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∈Rd 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.

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QIP 2022, 25th Annual Conference on Quantum Information Processing
Algorithms and Complexity

Chen, Y, & de Wolf, R.M. (2022). Quantum algorithms and lower bounds for linear regression with norm constraints.