We study the worst-case adaptive optimization problem with budget constraint that is useful for modeling various practical applications in artificial intelligence and machine learning. We investigate the near-optimality of greedy algorithms for this problem with both modular and non-modular cost functions. In both cases, we prove that two simple greedy algorithms are not near-optimal but the best between them is near-optimal if the utility function satisfies pointwise submodularity and pointwise cost-sensitive submodularity respectively. This implies a combined algorithm that is near-optimal with respect to the optimal algorithm that uses half of the budget. We discuss applications of our theoretical results and also report experiments comparing the greedy algorithms on the active learning problem.
The adaptive optimization problem in our paper is for the worst-case maximum coverage setting, where we look for a policy that maximizes the utility while maintaining its cost within a certain budget. This problem can also be considered in the worst-case min-cost setting @cite_9 , where we look for a policy that can achieve at least a certain value of utility while minimizing the total cost.
@cite_9
2909648661 2080379754 2164571150 171504317
We consider the problem of controlling a fully specified Markov decision process (MDP), also known as the planning problem, when the state space is very large and calculating the optimal policy is intractable. Instead, we pursue the more modest goal of optimizing over some small family of policies. Specifically, we show that the family of policies associated with a low-dimensional approximation of occupancy measures yields a tractable optimization. Moreover, we propose an efficient algorithm, scaling with the size of the subspace but not the state space, that is able to find a policy with low excess loss relative to the best policy in this class. To the best of our knowledge, such results did not exist in the literature previously. We bound excess loss in the average cost and discounted cost cases, which are treated separately. Preliminary experiments show the effectiveness of the proposed algorithms in a queueing application. Abstract The budgeted maximum coverage problem is: given a collection S of sets with associated costs defined over a domain of weighted elements, and a budget L , find a subset of S ′⫅ S such that the total cost of sets in S ′ does not exceed L , and the total weight of elements covered by S ′ is maximized. This problem is NP-hard. For the special case of this problem, where each set has unit cost, a (1−1 e ) -approximation is known. Yet, prior to this work, no approximation results were known for the general cost version. The contribution of this paper is a (1−1 e ) -approximation algorithm for the budgeted maximum coverage problem. We also argue that this approximation factor is the best possible, unless NP ⫅ DTIME (n O ( log log n) ) . One central issue in practically deploying network coding is the adaptive and economic allocation of network resource. We cast this as an optimization, where the net-utility-the difference between a utility derived from the attainable multicast throughput and the total cost of resource provisioning-is maximized. By employing the MAX of flows characterization of the admissible rate region for multicasting, this paper gives a novel reformulation of the optimization problem, which has a separable structure. The Lagrangian relaxation method is applied to decompose the problem into subproblems involving one destination each. Our specific formulation of the primal problem results in two key properties. First, the resulting subproblem after decomposition amounts to the problem of finding a shortest path from the source to each destination. Second, assuming the net-utility function is strictly concave, our proposed method enables a near-optimal primal variable to be uniquely recovered from a near-optimal dual variable. A numerical robustness analysis of the primal recovery method is also conducted. For ill-conditioned problems that arise, for instance, when the cost functions are linear, we propose to use the proximal method, which solves a sequence of well-conditioned problems obtained from the original problem by adding quadratic regularization terms. Furthermore, the simulation results confirm the numerical robustness of the proposed algorithms. Finally, the proximal method and the dual subgradient method can be naturally extended to provide an effective solution for applications with multiple multicast sessions With an abstraction of serving rate-adaptive sources on a broadcast-type wireless channel as a utility maximization problem, it is shown how one can design many intuitive online scheduling policies based upon the feedback that one obtains at the scheduler. Using a stochastic approximation argument it is then shown that the constructed algorithms converge to optimal solutions of the utility maximization problem over different sets which critically depend on the quality of the feedback information.