In a multi-class classification problem, how does the choice of loss function impact the gradient descent optimization process?

In a multi‑class problem, the loss function tells the algorithm how far the predicted probabilities are from the true labels, and this distance is turned into a gradient that moves the weights. A loss that is smooth and convex, like cross‑entropy, produces gradients that are proportional to the difference between predicted and actual class probabilities, making the steps in gradient descent stable and fast. If the loss is too steep or has many flat regions, the gradients can vanish or explode, causing the optimizer to stall or overshoot. For example, using a simple squared error loss on softmax outputs can create very small gradients for high‑confidence predictions, slowing learning compared to cross‑entropy. Therefore, choosing a loss that matches the output activation and provides well‑scaled gradients is essential for efficient convergence.