neural network optimization adam sgd

**Neural Network Optimizers** are **the algorithms that update model parameters based on computed gradients to minimize the training loss function — with the choice of optimizer (SGD, Adam, AdamW, LAMB) and its hyperparameters (learning rate, momentum, weight decay) directly determining convergence speed, final accuracy, and generalization quality of the trained model**. **Stochastic Gradient Descent (SGD):** - **Vanilla SGD**: θ_{t+1} = θ_t - η∇L(θ_t) — learning rate η scales gradient; noisy gradient estimates from mini-batches provide implicit regularization but cause slow convergence - **Momentum**: accumulate exponentially decayed gradient history — v_t = βv_{t-1} + ∇L(θ_t), θ_{t+1} = θ_t - ηv_t; β=0.9 typical; accelerates convergence in consistent gradient directions while dampening oscillations - **Nesterov Momentum**: evaluate gradient at the "look-ahead" position — computes gradient at θ_t - ηβv_{t-1} instead of θ_t; provides better convergence for convex objectives; slightly better in practice than standard momentum - **SGD + Momentum**: still achieves best generalization for many vision tasks — requires careful learning rate tuning and schedule but often produces models that generalize better than adaptive methods **Adaptive Learning Rate Methods:** - **Adam**: maintains per-parameter first moment (mean) and second moment (uncentered variance) of gradients — m_t = β₁m_{t-1} + (1-β₁)g_t, v_t = β₂v_{t-1} + (1-β₂)g_t²; update = η × m̂_t/(√v̂_t + ε) where m̂, v̂ are bias-corrected; default β₁=0.9, β₂=0.999, ε=1e-8 - **AdamW**: fixes weight decay implementation in Adam — standard Adam applies L2 regularization to gradient before adaptive scaling (incorrect), AdamW applies weight decay directly to weights after Adam step (correct); consistently outperforms Adam with L2 regularization - **AdaGrad**: accumulates squared gradients from all past steps — effective for sparse gradients (NLP embeddings) but learning rate monotonically decreases, eventually becoming too small to learn - **RMSProp**: AdaGrad with exponential moving average of squared gradients — prevents learning rate from shrinking to zero; predecessor to Adam; still used for RNN training in some settings **Large Batch Optimization:** - **LARS (Layer-wise Adaptive Rate Scaling)**: adjusts learning rate per layer based on weight-to-gradient norm ratio — enables training with batch sizes up to 32K without accuracy loss; used for large-batch ImageNet training - **LAMB (Layer-wise Adaptive Moments for Batch training)**: combines LARS-style layer adaptation with Adam — enables BERT pre-training with batch size 64K in 76 minutes; critical for distributed training efficiency - **Gradient Accumulation**: simulate large batch by accumulating gradients over multiple forward-backward passes — equivalent to large batch training without additional GPU memory; division by accumulation steps normalizes gradient scale **Optimizer selection is a foundational decision in deep learning training — AdamW has become the default for Transformer-based models (NLP, ViT), while SGD with momentum remains competitive for CNNs; understanding the tradeoffs between convergence speed, memory overhead, and generalization quality enables practitioners to choose the optimal optimizer for each architecture and dataset.**