Source-linked AI summary
Nonlinear Information Bottleneck
Artemy Kolchinsky, Brendan D. Tracey, David H. Wolpert
TL;DR
The paper addresses the limited applicability and difficult optimization of information bottleneck methods. It proposes nonlinear IB using information bounds and neural-network implementations, and reports superior solutions to competing methods on real-world datasets, under a continuous bottleneck assumption.
Problem
Existing IB methods were mainly developed for small discrete spaces or jointly Gaussian continuous variables, while general optimization remains difficult.
Method
Nonlinear IB supports discrete or continuous X and Y with arbitrary joint distributions and nonlinear maps, using a non-parametric mutual-information upper bound.
Results
Nonlinear IB effectively discovers the IB trade-off curve and finds solutions superior to competing methods, including variational IB, on real-world datasets.
Takeaways & Limitations
The approach yields intermediate representations with visibly tighter clusters in classification problems and performs effectively in regression.
Takeaways & Limitations
The method assumes a continuous-valued bottleneck variable M.
Abstract
from arXiv · showhide
Information bottleneck (IB) is a technique for extracting information in one random variable $X$ that is relevant for predicting another random variable $Y$. IB works by encoding $X$ in a compressed "bottleneck" random variable $M$ from which $Y$ can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete $X$ and $Y$ with small state spaces, and continuous $X$ and $Y$ with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous $X$ and $Y$, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed "variational IB" method on several real-world datasets.
I. INTRODUCTION
Information bottleneck seeks a compressed representation M of X that preserves information useful for predicting Y, balancing compression against prediction. The paper proposes nonlinear IB for broader distributions and variable types using information bounds optimized with neural-network methods.
- Motivation: IB finds a bottleneck M that compresses X while retaining information relevant to predicting Y.The trade-off is represented by low I(X; M) and high I(Y; M).
- Motivation: The IB curve specifies the trade-off between channel capacity and prediction accuracy.
- Limitations of existing methods: For deterministic or nearly deterministic Y=f(X), sweeping the standard IB Lagrangian can return the same M for every β and fail to explore the IB curve.
- Limitations of existing methods: Standard IB optimization is difficult because its objectives are non-convex and mutual-information terms can require intractable integrals.
- Limitations of existing methods: Consequently, prior IB solutions mainly addressed small discrete state spaces or jointly Gaussian continuous variables with linear maps.
- Proposed approach: Nonlinear IB handles discrete or continuous X and Y with arbitrary joint distributions, while permitting nonlinear encoding and decoding maps.
- Proposed approach: The method derives a lower bound on the IB objective using a variational lower bound for prediction and a non-parametric upper bound for compression.
- Proposed approach: The approach represents data with finite samples, parameterizes encoding and decoding conditional distributions, and supports implementation with standard neural-network techniques.
II. PROPOSED APPROACH
The proposed approach makes information bottleneck optimization tractable for continuous variables by parameterizing encoding and decoding maps and bounding the mutual-information terms. It also supports discrete variables, squared-IB objectives, and neural-network optimization, subject to computational limits.
- Scope and assumptions: The derivations assume continuous variables but extend to discrete variables by replacing integrals with sums.The encoding noise may be heteroscedastic because its covariance can depend on both parameters and the input outcome.
- Encoding and decoding: The method parameterizes an encoding map from X to bottleneck variable M and a variational decoding map from M to Y.The encoding uses Pθ(m|x), while the decoding uses Pφ(y|m).
- Information bounds: A variational lower bound replaces the prediction mutual-information term, becoming tight when the decoding map equals the optimal conditional distribution.The bound tightens as the KL divergence between the variational and optimal decoding maps decreases.
- Information bounds: A non-parametric upper bound estimates compression mutual information using pairwise KL divergences between Gaussian mixture components.The Gaussian KL divergence has a closed-form expression, and the bound becomes tight for equal-covariance, well-separated clusters.
