Distributionally Robust Logistic Regression (1509.09259v3)

Abstract: This paper proposes a distributionally robust approach to logistic regression. We use the Wasserstein distance to construct a ball in the space of probability distributions centered at the uniform distribution on the training samples. If the radius of this ball is chosen judiciously, we can guarantee that it contains the unknown data-generating distribution with high confidence. We then formulate a distributionally robust logistic regression model that minimizes a worst-case expected logloss function, where the worst case is taken over all distributions in the Wasserstein ball. We prove that this optimization problem admits a tractable reformulation and encapsulates the classical as well as the popular regularized logistic regression problems as special cases. We further propose a distributionally robust approach based on Wasserstein balls to compute upper and lower confidence bounds on the misclassification probability of the resulting classifier. These bounds are given by the optimal values of two highly tractable linear programs. We validate our theoretical out-of-sample guarantees through simulated and empirical experiments.

• The paper introduces a distributionally robust logistic regression model that optimizes worst-case logloss by constructing a Wasserstein ball around the empirical distribution.
• It offers theoretical guarantees for out-of-sample performance by linking regularization to distributionally robust optimization through measure concentration.
• The approach leads to a tractable reformulation via linear programming, allowing precise confidence bounds on misclassification risk.

Insights on "Distributionally Robust Logistic Regression"

The paper "Distributionally Robust Logistic Regression" by Soroosh Shafieezadeh-Abadeh, Peyman Mohajerin Esfahani, and Daniel Kuhn explores the application of distributionally robust optimization (DRO) to enhance the logistic regression model's performance, specifically targeting its susceptibility to overfitting when trained on sparse datasets. The authors utilize the Wasserstein metric to formulate ambiguity sets, which are central to the proposed framework.

Key Contributions

The paper introduces a distributionally robust logistic regression model by constructing a Wasserstein ball centered on the empirical distribution of training samples. The primary focus is to optimize the worst-case expected logloss function, where the worst-case scenarios are contemplated over all probability distributions within this Wasserstein ball. The authors achieve this through a computationally attractive and solvable optimization problem that encompasses classical and regularized logistic regression models as special cases. Key contributions from the paper include:

1. Tractable Reformulation: The authors provide a reformulation of the robust optimization problem that is computationally manageable. This reformulation leverages the dual norm of feature space to encapsulate regularization effects.
2. Theoretical Guarantees: By integrating measure concentration results, the paper offers probabilistic guarantees for the out-of-sample performance of the logistic regression classifier. This ensures that the solution derived is reliable beyond the training dataset.
3. Risk Estimation: The paper introduces methods to compute confidence bounds on the misclassification risk using distributionally robust techniques. These bounds are obtained through linear programming, adding further analytical depth to decision-makers who use the model.
4. Probabilistic Interpretation: An innovative aspect of this research lies in connecting regularization terms often used as ad hoc methods to probabilistic interpretations through the lens of DRO. The regularization coefficient is effectively linked to the radius of the Wasserstein ball, providing a meaningful interpretation of its impact on model robustness.

Technical Novelty and Implications

The use of the Wasserstein metric for constructing ambiguity sets is significant, as it takes advantage of measure concentration properties to create a robust framework. This choice allows for a seamless transition from empirical risk minimization to robust optimization, thus providing a spectrum of solutions that regularization parameters can navigate.

The implications of this research extend to both theory and practice. Theoretically, this work bridges a gap between regularization phenomena and robust optimization interpretations, offering a fresh perspective on the utilization of DRO in logistic regression. From a practical standpoint, the distributionally robust approach not only addresses overfitting but also provides insights into the reliability of predictions through risk assessment tools like worst-case misclassification probability bounds.

Future Research Directions

The model and findings presented pave the way for several future research avenues. One potential direction is the exploration of alternative uncertainty measures beyond the Wasserstein metric to construct ambiguity sets, potentially offering varying robustness characteristics. Another area is the extension of this robust framework to multiclass classification problems or to other types of generalized linear models, which would augment the applicability of the techniques proposed.

In conclusion, this paper contributes a robust and theoretically-grounded advancement to logistic regression modeling by integrating sophisticated distributionally robust optimization methodologies. The findings and methodologies not only enhance logistic regression's practical robustness but also deepen the theoretical understanding of model regularization through the DRO framework.