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Neural Networks & New Kinds
Compression is how I think about learning. The tighter a model can compress its inputs, the more structure it has actually found. Kolmogorov complexity makes this precise. It measures the length of the shortest program that produces a given output, which turns out to be the theoretical floor for any compressor.
The Ultimate Compressor
K(X) = length of the shortest program that outputs X. For any computable compressor C and all strings X:
K(X) ≤ |C(X)| + K(C) + O(1)
via the simulation argument: run C inside a universal machine
The Catch
K(X) is uncomputable. You can never know the true shortest program. But a deep network is a finite parallel computer that approximates it with bounded resources.
Why Neural Nets are Compressors
- Neural nets can simulate arbitrary programs.
- They are small computers, circuits wired by data.
- SGD searches over the space of programs they can express.
Micro-Kolmogorov Complexity
Fix an architecture, then fit a network with SGD. The bit-length of the resulting weights is a practical proxy for description length:
minf ∈ F [ loss(f) + λ · micro-K(f) ]
micro-K(f) ≈ bit-length of weights in a fixed architecture
Shorter description length → better generalization.
I'm deeply invested in methods that make learning systems compress harder and generalize further.