We present a polynomial time algorithm for online maximization of @math -submodular maximization. For online (nonmonotone) @math -submodular maximization, our algorithm achieves a tight approximate factor in an approximate regret. For online monotone @math -submodular maximization, our approximate-regret matches to the best-known approximation ratio, which is tight asymptotically as @math tends to infinity. Our approach is based on the Blackwell approachability theorem and online linear optimization.
Online learning of discrete structure is called . Efficient online algorithms were developed for various discrete structures, such as shortest paths and matroid basis @cite_8 @cite_13 . Most of these studies focused on optimizing reward loss functions (under a constraint), whereas our paper studies functions (without constraint).
@cite_13 @cite_8
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We consider online learning problems under a a partial observability model capturing situations where the information conveyed to the learner is between full information and bandit feedback. In the simplest variant, we assume that in addition to its own loss, the learner also gets to observe losses of some other actions. The revealed losses depend on the learner's action and a directed observation system chosen by the environment. For this setting, we propose the first algorithm that enjoys near-optimal regret guarantees without having to know the observation system before selecting its actions. Along similar lines, we also define a new partial information setting that models online combinatorial optimization problems where the feedback received by the learner is between semi-bandit and full feedback. As the predictions of our first algorithm cannot be always computed efficiently in this setting, we propose another algorithm with similar properties and with the benefit of always being computationally efficient, at the price of a slightly more complicated tuning mechanism. Both algorithms rely on a novel exploration strategy called implicit exploration, which is shown to be more efficient both computationally and information-theoretically than previously studied exploration strategies for the problem. In this paper, we study the generalization properties of online learning based stochastic methods for supervised learning problems where the loss function is dependent on more than one training sample (e.g., metric learning, ranking). We present a generic decoupling technique that enables us to provide Rademacher complexity-based generalization error bounds. Our bounds are in general tighter than those obtained by (2012) for the same problem. Using our decoupling technique, we are further able to obtain fast convergence rates for strongly convex pairwise loss functions. We are also able to analyze a class of memory efficient online learning algorithms for pairwise learning problems that use only a bounded subset of past training samples to update the hypothesis at each step. Finally, in order to complement our generalization bounds, we propose a novel memory efficient online learning algorithm for higher order learning problems with bounded regret guarantees. We consider an online decision problem over a discrete space in which the loss function is submodular. We give algorithms which are computationally efficient and are Hannan-consistent in both the full information and partial feedback settings. Online learning is the process of answering a sequence of questions given knowledge of the correct answers to previous questions and possibly additional available information. Answering questions in an intelligent fashion and being able to make rational decisions as a result is a basic feature of everyday life. Will it rain today (so should I take an umbrella)? Should I fight the wild animal that is after me, or should I run away? Should I open an attachment in an email message or is it a virus? The study of online learning algorithms is thus an important domain in machine learning, and one that has interesting theoretical properties and practical applications. This dissertation describes a novel framework for the design and analysis of online learning algorithms. We show that various online learning algorithms can all be derived as special cases of our algorithmic framework. This unified view explains the properties of existing algorithms and also enables us to derive several new interesting algorithms. Online learning is performed in a sequence of consecutive rounds, where at each round the learner is given a question and is required to provide an answer to this question. After predicting an answer, the correct answer is revealed and the learner suffers a loss if there is a discrepancy between his answer and the correct one. The algorithmic framework for online learning we propose in this dissertation stems from a connection that we make between the notions of regret in online learning and weak duality in convex optimization. Regret bounds are the common thread in the analysis of online learning algorithms. A regret bound measures the performance of an online algorithm relative to the performance of a competing prediction mechanism, called a competing hypothesis. The competing hypothesis can be chosen in hindsight from a class of hypotheses, after observing the entire sequence of question- answer pairs. Over the years, competitive analysis techniques have been refined and extended to numerous prediction problems by employing complex and varied notions of progress toward a good competing hypothesis. We propose a new perspective on regret bounds which is based on the notion of duality in convex optimization. Regret bounds are universal in the sense that they hold for any possible fixed hypothesis in a given hypothesis class. We therefore cast the universal bound as a lower bound