Explain Adam, AdamW, and Decoupled Weight Decay

Explain Adam and AdamW. Why is adding an L2 penalty to the loss generally different from decoupled weight decay under Adam? ### Constraints & Assumptions Use standard first- and second-moment estimates, bias correction, and a scalar learning rate. State whether a parameter group is subject to decay; treatment of biases and normalization parameters is a separate choice. ### Clarifying Questions Is regularization added to the gradient or applied directly to parameters? What learning-rate schedule and epsilon convention are used? Which parameters should decay? ### What a Strong Answer Covers Write the moment updates, distinguish the two decay mechanisms, and explain how adaptive gradient scaling changes an L2 gradient contribution. ### Follow-up Questions When can L2 regularization and weight decay be equivalent? Why does the decay coefficient interact with the learning-rate schedule? Does AdamW remove the need to tune regularization?