How Many Different Outputs Can a Transformer Generate?

Despite the remarkable success of transformer architectures in natural language processing, their scalability limitations remain poorly understood through systematic empirical analysis. This paper presents the first comprehensive large-scale evaluation of 118 transformer models across seven distinct architectural categories, revealing fundamental performance walls that manifest as hard deployment constraints. Our systematic benchmarking methodology uncovers a critical scalability crisis: while 88.1% of models successfully process sequences up to 512 tokens, this drops dramatically to 44.9% at 1024 tokens, with complete failure (0%) at 2048 tokens. Through rigorous analysis of loading times, memory consumption, and computational efficiency across sequence lengths from 128 to 2048 tokens, we demonstrate that compressed models achieve superior parameter efficiency (649.2 tokens/sec/M parameters) compared to large generative models (12.5 tokens/sec/M). Our findings challenge prevailing scaling assumptions and provide the first quantitative evidence that the theoretical O(n2) attention complexity translates into measurable performance walls. This work establishes new benchmarking methodologies for transformer evaluation and provides critical insights for practical deployment decisions in production environments.

We tightly characterize the VC dimension of depth-$L$ Transformers with a total of $W$ parameters, mapping an input sequence of length $T$ to a single output, establishing an upper bound of $O(L W \log (T W))$ and a nearly matching lower bound of $Ω(L W \log (T W / L))$. We further tightly characterize the sample complexity of chain-of-thought learning using such a Transformer, showing teacher forcing (i.e. selecting a predictor consistent with the entire chain-of-thought on training data) learns with sample complexity $O\left(L W \log \left(\left(T+T^{\prime}\right) W\right)\right)$ and that any learning rule that uses chain-of-thought data requires at least $Ω\left(L W \log \left(\left(T+T^{\prime}\right) W / L\right)\right)$ examples, where $T$ is the input length and $T^{\prime}$ is the number of autoregressive steps.

Recent advancements in transformer length generalization theory enable us to reliably predict when a transformer can learn to solve a task. In particular, the C-RASP hypothesis (a formalized version of the so-called RASP-l conjecture) posits that transformers length-generalize on a task if and only if a solution is expressible in the C-RASP language. While this hypothesis has strong empirical validation, theoretical problems arise from the fact that no computable length generalization bounds exist for C-RASP, alongside the discovery of seemingly contradictory experiments. To address these problems, we refine the C-RASP hypothesis utilizing the recently-proposed fragments C-RASP+ and C-RASP1. These fragments have computable length generalization bounds, though in the worst case requiring an extremely large (double exponential) sample size. It is an open question whether these sample size bounds are tight. In this paper, we resolve this open question by providing an exponentially tighter bound. In doing so, we show a polynomial length generalization bound for transformers if we adopt compressed strings, via a novel connection to power words. As an application, we show how this yields a fine-grained analysis of the C-RASP conjecture that resolves contradicting experimental evidence against it.