Abstract
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 [Zin03] introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover’s Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret \(O({\sqrt{T}})\), for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best single decision in hindsight.
In this paper, we give algorithms that achieve regret O(log(T)) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives). This mirrors what has been done for the special cases of prediction from expert advice by Kivinen and Warmuth [KW99], and Universal Portfolios by Cover [Cov91]. We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement.
The main new ideas give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field. Our analysis shows a surprising connection to follow-the-leader method, and builds on the recent work of Agarwal and Hazan [AH05]. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover’s algorithm and gradient descent.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Agarwal, A., Hazan, E.: Efficient algorithms for online game playing and universal portfolio management. In: ECCC, TR06-033 (2005)
Blum, A., Kalai, A.: Universal portfolios with and without transaction costs. In: COLT 1997: Proceedings of the tenth annual conference on Computational learning theory, pp. 309–313. ACM Press, New York (1997)
Brookes, M.: The matrix reference manual (2005), online: http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/intro.html
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, and Games. Cambridge University Press, Cambridge (2006)
Cover, T.: Universal portfolios. Math. Finance 1, 1–19 (1991)
Flaxman, A., Kalai, A.T., McMahan, H.B.: Online convex optimization in the bandit setting: gradient descent without a gradient. In: Proceedings of 16th SODA, pp. 385–394 (2005)
Hannan, J.: Approximation to bayes risk in repeated play. In: Dresher, M., Tucker, A.W., Wolfe, P. (eds.) Contributions to the Theory of Games, vol. III, pp. 97–139 (1957)
Kakade, S.: Personal communication (2005)
Kalai, A., Vempala, S.: Efficient algorithms for universal portfolios. J. Mach. Learn. Res. 3, 423–440 (2003)
Kalai, A., Vempala, S.: Efficient algorithms for on-line optimization. Journal of Computer and System Sciences 71(3), 291–307 (2005)
Kivinen, J., Warmuth, M.K.: Averaging expert predictions. In: Fischer, P., Simon, H.U. (eds.) EuroCOLT 1999. LNCS, vol. 1572, pp. 153–167. Springer, Heidelberg (1999)
Lovász, L., Vempala, S.: The geometry of logconcave functions and an o *(n 3) sampling algorithm. Technical Report MSR-TR-2003-04, Microsoft Research (2003)
Lovász, L., Vempala, S.: Simulated annealing in convex bodies and an 0*(n 4) volume algorithm. In: Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS), pp. 650–659 (2003)
Riedel, K.: A sherman-morrison-woodbury identity for rank augmenting matrices with application to centering. SIAM J. Mat. Anal. 12(1), 80–95 (1991)
Spall, J.: Introduction to Stochastic Search and Optimization. John Wiley & Sons, Inc., New York (2003)
Vaidya, P.M.: A new algorithm for minimizing convex functions over convex sets. Math. Program. 73(3), 291–341 (1996)
Zinkevich, M.: Online convex programming and generalized infinitesimal gradient ascent. In: Proceedings of the Twentieth International Conference (ICML), pp. 928–936 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hazan, E., Kalai, A., Kale, S., Agarwal, A. (2006). Logarithmic Regret Algorithms for Online Convex Optimization. In: Lugosi, G., Simon, H.U. (eds) Learning Theory. COLT 2006. Lecture Notes in Computer Science(), vol 4005. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11776420_37
Download citation
DOI: https://doi.org/10.1007/11776420_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-35294-5
Online ISBN: 978-3-540-35296-9
eBook Packages: Computer ScienceComputer Science (R0)