No, stochastic gradient descent (SGD) needs a differentiable loss function (differentiable with respect to the trainable parameters). The stochastic bit is that

No, stochastic gradient descent (SGD) needs a differentiable loss function (differentiable with respect to the trainable parameters). The stochastic bit is that you want to optimize the loss for the whole dataset but what you actually do is to do each gradient descent step on a small mini-batch of the dataset. There might be extensions where you have some non-differentiable part where you just assume some gradient in back-propagation. Like Gumble-Softmax (https://arxiv.org/abs/1611.01144 https://arxiv.org/abs/1611.01144). Or similar is the reparametrization trick in VAEs (https://arxiv.org/abs/1606.05908 https://arxiv.org/abs/1606.05908). But that are special cases. Maybe you are thinking about reinforcement learning?