Multi-step learning and underlying structure in statistical models

In multi-step learning, where a final learning task is accomplished via a sequence of intermediate learning tasks, the intuition is that successive steps or levels transform the initial data into representations more and more ``suited" to the final learning task. A related principle arises in transfer-learning where Baxter (2000) proposed a theoretical framework to study how learning multiple tasks transforms the inductive bias of a learner. The most widespread multi-step learning approach is semi-supervised learning with two steps: unsupervised, then supervised. Several authors (Castelli-Cover, 1996; Balcan-Blum, 2005; Niyogi, 2008; Ben-David et al, 2008; Urner et al, 2011) have analyzed SSL, with Balcan-Blum (2005) proposing a version of the PAC learning framework augmented by a ``compatibility function" to link concept class and unlabeled data distribution. We propose to analyze SSL and other multi-step learning approaches, much in the spirit of Baxter's framework, by defining a learning problem generatively as a joint statistical model on $X \times Y$. This determines in a natural way the class of conditional distributions that are possible with each marginal, and amounts to an abstract form of compatibility function. It also allows to analyze both discrete and non-discrete settings. As tool for our analysis, we define a notion of $\gamma$-uniform shattering for statistical models. We use this to give conditions on the marginal and conditional models which imply an advantage for multi-step learning approaches. In particular, we recover a more general version of a result of Poggio et al (2012): under mild hypotheses a multi-step approach which learns features invariant under successive factors of a finite group of invariances has sample complexity requirements that are additive rather than multiplicative in the size of the subgroups.