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A class of parameter-free online linear optimization algorithms is proposedthat harnesses the structure of an adversarial sequence by adapting to someside information. These algorithms combine the reduction technique of Orabonaand P{\'a}l (2016) for adapting coin betting algorithms for online linearoptimization with universal compression techniques in information theory forincorporating sequential side information to coin betting. Concrete examplesare studied in which the side information has a tree structure and consists ofquantized values of the previous symbols of the adversarial sequence, includingfixed-order and variable-order Markov cases. By modifying the context-treeweighting technique of Willems, Shtarkov, and Tjalkens (1995), the proposedalgorithm is further refined to achieve the best performance over all adaptivealgorithms with tree-structured side information of a given maximum order in acomputationally efficient manner.

Parameter-free Online Linear Optimization with Side Information via Universal Coin Betting