Online Learning with Adversaries: A Differential Inclusion Analysis
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We consider the measurement model Y = AX, where X and, hence, Y are random variables and A is an a priori known tall matrix. At each time instance, a sample of one of Y's coordinates is available, and the goal is to estimate μ := 𝔼[X] via these samples. However, the challenge is that a small but unknown subset of Y's coordinates are controlled by adversaries with infinite power: they can return any real number each time they are queried for a sample. For such an adversarial setting, we propose the first asynchronous online algorithm that converges to μ almost surely. We prove this result using a novel differential inclusion based two-timescale analysis. Two key highlights of our proof include: (a) the use of a novel Lyapunov function for showing that μ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping time theory to show that our algorithm's iterates are almost surely bounded.
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