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The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks

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arxiv 2306.17844 v2 pith:JF5DCC3V submitted 2023-06-30 cs.LG

classification cs.LG
keywords networksneuralalgorithmalgorithmseventasksadditionalgorithmic
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Do neural networks, trained on well-understood algorithmic tasks, reliably rediscover known algorithms for solving those tasks? Several recent studies, on tasks ranging from group arithmetic to in-context linear regression, have suggested that the answer is yes. Using modular addition as a prototypical problem, we show that algorithm discovery in neural networks is sometimes more complex. Small changes to model hyperparameters and initializations can induce the discovery of qualitatively different algorithms from a fixed training set, and even parallel implementations of multiple such algorithms. Some networks trained to perform modular addition implement a familiar Clock algorithm; others implement a previously undescribed, less intuitive, but comprehensible procedure which we term the Pizza algorithm, or a variety of even more complex procedures. Our results show that even simple learning problems can admit a surprising diversity of solutions, motivating the development of new tools for characterizing the behavior of neural networks across their algorithmic phase space.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 11 citations worldwide. Full citation record

  1. Learning Pseudorandom Numbers with Transformers: Permuted Congruential Generators, Curricula, and Interpretability

    cs.LG 2025-10 conditional novelty 7.0 of 10

    Transformers can in-context predict PCG outputs on unseen parameters; required context length scales as sqrt(m), and curriculum training with smaller moduli is necessary for large moduli.

  2. Grokking Is Conditional and Fragile: A Fully-Tractable, Multi-Seed Study at 12K Parameters

    cs.LG 2026-07 accept novelty 6.0 of 10

    In a fully tractable 12K Llama-style model, grokking is a conditional fragile phase transition gated by coverage (tracking modulus more than structure), weight decay, and floating-point reduction order, so evidence mu...

  3. Input Pathways Shape Few-Shot, Not Zero-Shot, Binding in Tiny Transformers: A Fully-Enumerable Study

    cs.LG 2026-07 accept novelty 6.0 of 10

    In information-matched tiny transformers, zero-shot compositional binding fails for every route, while few-shot efficiency is governed by input-pathway sharing and code readability.

  4. Distinct Computations Emerge From Compositional Curricula in In-Context Learning

    cs.LG 2025-06 conditional novelty 6.0 of 10

    When transformer models see easy component examples before a harder combined math problem in one prompt, they solve unseen versions of the combined problem and store intermediate steps internally, unlike models traine...

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