Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in the non-private case. In contrast, we explore how the privacy noise affects the optimization property. We show that for differentially private convex optimization, the utility guarantee of differentially private (stochastic) gradient descent is determined by an expected curvature rather than the minimum curvature. The expected curvature, which represents the average curvature over the optimization path, is usually much larger than the minimum curvature. By using the expected curvature, we show that gradient perturbation can achieve a significantly improved utility guarantee that can theoretically justify the advantage of gradient perturbation over other perturbation methods. Finally, our extensive experiments suggest that gradient perturbation with the advanced composition method indeed outperforms other perturbation approaches by a large margin, matching our theoretical findings.