The information bottleneck theory was recently used to study Deep Neural Networks. Shwartz-Ziv and Tishby proposed that the information bottleneck expresses the tradeoff between the mutual information measures \(I(X,T)\) and \(I(T,Y)\) (Shwartz-Ziv & Tishby, 2017).

\(I(X,T)\) and \(I(X,Y)\) quantifies the amount of information that the layer contains about the input and output respectively.

## Key Findings from (Shwartz-Ziv & Tishby, 2017)

- Most of the training epochs are spent on compression of the input to efficient representation, and not on fitting the training labels
- Th representation compression phase begins when training errors become small, and SGD epochs change from a fast drift to a smaller training error into a stochastic relaxation or random diffusion, constrained by the training error value.
- The converged layers lie on or very close to the Information Botteneck (IB) theoretical bound, and the maps from the input to any hidden layer and from this hidden layer to the output satisfy the IB self-consistent equations.
- The generalization-through-noise mechanism is unique to DNNs
- The training time is dramatically reduced when adding more layers,
*hence the main advantage of adding layers is computational*??

TL;DR: SGD has 2 phases: ERM, and representation compression. The latter phase is an explanation for the lack of overfitting in DL.

# Bibliography

Shwartz-Ziv, R., & Tishby, N., *Opening the black box of deep neural networks via information*, CoRR, *()*, (2017). ↩