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.
Combining the Information Bottleneck model with deep learning by replacing mutual information terms with deep neural nets has proved successful in areas ranging from generative modelling to interpreting deep neural networks. In this paper, we revisit the Deep Variational Information Bottleneck and the assumptions needed for its derivation. The two assumed properties of the data $X$, $Y$ and their latent representation $T$ take the form of two Markov chains $T-X-Y$ and $X-T-Y$. Requiring both to hold during the optimisation process can be limiting for the set of potential joint distributions $P…
In classic papers, Zellner demonstrated that Bayesian inference could be derived as the solution to an information theoretic functional. Below we derive a generalized form of this functional as a variational lower bound of a predictive information bottleneck objective. This generalized functional encompasses most modern inference procedures and suggests novel ones.
Zellner (1988) modeled statistical inference in terms of information processing and postulated the Information Conservation Principle (ICP) between the input and output of the information processing block, showing that this yielded Bayesian inference as the optimum information processing rule. Recently, Alemi (2019) reviewed Zellner's work in the context of machine learning and showed that the ICP could be seen as a special case of a more general optimum information processing criterion, namely the Predictive Information Bottleneck Objective. However, Alemi modeled the model training step in m…
We present a variational approximation to the information bottleneck of Tishby et al. (1999). This variational approach allows us to parameterize the information bottleneck model using a neural network and leverage the reparameterization trick for efficient training. We call this method "Deep Variational Information Bottleneck", or Deep VIB. We show that models trained with the VIB objective outperform those that are trained with other forms of regularization, in terms of generalization performance and robustness to adversarial attack.
Facebook AI Research（脸书人工智能研究院）；CUNY Graduate Center（纽约市立大学研究生中心）
We address the question of characterizing and finding optimal representations for supervised learning. Traditionally, this question has been tackled using the Information Bottleneck, which compresses the inputs while retaining information about the targets, in a decoder-agnostic fashion. In machine learning, however, our goal is not compression but rather generalization, which is intimately linked to the predictive family or decoder of interest (e.g. linear classifier). We propose the Decodable Information Bottleneck (DIB) that considers information retention and compression from the perspecti…
In this paper, we develop an unsupervised generative clustering framework that combines the Variational Information Bottleneck and the Gaussian Mixture Model. Specifically, in our approach, we use the Variational Information Bottleneck method and model the latent space as a mixture of Gaussians. We derive a bound on the cost function of our model that generalizes the Evidence Lower Bound (ELBO) and provide a variational inference type algorithm that allows computing it. In the algorithm, the coders' mappings are parametrized using neural networks, and the bound is approximated by Monte Carlo s…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Columbia University（哥伦比亚大学）
We propose a novel strategy for extracting features in supervised learning that can be used to construct a classifier which is more robust to small perturbations in the input space. Our method builds upon the idea of the information bottleneck by introducing an additional penalty term that encourages the Fisher information of the extracted features to be small, when parametrized by the inputs. By tuning the regularization parameter, we can explicitly trade off the opposing desiderata of robustness and accuracy when constructing a classifier. We derive the optimal solution to the robust informa…
Imperial College London（伦敦帝国学院）；CentraleSupélec-CNRS-Université Paris-Sud（中央理工-高等电力学院-法国国家科学研究中心-巴黎南大学）
The muti-layer information bottleneck (IB) problem, where information is propagated (or successively refined) from layer to layer, is considered. Based on information forwarded by the preceding layer, each stage of the network is required to preserve a certain level of relevance with regards to a specific hidden variable, quantified by the mutual information. The hidden variables and the source can be arbitrarily correlated. The optimal trade-off between rates of relevance and compression (or complexity) is obtained through a single-letter characterization, referred to as the rate-relevance re…
This tutorial paper focuses on the variants of the bottleneck problem taking an information theoretic perspective and discusses practical methods to solve it, as well as its connection to coding and learning aspects. The intimate connections of this setting to remote source-coding under logarithmic loss distortion measure, information combining, common reconstruction, the Wyner-Ahlswede-Korner problem, the efficiency of investment information, as well as, generalization, variational inference, representation learning, autoencoders, and others are highlighted. We discuss its extension to the di…
