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 Minimum Description Length (MDL)-like compression.
Controlled experiments provide evidence for this reinforcement--forgetting selection mechanism in scalar memories and a nonlinear shared network.
Joint reinforcement and attenuation interventions shift conditional retention, while matched exposure counts reveal forgetting-dependent effects of reinforcement timing and changes in the composition of the retained set.
Together, these results show that repeated reinforcement and persistent forgetting jointly provide a controllable source of inductive bias beyond neural architecture and scale.