arXiv:2610.39837v1 Announce Type: cross Abstract: Best Arm Identification (BAI) conventionally measures the cost of experimentation by the number of samples collected. In many operational settings, however, the cost of exploration relative to a trusted baseline already in place is equally important. We study BAI under both a fixed sampling horizon and a regret budget. Unlike generic resources considered in constrained pure exploration, regret has a distinctive structure: sampling better arms is less costly relative to the baseline. We theoretically show that the regret cost of exploration decreases across the phases of Sequential Halving, a widely used pure-exploration algorithm. This observation yields improved guarantees on the probability of identifying the best arm and motivates a more effective allocation of the regret budget. We propose Sequential Halving with Proportional Regret Rationing (SHPRR), which front-loads the regret budget to allow more exploration when many arms remain. Experiments on synthetic and real-world instances show that SHPRR improves best-arm identification performance over uniform regret allocation and existing baselines.