University of Florida（佛罗里达大学）；NEC Laboratories Europe（NEC欧洲实验室）
We introduce the matrix-based Renyi's $α$-order entropy functional to parameterize Tishby et al. information bottleneck (IB) principle with a neural network. We term our methodology Deep Deterministic Information Bottleneck (DIB), as it avoids variational inference and distribution assumption. We show that deep neural networks trained with DIB outperform the variational objective counterpart and those that are trained with other forms of regularization, in terms of generalization performance and robustness to adversarial attack.Code available at https://github.com/yuxi120407/DIB
We introduce a bottleneck method for learning data representations based on information deficiency, rather than the more traditional information sufficiency. A variational upper bound allows us to implement this method efficiently. The bound itself is bounded above by the variational information bottleneck objective, and the two methods coincide in the regime of single-shot Monte Carlo approximations. The notion of deficiency provides a principled way of approximating complicated channels by relatively simpler ones. We show that the deficiency of one channel with respect to another has an oper…
We view the Information Bottleneck Principle (IBP: Tishby et al., 1999; Schwartz-Ziv and Tishby, 2017) and Predictive Information Bottleneck Principle (PIBP: Still et al., 2007; Alemi, 2019) as special cases of a family of general information bottleneck objectives (IBOs). Each IBO corresponds to a particular constrained optimization problem where the constraints apply to: (a) the mutual information between the training data and the learned model parameters or extracted representation of the data, and (b) the mutual information between the learned model parameters or extracted representation of…
Racah Institute of Physics（拉卡物理研究所）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）；School of Computer Science（计算机科学学院）
The Information Bottleneck (IB) framework is a general characterization of optimal representations obtained using a principled approach for balancing accuracy and complexity. Here we present a new framework, the Dual Information Bottleneck (dualIB), which resolves some of the known drawbacks of the IB. We provide a theoretical analysis of the dualIB framework; (i) solving for the structure of its solutions (ii) unraveling its superiority in optimizing the mean prediction error exponent and (iii) demonstrating its ability to preserve exponential forms of the original distribution. To approach l…
The information bottleneck (IB) method is a technique for extracting information that is relevant for predicting the target random variable from the source random variable, which is typically implemented by optimizing the IB Lagrangian that balances the compression and prediction terms. However, the IB Lagrangian is hard to optimize, and multiple trials for tuning values of Lagrangian multiplier are required. Moreover, we show that the prediction performance strictly decreases as the compression gets stronger during optimizing the IB Lagrangian. In this paper, we implement the IB method from t…
This paper considers the information bottleneck (IB) problem of a Rayleigh fading multiple-input multiple-out (MIMO) channel. Due to the bottleneck constraint, it is impossible for the oblivious relay to inform the destination node of the perfect channel state information (CSI) in each channel realization. To evaluate the bottleneck rate, we provide an upper bound by assuming that the destination node can get the perfect CSI at no cost and two achievable schemes with simple symbol-by-symbol relay processing and compression. Numerical results show that the lower bounds obtained by the proposed …
We propose a new method for learning image attention masks in a semi-supervised setting based on the Information Bottleneck principle. Provided with a set of labeled images, the mask generation model is minimizing mutual information between the input and the masked image while maximizing the mutual information between the same masked image and the image label. In contrast with other approaches, our attention model produces a Boolean rather than a continuous mask, entirely concealing the information in masked-out pixels. Using a set of synthetic datasets based on MNIST and CIFAR10 and the SVHN …
We study the problem of distributed information bottleneck, in which multiple encoders separately compress their observations in a manner such that, collectively, the compressed signals preserve as much information as possible about another signal. The model generalizes Tishby's centralized information bottleneck method to the setting of multiple distributed encoders. We establish single-letter characterizations of the information-rate region of this problem for both i) a class of discrete memoryless sources and ii) memoryless vector Gaussian sources. Furthermore, assuming a sum constraint on …
Sharif University of Technology（谢里夫理工大学）；Technical University of Munich（慕尼黑工业大学）；Universite Paris-Est（巴黎东部大学）
