Research map / Learning theory and algorithms / Sample complexity
Statistical Query Lower Bounds for List-Decodable Linear Regression
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Require the output list to contain a regressor within ρ/2 of the target under the paper's SQ construction | List-decodable Gaussian linear regression · SQ model | query complexity: 2^{Ω(d^{c})} d^{-(2m+1)(1/2-c)} queries, or a query to STAT(Ω(d)^(-(2m+1)(1/4-c/2)) e^{O(m)}/σ) | Hurt |
| Match the first 2m conditional moments to the standard Gaussian | SQ-hard distribution construction · List-decodable Gaussian linear regression | pairwise correlation: γ ≤ d^{-Ω(m)}b | Helped |
| Use a reference label distribution with heavier tails, R(y)=N(0,1/α) | SQ-hard distribution construction · List-decodable Gaussian linear regression | validity of the contamination construction | Helped |
| Apply Gaussian smoothing with the Ornstein-Uhlenbeck operator to the moment-matched distribution | SQ-hard distribution construction · Conditional covariate distribution | χ² divergence: χ²(A_y, N(0,1)) ≤ α O(e^{y²(α-1/2)})/(1-ρ²) + O(e^{B²/2})/(1-ρ²) | Helped |
| Use a computationally inefficient information-theoretic list-decoding algorithm | List-decodable Gaussian linear regression · Gaussian covariates with α-fraction inliers | ℓ2 error: O((σ/α)√log(1/α)) | Helped |
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