Ο πίσω πολλαπλασιασμός

A neural-network training step commonly involves a forward computation, derivative computation, and parameter update. The forward computation forms predictions and a loss, making intermediate values available for the derivative calculation. Backpropagation works backward through the dependencies of this computational graph. It applies the chain rule to combine local derivatives rather than separately recomputing the entire prediction for every parameter. The result includes derivatives of the loss with respect to relevant weights. A derivative describes how the loss changes locally when a parameter changes; it is not a causal accounting of how much total error that parameter independently caused. Backpropagation also does not choose the learning rate or automatically guarantee that weights were updated. An optimizer performs the update using the computed gradients and its own rule. In a composed scalar path, local derivatives multiply: derivatives three and four along that path contribute a product of twelve. When a parameter influences a loss through more than one path, contributions from those paths add. For an illustrative pair of path contributions two and five, the combined derivative is seven. Efficient reverse computation uses graph structure and relevant intermediate values. Saving values consumes memory, and some implementations trade memory for recomputation instead. These choices change runtime and storage costs without changing the role of the derivative calculation. Training quality still depends on the objective, data, optimizer, and configuration rather than a guarantee supplied by backpropagation alone.