The Expressive Power of Transformers with Chain of Thought
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910406942916608 |
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| author | Merrill, William Sabharwal, Ashish |
| author_facet | Merrill, William Sabharwal, Ashish |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07923 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Expressive Power of Transformers with Chain of Thought Merrill, William Sabharwal, Ashish Machine Learning Computational Complexity Computation and Language Logic in Computer Science 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. |
| title | The Expressive Power of Transformers with Chain of Thought |
| topic | Machine Learning Computational Complexity Computation and Language Logic in Computer Science |
| url | https://arxiv.org/abs/2310.07923 |