The information plane (Tishby et al. arXiv:physics/0004057, Shwartz-Ziv et al. arXiv:1703.00810) has been proposed as an analytical tool for studying the learning dynamics of neural networks. It provides quantitative insight on how the model approaches the learned state by approximating a minimal sufficient statistics. In this paper we extend this tool to the domain of quantum learning models. In a second step, we study how the insight on how much the model compresses the input data (provided by the information plane) can be used to improve a learning algorithm. Specifically, we consider two ways to do so: via a multiplicative regularization of the loss function, or with a compression-gnostic scheduler of the learning rate (for algorithms based on gradient descent). Both ways turn out to be equivalent in our implementation. Finally, we benchmark the proposed learning algorithms on several classification and regression tasks using variational quantum circuits. The results demonstrate an improvement in test accuracy and convergence speed for both synthetic and real-world datasets. Additionally, with one example we analyzed the impact of the proposed modifications on the performances of neural networks in a classification task.
The Information Plane is a conceptual framework used to analyze the flow of information in neural networks, but traditional methods based on activations may not fully capture the dynamics of information processing. This paper introduces a new approach that uses layer conductance, a measure of sensitivity to input features, to enhance the Information Plane analysis. By incorporating gradient-based contributions, we provide a more precise characterization of information dynamics within the network. The proposed conductance-based Information Plane and a new Information Transformation Efficiency (ITE) metric are evaluated on pretrained ResNet50 and VGG16 models using the ImageNet dataset. Our results demonstrate the ability to identify critical hidden layers that contribute significantly to model performance and interpretability, giving insights into information compression, preservation, and utilization across layers. The conductance-based approach offers a granular perspective on feature attribution, enhancing our understanding of the decision-making processes within neural networks. Furthermore, our empirical findings challenge certain theoretical predictions of the Information Bottleneck theory, highlighting the complexities of information dynamics in real-world data scenarios. The proposed method not only advances our understanding of information dynamics in neural networks but also has the potential to significantly impact the broader field of Artificial Intelligence by enabling the development of more interpretable, efficient, and robust models.
We propose a novel, lightweight supervised dictionary learning framework for text classification based on data compression and representation. This two-phase algorithm initially employs the Lempel-Ziv-Welch (LZW) algorithm to construct a dictionary from text datasets, focusing on the conceptual significance of dictionary elements. Subsequently, dictionaries are refined considering label data, optimizing dictionary atoms to enhance discriminative power based on mutual information and class distribution. This process generates discriminative numerical representations, facilitating the training of simple classifiers such as SVMs and neural networks. We evaluate our algorithm's information-theoretic performance using information bottleneck principles and introduce the information plane area rank (IPAR) as a novel metric to quantify the information-theoretic performance. Tested on six benchmark text datasets, our algorithm competes closely with top models, especially in limited-vocabulary contexts, using significantly fewer parameters. \review{Our algorithm closely matches top-performing models, deviating by only ~2\% on limited-vocabulary datasets, using just 10\% of their parameters. However, it falls short on diverse-vocabulary datasets, likely due to the LZW algorithm's constraints with low-repetition data. This contrast highlights its efficiency and limitations across different dataset types.
The information bottleneck (IB) approach is popular to improve the generalization, robustness and explainability of deep neural networks. Essentially, it aims to find a minimum sufficient representation $\mathbf{t}$ by striking a trade-off between a compression term $I(\mathbf{x};\mathbf{t})$ and a prediction term $I(y;\mathbf{t})$, where $I(\cdot;\cdot)$ refers to the mutual information (MI). MI is for the IB for the most part expressed in terms of the Kullback-Leibler (KL) divergence, which in the regression case corresponds to prediction based on mean squared error (MSE) loss with Gaussian assumption and compression approximated by variational inference. In this paper, we study the IB principle for the regression problem and develop a new way to parameterize the IB with deep neural networks by exploiting favorable properties of the Cauchy-Schwarz (CS) divergence. By doing so, we move away from MSE-based regression and ease estimation by avoiding variational approximations or distributional assumptions. We investigate the improved generalization ability of our proposed CS-IB and demonstrate strong adversarial robustness guarantees. We demonstrate its superior performance on six real-world regression tasks over other popular deep IB approaches. We additionally observe that the solutions discovered by CS-IB always achieve the best trade-off between prediction accuracy and compression ratio in the information plane. The code is available at \url{https://github.com/SJYuCNEL/Cauchy-Schwarz-Information-Bottleneck}.
