Advanced Optimizers | Math for ML

Optimizer Showdown: Learning Curves

Compare how different optimizers train a neural network. Lower loss is better!

Training Dataset:

-

Adam Final Loss

-

SGD Final Loss

-

RMSprop Final Loss

-

Momentum Final Loss

Meet the Optimizers

🎯 Adam

Adaptive Moment Estimation

m = β₁·m + (1-β₁)·g
v = β₂·v + (1-β₂)·g²
θ -= η·m/(√v + ε)

✅ Pros: Fast convergence, handles sparse data, popular

❌ Cons: Can generalize worse, memory overhead

🚀 SGD + Momentum

Stochastic Gradient Descent with Momentum

v = β·v - η·g
θ += v

✅ Pros: Simple, often better generalization, proven

❌ Cons: Hyperparameter tuning, slow start

⚡ RMSprop

Root Mean Square Propagation

v = β·v + (1-β)·g²
θ -= η·g/√(v + ε)

✅ Pros: Adaptive rates, good for RNNs

❌ Cons: Less popular than Adam

🔄 AdamW

Adam with Decoupled Weight Decay

m = β₁·m + (1-β₁)·g
v = β₂·v + (1-β₂)·g²
θ -= η·m/(√v + ε) - λ·θ

✅ Pros: Better generalization than Adam

❌ Cons: More recent, slight overhead

⚙️ Adagrad

Adaptive Gradient

G += g²
θ -= η·g/√(G + ε)

✅ Pros: Classic, handles sparse features

❌ Cons: Learning rate keeps decreasing

🎬 Nesterov

Momentum with Lookahead

v = β·v - η·∇f(θ + β·v)
θ += v

✅ Pros: Faster convergence, theoretical guarantee

❌ Cons: Complex to implement

When to Use Which?

Task	Recommended	Why
Computer Vision (ResNets)	SGD + Momentum	Better generalization, proven track record
Transformers (NLP)	AdamW	Works out of box, better generalization
Recurrent Networks	RMSprop or Adam	Handles gradient explosion/vanishing
Sparse Data	Adagrad or Adam	Adaptive learning rates per parameter

💡 Pro Tips

🎯 Default choice: Start with Adam, tune if needed
📊 Production models: Use SGD+Momentum for best generalization
⚡ Learning rate schedule: Combine any optimizer with learning rate decay
🔍 Hyperparameter tuning: β₁=0.9, β₂=0.999 are good defaults
📈 Monitor: Watch both training and validation loss to detect overfitting

Advanced Optimization Algorithms