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Will Merrill

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NeurIPS Conference 2025 Conference Paper

A Little Depth Goes a Long Way: The Expressive Power of Log-Depth Transformers

Will Merrill
Ashish Sabharwal

Recent theoretical results show transformers cannot express sequential reasoning problems over long inputs, intuitively because their computational *depth* is bounded. However, prior work treats the depth as a constant, leaving it unclear to what degree bounded depth may suffice for solving problems over short inputs, or how increasing the transformer's depth affects its expressive power. We address these questions by analyzing transformers whose depth can grow minimally with context length $n$. We show even highly uniform transformers with depth $\Theta(\log n)$ can express two important problems: *recognizing regular languages*, which captures state tracking abilities and was known to be expressible only by an unconventional, non-uniform model of transformers, and *graph connectivity*, which underlies multi-step reasoning. Notably, both of these problems cannot be expressed by fixed-depth transformers under standard complexity conjectures, demonstrating the expressivity benefit of growing depth. Moreover, our theory quantitatively predicts how depth must grow with input length to express these problems, showing that depth scaling is more efficient than scaling width or chain-of-thought steps. Empirically, our detailed experiments designed to bridge the expressivity vs. learnability gap reveal that our theoretical depth requirements for regular language recognition closely match the practical depth requirements for successfully training transformers. Thus, our results clarify how depth affects a transformer's reasoning capabilities, and provide practical guidance for effective depth selection for sequential reasoning.

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NeurIPS Conference 2025 Conference Paper

Critical Batch Size Revisited: A Simple Empirical Approach to Large-Batch Language Model Training

Will Merrill
Shane Arora
Dirk Groeneveld
Hanna Hajishirzi

The right batch size is important when training language models at scale: a large batch size is necessary for fast training, but a batch size that is too large will harm token efficiency. To navigate this tradeoff, McCandlish et al. (2018) suggest that a critical batch size (CBS), below which training will not substantially degrade loss, can be estimated based on the gradient noise scale during training. While their method has been adopted in practice, e. g. , when training GPT-3, strong assumptions are required to justify gradient noise as a proxy for the CBS, which makes it unclear whether their approach should be trusted in practice, limiting its applicability. In this paper, we introduce a simple, empirical approach to directly measure the CBS and show how the CBS evolves over training. Applying our approach to the OLMo models, we find that CBS is near 0 at initialization, increases rapidly at first, and then plateaus as training progresses. Furthermore, we find that this trend holds across different model sizes (1B and 7B), suggesting CBS from small training runs can inform larger-scale training runs. Our findings about how the CBS changes over training motivate batch size warmup as a natural way to reliably train language models at large batch size: start the batch size small and increase it as the CBS grows. To validate this claim, we use batch size warmup to train OLMo 1B to slightly better loss than the original training run with 43% fewer gradient steps. This shows how our framework can be applied to reliably train language models at larger batch sizes, increasing data parallelism without compromising performance.

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NeurIPS Conference 2025 Conference Paper

Exact Expressive Power of Transformers with Padding

Will Merrill
Ashish Sabharwal

Chain of thought is a natural inference-time method for increasing the computational power of transformer-based large language models (LLMs), but comes at the cost of sequential decoding. Are there more efficient alternatives to expand a transformer's expressive power without adding parameters? We consider transformers with *padding* tokens as a form of parallelizable test-time compute. We show that averaging-hard-attention, masked-pre-norm transformers with polynomial padding recognize precisely the class $\mathsf{FO}$-uniform $\mathsf{TC}^0$ of extremely parallelizable problems. While the $\mathsf{TC}^0$ upper bound was known, proving a matching lower bound had been elusive. Further, our novel analysis reveals the precise expanded power of padded transformers when coupled with another form of inference-time compute, namely dynamically increasing depth via *looping*. Our core technical contribution is to show how padding helps bring the notions of *complete problems* and *reductions*, which have been a cornerstone of classical complexity theory, to the formal study of transformers. Armed with this new tool, we prove that padded transformers with $\mathrm{O}(\log^d n)$ looping on inputs of length $n$ recognize exactly the class $\mathsf{FO}$-uniform $\mathsf{TC}^d$ of moderately parallelizable problems. Thus, padding and looping together systematically expand transformers' expressive power: with polylogarithmic looping, polynomially padded transformers recognize precisely the class $\mathsf{FO}$-uniform $\mathsf{NC}$, the best that could be expected without losing parallelism (unless $\mathsf{NC} = \mathsf{P}$). Our results thus motivate further exploration of padding and looping as parallelizable alternatives to chain of thought for test-time compute.

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