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- W2789163852 abstract "The $n times d$ design matrix in a linear regression problem is given, but the response for each point is hidden unless explicitly requested. The goal is to observe only a small number $k ll n$ of the responses, and then produce a weight vector whose sum of square loss over all points is at most $1+epsilon$ times the minimum. A standard approach to this problem is to use i.i.d. leverage score sampling, but this approach is known to perform poorly when $k$ is small (e.g., $k = d$); in such cases, it is dominated by volume sampling, a joint sampling method that explicitly promotes diversity. How these methods compare for larger $k$ was not previously understood. We prove that volume sampling can have poor behavior for large $k$ - indeed worse than leverage score sampling. We also show how to repair volume sampling using a new padding technique. We prove that padded volume sampling has at least as good a tail bound as leverage score sampling: sample size $k=O(dlog d + d/epsilon)$ suffices to guarantee total loss at most $1+epsilon$ times the minimum with high probability. The main technical challenge is proving tail bounds for the sums of dependent random matrices that arise from volume sampling." @default.
- W2789163852 created "2018-03-06" @default.
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- W2789163852 date "2018-02-19" @default.
- W2789163852 modified "2023-09-27" @default.
- W2789163852 title "Tail bounds for volume sampled linear regression." @default.
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