The Information Bottleneck method is a learning technique that seeks a right balance between accuracy and generalization capability through a suitable tradeoff between compression complexity, measured by minimum description length, and distortion evaluated under logarithmic loss measure. In this paper, we study a variation of the problem, called scalable information bottleneck, where the encoder outputs multiple descriptions of the observation with increasingly richer features. The problem at hand is motivated by some application scenarios that require varying levels of accuracy depending on t…
Tsinghua University（清华大学）；Louisiana State University（路易斯安那州立大学）
In multi-agent deep reinforcement learning, extracting sufficient and compact information of other agents is critical to attain efficient convergence and scalability of an algorithm. In canonical frameworks, distilling of such information is often done in an implicit and uninterpretable manner, or explicitly with cost functions not able to reflect the relationship between information compression and utility in representation. In this paper, we present Information-Bottleneck-based Other agents' behavior Representation learning for Multi-agent reinforcement learning (IBORM) to explicitly seek lo…
A hallmark of chaotic dynamics is the loss of information with time. Although information loss is often expressed through a connection to Lyapunov exponents -- valid in the limit of high information about the system state -- this picture misses the rich spectrum of information decay across different levels of granularity. Here we show how machine learning presents new opportunities for the study of information loss in chaotic dynamics, with a double pendulum serving as a model system. We use the Information Bottleneck as a training objective for a neural network to extract information from the…
Sharif University of Technology（谢里夫理工大学）；Technical University of Munich（慕尼黑工业大学）；Universite Paris-Est（巴黎东部大学）
In the context of statistical learning, the Information Bottleneck method seeks a right balance between accuracy and generalization capability through a suitable tradeoff between compression complexity, measured by minimum description length, and distortion evaluated under logarithmic loss measure. In this paper, we study a variation of the problem, called scalable information bottleneck, in which the encoder outputs multiple descriptions of the observation with increasingly richer features. The model, which is of successive-refinement type with degraded side information streams at the decoder…
University of California, San Francisco（加利福尼亚大学旧金山分校）
Multiview data contain information from multiple modalities and have potentials to provide more comprehensive features for diverse machine learning tasks. A fundamental question in multiview analysis is what is the additional information brought by additional views and can quantitatively identify this additional information. In this work, we try to tackle this challenge by decomposing the entangled multiview features into shared latent representations that are common across all views and private representations that are specific to each single view. We formulate this feature disentanglement in…
Johns Hopkins University（约翰斯·霍普金斯大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We develop a novel method for ensuring fairness in machine learning which we term as the Renyi Fair Information Bottleneck (RFIB). We consider two different fairness constraints - demographic parity and equalized odds - for learning fair representations and derive a loss function via a variational approach that uses Renyi's divergence with its tunable parameter $α$ and that takes into account the triple constraints of utility, fairness, and compactness of representation. We then evaluate the performance of our method for image classification using the EyePACS medical imaging dataset, showing i…
This paper considers the distributed information bottleneck (D-IB) problem for a primitive Gaussian diamond channel with two relays and Rayleigh fading. Due to the bottleneck constraint, it is impossible for the relays to inform the destination node of the perfect channel state information (CSI) in each realization. To evaluate the bottleneck rate, we provide an upper bound by assuming that the destination node knows the CSI and the relays can cooperate with each other, and also three achievable schemes with simple symbol-by-symbol relay processing and compression. Numerical results show that …
This paper considers an information bottleneck problem with the objective of obtaining a most informative representation of a hidden feature subject to a Rényi entropy complexity constraint. The optimal bottleneck trade-off between relevance (measured via Shannon's mutual information) and Rényi entropy cost is defined and an iterative algorithm for finding approximate solutions is provided. We also derive an operational characterization for the optimal trade-off by demonstrating that the optimal Rényi entropy-relevance trade-off is achievable by a simple time-sharing scalar coding scheme and t…
Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems (e.g., Mrs. Gerber's Lemma), strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding and dependence dilution. Leveraging these connections, we prove a new c…
