Learning to Generalize Provably in Learning to Optimize (2302.11085v2)

Abstract: Learning to optimize (L2O) has gained increasing popularity, which automates the design of optimizers by data-driven approaches. However, current L2O methods often suffer from poor generalization performance in at least two folds: (i) applying the L2O-learned optimizer to unseen optimizees, in terms of lowering their loss function values (optimizer generalization, or generalizable learning of optimizers"); and (ii) the test performance of an optimizee (itself as a machine learning model), trained by the optimizer, in terms of the accuracy over unseen data (optimizee generalization, orlearning to generalize"). While the optimizer generalization has been recently studied, the optimizee generalization (or learning to generalize) has not been rigorously studied in the L2O context, which is the aim of this paper. We first theoretically establish an implicit connection between the local entropy and the Hessian, and hence unify their roles in the handcrafted design of generalizable optimizers as equivalent metrics of the landscape flatness of loss functions. We then propose to incorporate these two metrics as flatness-aware regularizers into the L2O framework in order to meta-train optimizers to learn to generalize, and theoretically show that such generalization ability can be learned during the L2O meta-training process and then transformed to the optimizee loss function. Extensive experiments consistently validate the effectiveness of our proposals with substantially improved generalization on multiple sophisticated L2O models and diverse optimizees. Our code is available at: https://github.com/VITA-Group/Open-L2O/tree/main/Model_Free_L2O/L2O-Entropy.

• The paper demonstrates how flatness-aware regularizers can provably enhance optimizee generalization by linking local entropy to Hessian metrics.
• It introduces a meta-training framework that minimizes the spectral norm of the Hessian and maximizes local entropy to achieve flatter loss landscapes.
• Empirical results show significant test accuracy improvements across diverse models, confirming the practical utility of the proposed method.

Insightful Overview of "Learning to Generalize Provably in Learning to Optimize"

The paper "Learning to Generalize Provably in Learning to Optimize" addresses a significant gap in the growing field of Learning to Optimize (L2O). While previous research has primarily focused on creating data-driven optimizers that can reduce objective functions efficiently, the generalization ability of these learned optimizers has not been rigorously studied, particularly concerning optimizee generalization. This paper aims to fill this void by investigating how an L2O framework can be designed to enhance the generalization of the optimizee, which reflects the model's performance on unseen test data.

Theoretical Connections

A notable theoretical contribution of this paper is the establishment of an implicit connection between local entropy and Hessian metrics. The authors theoretically demonstrate that Hessian, a common metric for landscape flatness, is bounded by a function of local entropy. By minimizing the negative local entropy, one can implicitly control the Hessian, thereby favoring solutions with flatter landscapes. This connection underpins the subsequent development of flatness-aware regularizers within the L2O framework.

Proposed Methodology: Flatness-Aware Regularizers

The authors propose incorporating two specific flatness-aware regularizers—based on Hessian and local entropy—into the meta-training of L2O optimizers. The Hessian regularizer aims to minimize the spectral norm of the Hessian, while the entropy regularizer focuses on maximizing local entropy metrics. These regularizers are integrated into the L2O training process to ensure that the resulting optimizers produce models with enhanced generalization on unseen data. The meta-training process is theoretically grounded to demonstrate that this generalization can be effectively learned and applied to improve the optimizee's testing performance.

Empirical Validation and Numerical Results

Extensive experiments conducted across various tasks validate the efficacy of the proposed approach. When integrated into existing state-of-the-art L2O models, the flatness-aware regularizers consistently improve both optimizer and optimizee generalization abilities. Notably, the proposed regularizers yield substantial test accuracy improvements, showcasing their practical utility on diverse models. This paper provides comprehensive empirical evidence supporting its theoretical claims, demonstrating numerical improvements that strengthen its contributions to the L2O literature.

Implications and Future Directions

The implications of this research are multifaceted. Practically, it provides a robust methodology for improving the generalization of models trained via L2O, making the application of these optimizers feasible in real-world scenarios. Theoretically, the connection between local entropy and Hessian opens new avenues for research into optimization landscapes and generalization in neural networks.

Future work could explore more computationally efficient approaches to estimate the proposed regularizers, given the computational overhead of calculating Hessians and local entropy. Additionally, extending this work to other optimization contexts beyond machine learning, where generalization of solutions is critical, may yield further impactful insights.

In summary, this paper contributes to the body of L2O literature by addressing optimizee generalization through innovative theoretical and practical techniques. Its methodology holds promise for enhancing generalization in machine learning models, supported by strong theoretical foundations and validated by experimental results.