Deriving Tighter PAC-Bayes Bounds with a Novel Divergence Measure

Core Concepts

This paper presents a novel high-probability PAC-Bayes bound that achieves a strictly tighter complexity measure than the standard Kullback-Leibler (KL) divergence. The new bound is based on a divergence measure called the Zhang-Cutkosky-Paschalidis (ZCP) divergence, which is shown to be orderwise better than the KL divergence in certain cases.

Abstract

The paper studies the problem of estimating the mean of a sequence of random elements f(θ, X1), ..., f(θ, Xn), where θ is a random parameter drawn from a data-dependent posterior distribution Pn. This problem is commonly approached through PAC-Bayes analysis, where a prior distribution P0 is chosen to capture the inductive bias of the learning problem.
The key contribution of the paper is to show that the standard choice of the Kullback-Leibler (KL) divergence as the complexity measure in PAC-Bayes bounds is suboptimal. The authors derive a new high-probability PAC-Bayes bound that uses a novel divergence measure called the Zhang-Cutkosky-Paschalidis (ZCP) divergence, which is shown to be strictly tighter than the KL divergence in certain cases.
The proof of the new bound is inspired by recent advances in regret analysis of gambling algorithms, which are used to derive concentration inequalities. The authors also show how the new bound can be relaxed to recover various known PAC-Bayes inequalities, such as the empirical Bernstein inequality and the Bernoulli KL-divergence bound.
The paper concludes by discussing the implications of the results, suggesting that there is much room for studying optimal rates of PAC-Bayes bounds and that the choice of the complexity measure is an important aspect that deserves further investigation.

Stats

The sample size is denoted by n.
The failure probability is denoted by δ.
The paper considers a sequence of random elements f(θ, X1), ..., f(θ, Xn), where θ is a random parameter drawn from a data-dependent posterior distribution Pn.
The generalization error is defined as ∆n(θ) = 1/n * Σ(f(θ, Xi) - E[f(θ, X1)]).

Quotes

"We challenge the tightness of the KL-divergence-based bounds by showing that it is possible to achieve a strictly tighter bound."
"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."
"Our analysis is inspired by a recent observation of Zhang et al. (2022), who pointed out an interesting phenomenon arising in regret analysis of online algorithms."

Key Insights Distilled From

Better-than-KL PAC-Bayes Bounds

by Ilja Kuzbors... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2402.09201.pdf

Deeper Inquiries

What other divergence measures could be explored to further improve the tightness of PAC-Bayes bounds

In the context of improving the tightness of PAC-Bayes bounds, one potential direction could be to explore other divergence measures beyond the KL divergence and the ZCP divergence. Some possibilities include:

Total Variation (TV) Divergence: TV divergence measures the discrepancy between two probability distributions. It could be interesting to investigate how incorporating TV divergence into PAC-Bayes bounds could lead to tighter concentration inequalities.

Jensen-Shannon Divergence: This divergence is a symmetrized and smoothed version of the KL divergence. By considering the Jensen-Shannon divergence, which combines elements of both KL and TV divergences, it may be possible to derive bounds that capture a more nuanced view of the complexity of the learning problem.

Hellinger Divergence: Hellinger divergence is another measure of the difference between two probability distributions. Exploring the use of Hellinger divergence in PAC-Bayes analysis could provide insights into alternative ways to quantify the complexity of the learning problem.

Chi-Squared Divergence: Chi-squared divergence is commonly used in statistical hypothesis testing. Investigating the applicability of chi-squared divergence in the context of PAC-Bayes bounds could offer a different perspective on concentration inequalities.

By exploring these and other divergence measures, researchers may uncover new insights and potentially discover divergence metrics that lead to even tighter PAC-Bayes bounds.

How can the insights from this work be extended to other statistical learning problems beyond the PAC-Bayes framework

The insights from this work on better-than-KL PAC-Bayes bounds can be extended to various other statistical learning problems beyond the PAC-Bayes framework. Some potential extensions include:

Regularization Techniques: The concept of optimal log wealth and the use of regret analysis in deriving concentration inequalities can be applied to regularization techniques in machine learning. By incorporating similar ideas, researchers can develop tighter bounds for regularization methods in various learning algorithms.

Online Learning Algorithms: The connection between online betting algorithms and statistical learning can be further explored in the context of online learning algorithms. Insights from this work can be leveraged to derive concentration inequalities for online learning problems with applications in reinforcement learning and sequential decision-making.

Optimization Theory: The optimization of log wealth in betting algorithms can be extended to optimization problems in machine learning. By framing optimization objectives in terms of log wealth, researchers can develop novel optimization algorithms with improved convergence rates and performance guarantees.

By applying the principles and methodologies introduced in this work to a broader range of statistical learning problems, researchers can advance the understanding of concentration inequalities and complexity measures in machine learning.

Are there any connections between the optimal log wealth concept used in this paper and other information-theoretic measures of complexity in machine learning

The concept of optimal log wealth used in this paper can be connected to other information-theoretic measures of complexity in machine learning in the following ways:

Mutual Information: Mutual information measures the amount of information shared between two random variables. By relating the optimal log wealth to mutual information, researchers can explore the information content captured by the log wealth optimization process in statistical learning problems.

Entropy: Entropy quantifies the uncertainty or randomness in a random variable. The relationship between optimal log wealth and entropy can provide insights into the level of uncertainty reduction achieved through the log wealth optimization strategy in betting algorithms.

Kolmogorov Complexity: Kolmogorov complexity measures the shortest description length of an object. By examining the Kolmogorov complexity of the optimal log wealth process, researchers can analyze the simplicity or complexity of the log wealth optimization algorithm in capturing the underlying data distribution.

By drawing connections between optimal log wealth and these information-theoretic measures, researchers can deepen their understanding of the complexity and information content in machine learning processes.