Recently, a novel lookup table based decoding method for binary low-density parity-check codes has attracted considerable attention. In this approach, mutual-information maximizing lookup tables replace the conventional operations of the variable nodes and the check nodes in message passing decoding. Moreover, the exchanged messages are represented by integers with very small bit width. A machine learning framework termed the information bottleneck method is used to design the corresponding lookup tables. In this paper, we extend this decoding principle from binary to non-binary codes. This is…
This paper investigates a multi-terminal source coding problem under a logarithmic loss fidelity which does not necessarily lead to an additive distortion measure. The problem is motivated by an extension of the Information Bottleneck method to a multi-source scenario where several encoders have to build cooperatively rate-limited descriptions of their sources in order to maximize information with respect to other unobserved (hidden) sources. More precisely, we study fundamental information-theoretic limits of the so-called: (i) Two-way Collaborative Information Bottleneck (TW-CIB) and (ii) th…
We study two dual settings of information processing. Let $ \mathsf{Y} \rightarrow \mathsf{X} \rightarrow \mathsf{W} $ be a Markov chain with fixed joint probability mass function $ \mathsf{P}_{\mathsf{X}\mathsf{Y}} $ and a mutual information constraint on the pair $ (\mathsf{W},\mathsf{X}) $. For the first problem, known as Information Bottleneck, we aim to maximize the mutual information between the random variables $ \mathsf{Y} $ and $ \mathsf{W} $, while for the second problem, termed as Privacy Funnel, our goal is to minimize it. In particular, we analyze the scenario for which $ \mathsf{…
The information bottleneck problem (IB) of jointly stationary Gaussian sources is considered. A water-filling solution for the IB rate is given in terms of its SNR spectrum and whose rate is attained via frequency domain test-channel realization. A time-domain realization of the IB rate, based on linear prediction, is also proposed, which lends itself to an efficient implementation of the corresponding remote source-coding problem. A compound version of the problem is addressed, in which the joint distribution of the source is not precisely specified but rather in terms of a lower bound on the…
This work considers a layered coding approach for efficient transmission of data over a wireless block fading channel without transmitter channel state information (CSI), which is connected to a limited capacity reliable link, known as the bottleneck channel. Two main approaches are considered, the first is an oblivious approach, where the sampled noisy observations are compressed and transmitted over the bottleneck channel without having any knowledge of the original information codebook. The second approach is a non-oblivious decode-forward (DF) relay where the sampled noisy data is decoded,…
We prove rigorously a source coding theorem that can probably be considered folklore, a generalization to arbitrary alphabets of a problem motivated by the Information Bottleneck method. For general random variables $(Y, X)$, we show essentially that for some $n \in \mathbb{N}$, a function $f$ with rate limit $\log|f| \le nR$ and $I(Y^n; f(X^n)) \ge nS$ exists if and only if there is a random variable $U$ such that the Markov chain $Y - X - U$ holds, $I(U; X) \le R$ and $I(U; Y) \ge S$. The proof relies on the well established discrete case and showcases a technique for lifting discrete coding…
Initiative for the Theoretical Sciences, The Graduate Center, CUNY（纽约市立大学研究生中心理论科学倡议）
Avoiding overfitting is a central challenge in machine learning, yet many large neural networks readily achieve zero training loss. This puzzling contradiction necessitates new approaches to the study of overfitting. Here we quantify overfitting via residual information, defined as the bits in fitted models that encode noise in training data. Information efficient learning algorithms minimize residual information while maximizing the relevant bits, which are predictive of the unknown generative models. We solve this optimization to obtain the information content of optimal algorithms for a lin…
University of Pennsylvania（宾夕法尼亚大学）；The Santa Fe Institute（圣塔菲研究所）
The fruits of science are relationships made comprehensible, often by way of approximation. While deep learning is an extremely powerful way to find relationships in data, its use in science has been hindered by the difficulty of understanding the learned relationships. The Information Bottleneck (IB) is an information theoretic framework for understanding a relationship between an input and an output in terms of a trade-off between the fidelity and complexity of approximations to the relationship. Here we show that a crucial modification -- distributing bottlenecks across multiple components …
In this work, we generalize the information bottleneck (IB) approach to the multi-view learning context. The exponentially growing complexity of the optimal representation motivates the development of two novel formulations with more favorable performance-complexity tradeoffs. The first approach is based on forming a stochastic consensus and is suited for scenarios with significant {\em representation overlap} between the different views. The second method, relying on incremental updates, is tailored for the other extreme scenario with minimal representation overlap. In both cases, we extend o…
