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By ensuring differential privacy in the learning algorithms, one can rigorously mitigate the risk of large models memorizing sensitive training data. In this paper, we study two algorithms for this purpose, i.e., DP-SGD and DP-NSGD, which first clip or normalize \textit{per-sample} gradients to bound the sensitivity and then add noise to obfuscate the exact information. We analyze the convergence behavior of these two algorithms in the non-convex optimization setting with two common assumptions and achieve a rate $\mathcal{O}\left(\sqrt[4]{\frac{d\log(1/δ)}{N^2ε^2}}\right)$ of the gradient norm for a $d$-dimensional model, $N$ samples and $(ε,δ)$-DP, which improves over previous bounds under much weaker assumptions. Specifically, we introduce a regularizing factor in DP-NSGD and show that it is crucial in the convergence proof and subtly controls the bias and noise trade-off. Our proof deliberately handles the per-sample gradient clipping and normalization that are specified for the private setting. Empirically, we demonstrate that these two algorithms achieve similar best accuracy while DP-NSGD is comparatively easier to tune than DP-SGD and hence may help further save the privacy budget when accounting the tuning effort.