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What One Cannot, Two Can: Two-Layer Transformers Provably Represent Induction Heads on Any-Order Markov Chains
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Shows that a two-layer, single-head transformer (embedding dim 6S+3) can represent the conditional k-gram model for any Markov order k, improving on prior three-layer single-head constructions, and gives a gradient-descent analysis for first-order chains.
Problem
Whether a two-layer single-head transformer can represent induction heads / conditional k-gram estimators for any-order Markov chains, since prior best constructions needed three layers (single head) or two layers with multiple heads.
Approach
Uses the MLP between the two attention layers (ReLU and layer normalization) to isolate the symbol at position n-k and reconstruct the hard-attention representation u_n, so the second layer acts as a k-th order induction head with one attention head per layer.
Key results
- Two-layer single-head transformer with embedding dimension 6S+3 represents the conditional k-gram model for any Markov order k, improving on the prior three-layer single-head construction.
- Two-stage gradient descent on a reduced two-layer model for first-order Markov chains converges to near-optimal loss with bound log(T+1)/(T+1)^{c*gamma}.
- Parameter count of the construction is 9(6S+3)^2 + (6S+3)(2T+2S+9), with a more efficient variant of 8(6S+3)^2 + ... noted.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use a two-layer transformer with one attention head per layer | conditional k-gram representation on any-order Markov chains · Markov-ICL, any Markov order k ≥ 1 | representation capability | Helped |
| reduce depth from three layers to two while keeping one head per layer | transformer representation of conditional k-grams · Markov-ICL, higher-order Markov processes | required transformer depth: three layers -> two layers | Helped |
| use two attention heads in the first layer and one in the second | transformer representation of conditional k-grams · Markov-ICL, any Markov order k | representation capability: three layers, one head -> two layers, two heads in layer one and one in layer two | Helped |
| train the reduced two-layer model with the two-stage gradient descent algorithm | two-layer transformer · first-order Markov chains | excess loss: ≲ \frac{\log(T + 1)}{(T + 1)^{c \gamma}} | Helped |
| use ReLU and layer normalization in the intermediate MLP to construct the complementary context representation | two-layer, single-head transformer · Markov-ICL, higher-order Markov processes | representation capability | Helped |
| use layer normalization on the second-layer query and key projections instead of intermediate MLP layers | two-layer, single-head transformer · Markov-ICL, any Markov order k | representation capability | No clear effect |
| replace the three-layer, single-head construction with a two-layer construction using two first-layer heads | transformer representation of conditional k-grams · Markov-ICL, any Markov order k | depth versus width: depth three, width one -> depth two, width two | Mixed |
Surprising findings
- The prior three-layer single-head construction is equivalent to a two-layer architecture with two heads in the first layer, revealing a depth-width tradeoff.
Limitations
- Gradient analysis only covers first-order Markov chains and a reduced model with treated-as-optimal MLP and layer norms omitted in stage one.
- Higher-order learning dynamics are left open.
- Representation result relies on large kappa limits and approximate optimality of the MLP.
Experimental setup
Datasets: synthetic random Markov chain sequences (Dirichlet prior over transition kernels)
Metrics: representation of conditional k-gram (theoretical) cross-entropy loss convergence bound attention pattern visualization
Baselines: three-layer single-head construction two-layer two-head construction single-layer transformer
Builds on: induction heads Markov-ICL three-layer construction relative positional encodings conditional k-gram model
Closest papers by meaning
- Unveiling Induction Heads: Provable Training Dynamics and Feature Learning in Transformers (NeurIPS 2024 · similarity 0.93)
- Transformers on Markov data: Constant depth suffices (NeurIPS 2024 · similarity 0.90)
- Towards Understanding How Transformers Learn In-context Through a Representation Learning Lens (NeurIPS 2024 · similarity 0.90)
- How Do Transformers Learn In-Context Beyond Simple Functions? A Case Study on Learning with Representations (ICLR 2024 · similarity 0.90)
- In-Context Learning through the Bayesian Prism (ICLR 2024 · similarity 0.90)
- Local to Global: Learning Dynamics and Effect of Initialization for Transformers (NeurIPS 2024 · similarity 0.89)
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