In a feedforward network, Transfer Entropy (TE) can be used to measure the influence that one layer has on another by quantifying the information transfer between them during training. According to the Information Bottleneck principle, a neural model's internal representation should compress the input data as much as possible while still retaining sufficient information about the output. Information Plane analysis is a visualization technique used to understand the trade-off between compression and information preservation in the context of the Information Bottleneck method by plotting the amount of information in the input data against the compressed representation. The claim that there is a causal link between information-theoretic compression and generalization, measured by mutual information, is plausible, but results from different studies are conflicting. In contrast to mutual information, TE can capture temporal relationships between variables. To explore such links, in our novel approach we use TE to quantify information transfer between neural layers and perform Information Plane analysis. We obtained encouraging experimental results, opening the possibility for further investigations.
End-to-end (E2E) training, optimizing the entire model through error backpropagation, fundamentally supports the advancements of deep learning. Despite its high performance, E2E training faces the problems of memory consumption, parallel computing, and discrepancy with the functionalities of the actual brain. Various alternative methods have been proposed to overcome these difficulties; however, no one can yet match the performance of E2E training, thereby falling short in practicality. Furthermore, there is no deep understanding regarding differences in the trained model properties beyond the performance gap. In this paper, we reconsider why E2E training demonstrates a superior performance through a comparison with layer-wise training, a non-E2E method that locally sets errors. On the basis of the observation that E2E training has an advantage in propagating input information, we analyze the information plane dynamics of intermediate representations based on the Hilbert-Schmidt independence criterion (HSIC). The results of our normalized HSIC value analysis reveal the E2E training ability to exhibit different information dynamics across layers, in addition to efficient information propagation. Furthermore, we show that this layer-role differentiation leads to the final representation following the information bottleneck principle. It suggests the need to consider the cooperative interactions between layers, not just the final layer when analyzing the information bottleneck of deep learning.
Most real-world applications that employ deep neural networks (DNNs) quantize them to low precision to reduce the compute needs. We present a method to improve the robustness of quantized DNNs to white-box adversarial attacks. We first tackle the limitation of deterministic quantization to fixed ``bins'' by introducing a differentiable Stochastic Quantizer (SQ). We explore the hypothesis that different quantizations may collectively be more robust than each quantized DNN. We formulate a training objective to encourage different quantized DNNs to learn different representations of the input image. The training objective captures diversity and accuracy via mutual information between ensemble members. Through experimentation, we demonstrate substantial improvement in robustness against $L_\infty$ attacks even if the attacker is allowed to backpropagate through SQ (e.g., > 50\% accuracy to PGD(5/255) on CIFAR10 without adversarial training), compared to vanilla DNNs as well as existing ensembles of quantized DNNs. We extend the method to detect attacks and generate robustness profiles in the adversarial information plane (AIP), towards a unified analysis of different threat models by correlating the MI and accuracy.
To avoid discriminatory uses of their data, organizations can learn to map them into a representation that filters out information related to sensitive attributes. However, all existing methods in fair representation learning generate a fairness-information trade-off. To achieve different points on the fairness-information plane, one must train different models. In this paper, we first demonstrate that fairness-information trade-offs are fully characterized by rate-distortion trade-offs. Then, we use this key result and propose SoFaiR, a single shot fair representation learning method that generates with one trained model many points on the fairness-information plane. Besides its computational saving, our single-shot approach is, to the extent of our knowledge, the first fair representation learning method that explains what information is affected by changes in the fairness / distortion properties of the representation. Empirically, we find on three datasets that SoFaiR achieves similar fairness-information trade-offs as its multi-shot counterparts.
