To address the needs of modeling uncertainty in sensitive machine learning
applications, the setup of distributionally robust optimization (DRO) seeks
good performance uniformly across a variety of tasks. The recent
multi-distribution learning (MDL) framework tackles this objective in a dynamic
interaction with the environment, where the learner has sampling access to each
target distribution. Drawing inspiration from the field of pure-exploration
multi-armed bandits, we provide distribution-dependent guarantees in the MDL
regime, that scale with suboptimality gaps and result in superior dependence on
the sample size when compared to the existing distribution-independent
analyses. We investigate two non-adaptive strategies, uniform and non-uniform
exploration, and present non-asymptotic regret bounds using novel tools from
empirical process theory. Furthermore, we devise an adaptive optimistic
algorithm, LCB-DR, that showcases enhanced dependence on the gaps, mirroring
the contrast between uniform and optimistic allocation in the multi-armed
bandit literature.

为了应对敏感机器学习应用中的不确定性建模需求，分布鲁棒优化（DRO）的设置在各种任务中寻求统一的良好性能。最近的多分布学习（MDL）框架以与环境的动态互动的方式解决了这一目标，在该框架中，学习者可以对每个目标分布进行采样访问。借鉴了纯探索多臂赌博机领域的观点，我们在 MDL 体制下提供了依赖于分布的保证，并且在与现有的分布无关分析相比，这种保证随着次优性差距的缩小而产生了优秀的样本大小依赖性。我们研究了两种非自适应策略：均匀探索和非均匀探索，并使用经验过程理论中的新工具提供了非渐进性后悔上界。此外，我们设计了一种自适应乐观算法 LCB-DR，展示了对差距的增强依赖性，类似于多臂赌博机文献中均匀分配和乐观分配之间的对比。