On a uniform law of large numbers for random sets and subdifferentials of random functions

P. Terán (2008). Statistics and Probability Letters 78, 42-49.


While spending a night in the (South) Tenerife Airport, and that means a lot of time to kill, I read a JMAA paper by Alexander Shapiro and Huifu Xu where they obtained a strong law of large numbers for subdifferentials of random functions as a tool for consistency analysis of stationary points in non-convex non-smooth stochastic optimization (math jargon rules, doesn't it?)

Their LLN, however sufficient for their purpose, depended on a sort of `blurring radius' parameter r>0 and so didn't cover the `exact' case r=0 unless the functions be additionally assumed to be continuous.

That looked like a perfect benchmark for the abstract LLN Ilya Molchanov and I had proved. It turned out that the strongest case r=0 held under upper semicontinuity (provided a separability condition on the range of the multifunction) or even weaker conditions, showing that continuity in fact played no role in the problem.


Up the line:
A law of large numbers in a metric space with a convex combination operation (w. Ilya Molchanov). You may download a (non-final) preprint copy from Ilya's website.

Down the line:
On consistency of stationary points of stochastic optimization problems in a Banach space (200x).

To download, click here. This is *the* good version; SPL has published their own version which I don't endorse in any way.