Remove the inverse-square-root sparsity factor from the minimax lower bound

Establish a stronger minimax regret lower bound for online sparse linear regression by proving that the factor O(1/√b) in the current lower bound can be removed.

Background

The paper derives a minimax regret lower bound for (d,b,b₀)-OSLR that scales as Ω(√(dT/(b(b+b₀)))); equivalently, the bound contains an O(1/√b) factor in its problem-dependent dependence. Although the paper characterizes the optimal Θ(√T) dependence on the time horizon, it acknowledges that substantial gaps remain between the lower and upper bounds in their dependence on d, b, and b₀.

The authors explicitly conjecture that the inverse-square-root dependence on the sparsity parameter b can be eliminated, which would yield a stronger lower bound and sharpen the information-theoretic characterization of online sparse linear regression.