Approximation Algorithms for Active Sequential Hypothesis Testing
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In the problem of active sequential hypotheses testing (ASHT), a learner seeks to identify the true hypothesis h^* from among a set of hypotheses H. The learner is given a set of actions and knows the outcome distribution of any action under any true hypothesis. While repeatedly playing the entire set of actions suffices to identify h^*, a cost is incurred with each action. Thus, given a target error δ>0, the goal is to find the minimal cost policy for sequentially selecting actions that identify h^* with probability at least 1 - δ. This paper provides the first approximation algorithms for ASHT, under two types of adaptivity. First, a policy is partially adaptive if it fixes a sequence of actions in advance and adaptively decides when to terminate and what hypothesis to return. Under partial adaptivity, we provide an O(s^-1(1+log_1/δ|H|)log (s^-1|H| log |H|))-approximation algorithm, where s is a natural separation parameter between the hypotheses. Second, a policy is fully adaptive if action selection is allowed to depend on previous outcomes. Under full adaptivity, we provide an O(s^-1log (|H|/δ)log |H|)-approximation algorithm. We numerically investigate the performance of our algorithms using both synthetic and real-world data, showing that our algorithms outperform a previously proposed heuristic policy.
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