Information-theoretic analysis of generalization capability of learning algorithms

We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems, and give theoretical guidelines for striking the right balance between data fit and generalization by controlling the input-output mutual information. We propose a number of methods for this purpose, among which are algorithms that regularize the ERM algorithm with relative entropy or with random noise. Our work extends and leads to nontrivial improvements on the recent results of Russo and Zou.Comment: Final version, accepted to NIPS 201

Average-Case Information Complexity of Learning

How many bits of information are revealed by a learning algorithm for a concept class of VC-dimension $d$? Previous works have shown that even for $d=1$ the amount of information may be unbounded (tend to $\infty$ with the universe size). Can it be that all concepts in the class require leaking a large amount of information? We show that typically concepts do not require leakage. There exists a proper learning algorithm that reveals $O(d)$ bits of information for most concepts in the class. This result is a special case of a more general phenomenon we explore. If there is a low information learner when the algorithm {\em knows} the underlying distribution on inputs, then there is a learner that reveals little information on an average concept {\em without knowing} the distribution on inputs

Quantization-Based Regularization for Autoencoders

Autoencoders and their variations provide unsupervised models for learning low-dimensional representations for downstream tasks. Without proper regularization, autoencoder models are susceptible to the overfitting problem and the so-called posterior collapse phenomenon. In this paper, we introduce a quantization-based regularizer in the bottleneck stage of autoencoder models to learn meaningful latent representations. We combine both perspectives of Vector Quantized-Variational AutoEncoders (VQ-VAE) and classical denoising regularization methods of neural networks. We interpret quantizers as regularizers that constrain latent representations while fostering a similarity-preserving mapping at the encoder. Before quantization, we impose noise on the latent codes and use a Bayesian estimator to optimize the quantizer-based representation. The introduced bottleneck Bayesian estimator outputs the posterior mean of the centroids to the decoder, and thus, is performing soft quantization of the noisy latent codes. We show that our proposed regularization method results in improved latent representations for both supervised learning and clustering downstream tasks when compared to autoencoders using other bottleneck structures.Comment: AAAI 202

SHADE: Information Based Regularization for Deep Learning

Regularization is a big issue for training deep neural networks. In this paper, we propose a new information-theory-based regularization scheme named SHADE for SHAnnon DEcay. The originality of the approach is to define a prior based on conditional entropy, which explicitly decouples the learning of invariant representations in the regularizer and the learning of correlations between inputs and labels in the data fitting term. Our second contribution is to derive a stochastic version of the regularizer compatible with deep learning, resulting in a tractable training scheme. We empirically validate the efficiency of our approach to improve classification performances compared to common regularization schemes on several standard architectures

Strengthened Information-theoretic Bounds on the Generalization Error

The following problem is considered: given a joint distribution $P_{XY}$ and an event $E$, bound $P_{XY}(E)$ in terms of $P_XP_Y(E)$ (where $P_XP_Y$ is the product of the marginals of $P_{XY}$) and a measure of dependence of $X$ and $Y$. Such bounds have direct applications in the analysis of the generalization error of learning algorithms, where $E$ represents a large error event and the measure of dependence controls the degree of overfitting. Herein, bounds are demonstrated using several information-theoretic metrics, in particular: mutual information, lautum information, maximal leakage, and $J_\infty$. The mutual information bound can outperform comparable bounds in the literature by an arbitrarily large factor.Comment: Submitted to ISIT 201

Chaining Mutual Information and Tightening Generalization Bounds

Bounding the generalization error of learning algorithms has a long history, which yet falls short in explaining various generalization successes including those of deep learning. Two important difficulties are (i) exploiting the dependencies between the hypotheses, (ii) exploiting the dependence between the algorithm's input and output. Progress on the first point was made with the chaining method, originating from the work of Kolmogorov, and used in the VC-dimension bound. More recently, progress on the second point was made with the mutual information method by Russo and Zou '15. Yet, these two methods are currently disjoint. In this paper, we introduce a technique to combine the chaining and mutual information methods, to obtain a generalization bound that is both algorithm-dependent and that exploits the dependencies between the hypotheses. We provide an example in which our bound significantly outperforms both the chaining and the mutual information bounds. As a corollary, we tighten Dudley's inequality when the learning algorithm chooses its output from a small subset of hypotheses with high probability.Comment: 20 pages, 1 figure; published at the NeurIPS 2018 conferenc

