Inspired by advertising markets, we consider large-scale sequential decision making problems in which a learner must deploy an algorithm to behave optimally under uncertainty. Although many of these problems can be modeled as contextual bandit problems, we argue that the tools and techniques for analyzing bandit problems with large numbers of actions and contexts can be greatly expanded. While convexity and metric-similarity assumptions on the process generating rewards have yielded some algorithms in existing literature, certain types of assumptions that have been fruitful in offline supervised learning settings have yet to even be considered. Notably missing, for example, is any kind of graphical model approach to assuming structured rewards, despite the success such assumptions have
achieved in inducing scalable learning and inference with high-dimensional distributions. Similarly, we observe that there are countless tools for understanding the relationship between a choice of model class in supervised learning, and the generalization error of the best fit from that class, such as the celebrated VC-theory. However, an analogous notion of dimensionality, which relates a generic structural assumption on rewards to regret rates in an online optimization problem, is not fully developed. The primary goal of this dissertation, therefore, will be to fill out the space of models, algorithms, and assumptions used in sequential decision making problems. Toward this end, we will develop a theory for bandit problems with structured rewards that permit a graphical model representation. We will give an efficient algorithm for regret-minimization in such a setting, and along the way will develop a deeper connection between online supervised learning and regret-minimization. This dissertation will also introduce a complexity measure for generic structural assumptions on reward functions, which we call the Haystack Dimension. We will prove that the Haystack Dimension characterizes the optimal rates achievable up to log factors. Finally, we will describe more application-oriented techniques for solving problems in advertising markets, which again demonstrate how methods from traditional disciplines, such as statistical survival analysis, can be leveraged to design novel algorithms for optimization in markets
