Learning Under Forgetting: Statistical Support-Selective Retention in Stochastic Training Dynamics

Fujie Gao, Zuyue Zhang, Gang Sun

Prior work has shown that neural networks exhibit implicit biases toward low-complexity structure (e.g., spectral bias), memorization dynamics, and compression-like effects during training, but a unified dynamical account of selective retention remains incomplete. We propose Repeated Reinforcement with Persistent Forgetting (RPF) dynamics, a minimal framework in which repeated exposure reinforces patterns and structures that recur in the data, while persistent forgetting attenuates learned information. This view treats forgetting not merely as a failure mode, but as a selection mechanism. We build the theory in three successive layers. First, in an independent-feature model, we derive an exposure-selective survival law and a support-dependent retention boundary characterizing which patterns persist under forgetting. Second, in a shared-parameter model, we show that forgetting induces spectral filtering over covariance modes, preserving strongly supported shared components while suppressing weak ones. Third, under small-step and norm/coding approximations, we show how RPF dynamics induce an implicit trade-off between data fitting and the cost of stored information, yielding…