Title: Sample Complexity and Representation Ability of Test-time Scaling Paradigms
Primary area: learning theory
Keywords: Large Language Models, Test-time Scaling, Sample Complexity, Representation Theory

Abstract: Test-time scaling paradigms have significantly advanced the capabilities of large language models (LLMs) on complex tasks. Despite their empirical success, theoretical understanding of the sample efficiency of various test-time strategies---such as self-consistency, best-of-$n$, and self-correction---remains limited. 
In this work, we first establish a separation result between two repeated sampling strategies: self-consistency requires $\Theta(1/\Delta^2)$ samples to produce the correct answer, while best-of-$n$ only needs $\Theta(1/\Delta)$, where $\Delta < 1$ denotes the probability gap between the correct and second most likely answers. 
Next, we present an expressiveness result for the self-correction approach with verifier feedback: it enables Transformers to simulate online learning over a pool of experts at test time. Therefore, a single Transformer architecture can provably solve multiple tasks without prior knowledge of the specific task associated with a user query, extending the representation theory of Transformers from single-task to multi-task settings. 
Finally, we empirically validate our theoretical results, demonstrating the practical effectiveness of self-correction methods.