Better-than-KL PAC-Bayes Bounds (2402.09201v2)

Abstract: Let $f(\theta, X_1),$ $ \dots,$ $ f(\theta, X_n)$ be a sequence of random elements, where $f$ is a fixed scalar function, $X_1, \dots, X_n$ are independent random variables (data), and $\theta$ is a random parameter distributed according to some data-dependent posterior distribution $P_n$. In this paper, we consider the problem of proving concentration inequalities to estimate the mean of the sequence. An example of such a problem is the estimation of the generalization error of some predictor trained by a stochastic algorithm, such as a neural network where $f$ is a loss function. Classically, this problem is approached through a PAC-Bayes analysis where, in addition to the posterior, we choose a prior distribution which captures our belief about the inductive bias of the learning problem. Then, the key quantity in PAC-Bayes concentration bounds is a divergence that captures the complexity of the learning problem where the de facto standard choice is the KL divergence. However, the tightness of this choice has rarely been questioned. In this paper, we challenge the tightness of the KL-divergence-based bounds by showing that it is possible to achieve a strictly tighter bound. In particular, we demonstrate new high-probability PAC-Bayes bounds with a novel and better-than-KL divergence that is inspired by Zhang et al. (2022). Our proof is inspired by recent advances in regret analysis of gambling algorithms, and its use to derive concentration inequalities. Our result is first-of-its-kind in that existing PAC-Bayes bounds with non-KL divergences are not known to be strictly better than KL. Thus, we believe our work marks the first step towards identifying optimal rates of PAC-Bayes bounds.

• The paper presents a novel divergence measure, ZCP, that achieves strictly tighter PAC-Bayes bounds compared to traditional KL divergence.
• It employs innovative change-of-measure techniques inspired by online algorithms and betting frameworks to derive new concentration inequalities.
• The findings imply significant improvements in estimating generalization errors and enhancing model training methodologies.

Advancing PAC-Bayes Bounds: Introducing a Tighter Divergence Measure

Introduction

The quest for understanding and improving the generalization error of predictors, especially those trained through stochastic algorithms like neural networks, has long been central to research in statistical learning theory. A pivotal aspect of this is the estimation of the generalization error which reflects how well a learning model performs on unseen data. Historically, PAC-Bayesian (Probably Approximately Correct-Bayesian) frameworks have been instrumental in providing bounds for these generalization errors, utilizing the Kullback-Leibler (KL) divergence as a measure of complexity between probability distributions - specifically, the data-dependent posterior and a prior distribution. However, the conventional reliance on KL divergence has unexplored avenues, particularly concerning the optimality of these bounds given the complexity measure used.

Exploration Beyond KL Divergence

The research presented in this paper challenges the traditional use of KL divergence in formulating PAC-Bayes bounds. By investigating alternative divergences, this work establishes a strictly tighter bound using a novel divergence measure inspired by recent findings in regret analysis. This alternative measure, termed the Zhang-Cutkosky-Paschalidis (ZCP) divergence, demonstrates a fundamentally tighter bound than those derived from KL divergence. The new bounds exhibit substantial deviations, laying the groundwork for re-evaluating the standard practices in PAC-Bayesian analysis.

Methodological Innovations

The derivation of new PAC-Bayes bounds in this paper involves an inventive change-of-measure analysis, not predicated on the regret analysis commonly used alongside KL divergence. The proof technique employed hinges on insights from online algorithms and betting frameworks, opening up novel pathways for deriving concentration inequalities. This approach showcases the robustness and adaptability of the proposed method in obtaining PAC-Bayes bounds divergent from traditional reliance on KL divergence.

Implications and Perspectives

The introduction of ZCP divergence as a superior alternative to KL divergence for PAC-Bayesian bounds brings to light the potential suboptimality of widely accepted measures of complexity in learning theory. The findings underscore the critical reassessment needed in the existing paradigms and suggest a broader range of possibilities for future exploration in achieving optimal rates for PAC-Bayes bounds. The capability of the ZCP divergence to provide tighter bounds hints at the untapped efficiencies lying beyond the conventional frameworks, suggesting an array of promising directions for further research.

The practical implications of this work are manifold, ranging from enhanced predictability and reliability of model performance estimates to potential improvements in model training methodologies. By offering a more nuanced understanding of the theoretical underpinnings of generalization errors in machine learning models, this research contributes to advancing the precision and effectiveness of statistical learning theories.

Concluding Remarks

This paper presents a compelling argument for the reconsideration of standard practices in PAC-Bayesian analysis by introducing a meticulously derived alternative to KL divergence. The demonstrated superiority of the ZCP divergence in formulating tighter PAC-Bayes bounds signifies a critical step forward in our collective endeavor to refine and optimize the theoretical frameworks underpinning machine learning models. The proposed divergence measure not only enriches the theoretical landscape but also heralds practical advancements in the pursuit of more accurate and reliable generalization error estimates. This research sets a new benchmark in the ongoing evolution of learning theory, spotlighting the continuous need for innovation and critical scrutiny in the quest for optimal statistical bounds.