Measuring model performance is a key issue for deep learning practitioners. However, we often lack the ability to explain why a specific architecture attains superior predictive accuracy for a given data set. Often, validation accuracy is used as a performance heuristic quantifying how well a network generalizes to unseen data, but it does not capture anything about the information flow in the model. Mutual information can be used as a measure of the quality of internal representations in deep learning models, and the information plane may provide insights into whether the model exploits the available information in the data. The information plane has previously been explored for fully connected neural networks and convolutional architectures. We present an architecture-agnostic method for tracking a network's internal representations during training, which are then used to create the mutual information plane. The method is exemplified for graph-based neural networks fitted on citation data. We compare how the inductive bias introduced in graph-based architectures changes the mutual information plane relative to a fully connected neural network.
This paper proposes an Information Bottleneck theory based filter pruning method that uses a statistical measure called Mutual Information (MI). The MI between filters and class labels, also called \textit{Relevance}, is computed using the filter's activation maps and the annotations. The filters having High Relevance (HRel) are considered to be more important. Consequently, the least important filters, which have lower Mutual Information with the class labels, are pruned. Unlike the existing MI based pruning methods, the proposed method determines the significance of the filters purely based on their corresponding activation map's relationship with the class labels. Architectures such as LeNet-5, VGG-16, ResNet-56\textcolor{myblue}{, ResNet-110 and ResNet-50 are utilized to demonstrate the efficacy of the proposed pruning method over MNIST, CIFAR-10 and ImageNet datasets. The proposed method shows the state-of-the-art pruning results for LeNet-5, VGG-16, ResNet-56, ResNet-110 and ResNet-50 architectures. In the experiments, we prune 97.98 \%, 84.85 \%, 76.89\%, 76.95\%, and 63.99\% of Floating Point Operation (FLOP)s from LeNet-5, VGG-16, ResNet-56, ResNet-110, and ResNet-50 respectively.} The proposed HRel pruning method outperforms recent state-of-the-art filter pruning methods. Even after pruning the filters from convolutional layers of LeNet-5 drastically (i.e. from 20, 50 to 2, 3, respectively), only a small accuracy drop of 0.52\% is observed. Notably, for VGG-16, 94.98\% parameters are reduced, only with a drop of 0.36\% in top-1 accuracy. \textcolor{myblue}{ResNet-50 has shown a 1.17\% drop in the top-5 accuracy after pruning 66.42\% of the FLOPs.} In addition to pruning, the Information Plane dynamics of Information Bottleneck theory is analyzed for various Convolutional Neural Network architectures with the effect of pruning.
From its first emergence in Wuhan, China in December, 2019 the COVID-19 pandemic has caused unprecedented health crisis throughout the world. The novel coronavirus disease is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) which belongs to the coronaviridae family. In this paper, a comparative genomic analysis of eight coronaviruses namely Human coronavirus OC43 (HCoV-OC43), Human coronavirus HKU1 (HCoV-HKU1), Human coronavirus 229E (HCoV-229E), Human coronavirus NL63 (HCoV-NL63), Severe acute respiratory syndrome coronavirus (SARS-CoV), Middle East respiratory syndrome-related coronavirus (MERS-CoV), Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and Bat coronavirus RaTG13 has been carried out using Chaos Game Representation and Fisher-Shannon Complexity (CGR-FSC) measure. Chaos Game Representation (CGR) is a unique alignment-free method to visualize one dimensional DNA sequence in a two-dimensional fractal-like pattern. The two-dimensional CGR pattern is then quantified by Fisher-Shannon Complexity (FSC) measure. The CGR-FSC can effectively identify the viruses uniquely and their similarity/dissimilarity can be revealed in the Fisher-Shannon Information Plane (FSIP).
