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Deep Learning Through the Information Bottleneck

By Bbenzon @bbenzon

Tishby began contemplating the information bottleneck around the time that other researchers were first mulling over deep neural networks, though neither concept had been named yet. It was the 1980s, and Tishby was thinking about how good humans are at speech recognition - a major challenge for AI at the time. Tishby realized that the crux of the issue was the question of relevance: What are the most relevant features of a spoken word, and how do we tease these out from the variables that accompany them, such as accents, mumbling and intonation? In general, when we face the sea of data that is reality, which signals do we keep?

"This notion of relevant information was mentioned many times in history but never formulated correctly," Tishby said in an interview last month. "For many years people thought information theory wasn't the right way to think about relevance, starting with misconceptions that go all the way to Shannon himself." [...]

Imagine X is a complex data set, like the pixels of a dog photo, and Y is a simpler variable represented by those data, like the word "dog." You can capture all the "relevant" information in X about Y by compressing X as much as you can without losing the ability to predict Y. In their 1999 paper, Tishby and co-authors Fernando Pereira, now at Google, and William Bialek, now at Princeton University, formulated this as a mathematical optimization problem. It was a fundamental idea with no killer application.

But, you know, the emic/etic distinction is about relevance. What are phonemes, they are "he most relevant features of a spoken word".

To the most recent experiments:

In their experiments, Tishby and Shwartz-Ziv tracked how much information each layer of a deep neural network retained about the input data and how much information each one retained about the output label. The scientists found that, layer by layer, the networks converged to the information bottleneck theoretical bound: a theoretical limit derived in Tishby, Pereira and Bialek's original paper that represents the absolute best the system can do at extracting relevant information. At the bound, the network has compressed the input as much as possible without sacrificing the ability to accurately predict its label.

Tishby and Shwartz-Ziv also made the intriguing discovery that deep learning proceeds in two phases: a short "fitting" phase, during which the network learns to label its training data, and a much longer "compression" phase, during which it becomes good at generalization, as measured by its performance at labeling new test data.

For instance, Lake said the fitting and compression phases that Tishby identified don't seem to have analogues in the way children learn handwritten characters, which he studies. Children don't need to see thousands of examples of a character and compress their mental representation over an extended period of time before they're able to recognize other instances of that letter and write it themselves. In fact, they can learn from a single example. Lake and his colleagues' models suggest the brain may deconstruct the new letter into a series of strokes - previously existing mental constructs - allowing the conception of the letter to be tacked onto an edifice of prior knowledge. "Rather than thinking of an image of a letter as a pattern of pixels and learning the concept as mapping those features" as in standard machine-learning algorithms, Lake explained, "instead I aim to build a simple causal model of the letter," a shorter path to generalization.

On 'deconstructing' letterforms into strokes, see the work of Mark Changizi [1].

For a technical account of this work, see Ravid Schwartz-Ziv and Naftali Tishby, Opening the black box of Deep Neural Networks via Information:

[1] Mark A. Changizi, Qiong Zhang, Hao Ye, and Shinsuke Shimojo, The Structures of Letters and Symbols throughout Human History Are Selected to Match Those Found in Objects in Natural Scenes, vol. 167, no. 5, The American Naturalist, May 2006

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