of O(1/√n) where n is the number of training samples."
"We study the solution to the KL-regularized expected tilted empirical risk minimization problem and derive an upper bound on the expected tilted generalization error with a convergence rate of O(1/n)."

Key Insights Distilled From

by Gholamali Am... at **arxiv.org** 10-01-2024

Deeper Inquiries

The generalization error analysis for positive tilting (γ > 0) in the context of tilted empirical risk minimization (TERM) can be extended by focusing on the characteristics of unbounded loss functions while ensuring that certain conditions are met. One approach is to relax the boundedness assumption on the loss function, as seen in the analysis for negative tilting. This involves establishing conditions under which the second moment of the loss function is bounded, which is crucial for deriving meaningful generalization bounds.
To achieve this, researchers can utilize techniques from the PAC-Bayesian framework and information-theoretic bounds, which have been effective in analyzing generalization errors under various conditions. Specifically, the analysis can leverage the properties of the tilted population risk and the tilted empirical risk, ensuring that the expected population risk remains finite. By employing tools such as Bernstein's inequality and the concentration of measure, one can derive upper and lower bounds on the tilted generalization error for positive tilting.
Moreover, it is essential to consider the implications of the choice of the tilting parameter γ. As γ approaches zero, the tilted empirical risk converges to the linear empirical risk, allowing for a smoother transition between the two frameworks. This convergence can be exploited to derive bounds that are asymptotically valid as the sample size increases, thus providing a robust framework for analyzing generalization errors in scenarios with unbounded loss functions.

The trade-off between robustness and generalization in the tilted empirical risk minimization (TERM) framework is a critical consideration, particularly when dealing with noisy data or outliers. In scenarios where the training dataset is affected by distributional shifts, such as the presence of outliers, the choice of the tilting parameter γ plays a pivotal role in balancing these two aspects.
When a negative tilt (γ < 0) is employed, TERM demonstrates robustness against noise and outliers, as it effectively down-weights the influence of these problematic samples on the learning process. This robustness is achieved at the potential cost of generalization, as the model may become overly conservative, leading to underfitting in the presence of clean data. Conversely, using a positive tilt (γ > 0) can enhance generalization by allowing the model to focus more on the majority class or the more prevalent patterns in the data. However, this may also increase sensitivity to outliers, thereby compromising robustness.
The implications of this trade-off are significant for practitioners. They must carefully select the tilting parameter γ based on the specific characteristics of their dataset and the desired outcomes. A well-chosen γ can lead to improved performance in terms of both robustness and generalization, while a poor choice may result in a model that either fails to generalize well to unseen data or is overly influenced by noise. Therefore, understanding this trade-off is essential for effectively applying TERM in real-world scenarios.

Tilted empirical risk minimization (TERM) offers a promising approach to tackle the problem of class imbalance in real-world datasets, which is a common challenge in machine learning. Class imbalance occurs when certain classes are underrepresented in the training data, leading to biased models that perform poorly on minority classes. TERM addresses this issue through its inherent design, which allows for the adjustment of the learning process based on the distribution of the classes.
By employing a positive tilt (γ > 0), TERM can be configured to give more weight to the minority class during the training phase. This is achieved by modifying the empirical risk function to emphasize the loss incurred on underrepresented classes, effectively tilting the risk landscape. As a result, the model becomes more sensitive to the minority class, improving its predictive performance on these instances.
Furthermore, TERM can incorporate fairness constraints, ensuring that the model not only performs well on the majority class but also maintains equitable performance across all classes. This is particularly important in applications where fairness is a critical concern, such as in healthcare or criminal justice.
In practice, implementing TERM for class imbalance involves selecting an appropriate tilting parameter γ that reflects the degree of imbalance in the dataset. Techniques such as cross-validation can be employed to optimize this parameter, ensuring that the model achieves a balance between generalization and robustness. By leveraging the capabilities of TERM, practitioners can develop more effective models that are better equipped to handle class imbalance, ultimately leading to improved outcomes in various applications.

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