Statistically Guaranteed Information Bottleneck via Multiple Hypothesis Testing

Core Concepts

The proposed IB-MHT methodology provides statistical guarantees on the information bottleneck constraints, while approximately minimizing the information loss, by leveraging multiple hypothesis testing.

Abstract

The content introduces the information bottleneck (IB) problem, which aims to extract a low-dimensional statistic T from an observation X that retains sufficient information about a correlated variable Y. Conventional approaches to solving the IB problem rely on heuristic tuning of hyperparameters, offering no guarantees that the learned features satisfy the information-theoretic constraints.
The paper proposes a new methodology called IB-MHT (Information Bottleneck via Multiple Hypothesis Testing) that wraps around existing IB solvers to provide statistical guarantees on the IB constraints. The key steps are:

Estimating an approximate Pareto frontier on the plane (I(T;Y), I(X;T)) using a portion of the available data.
Sequentially testing candidate hyperparameters in order of decreasing estimated I(T;Y) using a family-wise error rate (FWER) controlling algorithm on a separate portion of the data.
Selecting the hyperparameter λ* that minimizes the estimated I(X;T) among the hyperparameters that are likely to satisfy the IB constraint.

The proposed IB-MHT approach is demonstrated on both the classical IB problem formulation and the deterministic IB problem. The results show that IB-MHT can satisfy the IB constraint with high probability, while achieving comparable or better performance on the objective I(X;T) compared to conventional IB solvers.

Stats

The mutual information I(T;Y) is required to be greater than or equal to a threshold α with high probability.
The mutual information I(X;T) is to be minimized.

Quotes

"The information bottleneck (IB) problem is a widely studied framework in machine learning for extracting compressed features that are informative for downstream tasks."
"However, current approaches to solving the IB problem rely on a heuristic tuning of hyperparameters, offering no guarantees that the learned features satisfy information-theoretic constraints."

Key Insights Distilled From

Statistically Valid Information Bottleneck via Multiple Hypothesis Testing

by Amirmohammad... at arxiv.org 09-12-2024

https://arxiv.org/pdf/2409.07325.pdf

Deeper Inquiries

How can the proposed IB-MHT methodology be extended to handle continuous variables?

The proposed IB-MHT methodology, which currently focuses on discrete random variables, can be extended to handle continuous variables by employing techniques from non-parametric statistics and kernel density estimation. In the context of continuous variables, the mutual information estimates can be derived using methods such as the k-nearest neighbors (KNN) approach or kernel density estimators, which allow for the estimation of joint and marginal distributions without assuming a specific parametric form.
To implement this extension, the following steps can be taken:

Estimation of Mutual Information: Use KNN or kernel density estimation to compute the empirical joint distribution ( P_{UV} ) for continuous variables ( U ) and ( V ). The mutual information can then be estimated using the plug-in estimator adapted for continuous distributions.

Hypothesis Testing Framework: The multiple hypothesis testing framework can be adapted by utilizing continuous p-value calculations. This may involve defining a suitable test statistic based on the estimated mutual information and deriving p-values through resampling methods or bootstrap techniques.

Control of Type I Error: The family-wise error rate (FWER) control can be maintained by applying techniques such as the Bonferroni correction or the Benjamini-Hochberg procedure, which are effective in managing false discoveries in continuous settings.

Pareto Front Estimation: The estimation of the Pareto front can be achieved by evaluating the trade-offs between the estimated mutual information values for different hyperparameters, similar to the discrete case but using the continuous estimates.

By integrating these techniques, IB-MHT can effectively manage continuous variables while ensuring that the statistical guarantees on the mutual information constraints are preserved.

What are the potential limitations of the multiple hypothesis testing approach used in IB-MHT, and how can they be addressed?

The multiple hypothesis testing (MHT) approach used in IB-MHT, while providing statistical guarantees, has several potential limitations:

Assumption of Independence: The MHT framework often assumes that the hypotheses being tested are independent. In practice, hyperparameters may be correlated, leading to an increased risk of false discoveries. This can be addressed by employing methods that account for dependency structures, such as the use of graphical models or the application of techniques like the Holm-Bonferroni method, which adjusts for dependencies among tests.

Choice of Outage Level: The selection of the outage level ( \delta ) can significantly impact the performance of the IB-MHT. A conservative choice may lead to overly restrictive conditions, while a liberal choice may increase the risk of violating the constraints. To address this, a data-driven approach can be adopted to optimize the outage level based on the specific characteristics of the dataset and the desired trade-off between robustness and flexibility.

Computational Complexity: The sequential testing process can become computationally intensive, especially with a large number of hyperparameters. This can be mitigated by employing adaptive testing strategies that reduce the number of hypotheses tested based on preliminary results, or by using parallel computing techniques to expedite the testing process.

Overfitting Risk: The iterative nature of the MHT can lead to overfitting, particularly when the sample size is small. To counteract this, regularization techniques can be integrated into the hyperparameter selection process, ensuring that the selected model generalizes well to unseen data.

By addressing these limitations, the robustness and applicability of the IB-MHT methodology can be significantly enhanced, ensuring reliable performance across various scenarios.

How can the ideas behind IB-MHT be applied to optimize other information-theoretic metrics beyond the mutual information, such as Rényi divergences or f-divergences?

The principles underlying IB-MHT can be effectively adapted to optimize other information-theoretic metrics, such as Rényi divergences or f-divergences, by following a similar framework of hypothesis testing and Pareto optimization. Here’s how this can be achieved:

Metric Definition: First, define the target information-theoretic metric, such as Rényi divergence ( D_\alpha(P || Q) ) for a given order ( \alpha ), or f-divergences, which generalize various divergence measures. The optimization problem can then be framed to minimize the chosen divergence while satisfying certain constraints.

Estimation Techniques: Just as with mutual information, the estimation of Rényi divergences or f-divergences can be performed using empirical distributions derived from the data. Techniques such as kernel density estimation or plug-in estimators can be employed to compute these divergences from the observed data.

Hypothesis Testing Framework: Similar to the MHT approach in IB-MHT, a set of null hypotheses can be formulated for the chosen divergence metrics. For instance, one could test whether the divergence exceeds a specified threshold, analogous to testing mutual information constraints.

Pareto Front Estimation: The Pareto front can be constructed in the same manner as in IB-MHT, where the trade-offs between the divergence metric and other relevant quantities (e.g., model complexity or another divergence measure) are evaluated. This allows for the identification of hyperparameters that balance the trade-offs effectively.

Sequential Testing and FWER Control: The sequential testing approach can be adapted to control the family-wise error rate for the new divergence metrics, ensuring that the selected hyperparameters meet the desired statistical guarantees.

By leveraging these strategies, the IB-MHT framework can be generalized to optimize a wide range of information-theoretic metrics, enhancing its applicability in various domains such as machine learning, statistics, and information theory.