Emergence of Invariance and Disentanglement in Deep Representations (1706.01350v3)

Abstract: Using established principles from Statistics and Information Theory, we show that invariance to nuisance factors in a deep neural network 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 decompose the cross-entropy loss used during training and highlight the presence of an inherent overfitting term. We propose regularizing the loss by bounding such a term in two equivalent ways: One with a Kullbach-Leibler term, which relates to a PAC-Bayes perspective; the other using the information in the weights as a measure of complexity of a learned model, yielding a novel Information Bottleneck for the weights. Finally, we show that invariance and independence of the components of the representation learned by the network are bounded above and below by the information in the weights, and therefore are implicitly optimized during training. The theory enables us to quantify and predict sharp phase transitions between underfitting and overfitting of random labels when using our regularized loss, which we verify in experiments, and sheds light on the relation between the geometry of the loss function, invariance properties of the learned representation, and generalization error.

• The paper establishes that invariance to nuisance factors is linked to minimal information representations learned naturally through noise injection and layer stacking.
• It proposes a novel Information Bottleneck regularization of the cross-entropy loss, mitigating overfitting by bounding excess information.
• The study quantifies phase transitions between underfitting and overfitting, offering practical insights for enhancing neural network generalization.

Emergence of Invariance and Disentanglement in Deep Representations

Achille and Soatto present an exploration into the mechanisms by which invariance and disentanglement emerge in deep neural networks. Their work deftly utilizes the frameworks of Statistics and Information Theory, revealing critical insights into how neural networks can learn invariant representations despite the lack of explicit architectural enforcement.

Core Contributions

1. Invariance and Information Minimality: The authors establish that a representation's invariance to nuisance factors correlates directly with the representation's information minimality. They propose that neural network training, characterized by stacked layers and noise injection, naturally biases towards learning such minimal representations.
2. Regularization through Cross-Entropy Loss: The paper dissects the standard cross-entropy loss, identifying an inherent overfitting component. It proposes regularizing this loss by incorporating a bound on the overfitting term, introducing a novel Information Bottleneck (IB) approach tailored for weights, analogous to a Kullback-Leibler term from the PAC-Bayes viewpoint.
3. Phase Transitions and Generalization Error: Through theoretical and empirical analysis, the paper quantifies and predicts phase transitions between underfitting and overfitting when training with random labels. This is elegantly linked to the geometry of the loss function and the invariant properties of the learned representations.

Theoretical Insights

The work posits that a task-relevant representation is minimally sufficient and invariant to nuisances when the information content in the representation is minimized. This is mathematically framed and experimentally supported via a series of empirical validations. Specifically:

• Invariance and Minimality: Minimal information representation inherently reduces sensitivity to nuisances. The implication being, that not only do deep networks learn compact representations, but they also naturally discard irrelevant, task-unrelated variations.
• Disentanglement through Total Correlation: Total correlation serves as the actionable measure of disentanglement. Achille and Soatto theorize and validate that minimizing total correlation, a byproduct of reducing information in weights, leads to disentangled representations.

Empirical Validation

Key experiments illustrate the proposed theories. Notably, experiments demonstrate the impact of regularization on overfitting and the consequent representational invariance and disentanglement. Importantly, they establish that random labels demand more information capacity for accurate model representation, predicting a distinct transition phase for overfitting control.

Implications and Future Directions

Practically, this research suggests that implicit biases introduced during training, such as noise injection and layer stacking, contribute significantly to learning invariant and disentangled representations. Theoretically, this calls for a reevaluation of how network complexity is gauged in terms of information content rather than mere parameter count.

Looking ahead, this research invites further exploration into the refinement of training methods to enhance representation learning. Such advancements could significantly impact transfer learning, representation robustness, and the broader understanding of neural network function and optimization.

Conclusion

This work presents a critical theoretical framework for understanding representation learning in deep neural networks. Achille and Soatto's findings underscore the pivotal role of information theory in guiding the design and training of neural networks towards learning representations that are invariant, disentangled, and optimally informative. This contributes not only to the theoretical landscape but also holds considerable potential for practical advancements in AI scalability and effectiveness.