Variance-Reduced Long-Term Rehearsal Learning with Quadratic Programming Reformulation

In machine learning, a critical class of decision-making problems involves *Avoiding Undesired Future* (AUF): given a predicted undesired outcome, how can one make decision about actions to prevent it? Recently, the *rehearsal learning* framework has been proposed to address AUF problem. While existing methods offer reliable decisions for single-round success, this paper considers long-term settings that involve coordinating multiple future outcomes, which is often required in real-world tasks. Specifically, we generalize the AUF objective to characterize a long-term decision target that incorporates cross-temporal relations among variables. As directly optimizing the *AUF probability* $\mathbb{P}_{\operatorname{AUF}}$ over this objective remains challenging, we derive an explicit expression for the objective and further propose a quadratic programming (QP) reformulation that transforms the intractable probabilistic AUF optimization into a tractable one. Under mild assumptions, we show that solutions to the QP reformulation are equivalent to those of the original AUF optimization, based on which we develop two novel rehearsal learning methods for long-term decision-making: (i) a *greedy* method that maximizes the single-round $\mathbb{P}_{\operatorname{AUF}}$ at each step, and (ii) a *far-sighted* method that accounts for future consequences in each decision, yielding a higher overall $\mathbb{P}_{\operatorname{AUF}}$ through an $L/(L+1)$ variance reduction in the AUF objective. We further establish an $\mathcal{O}(1/\sqrt{N})$ excess risk bound for decisions based on estimated parameters, ensuring reliable practical applicability with finite data. Experiments validate the effectiveness of our approach.