Finite-Time Analysis of Gradient Descent for Shallow Transformers

Enes Arda, Semih Cayci, Atilla Eryilmaz
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:3988-3996, 2026.

Abstract

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 compare Transformers with recurrent architectures on an autoregressive task.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-arda26a, title = { Finite-Time Analysis of Gradient Descent for Shallow Transformers }, author = {Arda, Enes and Cayci, Semih and Eryilmaz, Atilla}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {3988--3996}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/arda26a/arda26a.pdf}, url = {https://proceedings.mlr.press/v300/arda26a.html}, abstract = { 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 compare Transformers with recurrent architectures on an autoregressive task. } }
Endnote
%0 Conference Paper %T Finite-Time Analysis of Gradient Descent for Shallow Transformers %A Enes Arda %A Semih Cayci %A Atilla Eryilmaz %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-arda26a %I PMLR %P 3988--3996 %U https://proceedings.mlr.press/v300/arda26a.html %V 300 %X 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 compare Transformers with recurrent architectures on an autoregressive task.
APA
Arda, E., Cayci, S. & Eryilmaz, A.. (2026). Finite-Time Analysis of Gradient Descent for Shallow Transformers . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:3988-3996 Available from https://proceedings.mlr.press/v300/arda26a.html.

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