Stochastic gradient descent (SGD) and its variants train deep neural networks (DNNs) effectively in practice, yet classical optimization theory does not fully explain this success: conventional analyses rest on differentiability, convexity, and smoothness, assumptions that DNN objectives routinely violate. We close this gap by generalizing convexity and smoothness, through Legendre functions and convex conjugation, into a unified optimization framework for DNN training. We introduce \(\mathcal{H}(ψ)\)-convexity and \(\mathcal{H}(Ψ)\)-smoothness, a single formalism that encompasses convex and non-convex as well as smooth and non-smooth objectives and reveals a natural duality between generalized convexity and smoothness. Building on these notions, we derive generalized gradient descent (GD) and generalized SGD via convex conjugation. We prove that generalized GD attains an optimal learning rate of exactly \(1\), and we establish rigorous gradient-energy-based convergence rates for both optimizers. We further recast DNN training as a composite optimization problem and show that its convergence hinges on jointly reducing the gradient energy and controlling the extreme eigenvalues of the structure matrix; a model capacity risk quantifies the residual error that no optimizer can remove. Experiments on six datasets (MNIST, Fashion-MNIST, CIFAR-10, CIFAR-100, TREC, and SST-2) and on convolutional, residual, and Transformer architectures, with five loss functions, are consistent with the predicted training dynamics.

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