Learning from weakly dependent data under Dobrushin's condition
This work addresses the problem of statistical learning for non-time-series dependent data, such as in network applications, but is incremental as it extends existing theory to new dependency structures.
The paper tackles learning from complexly dependent data, such as network or spatial data, by providing generalization and learnability bounds under Dobrushin's condition, showing that standard complexity measures suffice with only constant or logarithmic degradation compared to i.i.d. settings.
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.