adam optimizer

Adam optimizer combines momentum and adaptive learning rates, the default choice for most deep learning. **Algorithm**: Maintains exponential moving averages of gradient (m) and squared gradient (v). Update: w -= lr * m / (sqrt(v) + eps). **Key features**: Per-parameter learning rates adapt to gradient history. Momentum smooths updates. Bias correction for early steps. **Hyperparameters**: lr (learning rate, ~1e-4 to 3e-4 for LLMs), beta1 (momentum, 0.9), beta2 (squared gradient decay, 0.999), epsilon (stability, 1e-8). **Variants**: **AdamW**: Decouples weight decay from gradient update. Preferred for transformers. **Adafactor**: Memory-efficient, factorizes second moment. **8-bit Adam**: Quantized states for memory savings. **Memory cost**: 2 states per parameter (m, v) plus parameters = 3x parameter memory. **Comparison to SGD**: Adam converges faster early, SGD may generalize better with tuning. Adam is default. **For LLMs**: AdamW with beta1=0.9, beta2=0.95 common. Higher beta2 for stability. **Best practices**: Use AdamW for transformers, tune learning rate first, default betas usually fine.