Parameter-free Regret in High Probability with Heavy Tails
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We present new algorithms for online convex optimization over unbounded domains that obtain parameter-free regret in high-probability given access only to potentially heavy-tailed subgradient estimates. Previous work in unbounded domains considers only in-expectation results for sub-exponential subgradients. Unlike in the bounded domain case, we cannot rely on straight-forward martingale concentration due to exponentially large iterates produced by the algorithm. We develop new regularization techniques to overcome these problems. Overall, with probability at most δ, for all comparators 𝐮 our algorithm achieves regret Õ(𝐮 T^1/𝔭log (1/δ)) for subgradients with bounded 𝔭^th moments for some 𝔭∈ (1, 2].
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