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We present a new family of subgradient methods that dynamically incorporate knowledge of the geometry of the data observed in earlier iterations to perform more informative gradientbased learning. Metaphorically, the adaptation allows us to find needles in haystacks in the form of very predictive but rarely seen features. Our paradigm stems from recent(More)
In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., chooses a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters a sequence of (possibly unrelated) convex cost functions. Zinkevich (ICML 2003) introduced this framework, which models many natural repeated(More)
Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple(More)
We introduce an efficient algorithm for the problem of online linear optimization in the bandit setting which achieves the optimal O∗( √ T ) regret. The setting is a natural generalization of the nonstochastic multi-armed bandit problem, and the existence of an efficient optimal algorithm has been posed as an open problem in a number of recent papers. We(More)
We give a novel algorithm for stochastic strongly-convex optimization in the gradient oracle model which returns an O( 1 T )-approximate solution after T gradient updates. This rate of convergence is optimal in the gradient oracle model. This improves upon the previously known best rate of O( log(T ) T ), which was obtained by applying an online(More)
We propose an algorithm for approximately maximizing a concave function over the bounded semi-definite cone, which produces sparse solutions. Sparsity for SDP corresponds to low rank matrices, and is a important property for both computational as well as learning theoretic reasons. As an application, building on Aaronson’s recent work, we derive a linear(More)
We experimentally study on-line investment algorithms first proposed by Agarwal and Hazan and extended by Hazan et al. which achieve almost the same wealth as the best constant-rebalanced portfolio determined in hindsight. These algorithms are the first to combine optimal logarithmic regret bounds with efficient deterministic computability. They are based(More)
The problem of principle component analysis (PCA) is traditionally solved by spectral or algebraic methods. We show how computing the leading principal component could be reduced to solving a small number of well-conditioned convex optimization problems. This gives rise to a new efficient method for PCA based on recent advances in stochastic methods for(More)