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This paper attempts to establish the theoretical foundation for the emergingsuper-model paradigm via domain adaptation, where one first trains a verylarge-scale model, {\it i.e.}, super model (or foundation model in some otherpapers), on a large amount of data and then adapts it to various specificdomains. Super-model paradigms help reduce computational and data cost andcarbon emission, which is critical to AI industry, especially enormous smalland medium-sized enterprises. We model the super-model paradigm as a two-stagediffusion process: (1) in the pre-training stage, the model parameter diffusesfrom random initials and converges to a steady distribution; and (2) in thefine-tuning stage, the model parameter is transported to another steadydistribution. Both training stages can be mathematically modeled by theUhlenbeck-Ornstein process which converges to two Maxwell-Boltzmanndistributions, respectively, each of which characterizes the correspondingconvergent model. An $\mathcal O(1/\sqrt{N})$ generalization bound is thenestablished via PAC-Bayesian framework. The theory finds that thegeneralization error of the fine-tuning stage is dominant in domain adaptation.In addition, our theory suggests that the generalization is determined by a newmeasure that characterizes the domain discrepancy between the source domain andtarget domain, based on the covariance matrices and the shift of the convergedlocal minimum.

Super-model ecosystem: A domain-adaptation perspective