Long-term memory enables language-model agents to personalize future interactions, but a finite memory bank cannot preserve every extracted fact. Existing memory gating policies are typically shared across users and therefore fail to account for user-specific patterns of future reuse. In this paper, we study memory gating at the fact level. We first propose a retrieval-based labeling strategy that labels each fact by whether it is later retrieved and incorporated into responses. We then formulate online memory gating as contextual apple tasting with delayed one-sided feedback, and propose Warm-Start UCB, which combines offline learning from existing data with online adaptation to the current user. Our theoretical analysis establishes a high-probability regret upper bound of
O ~ ( T ε ∧ d T ) \tilde{O}(Tε\wedge d\sqrt{T}) O ~ ( T ε ∧ d T ) when the delay window is at most
O ( T ) O(\sqrt{T}) O ( T ) , where
T T T is the total number of rounds,
ε ε ε is the initialization error,
d d d is the dimension of the feature map, and
O ~ ( ⋅ ) \tilde{O}(\cdot) O ~ ( ⋅ ) hides constant and poly-logarithmic factors. The results of our experiment demonstrate that our method outperforms all baselines under different memory budgets and history lengths, establishing it as a promising personalized memory gating policy.