2024/09/20 by Ingvar Ziemann, Nikolai Matni, Ziemann, Ingvar +3 · 1 citation
Computer Science · Decision Sciences · Economics, Econometrics and Finance · #Complex Systems and Decision Making #Complex Systems and Time Series Analysis #Evolutionary Algorithms and Applications #FOS: Computer and information sciences #FOS: Electrical engineering #Machine Learning (cs.LG) #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.2409.13421
openalex publication_date 2024/09/20 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28
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 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.