- Objectives: Combining the bounds yields the tractable nonlinear IB objective, which adds an information-theoretic regularization term to supervised-learning objectives.The analogous squared-IB objective is used when the ordinary IB Lagrangian cannot explore the IB curve across β values.
- Optimization: Gradient-based optimization is practical with mini-batches, but evaluating the compression estimator requires O(n^2) operations and larger batches may be needed for high-dimensional bottlenecks.The estimator's connection to kernel-density estimation makes sample requirements grow rapidly with bottleneck dimensionality.
III. RELATION TO PRIOR WORK
This section positions the proposed method within prior work in machine learning. It introduces the paper’s comparison with related machine-learning approaches.
- The authors relate their proposed method to prior work in machine learning.
- The relation-to-prior-work discussion is presented as a separate section of the paper.
- The section frames subsequent comparisons with existing machine-learning methods.
A. Variational IB
Variational IB methods provide tractable lower bounds by using surrogate marginal distributions for the compression term. Compared with these methods, the proposed estimator is more computationally expensive but is expected to estimate true mutual information more accurately.
- Variational bounds: Variational IB methods use tractable lower bounds for continuous, potentially non-Gaussian variables and share the variational prediction bound used here.Their main difference from the proposed approach is how they bound Iθ(X; M).
- Surrogate distributions: These methods bound the compression term with a surrogate marginal distribution R over the bottleneck variable.Different proposals use choices including a standard multivariate normal, products of Student’s t-distributions, log-uniform distributions, and log-normal distributions.
- Encoding choices: Alternative encoders use additive Gaussian noise or multiplicative noise applied to deterministic functions.The noise choice in the multiplicative-noise approach depends on the neural-network formulation.
- Computational trade-off: O(n) operations make the variational bound more computationally efficient than the proposed estimator, which requires O(n^2) operations per mini-batch.The proposed non-parametric estimator is nevertheless expected to provide a better estimate of true mutual information.
- Evaluation: The paper compares its approach with variational IB in Section IV B.
B. Neural Networks and Kernel Density Entropy Estimates
The method incorporates a differentiable non-parametric mutual-information estimator directly into neural-network training. This connects the optimization procedure to kernel-density estimation while distinguishing it from methods that use such estimates only diagnostically.
- Estimator: The approach uses a differentiable upper bound on Iθ(X; M) that is related to non-parametric kernel-density estimators.
- Relation to prior work: The estimator connects neural-network optimization with prior work on non-parametric estimates of information-theoretic functions.
- Training objective: Rather than estimating held-out likelihood only after learning, the method trains the network by incorporating its non-parametric estimator into the objective.The estimator is therefore part of optimization rather than only a post-training diagnostic.
C. Auto-Encoders
The paper relates nonlinear IB to auto-encoders through constrained intermediate representations, coding penalties, and injected noise. Unlike denoising auto-encoders, nonlinear IB provides an optimality-based way to adapt the noise level.
- Auto-encoders reconstruct inputs through intermediate representations that can be constrained by dimensionality or information-theoretic penalties.
- Prior approaches separately penalized hidden-layer coding length or injected noise, whereas nonlinear IB combines both mechanisms.
- Denoising auto-encoders lack an optimal noise level because reducing noise always improves training-data prediction error.
- Variational auto-encoders optimize an ELBO containing prediction and compression terms, making the objective a special case of variational IB.
- Replacing the ELBO compression term with the nonlinear IB mutual-information estimator might improve compression but could complicate latent-space sampling.
IV. EXPERIMENTS
The experiments implement nonlinear IB with neural networks and compare it with variational IB across several datasets. The paper reports better performance and different learned representations for nonlinear IB.
- The experiments implement nonlinear IB using neural-network techniques and evaluate it on several datasets against variational IB.
- Compared with VIB, nonlinear IB achieves better performance and uncovers different kinds of representations.
