Abstract:
In the first part of this dissertation, we focus on identifying the advantage afforded by quantum algorithms in the field of learning theory, which began as a mathematical framework in the mid-20th century to understand how algorithms can generalize from data [VC71; VC74; Vap82; Val84]. We specifically focus on the supervised setting under unbounded label-noise. Classically, efficient algorithms for various learning problems in this particular setting are often allowed to query Membership Query (MQ) oracles, which have historically been criticised for being too strong, both in a theoretical and practical sense. A long-standing open question in learning theory originating with the seminal work of Ehrenfeucht and Haussler [EH89] is as follows: Do there exist efficient learning algorithms for Boolean functions that have polynomial-sized decision tree representations under unbounded label noise and without Membership query access, assuming uniform marginal distribution over the instances? In the first part of our thesis, we answer theabove question in the affirmative, by showing that there exists a quantum learningalgorithm for the same. In the second part of our thesis, we introduce techniques to analyze the generalization behavior of statistical learning algorithms (both quantum and classical) trained on non-i.i.d. data under bounded and unbounded label noise. Our proof techniques involve generalizing a specific variant of the well known Online-to-Batch conversion paradigm [LN23] to the setting where the underlying data is drawn from β and ϕmixing stochastic processes.
