State space models, emergence, and ergodicity: How many parameters are needed for stable predictions?
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929508138876928 |
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| author | Ziemann, Ingvar Matni, Nikolai Pappas, George J. |
| author_facet | Ziemann, Ingvar Matni, Nikolai Pappas, George J. |
| contents | How many parameters are required for a model to execute a given task? It has been argued that large language models, pre-trained via self-supervised learning, exhibit emergent capabilities such as multi-step reasoning as their number of parameters reach a critical scale. In the present work, we explore whether this phenomenon can analogously be replicated in a simple theoretical model. We show that the problem of learning linear dynamical systems -- a simple instance of self-supervised learning -- exhibits a corresponding phase transition. Namely, for every non-ergodic linear system there exists a critical threshold such that a learner using fewer parameters than said threshold cannot achieve bounded error for large sequence lengths. Put differently, in our model we find that tasks exhibiting substantial long-range correlation require a certain critical number of parameters -- a phenomenon akin to emergence. We also investigate the role of the learner's parametrization and consider a simple version of a linear dynamical system with hidden state -- an imperfectly observed random walk in $\mathbb{R}$. For this situation, we show that there exists no learner using a linear filter which can succesfully learn the random walk unless the filter length exceeds a certain threshold depending on the effective memory length and horizon of the problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13421 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | State space models, emergence, and ergodicity: How many parameters are needed for stable predictions? Ziemann, Ingvar Matni, Nikolai Pappas, George J. Machine Learning Systems and Control How many parameters are required for a model to execute a given task? It has been argued that large language models, pre-trained via self-supervised learning, exhibit emergent capabilities such as multi-step reasoning as their number of parameters reach a critical scale. In the present work, we explore whether this phenomenon can analogously be replicated in a simple theoretical model. We show that the problem of learning linear dynamical systems -- a simple instance of self-supervised learning -- exhibits a corresponding phase transition. Namely, for every non-ergodic linear system there exists a critical threshold such that a learner using fewer parameters than said threshold cannot achieve bounded error for large sequence lengths. Put differently, in our model we find that tasks exhibiting substantial long-range correlation require a certain critical number of parameters -- a phenomenon akin to emergence. We also investigate the role of the learner's parametrization and consider a simple version of a linear dynamical system with hidden state -- an imperfectly observed random walk in $\mathbb{R}$. For this situation, we show that there exists no learner using a linear filter which can succesfully learn the random walk unless the filter length exceeds a certain threshold depending on the effective memory length and horizon of the problem. |
| title | State space models, emergence, and ergodicity: How many parameters are needed for stable predictions? |
| topic | Machine Learning Systems and Control |
| url | https://arxiv.org/abs/2409.13421 |