Before revision:  The upper bound is a polynomial in 1/ \epsilon improvement when d \in [3,1/ \epsilon^2) (for all kernels except the Gaussian kernel which had a previous upper bound of O((1/\epsilon) \log^d (1/\epsilon)) ) and the lower bound is the first known lower bound to depend on d for this problem.
After revision:  The upper bound is a polynomial in 1/ \varepsilon) improvement when d \in [3,1/ \epsilon^2) (for all kernels except the Gaussian kernel which had a previous upper bound of O((1/\epsilon) \log^d (1/\epsilon)) ) and the lower bound is the first known lower bound to depend on d for this problem.