Convex Optimization
Efficient algorithms for online decision problems
Journal of Computer and System Sciences - Special issue: Learning theory 2003
Online trading algorithms and robust option pricing
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Prediction, Learning, and Games
Prediction, Learning, and Games
Efficient algorithms for online convex optimization and their applications
Efficient algorithms for online convex optimization and their applications
Improved second-order bounds for prediction with expert advice
Machine Learning
Regret to the best vs. regret to the average
Machine Learning
Pricing exotic derivatives using regret minimization
SAGT'11 Proceedings of the 4th international conference on Algorithmic game theory
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In this work, we lower bound the individual sequence anytime regret of a large family of online algorithms. This bound depends on the quadratic variation of the sequence, QT, and the learning rate. Nevertheless, we show that any learning rate that guarantees a regret upper bound of $O(\sqrt{Q_T})$ necessarily implies an $\Omega(\sqrt{Q_T})$ anytime regret on any sequence with quadratic variation QT. The algorithms we consider are linear forecasters whose weight vector at time t+1 is the gradient of a concave potential function of cumulative losses at time t. We show that these algorithms include all linear Regularized Follow the Leader algorithms. We prove our result for the case of potentials with negative definite Hessians, and potentials for the best expert setting satisfying some natural regularity conditions. In the best expert setting, we give our result in terms of the translation-invariant relative quadratic variation. We apply our lower bounds to Randomized Weighted Majority and to linear cost Online Gradient Descent. We show that bounds on anytime regret imply a lower bound on the price of "at the money" call options in an arbitrage-free market. Given a lower bound Q on the quadratic variation of a stock price, we give an $\Omega(\sqrt{Q})$ lower bound on the option price, for QQ) result given in [4].