A Generalized McDiarmid Inequality: Sample Reuse and Non-Uniform Unbounded Differences

Jeremiah Birrell

We develop a novel generalization of McDiarmid's concentration inequality and use it to obtain a uniform law of large numbers result and error bounds on stochastic optimization problems which simultaneously incorporate two key features: 1) They capture the benefit of training sample reuse in algorithms that also employ easily-sampled auxiliary random variables. 2) They allow for one-component mean-difference bounding functions which are both unbounded in the component of interest and have non-uniform dependence on another component. We demonstrate the utility of our method by deriving concentration inequalities for score-based generative modeling. This application naturally leverages both key features of our new tools: 1) It utilizes auxiliary Gaussian random variables in conjunction with the training samples; accounting for training sample reuse results in tighter error bounds by effectively integrating out the dependence on the auxiliary random variables. 2) The ability to handle non-uniform mean-difference bounding functions allows us to treat unbounded score models and data distributions with unbounded support.