Cauchy-Schwarz Divergence Information Bottleneck for Regression

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}.

MIMIR: Masked Image Modeling for Mutual Information-based Adversarial Robustness

Vision Transformers (ViTs) achieve superior performance on various tasks compared to convolutional neural networks (CNNs), but ViTs are also vulnerable to adversarial attacks. Adversarial training is one of the most successful methods to build robust CNN models. Thus, recent works explored new methodologies for adversarial training of ViTs based on the differences between ViTs and CNNs, such as better training strategies, preventing attention from focusing on a single block, or discarding low-attention embeddings. However, these methods still follow the design of traditional supervised adversarial training, limiting the potential of adversarial training on ViTs. This paper proposes a novel defense method, MIMIR, which aims to build a different adversarial training methodology by utilizing Masked Image Modeling at pre-training. We create an autoencoder that accepts adversarial examples as input but takes the clean examples as the modeling target. Then, we create a mutual information (MI) penalty following the idea of the Information Bottleneck. Among the two information source inputs and corresponding adversarial perturbation, the perturbation information is eliminated due to the constraint of the modeling target. Next, we provide a theoretical analysis of MIMIR using the bounds of the MI penalty. We also design two adaptive attacks when the adversary is aware of the MIMIR defense and show that MIMIR still performs well. The experimental results show that MIMIR improves (natural and adversarial) accuracy on average by 4.19\% on CIFAR-10 and 5.52\% on ImageNet-1K, compared to baselines. On Tiny-ImageNet, we obtained improved natural accuracy of 2.99\% on average and comparable adversarial accuracy. Our code and trained models are publicly available\footnote{\url{https://anonymous.4open.science/r/MIMIR-5444/README.md}}.

Continual Invariant Risk Minimization

Empirical risk minimization can lead to poor generalization behavior on unseen environments if the learned model does not capture invariant feature representations. Invariant risk minimization (IRM) is a recent proposal for discovering environment-invariant representations. IRM was introduced by Arjovsky et al. (2019) and extended by Ahuja et al. (2020). IRM assumes that all environments are available to the learning system at the same time. With this work, we generalize the concept of IRM to scenarios where environments are observed sequentially. We show that existing approaches, including those designed for continual learning, fail to identify the invariant features and models across sequentially presented environments. We extend IRM under a variational Bayesian and bilevel framework, creating a general approach to continual invariant risk minimization. We also describe a strategy to solve the optimization problems using a variant of the alternating direction method of multiplier (ADMM). We show empirically using multiple datasets and with multiple sequential environments that the proposed methods outperform or is competitive with prior approaches.

Revisiting the Robustness of the Minimum Error Entropy Criterion: A Transfer Learning Case Study

Coping with distributional shifts is an important part of transfer learning methods in order to perform well in real-life tasks. However, most of the existing approaches in this area either focus on an ideal scenario in which the data does not contain noises or employ a complicated training paradigm or model design to deal with distributional shifts. In this paper, we revisit the robustness of the minimum error entropy (MEE) criterion, a widely used objective in statistical signal processing to deal with non-Gaussian noises, and investigate its feasibility and usefulness in real-life transfer learning regression tasks, where distributional shifts are common. Specifically, we put forward a new theoretical result showing the robustness of MEE against covariate shift. We also show that by simply replacing the mean squared error (MSE) loss with the MEE on basic transfer learning algorithms such as fine-tuning and linear probing, we can achieve competitive performance with respect to state-of-the-art transfer learning algorithms. We justify our arguments on both synthetic data and 5 real-world time-series data.

Coping with Change: Learning Invariant and Minimum Sufficient Representations for Fine-Grained Visual Categorization

Fine-grained visual categorization (FGVC) is a challenging task due to similar visual appearances between various species. Previous studies always implicitly assume that the training and test data have the same underlying distributions, and that features extracted by modern backbone architectures remain discriminative and generalize well to unseen test data. However, we empirically justify that these conditions are not always true on benchmark datasets. To this end, we combine the merits of invariant risk minimization (IRM) and information bottleneck (IB) principle to learn invariant and minimum sufficient (IMS) representations for FGVC, such that the overall model can always discover the most succinct and consistent fine-grained features. We apply the matrix-based R{\'e}nyi's $\alpha$-order entropy to simplify and stabilize the training of IB; we also design a ``soft" environment partition scheme to make IRM applicable to FGVC task. To the best of our knowledge, we are the first to address the problem of FGVC from a generalization perspective and develop a new information-theoretic solution accordingly. Extensive experiments demonstrate the consistent performance gain offered by our IMS.

The Conditional Cauchy-Schwarz Divergence with Applications to Time-Series Data and Sequential Decision Making

The Cauchy-Schwarz (CS) divergence was developed by Pr\'{i}ncipe et al. in 2000. In this paper, we extend the classic CS divergence to quantify the closeness between two conditional distributions and show that the developed conditional CS divergence can be simply estimated by a kernel density estimator from given samples. We illustrate the advantages (e.g., the rigorous faithfulness guarantee, the lower computational complexity, the higher statistical power, and the much more flexibility in a wide range of applications) of our conditional CS divergence over previous proposals, such as the conditional KL divergence and the conditional maximum mean discrepancy. We also demonstrate the compelling performance of conditional CS divergence in two machine learning tasks related to time series data and sequential inference, namely the time series clustering and the uncertainty-guided exploration for sequential decision making.