A. Implementation
The implementation uses stochastic neural-network encoders, deterministic neural-network decoders, and a mutual-information upper bound optimized through squared IB objectives. A homoscedastic noise model is used for both nonlinear IB and VIB, with practical initialization constraints.
- The decoder maps bottleneck states to Y through deterministic neural-network layers and uses cross-entropy for classification or MSE for regression.
- The stochastic encoder maps X to bottleneck states by adding Gaussian noise to a possibly nonlinear neural-network function.
- The implementation uses five linear bottleneck neurons, Adam optimization, mini-batches of 256, and early stopping on a validation split.
- A homoscedastic covariance model, Σθ(x) = σ2I, simplifies the mutual-information bound and is applied to both nonlinear IB and VIB.
- Noise initialization matters: very small σ2 prevents learning compression, whereas very large noise can collapse optimization to a maximally compressed model.
- The experiments optimize squared-IB bounds because they produce quantitatively similar results while being more numerically robust across the full IB curve.
B. Results
Across MNIST, FashionMNIST, and California housing, nonlinear IB achieves higher prediction at matched compression than VIB. Its bottleneck representations are also tighter or nearly one-dimensional in the reported analyses.
- MNIST: ≈0.1 bits better prediction was achieved near the MNIST corner point, where I(X; M) = I(Y ; M) ≈ log 10.
- MNIST: On MNIST, nonlinear IB produced tighter class clusters than VIB, with bottleneck states carrying almost no information beyond digit identity.
- FashionMNIST: FashionMNIST yielded ≈0.15 additional bits of prediction for nonlinear IB at the same compression across a range of compression values.
- FashionMNIST: FashionMNIST representations formed separated clusters for both methods, but nonlinear IB produced significantly tighter clusters.
- California housing: On California housing, nonlinear IB again achieved better prediction than VIB at matched compression on both training and testing data.
- California housing: The nonlinear IB housing representation was almost perfectly one-dimensional, with the first principal component explaining 99.8% of variance.
- Overall comparison: Nonlinear IB consistently achieves better prediction than VIB at the same compression level on training and testing data.
V. CONCLUSION
Nonlinear IB extends information-bottleneck analysis to general discrete or continuous inputs and outputs, arbitrary joint distributions, and nonlinear encoding and decoding maps. Across classification and regression tasks, it discovers the IB tradeoff curve and finds solutions superior to competing methods, while revealing compact intermediate representations.
- V. CONCLUSION: Nonlinear IB explores the IB tradeoff curve for discrete or continuous inputs and outputs, assuming a continuous bottleneck variable.It permits arbitrary, including non-Gaussian, joint input-output distributions.
- V. CONCLUSION: A new tractable and differentiable bound on the IB objective provides the method’s generality.
- V. CONCLUSION: The method can be implemented with off-the-shelf neural-network software and applied to standard classification and regression problems.
- V. CONCLUSION: Nonlinear IB effectively discovers the tradeoff curve and finds solutions superior to competing methods.
- V. CONCLUSION: Its intermediate representations form tighter classification clusters and a one-dimensional representation in regression.
- V. CONCLUSION: The efficacy of increased compression for generalization or adversarial robustness remains future work.
SUPPLEMENTARY MATERIAL
The supplementary experiments optimize bounds on the regular IB Lagrangian for the same three datasets used in the main text. They keep the main-text parameters unchanged except for the objective and compare nonlinear IB with VIB.
- SUPPLEMENTARY MATERIAL: The experiments use MNIST classification, FashionMNIST classification, and California housing prices regression.
- SUPPLEMENTARY MATERIAL: The supplementary material optimizes bounds on the regular IB Lagrangian defined by nonlinear IB and an extension of the VIB objective.
- SUPPLEMENTARY MATERIAL: All parameters remain the same as in the main text except for the change in objective.
- SUPPLEMENTARY MATERIAL: Figure S1 reports nonlinear IB and VIB performance when optimizing bounds on the regular IB objective.