Emergence of Invariance and Disentanglement in Deep Representations

By Alessandro Achille, Stefano Soatto. 2017

566 citations recorded by the source corpus.

Using established principles from Information Theory and Statistics, we show that in a deep neural network invariance to nuisance factors is equivalent to information minimality of the learned representation, and that stacking layers and injecting noise during training naturally bias the network towards learning invariant representations. We then show that, in order to avoid memorization, we need to limit the quantity of information stored in the weights, which leads to a novel usage of the Information Bottleneck Lagrangian on the weights as a learning criterion. This also has an alternative interpretation as minimizing a PAC-Bayesian bound on the test error. Finally, we exploit a duality between weights and activations induced by the architecture, to show that the information in the weights bounds the minimality and Total Correlation of the layers, therefore showing that regularizing the weights explicitly or implicitly, using SGD, not only helps avoid overfitting, but also fosters invariance and disentangling of the learned representation. The theory also enables predicting sharp phase transitions between underfitting and overfitting random labels at precise information values, and sheds light on the relation between the geometry of the loss function, in particular so-called “flat minima,” and generalization.

Subfields: Artificial Intelligence, Computer Vision and Pattern Recognition, Representation & Self-Supervised Learning

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Citations in the corpus

Towards a Definition of Disentangled Representations (2018, background): “The term “disentangling" is used frequently in this line of research, including later work in a deep learning framework [2], but in this context it is meant more in the sense of disentangling (and throwing away) nuisance variables and preserving task-relevant information.”
Excessive Invariance Causes Adversarial Vulnerability (2018, background): “The role of invariance has been studied extensively in the context of deep neural networks (Bruna & Mallat, 2013; Mallat, 2016; Achille & Soatto, 2018; Behrmann et al., 2018; Cadena et al., 2018).”
i-RevNet: Deep Invertible Networks (2018, background): “It is widely believed that this process is based on progressively discarding uninformative variability about the input with respect to the problem at hand (Dosovitskiy & Brox, 2016; Mahendran & Vedaldi, 2016; Shwartz-Ziv & Tishby, 2017; Achille & Soatto, 2017).”
Where is the Information in a Deep Neural Network? (2019, methodology): “This is the missing link to invariance since, as shown by [3], a sufﬁ-cient representation is invariant if and only if it is minimal.”
A Bayesian Perspective on Generalization and Stochastic Gradient Descent (2017, neutral): “The Bayesian evidence can be reframed in the language of information theory, whereby Occam’s factor penalizes the amount of information the model must learn about the parameters to accurately model the training data (Hinton & Van Camp, 1993; Achille & Soatto, 2017; Shwartz-Ziv & Tishby, 2017).”
Task2Vec: Task Embedding for Meta-Learning (2019, background): “In particular, “ﬂat minima” correspond to weights that have, on average, low (Fisher) information [5, 13].”
Adversarial Latent Autoencoders (2020, background): “Indeed, we recall that a dis-entangled representation [45, 42, 1] refers to a space consisting of linear subspaces, each of which is responsible for one factor of variation.”
Mutual Information Neural Estimation (2018, methodology): “This is done by minimizing the IB Lagrangian, which appears as a standard cross-entropy loss augmented with a regularizer promoting minimality of the representation (Achille & Soatto, 2017).”
Stochastic Gradient Descent Performs Variational Inference, Converges to Limit Cycles for Deep Networks (2017, background): “Minimizing this functional, explicitly, has been shown to lead to invariant representations (Achille and Soatto, 2017).”
Intrinsic dimension of data representations in deep neural networks (2019, background): “One can speculate that, in a trained deep network, the first layers prune the irrelevant features, formatting the representation for the more advanced processing carried out by the last layers [20].”
An Information-Theoretic Analysis of Deep Latent-Variable Models (2017, background): “The information bottleneck framework (Tishby et al., 1999; Shamir et al., 2010; Tishby & Zaslavsky, 2015; Alemi et al., 2017; Achille & Soatto, 2016; 2017) allows a model to smoothly trade off the minimality of the learned representation ( Z ) from data ( X ) by minimizing their mutual information,…”
Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle (2018, background): “The authors of [21] focused on formulating a similar set of desired properties in terms of information-theoretic objectives.”
The Information Complexity of Learning Tasks, their Structure and their Distance (2019, background): “Nevertheless, it is interesting that the information in the parameters specializes to Shannon’s mutual information in the weights [4].”
Why ResNet Works? Residuals Generalize (2019, unknown): “Section VII concludes this article.”
Language is Power: Representing States Using Natural Language in Reinforcement Learning. (2019, background): “Our results suggest that representation of the state using natural language can achieve better performance, even on difﬁcult tasks, or tasks in which the description of the state is saturated with task-nuisances [Achille and Soatto, 2018].”
A3T: Adversarially Augmented Adversarial Training (2018, background): “Achille and Soatto (2017) takes the view from information theory to a regularizing term added to the classiﬁer loss, which ﬁlters out the information of a given class of ‘nuisances’, i.e variations of the inputs that are irrelevant to the task.”