Optimal Sketching Bounds for Sparse Linear Regression
This work addresses the computational efficiency of sketching algorithms for sparse regression, which is important for machine learning practitioners dealing with high-dimensional data, though it is incremental in extending known techniques to new loss functions.
The paper tackles the problem of determining optimal sketching bounds for sparse linear regression under various loss functions, establishing tight bounds for sparse ℓ2 norm regression with Θ(k log(d/k)/ε²) rows and showing a separation from sparse recovery, while also providing first known bounds for hinge-like loss functions and LASSO regression.
We study oblivious sketching for $k$-sparse linear regression under various loss functions such as an $\ell_p$ norm, or from a broad class of hinge-like loss functions, which includes the logistic and ReLU losses. We show that for sparse $\ell_2$ norm regression, there is a distribution over oblivious sketches with $Θ(k\log(d/k)/\varepsilon^2)$ rows, which is tight up to a constant factor. This extends to $\ell_p$ loss with an additional additive $O(k\log(k/\varepsilon)/\varepsilon^2)$ term in the upper bound. This establishes a surprising separation from the related sparse recovery problem, which is an important special case of sparse regression. For this problem, under the $\ell_2$ norm, we observe an upper bound of $O(k \log (d)/\varepsilon + k\log(k/\varepsilon)/\varepsilon^2)$ rows, showing that sparse recovery is strictly easier to sketch than sparse regression. For sparse regression under hinge-like loss functions including sparse logistic and sparse ReLU regression, we give the first known sketching bounds that achieve $o(d)$ rows showing that $O(μ^2 k\log(μn d/\varepsilon)/\varepsilon^2)$ rows suffice, where $μ$ is a natural complexity parameter needed to obtain relative error bounds for these loss functions. We again show that this dimension is tight, up to lower order terms and the dependence on $μ$. Finally, we show that similar sketching bounds can be achieved for LASSO regression, a popular convex relaxation of sparse regression, where one aims to minimize $\|Ax-b\|_2^2+λ\|x\|_1$ over $x\in\mathbb{R}^d$. We show that sketching dimension $O(\log(d)/(λ\varepsilon)^2)$ suffices and that the dependence on $d$ and $λ$ is tight.