The Expressive Power of Transformers with Chain of Thought
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Keywords: The Expressive Power of Transformers with Chain of Thought
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TL;DR: We consider the expressive power of chain-of-thought transformers with different length reasoning chains, characterizing the additional reasoning power gained as a function of the length of the chain.
Abstract: Recent theoretical work has identified surprisingly simple reasoning problems, such as checking if two nodes in a graph are connected or simulating finite-state machines, that are provably unsolvable by standard transformers that answer immediately after reading their input. However, in practice, transformers' reasoning can be improved by allowing them to use a "chain of thought" or "scratchpad", i.e., generate and condition on a sequence of intermediate tokens before answering. Motivated by this, we ask: *Does such intermediate generation fundamentally extend the computational power of a decoder-only transformer?* We show that the answer is *yes*, but the amount of increase depends crucially on the amount of intermediate generation. For instance, we find that transformer decoders with a logarithmic number of decoding steps (w.r.t. the input length) push the limits of standard transformers only slightly, while a linear number of decoding steps, assuming projected pre-norm (a slight generalization of standard pre-norm), adds a clear new ability (under standard complexity conjectures): recognizing all regular languages. Our results also imply that linear steps keep transformer decoders within context-sensitive languages, and polynomial steps with generalized pre-norm make them recognize exactly the class of polynomial-time solvable problems—the first exact characterization of a type of transformers in terms of standard complexity classes. Together, this provides a nuanced framework for understanding how the length of a transformer’s chain of thought or scratchpad impacts its reasoning power.
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Primary Area: general machine learning (i.e., none of the above)
Submission Number: 1377
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