In this paper, we consider the problem of preserving privacy in the online learning setting. We study the problem in the online convex programming (OCP) framework---a popular online learning setting with several interesting theoretical and practical implications---while using differential privacy as the formal privacy measure. For this problem, we distill two critical attributes that a private OCP algorithm should have in order to provide reasonable privacy as well as utility guarantees: 1) linearly decreasing sensitivity, i.e., as new data points arrive their effect on the learning model decreases, 2) sub-linear regret bound---regret bound is a popular goodness/utility measure of an online learning algorithm. Given an OCP algorithm that satisfies these two conditions, we provide a general framework to convert the given algorithm into a privacy preserving OCP algorithm with good (sub-linear) regret. We then illustrate our approach by converting two popular online learning algorithms into their differentially private variants while guaranteeing sub-linear regret ($O(\sqrt{T})$). Next, we consider the special case of online linear regression problems, a practically important class of online learning problems, for which we generalize an approach by Dwork et al. to provide a differentially private algorithm with just $O(\log^{1.5} T)$ regret. Finally, we show that our online learning framework can be used to provide differentially private algorithms for offline learning as well. For the offline learning problem, our approach obtains better error bounds as well as can handle larger class of problems than the existing state-of-the-art methods Chaudhuri et al.

Learning theory has largely focused on two main learning scenarios. The first is the classical statistical setting where instances are drawn i.i.d. from a fixed distribution and the second scenario is the online learning, completely adversarial scenario where adversary at every time step picks the worst instance to provide the learner with. It can be argued that in the real world neither of these assumptions are reasonable. It is therefore important to study problems with a range of assumptions on data. Unfortunately, theoretical results in this area are scarce, possibly due to absence of general tools for analysis. Focusing on the regret formulation, we define the minimax value of a game where the adversary is restricted in his moves. The framework captures stochastic and non-stochastic assumptions on data. Building on the sequential symmetrization approach, we define a notion of distribution-dependent Rademacher complexity for the spectrum of problems ranging from i.i.d. to worst-case. The bounds let us immediately deduce variation-type bounds. We then consider the i.i.d. adversary and show equivalence of online and batch learnability. In the supervised setting, we consider various hybrid assumptions on the way that x and y variables are chosen. Finally, we consider smoothed learning problems and show that half-spaces are online learnable in the smoothed model. In fact, exponentially small noise added to adversary's decisions turns this problem with infinite Littlestone's dimension into a learnable problem.

We study online learnability of a wide class of problems, extending the results of (Rakhlin, Sridharan, Tewari, 2010) to general notions of performance measure well beyond external regret. Our framework simultaneously captures such well-known notions as internal and general Phi-regret, learning with non-additive global cost functions, Blackwell's approachability, calibration of forecasters, adaptive regret, and more. We show that learnability in all these situations is due to control of the same three quantities: a martingale convergence term, a term describing the ability to perform well if future is known, and a generalization of sequential Rademacher complexity, studied in (Rakhlin, Sridharan, Tewari, 2010). Since we directly study complexity of the problem instead of focusing on efficient algorithms, we are able to improve and extend many known results which have been previously derived via an algorithmic construction.

Sequential prediction problems such as imitation learning, where future observations depend on previous predictions (actions), violate the common i.i.d. assumptions made in statistical learning. This leads to poor performance in theory and often in practice. Some recent approaches provide stronger guarantees in this setting, but remain somewhat unsatisfactory as they train either non-stationary or stochastic policies and require a large number of iterations. In this paper, we propose a new iterative algorithm, which trains a stationary deterministic policy, that can be seen as a no regret algorithm in an online learning setting. We show that any such no regret algorithm, combined with additional reduction assumptions, must find a policy with good performance under the distribution of observations it induces in such sequential settings. We demonstrate that this new approach outperforms previous approaches on two challenging imitation learning problems and a benchmark sequence labeling problem.

Online learning has become increasingly popular on handling massive data. The sequential nature of online learning, however, requires a centralized learner to store data and update parameters. In this paper, we consider online learning with {\em distributed} data sources. The autonomous learners update local parameters based on local data sources and periodically exchange information with a small subset of neighbors in a communication network. We derive the regret bound for strongly convex functions that generalizes the work by Ram et al. (2010) for convex functions. Most importantly, we show that our algorithm has \emph{intrinsic} privacy-preserving properties, and we prove the sufficient and necessary conditions for privacy preservation in the network. These conditions imply that for networks with greater-than-one connectivity, a malicious learner cannot reconstruct the subgradients (and sensitive raw data) of other learners, which makes our algorithm appealing in privacy sensitive applications.

We develop a theory of online learning by defining several complexity measures. Among them are analogues of Rademacher complexity, covering numbers and fat-shattering dimension from statistical learning theory. Relationship among these complexity measures, their connection to online learning, and tools for bounding them are provided. We apply these results to various learning problems. We provide a complete characterization of online learnability in the supervised setting.

When learning models that are represented in matrix forms, enforcing a low-rank constraint can dramatically improve the memory and run time complexity, while providing a natural regularization of the model. However, naive approaches for minimizing functions over the set of low-rank matrices are either prohibitively time consuming (repeated singular value decomposition of the matrix) or numerically unstable (optimizing a factored representation of the low rank matrix). We build on recent advances in optimization over manifolds, and describe an iterative online learning procedure, consisting of a gradient step, followed by a second-order retraction back to the manifold. While the ideal retraction is hard to compute, and so is the projection operator that approximates it, we describe another second-order retraction that can be computed efficiently, with run time and memory complexity of O((n+m)k) for a rank-k matrix of dimension m x n, given rank one gradients. We use this algorithm, LORETA, to learn a matrix-form similarity measure over pairs of documents represented as high dimensional vectors. LORETA improves the mean average precision over a passive- aggressive approach in a factorized model, and also improves over a full model trained over pre-selected features using the same memory requirements. LORETA also showed consistent improvement over standard methods in a large (1600 classes) multi-label image classification task.

In the last years, knowledge technologies have been exploited for self-regulation functionalities inside e-learning systems. The definition of integrated system suitably scaffolding learners to improve their experi- ence is still lacking though. In this work, we propose an innovative Web-based educational environment that sustains metacognitive self-regulated learning processes upon Semantic Web and Social Web methods and technologies.

Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize value of information? These applications exhibit strong diminishing returns: Selection of redundant ads and information sources decreases their marginal utility. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our algorithm is equipped with strong theoretical guarantees, with a performance ratio that converges to the optimal constant of 1-1/e. We empirically evaluate our algorithms on two real-world online optimization problems on the web: ad allocation with submodular utilities, and dynamically ranking blogs to detect information cascades.

We propose a general method called truncated gradient to induce sparsity in the weights of online-learning algorithms with convex loss. This method has several essential properties. First, the degree of sparsity is continuous---a parameter controls the rate of sparsification from no sparsification to total sparsification. Second, the approach is theoretically motivated, and an instance of it can be regarded as an online counterpart of the popular $L_1$-regularization method in the batch setting. We prove that small rates of sparsification result in only small additional regret with respect to typical online-learning guarantees. Finally, the approach works well empirically. We apply it to several datasets and find that for datasets with large numbers of features, substantial sparsity is discoverable.