Optimization algorithms gradient descent variants

Master optimization algorithms for machine learning including gradient descent variants and advanced optimization techniques. Gradient descent fundamentals: 1. Batch gradient descent: full dataset computation, stable convergence, slow for large datasets. 2. Stochastic gradient descent (SGD): single sample updates, noisy gradients, faster convergence. 3. Mini-batch gradient descent: compromise between batch and SGD, batch size 32-512. Advanced optimizers: 1. Momentum: velocity accumulation, β=0.9, overcomes local minima, accelerated convergence. 2. Adam: adaptive learning rates, β1=0.9, β2=0.999, bias correction, most popular choice. 3. RMSprop: adaptive learning rate, root mean square propagation, good for RNNs. Learning rate scheduling: 1. Step decay: reduce LR by factor (0.1) every epoch, plateau detection. 2. Cosine annealing: cyclical learning rate, warm restarts, exploration vs exploitation. 3. Exponential decay: gradual reduction, smooth convergence, fine-tuning applications. Second-order methods: 1. Newton's method: Hessian matrix, quadratic convergence, computational expensive. 2. Quasi-Newton methods: BFGS, L-BFGS for large-scale problems, approximated Hessian. 3. Natural gradients: Fisher information matrix, geometric optimization, natural parameter space. Regularization integration: 1. L1/L2 regularization: weight decay, sparsity promotion, overfitting prevention. 2. Elastic net: combined L1/L2, feature selection, ridge regression benefits. 3. Dropout: stochastic regularization, ensemble effect, neural network specific. Hyperparameter optimization: grid search, random search, Bayesian optimization, learning rate range test, cyclical learning rates, adaptive batch sizes for optimal convergence speed and stability.