Set structured global empirical risk minimizers are rate optimal in general dimensions
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具有集合结构的全局经验风险最小化估计在一般维度下达到最优速率
- 研究问题
- 当熵积分发散时，全局经验风险最小化估计是否必然具有次优收敛速率，额外的集合结构能否消除上下界之间的缺口？
- 方法与思路
- 作者通过可测集合类上的经验过程刻画模型复杂度，建立匹配的上下界，并研究图像、分类、保序回归及凹密度估计。
- 材料陈述的贡献
- 摘要称，在具有适当集合结构的所考察模型中，即使熵积分发散，全局经验风险最小化估计也能达到匹配下界的最优速率。
- 阅读关注点
- 应核对集合结构的精确定义、各模型的损失与维度条件、经验过程上下界，以及所称最优速率对应的参数空间。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；摘要未列出各模型条件和速率，不能将结论推广到所有熵积分发散的模型。
原文摘要
Entropy integrals are widely used as a powerful empirical process tool to obtain upper bounds for the rates of convergence of global empirical risk minimizers (ERMs), in standard settings such as density estimation and regression. The upper bound for the convergence rates thus obtained typically matches the minimax lower bound when the entropy integral converges, but admits a strict gap compared to the lower bound when it diverges. Birgé and Massart (Probab. Theory Related Fields 97 (1993) 113–150) provided a striking example showing that such a gap is real with the entropy structure alone: for a variant of the natural Hölder class with low regularity, the global ERM actually converges at the rate predicted by the entropy integral that substantially deviates from the lower bound. The counter-example has spawned a long-standing negative position on the use of global ERMs in the regime where the entropy integral diverges, as they are heuristically believed to converge at a suboptimal rate in a variety of models. The present paper demonstrates that this gap can be closed if the models admit certain degree of “set structures” in addition to the entropy structure. In other words, the global ERMs in such set structured models will indeed be rate-optimal, matching the lower bound even when the entropy integral diverges. The models with set structures we investigate include (i) image and edge estimation, (ii) binary classification, (iii) multiple isotonic regression, (iv) s-concave density estimation, all in general dimensions when the entropy integral diverges. Here, set structures are interpreted broadly in the sense that the complexity of the underlying models can be essentially captured by the size of the empirical process over certain class of measurable sets, for which matching upper and lower bounds are obtained to facilitate the derivation of sharp convergence rates for the associated global ERMs.