Tight Bounds for Jensen's Gap with Applications to Variational Inference
This work addresses the intractability of log-likelihood in models like VAEs, offering incremental improvements in bounding techniques for variational inference.
The paper tackles the problem of estimating Jensen's gap, particularly for logarithmic functions in variational inference, by proposing new tight bounds that accommodate various assumptions and provide analytical and empirical evidence.
Since its original formulation, Jensen's inequality has played a fundamental role across mathematics, statistics, and machine learning, with its probabilistic version highlighting the nonnegativity of the so-called Jensen's gap, i.e., the difference between the expectation of a convex function and the function at the expectation. Of particular importance is the case when the function is logarithmic, as this setting underpins many applications in variational inference, where the term variational gap is often used interchangeably. Recent research has focused on estimating the size of Jensen's gap and establishing tight lower and upper bounds under various assumptions on the underlying function and distribution, driven by practical challenges such as the intractability of log-likelihood in graphical models like variational autoencoders (VAEs). In this paper, we propose new, general bounds for Jensen's gap that accommodate a broad range of assumptions on both the function and the random variable, with special attention to exponential and logarithmic cases. We provide both analytical and empirical evidence for the performance of our method. Furthermore, we relate our bounds to the PAC-Bayes framework, providing new insights into generalization performance in probabilistic models.