Causal Recurrent Variational Autoencoder for Medical Time Series Generation

We propose causal recurrent variational autoencoder (CR-VAE), a novel generative model that is able to learn a Granger causal graph from a multivariate time series x and incorporates the underlying causal mechanism into its data generation process. Distinct to the classical recurrent VAEs, our CR-VAE uses a multi-head decoder, in which the $p$-th head is responsible for generating the $p$-th dimension of $\mathbf{x}$ (i.e., $\mathbf{x}^p$). By imposing a sparsity-inducing penalty on the weights (of the decoder) and encouraging specific sets of weights to be zero, our CR-VAE learns a sparse adjacency matrix that encodes causal relations between all pairs of variables. Thanks to this causal matrix, our decoder strictly obeys the underlying principles of Granger causality, thereby making the data generating process transparent. We develop a two-stage approach to train the overall objective. Empirically, we evaluate the behavior of our model in synthetic data and two real-world human brain datasets involving, respectively, the electroencephalography (EEG) signals and the functional magnetic resonance imaging (fMRI) data. Our model consistently outperforms state-of-the-art time series generative models both qualitatively and quantitatively. Moreover, it also discovers a faithful causal graph with similar or improved accuracy over existing Granger causality-based causal inference methods. Code of CR-VAE is publicly available at https://github.com/hongmingli1995/CR-VAE.

CI-GNN: A Granger Causality-Inspired Graph Neural Network for Interpretable Brain Network-Based Psychiatric Diagnosis

There is a recent trend to leverage the power of graph neural networks (GNNs) for brain-network based psychiatric diagnosis, which,in turn, also motivates an urgent need for psychiatrists to fully understand the decision behavior of the used GNNs. However, most of the existing GNN explainers are either post-hoc in which another interpretive model needs to be created to explain a well-trained GNN, or do not consider the causal relationship between the extracted explanation and the decision, such that the explanation itself contains spurious correlations and suffers from weak faithfulness. In this work, we propose a granger causality-inspired graph neural network (CI-GNN), a built-in interpretable model that is able to identify the most influential subgraph (i.e., functional connectivity within brain regions) that is causally related to the decision (e.g., major depressive disorder patients or healthy controls), without the training of an auxillary interpretive network. CI-GNN learns disentangled subgraph-level representations {\alpha} and \b{eta} that encode, respectively, the causal and noncausal aspects of original graph under a graph variational autoencoder framework, regularized by a conditional mutual information (CMI) constraint. We theoretically justify the validity of the CMI regulation in capturing the causal relationship. We also empirically evaluate the performance of CI-GNN against three baseline GNNs and four state-of-the-art GNN explainers on synthetic data and two large-scale brain disease datasets. We observe that CI-GNN achieves the best performance in a wide range of metrics and provides more reliable and concise explanations which have clinical evidence.

Robust and Fast Measure of Information via Low-rank Representation

The matrix-based R\'enyi's entropy allows us to directly quantify information measures from given data, without explicit estimation of the underlying probability distribution. This intriguing property makes it widely applied in statistical inference and machine learning tasks. However, this information theoretical quantity is not robust against noise in the data, and is computationally prohibitive in large-scale applications. To address these issues, we propose a novel measure of information, termed low-rank matrix-based R\'enyi's entropy, based on low-rank representations of infinitely divisible kernel matrices. The proposed entropy functional inherits the specialty of of the original definition to directly quantify information from data, but enjoys additional advantages including robustness and effective calculation. Specifically, our low-rank variant is more sensitive to informative perturbations induced by changes in underlying distributions, while being insensitive to uninformative ones caused by noises. Moreover, low-rank R\'enyi's entropy can be efficiently approximated by random projection and Lanczos iteration techniques, reducing the overall complexity from $\mathcal{O}(n^3)$ to $\mathcal{O}(n^2 s)$ or even $\mathcal{O}(ns^2)$, where $n$ is the number of data samples and $s \ll n$. We conduct large-scale experiments to evaluate the effectiveness of this new information measure, demonstrating superior results compared to matrix-based R\'enyi's entropy in terms of both performance and computational efficiency.

Information-Theoretic Hashing for Zero-Shot Cross-Modal Retrieval

Zero-shot cross-modal retrieval (ZS-CMR) deals with the retrieval problem among heterogenous data from unseen classes. Typically, to guarantee generalization, the pre-defined class embeddings from natural language processing (NLP) models are used to build a common space. In this paper, instead of using an extra NLP model to define a common space beforehand, we consider a totally different way to construct (or learn) a common hamming space from an information-theoretic perspective. We term our model the Information-Theoretic Hashing (ITH), which is composed of two cascading modules: an Adaptive Information Aggregation (AIA) module; and a Semantic Preserving Encoding (SPE) module. Specifically, our AIA module takes the inspiration from the Principle of Relevant Information (PRI) to construct a common space that adaptively aggregates the intrinsic semantics of different modalities of data and filters out redundant or irrelevant information. On the other hand, our SPE module further generates the hashing codes of different modalities by preserving the similarity of intrinsic semantics with the element-wise Kullback-Leibler (KL) divergence. A total correlation regularization term is also imposed to reduce the redundancy amongst different dimensions of hash codes. Sufficient experiments on three benchmark datasets demonstrate the superiority of the proposed ITH in ZS-CMR. Source code is available in the supplementary material.