A spectral least-squares-type method for heavy-tailed corrupted regression with unknown covariance & heterogeneous noise
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We revisit heavy-tailed corrupted least-squares linear regression assuming to have a corrupted n-sized label-feature sample of at most ϵ n arbitrary outliers. We wish to estimate a p-dimensional parameter b^* given such sample of a label-feature pair (y,x) satisfying y=⟨ x,b^*⟩+ξ with heavy-tailed (x,ξ). We only assume x is L^4-L^2 hypercontractive with constant L>0 and has covariance matrix Σ with minimum eigenvalue 1/μ^2>0 and bounded condition number κ>0. The noise ξ can be arbitrarily dependent on x and nonsymmetric as long as ξ x has finite covariance matrix Ξ. We propose a near-optimal computationally tractable estimator, based on the power method, assuming no knowledge on (Σ,Ξ) nor the operator norm of Ξ. With probability at least 1-δ, our proposed estimator attains the statistical rate μ^2‖Ξ‖^1/2(p/n+log(1/δ)/n+ϵ)^1/2 and breakdown-point ϵ≲1/L^4κ^2, both optimal in the ℓ_2-norm, assuming the near-optimal minimum sample size L^4κ^2(plog p + log(1/δ))≲ n, up to a log factor. To the best of our knowledge, this is the first computationally tractable algorithm satisfying simultaneously all the mentioned properties. Our estimator is based on a two-stage Multiplicative Weight Update algorithm. The first stage estimates a descent direction v̂ with respect to the (unknown) pre-conditioned inner product ⟨Σ(·),·⟩. The second stage estimate the descent direction Σv̂ with respect to the (known) inner product ⟨·,·⟩, without knowing nor estimating Σ.
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