Exploding variance of means of exponentials: least-squares to the rescue (francisbach.com)

Recent discussions in the AI/ML community have highlighted the challenges of estimating log-sum-exp functions, which are common in various machine learning applications, such as reinforcement learning and probabilistic modeling. These functions often lead to significant computational difficulties due to exploding variances, particularly when potential functions yield large values. Researchers face the dual challenge of retaining the inherent benefits of log-sum-exp while mitigating its computational burdens. The article explores a compelling solution using least-squares regression, which offers simplicity through closed-form solutions and stable variance control, standing in contrast to traditional optimization techniques that may yield high approximation errors. The author proposes an innovative integration of least-squares methodologies with maximum likelihood estimation frameworks to stabilize the estimation process. This is achieved by representing the Kullback-Leibler divergence through alternative variational formulations that leverage least-squares characteristics. New integral equations developed within this framework present a pathway to potentially reduce the statistical instability encountered with direct empirical averages from probability distributions. This advancement is not only poised to enhance computational efficiency in diverse machine learning tasks, but it also invites further exploration of how least-squares techniques can be generalized across different divergence measures, potentially transforming approaches to complex estimation problems in AI/ML.