Multi-label audio tagging consists of assigning sets of tags to audio recordings. At inference time, thresholds are applied on the confidence scores outputted by a probabilistic classifier, in order to decide which classes are detected active. In this work, we consider having at disposal a trained classifier and we seek to automatically optimize the decision thresholds according to a performance metric of interest, in our case F-measure (micro-F1). We propose a new method, called SGL-Thresh for Surrogate Gradient Learning of Thresholds, that makes use of gradient descent. Since F1 is not differentiable, we propose to approximate the thresholding operation gradients with the gradients of a sigmoid function. We report experiments on three datasets, using state-of-the-art pre-trained deep neural networks. In all cases, SGL-Thresh outperformed three other approaches: a default threshold value (defThresh), an heuristic search algorithm and a method estimating F1 gradients numerically. It reached 54.9\% F1 on AudioSet eval, compared to 50.7% with defThresh. SGL-Thresh is very fast and scalable to a large number of tags. To facilitate reproducibility, data and source code in Pytorch are available online: https://github.com/topel/SGL-Thresh