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Building useful predictive models often involves learning from sensitive data. Training models with differential privacy can guarantee the privacy of such sensitive data. For convex optimization tasks, several differentially private algorithms are known, but none has yet been deployed in practice. In this work, we make two major contributions towards practical differentially private convex optimization. First, we present Approximate Minima Perturbation, a novel algorithm that can leverage any off-the-shelf optimizer. We show that it can be employed without any hyperparameter tuning, thus making it an attractive technique for practical deployment. Second, we perform an extensive empirical evaluation of the state-of-the-art algorithms for differentially private convex optimization, on a range of publicly available benchmark datasets, and real-world datasets obtained through an industrial collaboration. We release open-source implementations of all the differentially private convex optimization algorithms considered, and benchmarks on as many as nine public datasets, four of which are high-dimensional.