TY - JOUR

T1 - Logarithmic regret algorithms for online convex optimization

AU - Hazan, Elad

AU - Agarwal, Amit

AU - Kale, Satyen

N1 - Funding Information:
E. Hazan and S. Kale supported by Sanjeev Arora’s NSF grants MSPA-MCS 0528414, CCF 0514993, ITR 0205594.

PY - 2007/12

Y1 - 2007/12

N2 - 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 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(√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 (EuroCOLT 1999), and Universal Portfolios by Cover (Math. Finance 1:1-19, 1991). 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 between the natural follow-the-leader approach and the Newton method. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent.

AB - 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 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(√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 (EuroCOLT 1999), and Universal Portfolios by Cover (Math. Finance 1:1-19, 1991). 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 between the natural follow-the-leader approach and the Newton method. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent.

KW - Online learning

KW - Online optimization

KW - Portfolio management

KW - Regret minimization

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U2 - 10.1007/s10994-007-5016-8

DO - 10.1007/s10994-007-5016-8

M3 - Article

AN - SCOPUS:35348918820

SN - 0885-6125

VL - 69

SP - 169

EP - 192

JO - Machine Learning

JF - Machine Learning

IS - 2-3

ER -