We revisit McDiarmid's inequality, a central tool in statistical learning theory and algorithm analysis. We show that large deviations above the mean are exponentially unlikely for functions of independent random variables that are almost surely bounded above and satisfy a conditional moment assumption of order $p\in(1,2]$. When $p<2$, the result applies even when the function has infinite variance. Our proof relies on a new tensorization inequality for the Kullback--Leibler divergence, termed reverse KL tensorization, which may be of independent interest. Applications include concentration of the sample mean and median-of-means estimator, and one-sided uniform risk bounds for unbounded losses.

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