Online Sparse Linear Regression
This work addresses computational hardness in online learning for sparse linear regression, which is incremental as it builds on prior open problems.
The paper tackles the online sparse linear regression problem by providing an inefficient algorithm with regret bounded by $ ilde{O}(\sqrt{T})$ and proving that no polynomial-time algorithm can achieve sublinear regret unless $ ext{NP} \subseteq ext{BPP}$, resolving an open problem from COLT 2014.
We consider the online sparse linear regression problem, which is the problem of sequentially making predictions observing only a limited number of features in each round, to minimize regret with respect to the best sparse linear regressor, where prediction accuracy is measured by square loss. We give an inefficient algorithm that obtains regret bounded by $\tilde{O}(\sqrt{T})$ after $T$ prediction rounds. We complement this result by showing that no algorithm running in polynomial time per iteration can achieve regret bounded by $O(T^{1-δ})$ for any constant $δ> 0$ unless $\text{NP} \subseteq \text{BPP}$. This computational hardness result resolves an open problem presented in COLT 2014 (Kale, 2014) and also posed by Zolghadr et al. (2013). This hardness result holds even if the algorithm is allowed to access more features than the best sparse linear regressor up to a logarithmic factor in the dimension.