We introduce Local Reinforcement-Based Selection of Auxiliary Objectives (LRSAO), a novel approach that selects auxiliary objectives using reinforcement learning (RL) to support the optimization process of an evolutionary algorithm (EA) as in EA+RL framework and furthermore incorporates the ability to unlearn previously used objectives. By modifying the reward mechanism to penalize moves that do no increase the fitness value and relying on the local auxiliary objectives, LRSAO dynamically adapts its selection strategy to optimize performance according to the landscape and unlearn previous objectives when necessary. We analyze and evaluate LRSAO on the black-box complexity version of the non-monotonic Jump function, with gap parameter $\ell$, where each auxiliary objective is beneficial at specific stages of optimization. The Jump function is hard to optimize for evolutionary-based algorithms and the best-known complexity for reinforcement-based selection on Jump was $O(n^2 \log(n) / \ell)$. Our approach improves over this result to achieve a complexity of $Θ(n^2 / \ell^2 + n \log(n))$ resulting in a significant improvement, which demonstrates the efficiency and adaptability of LRSAO, highlighting its potential to outperform traditional methods in complex optimization scenarios.