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We analyse the complexity of learning first-order queries in a model-theoretic framework for supervised learning introduced by (Grohe and Tur\'an, TOCS 2004). Previous research on the complexity of learning in this framework focussed on the question of when learning is possible in time sublinear in the background structure. Here we study the parameterized complexity of the learning problem. We have two main results. The first is a hardness result, showing that learning first-order queries is atarXiv:2102.12201v2 fatcat:qe6rwrwzjbdshfxnq4s5f57gja