Online learning constitutes a mathematical and compelling framework to analyze sequential decision making problems in adversarial environments. The learner repeatedly chooses an action, the environment responds with an outcome, and then the learner receives a reward for the played action. The goal of the learner is to maximize his total reward. However, there are situations in which, in addition to maximizing the cumulative reward, there are some additional constraints on the sequence of decisions that must be satisfied on average by the learner. In this paper we study an extension to the online learning where the learner aims to maximize the total reward given that some additional constraints need to be satisfied. By leveraging on the theory of Lagrangian method in constrained optimization, we propose Lagrangian exponentially weighted average (LEWA) algorithm, which is a primal-dual variant of the well known exponentially weighted average algorithm, to efficiently solve constrained online decision making problems. Using novel theoretical analysis, we establish the regret and the violation of the constraint bounds in full information and bandit feedback models.
As is well known, a wide range of literature deals with the online decision making problem without constraints and there exist a number of regret-minimizing algorithms that have the optimal regret bound. The most well-known and successful work is probably the Hedge algorithm @cite_14 , which was a direct generalization of Littlestone and Warmuth's Weighted Majority (WM) algorithm @cite_4 . Other recent studies include the improved theoretical bounds and the parameter-free hedging algorithm @cite_19 and adaptive Hedge @cite_20 for decision-theoretic online learning. We refer readers to the @cite_11 for an in-depth discussion of this subject.
@cite_14 @cite_4 @cite_19 @cite_20 @cite_11
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We apply the exponential weight algorithm, introduced and Littlestone and Warmuth [26]and by Vovk [35]to the problem of predicting a binary sequence almost as well as the best biased coin. We first show that for the case of the logarithmic loss, the derived algorithm is equivalent to the Bayes algorithm with Jeffreys prior, that was studied by Xie and Barron [38]under probabilistic assumptions. We derive a uniform bound on the regret which holds for any sequence. We also show that if the empirical distribution of the sequence is bounded away from 0 and from 1, then, as the length of the sequence increases to infinity, the difference between this bound and a corresponding bound on the average case regret of the same algorithm (which is asymptotically optimal in that case) is only 1 2. We show that this gap of 1 2 is necessary by calculating the regret of the min–max optimal algorithm for this problem and showing that the asymptotic upper bound is tight. We also study the application of this algorithm to the square loss and show that the algorithm that is derived in this case is different from the Bayes algorithm and is better than it for prediction in the worstcase. We study the online decision problem in which the set of available actions varies over time, also called the sleeping experts problem. We consider the setting in which the performance comparison is made with respect to the best ordering of actions in hindsight. In this article, both the payoff function and the availability of actions are adversarial. [2010] gave a computationally efficient no-regret algorithm in the setting in which payoffs are stochastic. [2009] gave an efficient no-regret algorithm in the setting in which action availability is stochastic. However, the question of whether there exists a computationally efficient no-regret algorithm in the adversarial setting was posed as an open problem by [2010]. We show that such an algorithm would imply an algorithm for PAC learning DNF, a long-standing important open problem. We also consider the setting in which the number of available actions is restricted and study its relation to agnostic-learning monotone disjunctions over examples with bounded Hamming weight. In an online decision problem, one makes a sequence of decisions without knowledge of the future. Each period, one pays a cost based on the decision and observed state. We give a simple approach for doing nearly as well as the best single decision, where the best is chosen with the benefit of hindsight. A natural idea is to follow the leader, i.e. each period choose the decision which has done best so far. We show that by slightly perturbing the totals and then choosing the best decision, the expected performance is nearly as good as the best decision in hindsight. Our approach, which is very much like Hannan's original game-theoretic approach from the 1950s, yields guarantees competitive with the more modern exponential weighting algorithms like Weighted Majority. More importantly, these follow-the-leader style algorithms extend naturally to a large class of structured online problems for which the exponential algorithms are inefficient. We address online linear optimization problems when the possible actions of the decision maker are represented by binary vectors. The regret of the decision maker is the difference between her realized loss and the minimal loss she would have achieved by picking, in hindsight, the best possible action. Our goal is to understand the magnitude of the best possible minimax regret. We study the problem under three different assumptions for the feedback the decision maker receives: full information, and the partial information models of the so-called “semi-bandit” and “bandit” problems. In the full information case we show that the standard exponentially weighted average forecaster is a provably suboptimal strategy. For the semi-bandit model, by combining the Mirror Descent algorithm and the INF Implicitely Normalized Forecaster strategy, we are able to prove the first optimal bounds. Finally, in the bandit case we discuss existing results in light of a new lower bound, and suggest a conjecture on the optimal regret in that case.