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We revisit heavy-tailed corrupted least-squares linear regression assuming tohave a corrupted $n$-sized label-feature sample of at most $\epsilon n$arbitrary outliers. We wish to estimate a $p$-dimensional parameter $b^*$ givensuch sample of a label-feature pair $(y,x)$ satisfying $y=\langlex,b^*\rangle+\xi$ with heavy-tailed $(x,\xi)$. We only assume $x$ is $L^4-L^2$hypercontractive with constant $L>0$ and has covariance matrix $\Sigma$ withminimum eigenvalue $1/\mu^2>0$ and bounded condition number $\kappa>0$. Thenoise $\xi$ can be arbitrarily dependent on $x$ and nonsymmetric as long as$\xi x$ has finite covariance matrix $\Xi$. We propose a near-optimalcomputationally tractable estimator, based on the power method, assuming noknowledge on $(\Sigma,\Xi)$ nor the operator norm of $\Xi$. With probability atleast $1-\delta$, our proposed estimator attains the statistical rate$\mu^2\Vert\Xi\Vert^{1/2}(\frac{p}{n}+\frac{\log(1/\delta)}{n}+\epsilon)^{1/2}$and breakdown-point $\epsilon\lesssim\frac{1}{L^4\kappa^2}$, both optimal inthe $\ell_2$-norm, assuming the near-optimal minimum sample size$L^4\kappa^2(p\log p + \log(1/\delta))\lesssim n$, up to a log factor. To thebest of our knowledge, this is the first computationally tractable algorithmsatisfying simultaneously all the mentioned properties. Our estimator is basedon a two-stage Multiplicative Weight Update algorithm. The first stageestimates a descent direction $\hat v$ with respect to the (unknown)pre-conditioned inner product $\langle\Sigma(\cdot),\cdot\rangle$. The secondstage estimate the descent direction $\Sigma\hat v$ with respect to the (known)inner product $\langle\cdot,\cdot\rangle$, without knowing nor estimating$\Sigma$.

A spectral least-squares-type method for heavy-tailed corrupted regression with unknown covariance \& heterogeneous noise