The outstanding performance of deep learning in various fields has been a fundamental query, which can be potentially examined using information theory that interprets the learning process as the transmission and compression of information. Information plane analyses of the mutual information between the input-hidden-output layers demonstrated two distinct learning phases of fitting and compression. It is debatable if the compression phase is necessary to generalize the input-output relations extracted from training data. In this study, we investigated this through experiments with various species of autoencoders and evaluated their information processing phase with an accurate kernel-based estimator of mutual information. Given sufficient training data, vanilla autoencoders demonstrated the compression phase, which was amplified after imposing sparsity regularization for hidden activities. However, we found that the compression phase is not universally observed in different species of autoencoders, including variational autoencoders, that have special constraints on network weights or manifold of hidden space. These types of autoencoders exhibited perfect generalization ability for test data without requiring the compression phase. Thus, we conclude that the compression phase is not necessary for generalization in representation learning.
The Information bottleneck (IB) method enables optimizing over the trade-off between compression of data and prediction accuracy of learned representations, and has successfully and robustly been applied to both supervised and unsupervised representation learning problems. However, IB has several limitations. First, the IB problem is hard to optimize. The IB Lagrangian $\mathcal{L}_{IB}:=I(X;Z)-\beta I(Y;Z)$ is non-convex and existing solutions guarantee only local convergence. As a result, the obtained solutions depend on initialization. Second, the evaluation of a solution is also a challenging task. Conventionally, it resorts to characterizing the information plane, that is, plotting $I(Y;Z)$ versus $I(X;Z)$ for all solutions obtained from different initial points. Furthermore, the IB Lagrangian has phase transitions while varying the multiplier $\beta$. At phase transitions, both $I(X;Z)$ and $I(Y;Z)$ increase abruptly and the rate of convergence becomes significantly slow for existing solutions. Recent works with IB adopt variational surrogate bounds to the IB Lagrangian. Although allowing efficient optimization, how close are these surrogates to the IB Lagrangian is not clear. In this work, we solve the IB Lagrangian using augmented Lagrangian methods. With augmented variables, we show that the IB objective can be solved with the alternating direction method of multipliers (ADMM). Different from prior works, we prove that the proposed algorithm is consistently convergent, regardless of the value of $\beta$. Empirically, our gradient-descent-based method results in information plane points that are comparable to those obtained through the conventional Blahut-Arimoto-based solvers and is convergent for a wider range of the penalty coefficient than previous ADMM solvers.
A wide range of machine learning applications such as privacy-preserving learning, algorithmic fairness, and domain adaptation/generalization among others, involve learning invariant representations of the data that aim to achieve two competing goals: (a) maximize information or accuracy with respect to a target response, and (b) maximize invariance or independence with respect to a set of protected features (e.g., for fairness, privacy, etc). Despite their wide applicability, theoretical understanding of the optimal tradeoffs -- with respect to accuracy, and invariance -- achievable by invariant representations is still severely lacking. In this paper, we provide an information theoretic analysis of such tradeoffs under both classification and regression settings. More precisely, we provide a geometric characterization of the accuracy and invariance achievable by any representation of the data; we term this feasible region the information plane. We provide an inner bound for this feasible region for the classification case, and an exact characterization for the regression case, which allows us to either bound or exactly characterize the Pareto optimal frontier between accuracy and invariance. Although our contributions are mainly theoretical, a key practical application of our results is in certifying the potential sub-optimality of any given representation learning algorithm for either classification or regression tasks. Our results shed new light on the fundamental interplay between accuracy and invariance, and may be useful in guiding the design of future representation learning algorithms.
The attention that deep learning has garnered from the academic community and industry continues to grow year over year, and it has been said that we are in a new golden age of artificial intelligence research. However, neural networks are still often seen as a "black box" where learning occurs but cannot be understood in a human-interpretable way. Since these machine learning systems are increasingly being adopted in security contexts, it is important to explore these interpretations. We consider an Android malware traffic dataset for approaching this problem. Then, using the information plane, we explore how homeomorphism affects learned representation of the data and the invariance of the mutual information captured by the parameters on that data. We empirically validate these results, using accuracy as a second measure of similarity of learned representations. Our results suggest that although the details of learned representations and the specific coordinate system defined over the manifold of all parameters differ slightly, the functional approximations are the same. Furthermore, our results show that since mutual information remains invariant under homeomorphism, only feature engineering methods that alter the entropy of the dataset will change the outcome of the neural network. This means that for some datasets and tasks, neural networks require meaningful, human-driven feature engineering or changes in architecture to provide enough information for the neural network to generate a sufficient statistic. Applying our results can serve to guide analysis methods for machine learning engineers and suggests that neural networks that can exploit the convolution theorem are equally accurate as standard convolutional neural networks, and can be more computationally efficient.
