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Searching with Measurement Dependent Noise

Consider a target moving at a constant velocity on a unit-circumference circle, starting at an arbitrary location. To acquire the target, any region of the circle can be probed to obtain a noisy measurement of the target's presence, where the noise level increases with the size of the probed region. We are interested in the expected time required to find the target to within some given resolution and error probability. For a known velocity, we characterize the optimal tradeoff between time and resolution, and show that in contrast to the well studied case of constant measurement noise, measurement dependent noise incurs a multiplicative gap in the targeting rate between adaptive and non-adaptive search strategies. Moreover, our adaptive strategy attains the optimal rate-reliability tradeoff. We further show that for optimal non-adaptive search, accounting for an unknown velocity incurs a factor of at least two in the targeting rate.