An Optimal Transport View on Generalization

We derive upper bounds on the generalization error of learning algorithms based on their \emph{algorithmic transport cost}: the expected Wasserstein distance between the output hypothesis and the output hypothesis conditioned on an input example. The bounds provide a novel approach to study the generalization of learning algorithms from an optimal transport view and impose less constraints on the loss function, such as sub-gaussian or bounded. We further provide several upper bounds on the algorithmic transport cost in terms of total variation distance, relative entropy (or KL-divergence), and VC dimension, thus further bridging optimal transport theory and information theory with statistical learning theory. Moreover, we also study different conditions for loss functions under which the generalization error of a learning algorithm can be upper bounded by different probability metrics between distributions relating to the output hypothesis and/or the input data. Finally, under our established framework, we analyze the generalization in deep learning and conclude that the generalization error in deep neural networks (DNNs) decreases exponentially to zero as the number of layers increases. Our analyses of generalization error in deep learning mainly exploit the hierarchical structure in DNNs and the contraction property of $f$-divergence, which may be of independent interest in analyzing other learning models with hierarchical structure.Comment: 27 pages, 2 figures, 1 tabl

An Information-Theoretic View for Deep Learning

Deep learning has transformed computer vision, natural language processing, and speech recognition\cite{badrinarayanan2017segnet, dong2016image, ren2017faster, ji20133d}. However, two critical questions remain obscure: (1) why do deep neural networks generalize better than shallow networks; and (2) does it always hold that a deeper network leads to better performance? Specifically, letting $L$ be the number of convolutional and pooling layers in a deep neural network, and $n$ be the size of the training sample, we derive an upper bound on the expected generalization error for this network, i.e., \begin{eqnarray*} \mathbb{E}[R(W)-R_S(W)] \leq \exp{\left(-\frac{L}{2}\log{\frac{1}{\eta}}\right)}\sqrt{\frac{2\sigma^2}{n}I(S,W) } \end{eqnarray*} where $\sigma >0$ is a constant depending on the loss function, $0<\eta<1$ is a constant depending on the information loss for each convolutional or pooling layer, and $I(S, W)$ is the mutual information between the training sample $S$ and the output hypothesis $W$. This upper bound shows that as the number of convolutional and pooling layers $L$ increases in the network, the expected generalization error will decrease exponentially to zero. Layers with strict information loss, such as the convolutional layers, reduce the generalization error for the whole network; this answers the first question. However, algorithms with zero expected generalization error does not imply a small test error or $\mathbb{E}[R(W)]$. This is because $\mathbb{E}[R_S(W)]$ is large when the information for fitting the data is lost as the number of layers increases. This suggests that the claim `the deeper the better' is conditioned on a small training error or $\mathbb{E}[R_S(W)]$. Finally, we show that deep learning satisfies a weak notion of stability and the sample complexity of deep neural networks will decrease as $L$ increases.Comment: Add details in the proof of Theorem

Tightening Mutual Information Based Bounds on Generalization Error

An information-theoretic upper bound on the generalization error of supervised learning algorithms is derived. The bound is constructed in terms of the mutual information between each individual training sample and the output of the learning algorithm. The bound is derived under more general conditions on the loss function than in existing studies; nevertheless, it provides a tighter characterization of the generalization error. Examples of learning algorithms are provided to demonstrate the the tightness of the bound, and to show that it has a broad range of applicability. Application to noisy and iterative algorithms, e.g., stochastic gradient Langevin dynamics (SGLD), is also studied, where the constructed bound provides a tighter characterization of the generalization error than existing results. Finally, it is demonstrated that, unlike existing bounds, which are difficult to compute and evaluate empirically, the proposed bound can be estimated easily in practice

Learning Invariant Feature Representation to Improve Generalization across Chest X-ray Datasets

Chest radiography is the most common medical image examination for screening and diagnosis in hospitals. Automatic interpretation of chest X-rays at the level of an entry-level radiologist can greatly benefit work prioritization and assist in analyzing a larger population. Subsequently, several datasets and deep learning-based solutions have been proposed to identify diseases based on chest X-ray images. However, these methods are shown to be vulnerable to shift in the source of data: a deep learning model performing well when tested on the same dataset as training data, starts to perform poorly when it is tested on a dataset from a different source. In this work, we address this challenge of generalization to a new source by forcing the network to learn a source-invariant representation. By employing an adversarial training strategy, we show that a network can be forced to learn a source-invariant representation. Through pneumonia-classification experiments on multi-source chest X-ray datasets, we show that this algorithm helps in improving classification accuracy on a new source of X-ray dataset.Comment: Accepted to Machine Learning in Medical Imaging (MLMI 2020), in conjunction with MICCAI 2020, Oct. 4, 202