FINITE-TIME ANALYSIS OF GRADIENT DESCENT FOR SHALLOW TRANSFORMERS
Finite-Time Analysis of Gradient Descent for Shallow Transformers
Understanding why Transformers perform so well remains challenging due to their non-convex optimization landscape. In this work, we analyze a shallow Transformer with m independent heads trained by projected gradient descent in the kernel regime. Our analysis reveals two main findings: (i) the width required for nonasymptotic guarantees scales only logarithmically with the sample size n, and (ii) the optimization error is independent of the sequence length T. This contrasts sharply with recurrent architectures, where the optimization error can grow exponentially with T. The trade-off is memory: to keep the full context, the Transformer's memory requirement grows with the sequence length. We validate our theoretical results numerically in a teacher-student setting and confirm the predicted scaling laws for Transformers.
Published on 23 Jan 2026 by Enes Arda, Semih Caycі, Atilla Eryilmaz.
Executive Impact for Enterprise AI Leaders
This research provides critical insights for organizations deploying Transformers, offering a clearer understanding of their performance characteristics and resource implications.
Key Takeaways for Enterprise AI Leaders
- Breaks new ground by providing the first finite-time analysis of gradient descent for shallow Transformers, addressing the non-convex optimization landscape.
- Reveals that the required network width scales logarithmically with sample size, significantly improving over previous cubic overparameterization requirements for similar models.
- Demonstrates that the optimization error is independent of the sequence length (T), a critical advantage over recurrent neural networks where error can grow exponentially with T.
- Identifies a key trade-off: Transformer memory requirements scale linearly with sequence length to maintain full context, distinguishing it from recurrent models.
- Validates theoretical predictions with numerical experiments in a teacher-student setting, confirming the scaling laws.
Width Scaling
Opt. Error Dependence
Memory Scaling
Deep Analysis & Enterprise Applications
Select a topic to dive deeper, then explore the specific findings from the research, rebuilt as interactive, enterprise-focused modules.
The NTK framework linearizes networks around initialization, connecting gradient descent to kernel gradient flow, yielding optimization and generalization guarantees in overparameterized regimes. For Transformers, NTK limits have been derived under various scalings and infinite-width limits, but this work provides a finite-width analysis with realistic scaling. It bypasses the strict positive definiteness requirement of previous NTK-based proofs.
This paper analyzes a shallow multi-head Transformer. A key finding is that the attention gradient norms are independent of the sequence length (T). This contrasts with RNNs, where analogous gradients can grow exponentially with T, leading to improved optimization stability for Transformers over long sequences. The analysis preserves the genuine attention nonlinearity and allows independent heads, avoiding common simplifications.
A significant theoretical result is that the width required for nonasymptotic training guarantees scales only logarithmically with the sample size (n). This is a substantial improvement over prior works that required cubic overparameterization (d ≥ n), making Transformers more practical for real-world scenarios.
Log(n)
Width Scaling for Nonasymptotic Guarantees
Enterprise Process Flow
Projected Gradient Descent
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Near-Initialization Regime
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Transportation Mappings
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NTK Linearization
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T-Independent Optimization
| Feature | Transformer (This Work) | Recurrent Architectures (e.g., IndRNN) |
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Scaling Law Validation: Teacher-Student Setting
Our theoretical predictions for Transformer scaling laws were validated numerically in a teacher-student setting. This involved training a shallow Transformer with multiple independent heads using projected gradient descent. Key metrics such as linearization error, approximation error, and minimum training loss consistently showed a m-1/2 decay with width (m), matching our theory. This robust scaling across varying widths confirms the practical applicability of the theoretical bounds and the stability of the training process near initialization.
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