The nervous system encodes continuous information from the environment in the form of discrete spikes, and then decodes these to produce smooth motor actions. Understanding how spikes integrate, represent, and process information to produce behavior is one of the greatest challenges in neuroscience. Information theory has the potential to help us address this challenge. Informational analyses of deep and feed-forward artificial neural networks solving static input-output tasks, have led to the proposal of the \emph{Information Bottleneck} principle, which states that deeper layers encode more …
This paper considers the information bottleneck (IB) problem of a Rayleigh fading multiple-input multiple-out (MIMO) channel with an oblivious relay. The relay is constrained to operate without knowledge of the codebooks, i.e., it performs oblivious processing. Moreover, due to the bottleneck constraint, it is impossible for the relay to inform the destination node of the perfect channel state information (CSI) in each channel realization. To evaluate the bottleneck rate, we first provide an upper bound by assuming that the destination node can get the perfect CSI at no cost. Then, we provide …
Extracting relevant information from data is crucial for all forms of learning. The information bottleneck (IB) method formalizes this, offering a mathematically precise and conceptually appealing framework for understanding learning phenomena. However the nonlinearity of the IB problem makes it computationally expensive and analytically intractable in general. Here we derive a perturbation theory for the IB method and report the first complete characterization of the learning onset, the limit of maximum relevant information per bit extracted from data. We test our results on synthetic probabi…
We formulate and analyze the compound information bottleneck programming. In this problem, a Markov chain $ \mathsf{X} \rightarrow \mathsf{Y} \rightarrow \mathsf{Z} $ is assumed with fixed marginal distributions $\mathsf{P}_{\mathsf{X}}$ and $\mathsf{P}_{\mathsf{Y}}$, and the mutual information between $ \mathsf{X} $ and $ \mathsf{Z} $ is sought to be maximized over the choice of conditional probability of $\mathsf{Z}$ given $\mathsf{Y}$ from a given class, under the \textit{worst choice} of the joint probability of the pair $(\mathsf{X},\mathsf{Y})$ from a different class. We consider several…
We investigate a privacy-signaling game problem in which a sender with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The sender aims to hide one of these random vectors and convey the other one whereas the objective of the receiver is to accurately estimate both of the random vectors. We analyze these conflicting objectives in a game theoretic framework with quadratic costs where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg (Bayesian persuasion) equilibria. We show that a payoff dominant Nash equi…
Deep Variational Multivariate Information Bottleneck -- A Framework for Variational Losses
Department of Physics Emory University（埃默里大学物理系）；Departments of Physics and Biology Initiative in Theory and Modeling of Living Systems Emory University（埃默里大学物理与生物学理论与活体系统建模研究所）；Georgia Institute of Technology（佐治亚理工学院）
Lossy compression and clustering fundamentally involve a decision about what features are relevant and which are not. The information bottleneck method (IB) by Tishby, Pereira, and Bialek formalized this notion as an information-theoretic optimization problem and proposed an optimal tradeoff between throwing away as many bits as possible, and selectively keeping those that are most important. In the IB, compression is measure my mutual information. Here, we introduce an alternative formulation that replaces mutual information with entropy, which we call the deterministic information bottleneck…
University of Geneva（日内瓦大学）；Harvard University（哈佛大学）；Imperial College London（帝国理工学院）
Bottleneck problems are an important class of optimization problems that have recently gained increasing attention in the domain of machine learning and information theory. They are widely used in generative models, fair machine learning algorithms, design of privacy-assuring mechanisms, and appear as information-theoretic performance bounds in various multi-user communication problems. In this work, we propose a general family of optimization problems, termed as complexity-leakage-utility bottleneck (CLUB) model, which (i) provides a unified theoretical framework that generalizes most of the …
The new 5G communications standard increases data rates and supports low-latency communication that places constraints on the computational complexity of channel decoders. 5G low-density parity-check (LDPC) codes have the so-called protograph-based raptor-like (PBRL) structure which offers inherent rate-compatibility and excellent performance. Practical LDPC decoder implementations use message-passing decoding with finite precision, which becomes coarse as complexity is more severely constrained. Performance degrades as the precision becomes more coarse. Recently, the information bottleneck (I…
