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Consensus Is All You Get: The Role of Attention in Transformers
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Shows via a continuous-time control-theoretic model of attention on ellipsoids/spheres that tokens converge to consensus (collapse) under several assumptions, and checks this empirically on GPT-2 XL and GPT-Neo 2.7B.
Problem
Understanding why transformer tokens cluster (rank/representation collapse) as depth grows, with rigorous asymptotic guarantees for the attention dynamics.
Approach
Models attention as a continuous-time dynamical system on an ellipsoid (layer normalization as projection) and applies consensus dynamics and input-to-state stability from control theory, instead of mean-field or stochastic methods, to prove token convergence to a single cluster under several assumptions on Q, K, U and number of heads.
Key results
- Theorem 3.2: for full attention with any number of heads, tokens converge to consensus when starting in an open hemisphere, given bounded and uniformly continuous P=Q^T K and identity U.
- Theorem 4.2: for causal attention with identity U, the consensus set is asymptotically stable with a domain of attraction covering almost all initial conditions.
- GPT-2 XL and GPT-Neo 2.7B layer-wise cosine-consensus error E converges toward 0 across depth when feedforward layers are removed, and still converges with them present.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Include feedforward layers rather than remove them (ablation) | GPT-2 XL and GPT-Neo 2.7B · Repeated model passes on 100 random prompts | rate of convergence of average cosine-similarity error E | Mixed |
| Include feedforward layers rather than remove them (ablation) | GPT-2 XL and GPT-Neo 2.7B with random weight matrices · Repeated model passes on 100 random prompts | rate of convergence of average cosine-similarity error E | Hurt |
Surprising findings
- Consensus still occurs in GPT-2 and GPT-Neo with random weight matrices, suggesting collapse is a structural property of the architecture rather than of learned weights.
- Feedforward layers may slow convergence in GPT-Neo while speeding it up in GPT-2.
Limitations
- Theory holds under strong assumptions (bounded P, identity or single-head U); the symmetric-U result is proved only for a single head.
- Theoretical model excludes feedforward layers and uses continuous-time approximation.
- Experiments repeat tokens through the model to simulate increased depth, which deviates from standard usage.
Experimental setup
Datasets: 100 random prompts of 200 GPT-2 tokenizer tokens each (synthetic)
Metrics: cosine-similarity token consensus error E Lyapunov function V
Baselines: Geshkovski et al. 2023a Karagodin et al. 2024
Builds on: Geshkovski et al. 2023 mathematical model of attention Karagodin et al. 2024 consensus dynamics on manifolds
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