  Laws of large numbers guarantee that given a large enough sample from some
population, the measure of any fixed sub-population is well-estimated by its
frequency in the sample. We study laws of large numbers in sampling processes
that can affect the environment they are acting upon and interact with it.
Specifically, we consider the sequential sampling model proposed by Ben-Eliezer
and Yogev (2020), and characterize the classes which admit a uniform law of
large numbers in this model: these are exactly the classes that are
\emph{online learnable}. Our characterization may be interpreted as an online
analogue to the equivalence between learnability and uniform convergence in
statistical (PAC) learning.
  The sample-complexity bounds we obtain are tight for many parameter regimes,
and as an application, we determine the optimal regret bounds in online
learning, stated in terms of \emph{Littlestone's dimension}, thus resolving the
main open question from Ben-David, P\'al, and Shalev-Shwartz (2009), which was
also posed by Rakhlin, Sridharan, and Tewari (2015).
