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5 Note that the probability mass assigned to any inactive instance is zero. The probability distribution p t determines the action of SAOL at time t. Namely, we have x t = x t (I) with probability p t (I). A pseudo-code of SAOL is detailed in Algorithm 1. Algorithm 1 Strongly Adaptive Online Learner (with blackbox algorithm B) { 1/2 I = [1, 1] Initialize: w 1 (I) = 0 o.w. for t = 1 to T do Let W t = I ACTIVE(t) w t(i) Choose I ACTIVE(t) w.p. p t (I) = wt(i) W t Predict x t (I) Update weights according to Equation (2) end for 2.1. Proof Sketch of Theorem 1 In this section we sketch the proof of Theorem 1. A full proof is detailed in Appendix 1. The analysis of SAOL is divided into two parts. The first challenge is to prove the theorem for the intervals in I (see Lemma 2). Then, the theorem should be extended to any interval (end of Appendix 1). Let us start with the first task. Our first observation is that for every interval I, the regret of SAOL during the interval I is equal to (SAOL s regret relatively to B I + the regret of B I ) (3) (during the interval I). Since the regret of B I during the interval I is already guaranteed to be small (Equation (1)), the problem of ensuring low regret during each of the intervals in I is reduced to the problem of ensuring low regret with respect to each of the B I s. We next prove that the regret of SAOL with respect to the B I s is small. Our analysis is similar to the proof of (Blum & Mansour, 2007)[Theorem 16]. Both of these proofs are similar to the analysis of the Multiplicative Weights Update (MW) method. The main idea is to define a potential function and relate it both to the loss of the learner and the loss of the best expert. To this end, we start by defining pseudo-weights over the experts (the B I s). With a slight abuse of notation, we define I(t) = 1 [t I]. For any I = [q, s] I, the pseudoweight of B I is defined by: 0 t < q 1 t = q w t (I) = w t 1 (I) (1 + η I r t 1 (I)) q < t s + 1 w s (I) t > s + 1 Note that w t (I) = η I I(t) w t (I). The potential function we consider is the overall pseudoweight at time t, Wt = I I w t(i). The following lemma, whose proof is given in the appendix, is a useful consequence of our definitions. Lemma 1 For every t 1, W t t(log(t) + 1). Through straightforward calculations, we conclude the proof of Theorem 1 for any interval in I. Lemma 2 For every I = [q, s] I, s r t (I) 5 log(s + 1) I. t=q Hence, according to Equation (3), R SAOL B(I) C I α + 5 log(s + 1) I The proof is given in the appendix. The extension of the theorem to any interval relies on some useful properties of the set I (see Lemma 1.1 in the appendix). Roughly speaking, any interval I [T ] can be partitioned into two sequences of intervals from I, such that the lengths of the intervals in each sequence decay at an exponential rate (Lemma 1.2 in the appendix). The theorem now follows by bounding the regret during the interval I by the sum of the regrets during the intervals in the above two sequences, and by using the fact that the lengths decay exponentially. 3. Strongly Adaptive Regret Is Stronger Than Tracking Regret In this section we relate the notion of strong adaptivity to that of tracking regret, and show that algorithms with small strongly adaptive regret also have small tracking regret. Let us briefly review the problem of tracking. For simplicity, we focus on context-less learning problems, and on the case where the set of strategies coincides with the decision space (though the result can be straightforwardly generalized). Fix a decision space D and a family L of loss functions. A compound action is a sequence σ = (σ 1,..., σ T ) D T. Since there is no hope in competing w.r.t. all sequences 3, a typical restriction of the problem is to bound the number of switches in each sequence. For a positive integer m, 3 It is easy to prove a lower bound of order T for this problem

7 segment I of size T k = Ω ( ) ( T ɛ ɛ ) 2, the regret of A is Ω T 2 which grows faster than I 1 ɛ poly(log T ) Lemma 3 Let A be an algorithm with regret bounded R A (T ) k = k(t ), Then, there exists an interval I [T ] of size Ω(T/k) with R A (I) = Ω( I ). Proof Assume for simplicity that 4k divides T. Consider the environment E 0, in which t, l t (e 1 ) = 0.5, l t (e 2 ) = 1. Let U [T ] be the (possibly random) set of time slots in which the algorithm chooses e 2 when the environment is E 0. Since the regret is at most k, we have E[ U ] 2k. It follows that for some segment I [T ] of size T 4k we have E[ U I ] 1 2. Indeed, otherwise, if [T ] = I 1... I 4k is the partition of the interval [T ] into 4k disjoint and consecutive intervals of size T 4k we will have E[ U ] = 4k j=1 E[ U I j ] > 2k. Now, since U I is a non-negative integer, w.p. 1 2 we have U I = 0. Namely, w.p. 1 2 A does not inspect e 2 during the interval I when it runs against E 0. Consider now the environment E that is identical to E 0, besides that t I, l t (e 2 ) = 0. By the argument above, w.p. 1 2, the operation of A on E is identical to its operation on E 0. In particular, the regret on I when A plays against E is, w.p. 1 2, I 2, and in total, I. Acknowledgments We thank Yishay Mansour and Sergiu Hart for helpful discussions. This work is supported by the Intel Collaborative Research Institute for Computational Intelligence (ICRI- CI). A. Daniely is supported by the Google Europe Fellowship in Learning Theory. References Auer, Peter, Cesa-Bianchi, Nicolo, Freund, Yoav, and Schapire, Robert E. The nonstochastic multiarmed bandit problem. SIAM Journal on Computing, 32(1):48 77, Blum, Avrim and Mansour, Yishay. From external to internal regret. Journal of Machine Learning, Bousquet, Olivier and Warmuth, Manfred K. Tracking a small set of experts by mixing past posteriors. The Journal of Machine Learning Research, 3: , Cesa-Bianchi, Nicolo, Freund, Yoav, Haussler, David, Helmbold, David P, Schapire, Robert E, and Warmuth, Manfred K. How to use expert advice. Journal of the ACM (JACM), 44(3): , Strongly Adaptive Online Learning Cesa-Bianchi, Nicolo, Gaillard, Pierre, Lugosi, Gábor, and Stoltz, Gilles. A new look at shifting regret. CoRR, abs/ , Freund, Yoav, Schapire, Robert E, Singer, Yoram, and Warmuth, Manfred K. Using and combining predictors that specialize. In Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp ACM, Online opti- arxiv preprint Hall, Eric C and Willett, Rebecca M. mization in dynamic environments. arxiv: , Hazan, Elad and Seshadhri, C. Adaptive algorithms for online decision problems. In Electronic Colloquium on Computational Complexity (ECCC), volume 14, Herbster, Mark and Warmuth, Manfred K. Tracking the best expert. Machine Learning, 32(2): , Jadbabaie, Ali, Rakhlin, Alexander, Shahrampour, Shahin, and Sridharan, Karthik. Online optimization: Competing with dynamic comparators. arxiv preprint arxiv: , Rakhlin, Sasha and Sridharan, Karthik. Optimization, learning, and games with predictable sequences. In Advances in Neural Information Processing Systems, pp , Shalev-Shwartz, Shai. Online learning and online convex optimization. Foundations and Trends in Machine Learning, 4(2): , Zinkevich, Martin. Online convex programming and generalized infinitesimal gradient ascent

Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

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