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The passage around this formula

If a training job cannot simply be re-run task-by-task the way a MapReduce job could, recovering from a failure depends entirely on how recently and how cheaply its state was saved. Checkpointing research therefore had to become a first-class systems problem in its own right rather than a background convenience, and its mathematics is old. A 2024 re-derivation of the classical result on checkpoint scheduling shows that the loss-minimizing interval between checkpoints is proportional to the square root of the product of checkpoint save time and mean time to failure, $\tau_{\mathrm{opt}} \approx \sqrt{2\,\delta\,M}$. with $\delta$ the time cost of writing one checkpoint and M the mean time between failures — and the paper…

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