C05 — Information Bottleneck

This framework asks: How much does a circuit compress its input, and how much task-relevant information survives that compression?

The Information Bottleneck (IB) principle characterizes the optimal tradeoff between compression and prediction. A circuit that achieves low description length while retaining high MI with the task output has found an efficient representation. This connects directly to effective model complexity: circuits with fewer active components that still achieve high task performance sit on the IB frontier.

In practice, the local learning coefficient (LLC) measures a related quantity — the effective dimensionality of the loss landscape around a solution. Circuits on the IB frontier should have lower LLC, indicating they use fewer effective parameters.

Theoretical grounding

Source	Year	Key contribution
Tishby et al., “The Information Bottleneck Method”	1999	Original IB formulation
Shwartz-Ziv & Tishby, “Opening the Black Box of Deep Neural Networks”	2017	IB applied to deep network layers
Saxe et al., “On the Information Bottleneck Theory of Deep Learning”	2018	Critical analysis and phase transitions
Lau et al., “Quantifying Local Learning Coefficient”	2023	LLC as effective complexity measure
Geiger et al., “Causal Abstractions of Neural Networks”	2021	Minimal sufficient representations in circuits

Core concept

The IB objective finds a compressed representation ( T ) of input ( X ) that maximizes information about target ( Y ):

[ \max_{p(t|x)} ; I(T; Y) - \beta , I(T; X) ]

The Lagrange multiplier ( \beta ) controls the compression-prediction tradeoff. At ( \beta \to 0 ), maximal compression (T is trivial); at ( \beta \to \infty ), T preserves all of X.

For a circuit with ( k ) active components, we can frame the circuit itself as the bottleneck: ( I(\text{circuit}; Y) ) is the task performance, and ( I(\text{circuit}; X) ) relates to the circuit’s capacity (proportional to ( k ) and the effective dimensionality). The LLC provides a tractable proxy for this capacity.

Instruments under C05

LLC Script (10_llc.py)

Estimates the local learning coefficient around the trained circuit, measuring the effective number of parameters the model uses. Lower LLC indicates a more compressed (bottlenecked) solution.

What it establishes: Whether the circuit operates efficiently on the IB frontier — high performance with low effective complexity. What it does not establish: The explicit IB curve or whether alternative circuits with better tradeoffs exist.

Usage:

uv run python 10_llc.py --tasks ioi sva

Reading the scores

| Pattern | What it means | |---|---|---| | Low LLC + high task performance | Efficient circuit on the IB frontier | | High LLC + high performance | Over-parameterized; compression possible | | Low LLC + low performance | Under-powered; too compressed | | LLC drops when removing a head | That head was not contributing effective complexity |

Connection to other frameworks

The IB perspective complements C01 (MI) by adding the compression axis: it is not enough for a component to carry information — it must do so efficiently. C04 (PID) redundancy identifies where compression is possible (redundant heads can be removed without information loss). The structural pillar measures circuit size directly, while IB provides the information-theoretic justification for minimality.