This paper considers an empirical risk minimization problem under
heavy-tailed settings, where data does not have finite variance, but only has
$p$-th moment with $p \in (1,2)$. Instead of using estimation procedure based
on truncated observed data, we choose the optimizer by minimizing the risk
value. Those risk values can be robustly estimated via using the remarkable
Catoni's method (Catoni, 2012). Thanks to the structure of Catoni-type
influence functions, we are able to establish excess risk upper bounds via
using generalized generic chaining methods. Moreover, we take computational
issues into consideration. We especially theoretically investigate two types of
optimization methods, robust gradient descent algorithm and empirical
risk-based methods. With an extensive numerical study, we find that the
optimizer based on empirical risks via Catoni-style estimation indeed shows
better performance than other baselines. It indicates that estimation directly
based on truncated data may lead to unsatisfactory results.

本文研究了在重尾设置下的经验风险最小化问题，通过使用 Catoni 方法对风险值进行稳健估计，利用广义泛函链方法建立了超额风险上限界，并通过数值研究表明，基于 Catoni 风格估计的经验风险优化方法比其他基线方法表现更好。