Machine learning has enjoyed astounding practical success in a wide range of applications in recent years-practical success that often hurries ahead of our theoretical understanding. The standard framework for machine learning theory assumes full supervision, that is, training data consists of correctly labeled i.i.d. examples from the same task that the learned classifier is supposed to be applied to. However, many practical applications successfully make use of the sheer abundance of data that is currently produced. Such data may not be labeled or may be collected from various sources.

The focus of this thesis is to provide theoretical analysis of machine learning regimes where the learner is given such (possibly large amounts) of non-perfect training data. In particular, we investigate the benefits and limitations of learning with unlabeled data in semi-supervised learning and active learning as well as benefits and limitations of learning from data that has been generated by a task that is different from the target task (domain adaptation learning). For all three settings, we propose Probabilistic Lipschitzness to model the relatedness between the labels and the underlying domain space, and we discuss our suggested notion by comparing it to other common data assumptions.