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Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information in a speedy manner about an underlying phenomena of interest while accounting for the cost of data collection. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively (re-)evaluate the tradeoff between the cost of various sensing actions and the precision of their outcomes. In this paper, using results in dynamic programming, a lower bound for the optimal total cost is established. Moreover, an upper bound is obtained using a heuristic policy for dynamic selection of actions. Using the obtained bounds, the closed loop (feedback) gain is shown to be at least logarithmic in the penalty associated with wrong declarations. Furthermore, it is shown that the proposed heuristic achieves asymptotic optimality in many practically relevant problems such as variable-length coding with feedback and noisy dynamic search.