  Statistical learning theory has largely focused on learning and
generalization given independent and identically distributed (i.i.d.) samples.
Motivated by applications involving time-series data, there has been a growing
literature on learning and generalization in settings where data is sampled
from an ergodic process. This work has also developed complexity measures,
which appropriately extend the notion of Rademacher complexity to bound the
generalization error and learning rates of hypothesis classes in this setting.
Rather than time-series data, our work is motivated by settings where data is
sampled on a network or a spatial domain, and thus do not fit well within the
framework of prior work. We provide learning and generalization bounds for data
that are complexly dependent, yet their distribution satisfies the standard
Dobrushin's condition. Indeed, we show that the standard complexity measures of
Gaussian and Rademacher complexities and VC dimension are sufficient measures
of complexity for the purposes of bounding the generalization error and
learning rates of hypothesis classes in our setting. Moreover, our
generalization bounds only degrade by constant factors compared to their i.i.d.
analogs, and our learnability bounds degrade by log factors in the size of the
training set.
