We use online convex optimization (OCO) for setpoint tracking with uncertain, flexible loads. We consider full feedback from the loads, bandit feedback, and two intermediate types of feedback: partial bandit where a subset of the loads are individually observed and the rest are observed in aggregate, and Bernoulli feedback where in each round the aggregator receives either full or bandit feedback according to a known probability. We give sublinear regret bounds in all cases. We numerically evaluate our algorithms on examples with thermostatically controlled loads and electric vehicles.
Online learning @cite_15 @cite_17 and online convex optimization @cite_9 @cite_25 @cite_0 have already seen a number of applications in DR. Several variants of the multi-armed bandit framework have been used to curtail flexible loads in @cite_33 @cite_27 @cite_4 @cite_34 . Reference @cite_1 used adversarial bandits to shift load while learning load parameters.
@cite_4 @cite_33 @cite_9 @cite_1 @cite_0 @cite_27 @cite_15 @cite_34 @cite_25 @cite_17
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We describe a general framework for online adaptation of optimization hyperparameters by hot swapping' their values during learning. We investigate this approach in the context of adaptive learning rate selection using an explore-exploit strategy from the multi-armed bandit literature. Experiments on a benchmark neural network show that the hot swapping approach leads to consistently better solutions compared to well-known alternatives such as AdaDelta and stochastic gradient with exhaustive hyperparameter search. In the multi-armed bandit problem, an online algorithm must choose from a set of strategies in a sequence of n trials so as to minimize the total cost of the chosen strategies. While nearly tight upper and lower bounds are known in the case when the strategy set is finite, much less is known when there is an infinite strategy set. Here we consider the case when the set of strategies is a subset of ℝd, and the cost functions are continuous. In the d = 1 case, we improve on the best-known upper and lower bounds, closing the gap to a sublogarithmic factor. We also consider the case where d > 1 and the cost functions are convex, adapting a recent online convex optimization algorithm of Zinkevich to the sparser feedback model of the multi-armed bandit problem. Motivated by the engineering problem of cellular coverage optimization, we propose a novel multiarmed bandit model called generalized global bandit. We develop a series of greedy algorithms that have the capability to handle nonmonotonic but decomposable reward functions, multidimensional global parameters, and switching costs. The proposed algorithms are rigorously analyzed under the multiarmed bandit framework, where we show that they achieve bounded regret, and hence, they are guaranteed to converge to the optimal arm in finite time. The algorithms are then applied to the cellular coverage optimization problem to achieve the optimal tradeoff between sufficient small cell coverage and limited macroleakage without prior knowledge of the deployment environment. The performance advantage of the new algorithms over existing bandits solutions is revealed analytically and further confirmed via numerical simulations. The key element behind the performance improvement is a more efficient “trial and error” mechanism, in which any trial will help improve the knowledge of all candidate power levels. Multi-armed bandit problems are the predominant theoretical model of exploration-exploitation tradeoffs in learning, and they have countless applications ranging from medical trials, to communication networks, to Web search and advertising. In many of these application domains the learner may be constrained by one or more supply (or budget) limits, in addition to the customary limitation on the time horizon. The literature lacks a general model encompassing these sorts of problems. We introduce such a model, called "bandits with knapsacks", that combines aspects of stochastic integer programming with online learning. A distinctive feature of our problem, in comparison to the existing regret-minimization literature, is that the optimal policy for a given latent distribution may significantly outperform the policy that plays the optimal fixed arm. Consequently, achieving sub linear regret in the bandits-with-knapsacks problem is significantly more challenging than in conventional bandit problems. We present two algorithms whose reward is close to the information-theoretic optimum: one is based on a novel "balanced exploration" paradigm, while the other is a primal-dual algorithm that uses multiplicative updates. Further, we prove that the regret achieved by both algorithms is optimal up to polylogarithmic factors. We illustrate the generality of the problem by presenting applications in a number of different domains including electronic commerce, routing, and scheduling. As one example of a concrete application, we consider the problem of dynamic posted pricing with limited supply and obtain the first algorithm whose regret, with respect to the optimal dynamic policy, is sub linear in the supply.