勾配降下法

For a differentiable scalar objective, the gradient identifies the direction of steepest local increase under the usual Euclidean norm. Plain gradient descent moves in the opposite direction. Its update is new weight equals old weight minus learning rate times gradient. In a scalar illustration with old weight two, gradient three, and learning rate one-tenth, the new weight is one-point-seven. That formula describes the proposed update, not a guarantee that every finite step reduces loss. A learning rate that is too large can overshoot or lead to divergence, and a very small rate can make progress slow. Full-batch gradient descent uses the selected training objective over the whole dataset. Stochastic and mini-batch methods instead use gradients from sampled examples or batches. The batch gradient is exact for that batch calculation but estimates the full-data gradient; the resulting variability can help or hurt depending on the problem. Momentum combines information from earlier gradients to influence the current update. Adam maintains per-parameter moment estimates that adapt updates, so it should not be confused with plain unmodified gradient descent. Nonconvex loss surfaces also contain stationary points. A zero gradient is not proof of a globally lowest objective: local minima, maxima, and saddle points can have zero gradient. Training therefore needs monitoring of the objective and relevant evaluation metrics rather than assuming that a named optimizer or a stationary point establishes success.