``The right to be forgotten'' ensured by laws for user data privacy becomes increasingly important. Machine unlearning aims to efficiently remove the effect of certain data points on the trained model parameters so that it can be approximately the same as if one retrains the model from scratch. This work proposes stochastic gradient Langevin unlearning, the first unlearning framework based on noisy stochastic gradient descent (SGD) with privacy guarantees for approximate unlearning problems under convexity assumption. Our results show that mini-batch gradient updates provide a superior privacy-complexity trade-off compared to the full-batch counterpart. There are numerous algorithmic benefits of our unlearning approach, including complexity saving compared to retraining, and supporting sequential and batch unlearning. To examine the privacy-utility-complexity trade-off of our method, we conduct experiments on benchmark datasets compared against prior works. Our approach achieves a similar utility under the same privacy constraint while using $2\%$ and $10\%$ of the gradient computations compared with the state-of-the-art gradient-based approximate unlearning methods for mini-batch and full-batch settings, respectively.

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