Promotion is Rent-or-Buy: Exact Ledgers and Competitive Limits for Memory Consolidation in Continual Agents

A continual learning agent must decide when to move an item from its memory into its model weights. Current systems make this choice by heuristic, with no guarantees. We cast the decision as a rent-or-buy problem: keeping an item in memory costs a small rent on every query, while promoting it into the weights costs a one-time price. The classical answer is the break-even rule (rent until the total rent paid equals the price, then buy). Promotion, however, has a side effect absent from the classical problem, which we call crowding: each promoted item makes the shared classifier slightly worse at separating nearby concepts. Our guarantees depend on how crowding is charged. If it is charged once, at the moment of promotion, the break-even rule stays within a constant factor of the optimal schedule. If it is charged on every query, as in deployed systems, no promote-only policy achieves a bounded competitive ratio, even with just two items. Allowing the agent to demote items back out of the weights restores a bounded ratio. We test these predictions in six pre-registered experiments on synthetic data and on features from real vision and text datasets, and we report the cases where the results went against our hypotheses.