Computational-Statistical Gaps for Improper Learning in Sparse Linear Regression
Rares-Darius Buhai, Jingqiu Ding, Stefan Tiegel
TL;DR
The paper investigates computational-statistical gaps for improper learning in sparse linear regression with Gaussian random design, showing that any polynomial-time improper learner likely requires $Ω(k^2)$ samples, far above the information-theoretic $Θ(k \log(d/k))$ samples. The authors build a chain of reductions from a negative-spike sparse PCA (Wishart) problem to sparse linear regression, and bolster the argument with low-degree and statistical-query lower bounds. To handle variance issues, they introduce a paired sparse PCA framework that enforces symmetric noise, enabling a robust reduction to the training-error regime. The results highlight a fundamental k-to-k^2 gap in the random-design setting and situate the findings among concurrent hardness results, while contrasting with known efficient algorithms under restrictive design conditions such as RE or RIP. Overall, the work advances our understanding of when computational constraints prevent achieving information-theoretic performance in improper sparse learning tasks.
Abstract
We study computational-statistical gaps for improper learning in sparse linear regression. More specifically, given $n$ samples from a $k$-sparse linear model in dimension $d$, we ask what is the minimum sample complexity to efficiently (in time polynomial in $d$, $k$, and $n$) find a potentially dense estimate for the regression vector that achieves non-trivial prediction error on the $n$ samples. Information-theoretically this can be achieved using $Θ(k \log (d/k))$ samples. Yet, despite its prominence in the literature, there is no polynomial-time algorithm known to achieve the same guarantees using less than $Θ(d)$ samples without additional restrictions on the model. Similarly, existing hardness results are either restricted to the proper setting, in which the estimate must be sparse as well, or only apply to specific algorithms. We give evidence that efficient algorithms for this task require at least (roughly) $Ω(k^2)$ samples. In particular, we show that an improper learning algorithm for sparse linear regression can be used to solve sparse PCA problems (with a negative spike) in their Wishart form, in regimes in which efficient algorithms are widely believed to require at least $Ω(k^2)$ samples. We complement our reduction with low-degree and statistical query lower bounds for the sparse PCA problems from which we reduce. Our hardness results apply to the (correlated) random design setting in which the covariates are drawn i.i.d. from a mean-zero Gaussian distribution with unknown covariance.