Many problems in computational statistics and machine learning can be formulated as approximating a target distribution by minimizing a discrepancy over a restricted class of candidate distributions. This minimum-discrepancy viewpoint appears, for example, in numerical integration, sampling, generative modelling, and statistical inference, but these instances are often studied separately in the literature. This thesis develops the first unified treatment of these many problems through the minimization of the maximum mean discrepancy (MMD), a kernel-based discrepancy between probability measures. The central theme is that MMD is not only a statistically meaningful distance between probability distributions, but also a tractable optimization objective to design practical and efficient algorithms. This thesis summarizes the recent theoretical and empirical advancements of such minimum-MMD based algorithms.

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