Lasso and Ridge are important minimization problems in machine learning and statistics. They are versions of linear regression with squared loss where the vector θ ∈ ℝ^d of coefficients is constrained in either 𝓁₁-norm (for Lasso) or in 𝓁₂-norm (for Ridge). We study the complexity of quantum algorithms for finding ε-minimizers for these minimization problems. We show that for Lasso we can get a quadratic quantum speedup in terms of d by speeding up the cost-per-iteration of the Frank-Wolfe algorithm, while for Ridge the best quantum algorithms are linear in d, as are the best classical algorithms. As a byproduct of our quantum lower bound for Lasso, we also prove the first classical lower bound for Lasso that is tight up to polylog-factors.
@InProceedings{chen_et_al:LIPIcs.ICALP.2023.38, author = {Chen, Yanlin and de Wolf, Ronald}, title = {{Quantum Algorithms and Lower Bounds for Linear Regression with Norm Constraints}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {38:1--38:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.38}, URN = {urn:nbn:de:0030-drops-180907}, doi = {10.4230/LIPIcs.ICALP.2023.38}, annote = {Keywords: Quantum algorithms, Regularized linear regression, Lasso, Ridge, Lower bounds} }
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