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Can Looped Transformers Learn to Implement Multi-step Gradient Descent for In-context Learning?
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Shows that the global minimizer of the population loss for a linear looped Transformer implements multi-step preconditioned gradient descent with a near-inverse-covariance preconditioner, and proves gradient flow converges despite non-convexity via a novel gradient dominance condition.
Problem
Whether training Transformers (specifically looped Transformers with weight sharing) can converge to algorithmic solutions, i.e., multi-step gradient descent, for in-context linear regression.
Approach
Analyzes a linear looped Transformer (one shared linear attention layer applied L times) for in-context linear regression with Gaussian data; characterizes the global minimizer as near (Sigma*)^-1 preconditioned GD and proves a gradient dominance condition with power (2L-1)/L giving gradient flow convergence.
Key results
- Global minimizer of population loss has u=0 and preconditioner A within (1±8δd^{1/(2L)}) of (Sigma*)^-1, with δ=(8Ld/√n)^{1/(2L)}.
- Gradient flow reaches loss ≤ ξ after time polynomial in 1/ξ, under a gradient dominance condition with power (2L-1)/L.
- Experiments with d=10 show looped models with L loops reach loss comparable to L-layer non-looped models and loss decreases with more test-time loops.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase the number of training loops | Looped linear-attention Transformer · In-context linear regression (identity covariance) | loss | Helped |
| Use one attention layer looped L times instead of an L-layer non-looped model | Linear-attention Transformer · In-context linear regression | loss | No clear effect |
| Evaluate with more loops than used during training | Looped model trained with 2 or 5 loops · In-context linear regression, in-distribution | loss | Helped |
| Evaluate with more loops than used during training | Looped model trained with 2 or 5 loops · In-context linear regression, out-of-distribution covariance | loss | Helped |
| Evaluate on a different covariance rather than identity covariance | Looped model trained with identity covariance · In-context linear regression, out-of-distribution | loss | Hurt |
| Train with different numbers of in-context samples, including n closer to or smaller than d | Looped model · In-context linear regression | loss | No clear effect |
Surprising findings
- Training a single-layer model looped only once (1 loop) did not converge in the OOD setup, per the paper's experiments.
Limitations
- Analysis restricted to linear attention and Gaussian data with a specific parameterization (u parameter, shared Q and P).
- Convergence results for training without weight sharing and for nonlinear attention remain open.
- Experiments are small-scale synthetic (d=10).
Experimental setup
Datasets: synthetic Gaussian in-context linear regression data
Benchmarks: synthetic in-context linear regression
Metrics: in-context regression loss distance of preconditioner to identity
Baselines: multi-layer linear Transformer
Builds on: linear self-attention gradient descent expressivity (von Oswald et al.) Ahn et al. one-layer linear attention preconditioned GD looped transformers
Closest papers by meaning
- Linear Transformers are Versatile In-Context Learners (NeurIPS 2024 · similarity 0.93)
- Transformers Learn to Achieve Second-Order Convergence Rates for In-Context Linear Regression (NeurIPS 2024 · similarity 0.92)
- Transformers Learn to Implement Multi-step Gradient Descent with Chain of Thought (ICLR 2025 · similarity 0.90)
- How Transformers Utilize Multi-Head Attention in In-Context Learning? A Case Study on Sparse Linear Regression (NeurIPS 2024 · similarity 0.90)
- One Step of Gradient Descent is Provably the Optimal In-Context Learner with One Layer of Linear Self-Attention (ICLR 2024 · similarity 0.89)
- In-context Convergence of Transformers (ICML 2024 · similarity 0.89)
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