Information theoretic limits of robust sub-Gaussian mean estimation under star-shaped constraints

We obtain the minimax rate for a mean location model with a bounded star-shaped set K⊆Rn constraint on the mean, in an adversarially corrupted data setting with Gaussian noise. We assume an unknown fraction ϵ≤1/2−κ for some fixed κ∈(0,1/2] of N observations are arbitrarily corrupted. We obtain a minimax risk up to proportionality constants under the squared ℓ2 loss of max(η∗2,σ2ϵ2)∧d2 with η∗=sup{η≥0:Nη2 σ2≤logMKloc(η,c)}, where logMKloc(η,c) denotes the local entropy of the set K, d is the diameter of K, σ2 is the variance and c is some sufficiently large absolute constant. A variant of our algorithm achieves the same rate for settings with known or symmetric sub-Gaussian noise, with a smaller breakdown point, still of constant order. We further study the case of unknown sub-Gaussian noise and show that the rate is slightly slower: max(η∗2,σ2ϵ2log(1/ϵ))∧d2. We generalize our results to the case when K is star-shaped but unbounded.