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

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 distribution on the set \(\mathcal{X}\). This has many applications throughout data science, often through the normalization of probabilistic models , but also as a smooth approximation to the maximum , in transformers through its derivatives, or in reinforcement learning when using entropy regularization [ 19 ]. Sometimes the set \(\mathcal{X}\) is finite (potentially big) and the integral can be done by explicit summing, but often an exact computation is infeasible, and sampling from the probability distribution \(q\) is used instead.