Tight Bounds on Jensen's Gap: Novel Approach with Applications in Generative Modeling

Among various mathematical tools of particular interest are those that provide a common basis for researchers in different scientific fields. One of them is Jensen's inequality, which states that the expectation of a convex function is greater than or equal to the function evaluated at the expectation. The resulting difference, known as Jensen's gap, became the subject of investigation by both the statistical and machine learning communities. Among many related topics, finding lower and upper bounds on Jensen's gap (under different assumptions on the underlying function and distribution) has recently become a problem of particular interest. In our paper, we take another step in this direction by providing a novel general and mathematically rigorous technique, motivated by the recent results of Struski et al. (2023). In addition, by studying in detail the case of the logarithmic function and the log-normal distribution, we explore a method for tightly estimating the log-likelihood of generative models trained on real-world datasets. Furthermore, we present both analytical and experimental arguments in support of the superiority of our approach in comparison to existing state-of-the-art solutions, contingent upon fulfillment of the criteria set forth by theoretical studies and corresponding experiments on synthetic data.