Online sparse linear regression is the task of applying linear regression analysis to examples arriving sequentially subject to a resource constraint that a limited number of features of examples can be observed. Despite its importance in many practical applications, it has been recently shown that there is no polynomialtime sublinear-regret algorithm unless NP BPP, and only an exponential-time sublinear-regret algorithm has been found. In this paper, we introduce mild assumptions to solve the problem.

Online portfolio selection is a sequential decision-making problem in which a learner repetitively selects a portfolio over a set of assets, aiming to maximize long-term return. In this paper, we study the problem with the cardinality constraint that the number of assets in a portfolio is restricted to be at most k, and consider two scenarios: (i) in the full-feedback setting, the learner can observe price relatives (rates of return to cost) for all assets, and (ii) in the bandit-feedback setting, the learner can observe price relatives only for invested assets. We propose efficient algorithms for these scenarios, which achieve sublinear regrets. We also provide regret (statistical) lower bounds for both scenarios which nearly match the upper bounds when k is a constant. In addition, we give a computational lower bound, which implies that no algorithm maintains both computational efficiency, as well as a small regret upper bound.

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Online sparse linear regression is the task of applying linear regression analysis to examples arriving sequentially subject to a resource constraint that a limited number of features of examples can be observed. Despite its importance in many practical applications, it has been recently shown that there is no polynomialtime sublinear-regret algorithm unless NP BPP, and only an exponential-time sublinear-regret algorithm has been found. In this paper, we introduce mild assumptions to solve the problem.