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 large scale problems, we present a novel variational formulation of the dualIB for Deep Neural Networks. In experiments on several data-sets, we compare it to a variational form of the IB. This exposes superior Information Plane properties of the dualIB and its potential in improvement of the error.
The training dynamics of hidden layers in deep learning are poorly understood in theory. Recently, the Information Plane (IP) was proposed to analyze them, which is based on the information-theoretic concept of mutual information (MI). The Information Bottleneck (IB) theory predicts that layers maximize relevant information and compress irrelevant information. Due to the limitations in MI estimation from samples, there is an ongoing debate about the properties of the IP for the supervised learning case. In this work, we derive a theoretical convergence for the IP of autoencoders. The theory predicts that ideal autoencoders with a large bottleneck layer size do not compress input information, whereas a small size causes compression only in the encoder layers. For the experiments, we use a Gram-matrix based MI estimator recently proposed in the literature. We propose a new rule to adjust its parameters that compensates scale and dimensionality effects. Using our proposed rule, we obtain experimental IPs closer to the theory. Our theoretical IP for autoencoders could be used as a benchmark to validate new methods to estimate MI in neural networks. In this way, experimental limitations could be recognized and corrected, helping with the ongoing debate on the supervised learning case.
We review the current literature concerned with information plane analyses of neural network classifiers. While the underlying information bottleneck theory and the claim that information-theoretic compression is causally linked to generalization are plausible, empirical evidence was found to be both supporting and conflicting. We review this evidence together with a detailed analysis of how the respective information quantities were estimated. Our survey suggests that compression visualized in information planes is not necessarily information-theoretic, but is rather often compatible with geometric compression of the latent representations. This insight gives the information plane a renewed justification. Aside from this, we shed light on the problem of estimating mutual information in deterministic neural networks and its consequences. Specifically, we argue that even in feed-forward neural networks the data processing inequality need not hold for estimates of mutual information. Similarly, while a fitting phase, in which the mutual information between the latent representation and the target increases, is necessary (but not sufficient) for good classification performance, depending on the specifics of mutual information estimation such a fitting phase need not be visible in the information plane.
Certain biological neurons demonstrate a remarkable capability to optimally compress the history of sensory inputs while being maximally informative about the future. In this work, we investigate if the same can be said of artificial neurons in recurrent neural networks (RNNs) trained with maximum likelihood. Empirically, we find that RNNs are suboptimal in the information plane. Instead of optimally compressing past information, they extract additional information that is not relevant for predicting the future. We show that constraining past information by injecting noise into the hidden state can improve RNNs in several ways: optimality in the predictive information plane, sample quality, heldout likelihood, and downstream classification performance.
Analyzing deep neural networks (DNNs) via information plane (IP) theory has gained tremendous attention recently as a tool to gain insight into, among others, their generalization ability. However, it is by no means obvious how to estimate mutual information (MI) between each hidden layer and the input/desired output, to construct the IP. For instance, hidden layers with many neurons require MI estimators with robustness towards the high dimensionality associated with such layers. MI estimators should also be able to naturally handle convolutional layers, while at the same time being computationally tractable to scale to large networks. None of the existing IP methods to date have been able to study truly deep Convolutional Neural Networks (CNNs), such as the e.g.\ VGG-16. In this paper, we propose an IP analysis using the new matrix--based R\'enyi's entropy coupled with tensor kernels over convolutional layers, leveraging the power of kernel methods to represent properties of the probability distribution independently of the dimensionality of the data. The obtained results shed new light on the previous literature concerning small-scale DNNs, however using a completely new approach. Importantly, the new framework enables us to provide the first comprehensive IP analysis of contemporary large-scale DNNs and CNNs, investigating the different training phases and providing new insights into the training dynamics of large-scale neural networks.