Risk-Monotonicity in Statistical Learning

Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the populating risk to decrease (better performance) monotonically with increasing data points. It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms such as empirical risk minimization. Non-monotonic behaviour of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent. These problems highlight bewilderment in our understanding of learning algorithms and generalization. It is, therefore, crucial to pursue this concern and provide a characterization of such behaviour. In this paper, we derive the first consistent and risk-monotonic algorithms for a general statistical learning setting under weak assumptions, consequently resolving an open problem (Viering et al. 2019) on how to avoid non-monotonic behaviour of risk curves. Our work makes a significant contribution to the topic of risk-monotonicity, which may be key in resolving empirical phenomena such as double descent.