  We initiate a formal study of reproducibility in optimization. We define a
quantitative measure of reproducibility of optimization procedures in the face
of noisy or error-prone operations such as inexact or stochastic gradient
computations or inexact initialization. We then analyze several convex
optimization settings of interest such as smooth, non-smooth, and
strongly-convex objective functions and establish tight bounds on the limits of
reproducibility in each setting. Our analysis reveals a fundamental trade-off
between computation and reproducibility: more computation is necessary (and
sufficient) for better reproducibility.
