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The combination of lightly supervised pre-training and online fine-tuning has played a key role in recent AI developments. These new learning pipelines call for new theoretical frameworks. In this paper, we formalize key aspects of weakly supervised and active learning with a simple problem: the estimation of the mode of a distribution with partial feedback. We showcase how entropy coding allows for optimal information acquisition from partial feedback, develop coarse sufficient statistics for mode identification, and adapt bandit algorithms to our new setting. Finally, we combine those contributions into a statistically and computationally efficient solution to our original problem.