The hidden state threat model of differential privacy (DP) assumes that the adversary has access only to the final trained machine learning (ML) model, without seeing intermediate states during training. Current privacy analyses under this model, however, are limited to convex optimization problems, reducing their applicability to multi-layer neural networks, which are essential in modern deep learning applications. Additionally, the most successful applications of the hidden state privacy analyses in classification tasks have been for logistic regression models. We demonstrate that it is possible to privately train convex problems with privacy-utility trade-offs comparable to those of one hidden-layer ReLU networks trained with DP stochastic gradient descent (DP-SGD). We achieve this through a stochastic approximation of a dual formulation of the ReLU minimization problem which results in a strongly convex problem. This enables the use of existing hidden state privacy analyses, providing accurate privacy bounds also for the noisy cyclic mini-batch gradient descent (NoisyCGD) method with fixed disjoint mini-batches. Our experiments on benchmark classification tasks show that NoisyCGD can achieve privacy-utility trade-offs comparable to DP-SGD applied to one-hidden-layer ReLU networks. Additionally, we provide theoretical utility bounds that highlight the speed-ups gained through the convex approximation.