Can a Transformer Represent a Kalman Filter?
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913355495636992 |
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| author | Goel, Gautam Bartlett, Peter |
| author_facet | Goel, Gautam Bartlett, Peter |
| contents | Transformers are a class of autoregressive deep learning architectures which have recently achieved state-of-the-art performance in various vision, language, and robotics tasks. We revisit the problem of Kalman Filtering in linear dynamical systems and show that Transformers can approximate the Kalman Filter in a strong sense. Specifically, for any observable LTI system we construct an explicit causally-masked Transformer which implements the Kalman Filter, up to a small additive error which is bounded uniformly in time; we call our construction the Transformer Filter. Our construction is based on a two-step reduction. We first show that a softmax self-attention block can exactly represent a Nadaraya-Watson kernel smoothing estimator with a Gaussian kernel. We then show that this estimator closely approximates the Kalman Filter. We also investigate how the Transformer Filter can be used for measurement-feedback control and prove that the resulting nonlinear controllers closely approximate the performance of standard optimal control policies such as the LQG controller. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06937 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Can a Transformer Represent a Kalman Filter? Goel, Gautam Bartlett, Peter Machine Learning Transformers are a class of autoregressive deep learning architectures which have recently achieved state-of-the-art performance in various vision, language, and robotics tasks. We revisit the problem of Kalman Filtering in linear dynamical systems and show that Transformers can approximate the Kalman Filter in a strong sense. Specifically, for any observable LTI system we construct an explicit causally-masked Transformer which implements the Kalman Filter, up to a small additive error which is bounded uniformly in time; we call our construction the Transformer Filter. Our construction is based on a two-step reduction. We first show that a softmax self-attention block can exactly represent a Nadaraya-Watson kernel smoothing estimator with a Gaussian kernel. We then show that this estimator closely approximates the Kalman Filter. We also investigate how the Transformer Filter can be used for measurement-feedback control and prove that the resulting nonlinear controllers closely approximate the performance of standard optimal control policies such as the LQG controller. |
| title | Can a Transformer Represent a Kalman Filter? |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2312.06937 |