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Computer Science > Machine Learning

arXiv:2505.12514 (cs)
[Submitted on 18 May 2025]

Title:Reasoning by Superposition: A Theoretical Perspective on Chain
of Continuous Thought

Authors:Hanlin Zhu, Shibo Hao, Zhiting Hu, Jiantao Jiao, Stuart
Russell, Yuandong Tian
View a PDF of the paper titled Reasoning by Superposition: A
Theoretical Perspective on Chain of Continuous Thought, by Hanlin Zhu
and 5 other authors
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    Abstract:Large Language Models (LLMs) have demonstrated
    remarkable performance in many applications, including
    challenging reasoning problems via chain-of-thoughts (CoTs)
    techniques that generate ``thinking tokens'' before answering the
    questions. While existing theoretical works demonstrate that CoTs
    with discrete tokens boost the capability of LLMs, recent work on
    continuous CoTs lacks a theoretical understanding of why it
    outperforms discrete counterparts in various reasoning tasks such
    as directed graph reachability, a fundamental graph reasoning
    problem that includes many practical domain applications as
    special cases. In this paper, we prove that a two-layer
    transformer with $D$ steps of continuous CoTs can solve the
    directed graph reachability problem, where $D$ is the diameter of
    the graph, while the best known result of constant-depth
    transformers with discrete CoTs requires $O(n^2)$ decoding steps
    where $n$ is the number of vertices ($D<n$). In our construction,
    each continuous thought vector is a superposition state that
    encodes multiple search frontiers simultaneously (i.e., parallel
    breadth-first search (BFS)), while discrete CoTs must choose a
    single path sampled from the superposition state, which leads to
    sequential search that requires many more steps and may be
    trapped into local solutions. We also performed extensive
    experiments to verify that our theoretical construction aligns
    well with the empirical solution obtained via training dynamics.
    Notably, encoding of multiple search frontiers as a superposition
    state automatically emerges in training continuous CoTs, without
    explicit supervision to guide the model to explore multiple paths
    simultaneously.

Comments: 26 pages, 7 figures
Subjects: Machine Learning (cs.LG)
Cite as:  arXiv:2505.12514 [cs.LG]
          (or arXiv:2505.12514v1 [cs.LG] for this version)
          https://doi.org/10.48550/arXiv.2505.12514
          Focus to learn more
          arXiv-issued DOI via DataCite

Submission history

From: Hanlin Zhu [view email]
[v1] Sun, 18 May 2025 18:36:53 UTC (1,903 KB)
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