For $V : \mathbb{R}^d \to \mathbb{R}$ coercive, we study the convergence rate for the $L^1$-distance of the empiric minimizer, which is the true minimum of the function $V$ sampled with noise with a finite number $n$ of samples, to the minimum of $V$. We show that in general, for unbounded functions with fast growth, the convergence rate is bounded above by $a_n n^{-1/q}$, where $q$ is the dimension of the latent random variable and where $a_n = o(n^\varepsilon)$ for every $\varepsilon > 0$. We then present applications to optimization problems arising in Machine Learning and in Monte Carlo simulation.

翻译：对于强制函数 $V : \mathbb{R}^d \to \mathbb{R}$，我们研究了经验极小化子（即在有限 $n$ 个含噪声样本下函数 $V$ 的真实最小值）到 $V$ 的最小值的 $L^1$ 距离的收敛速率。我们证明，一般情况下，对于具有快速增长的无界函数，该收敛速率的上界为 $a_n n^{-1/q}$，其中 $q$ 是潜在随机变量的维度，且对任意 $\varepsilon > 0$ 有 $a_n = o(n^\varepsilon)$。随后，我们给出了该结果在机器学习及蒙特卡洛模拟中优化问题上的应用。