Exploding variance of means of exponentials: least-squares to the rescue

Machine Learning Research Blog Francis Bach Menu Home About Home page Menu Exploding variance of means of exponentials: least-squares to the rescue Posted on September 25, 2026 by Francis Bach A common task in machine learning is to estimate or optimize “log-sum-exp” functions with (potentially continuously) many terms such as $$ \log \Big( \int_{\mathcal{X}} e^{v(x)} dq(x) \Big),$$ where \(v: \mathcal{X} \to \mathbb{R}\) is some potential function, and \(q\) is a probability distrib