2024/12/05 by Prasadan, Akshay, Neykov, Matey · 1 citation
#FOS: Mathematics #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2412.03832
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)\wedge d2 with η^* = sup \η≥ 0 : (Nη2)/(σ2) ≤ log MKloc(η,c)\, where log MKloc(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))\wedge d2. We generalize our results to the case when K is star-shaped but unbounded.