23.2 Stochastic optimization techniques

Stochastic optimization tackles problems with uncertain variables, aiming to maximize or minimize expected outcomes. It's crucial in fields like finance and logistics, where randomness plays a big role in decision-making.

Techniques like stochastic gradient descent help solve these problems efficiently. By using random subsets of data, these methods can handle large-scale issues quickly, though they may sacrifice some accuracy for speed.

Stochastic Optimization Techniques

Formulation of stochastic optimization problems

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• Stochastic optimization problems involve objective functions that depend on random variables (demand uncertainty) and constraints that may involve random variables (resource availability)
  • Goal is to minimize or maximize the expected value of the objective function (expected profit or cost)
• Problem formulation requires defining decision variables (production quantities), specifying the objective function (may involve expectations, probabilities, or moments of random variables), and identifying constraints
  • Deterministic constraints are fixed and known with certainty (budget limits)
  • Probabilistic constraints, also known as chance constraints, hold with a certain probability (meeting demand with 95% confidence)
• Examples of stochastic optimization problems include portfolio optimization with uncertain returns (stock prices), inventory management with stochastic demand (retail sales), and resource allocation under uncertainty (project scheduling)

Implementation of stochastic gradient descent

• Stochastic gradient descent (SGD) is an iterative optimization algorithm suitable for large-scale problems with many training examples (image classification datasets)
• Algorithm steps:
  1. Initialize the model parameters (weights and biases)
  2. Repeat until convergence:
     • Randomly select a subset (mini-batch) of training examples (64 images)
     • Compute the gradient of the objective function with respect to the model parameters using the mini-batch (backpropagation)
     • Update the model parameters using the computed gradient and a learning rate (weight update rule)
• Learning rate determines the step size in the parameter update and can be fixed or adaptive, decreasing over time (learning rate decay)
• Mini-batch size is the number of training examples used in each iteration, representing a trade-off between computation efficiency and convergence speed (larger batches converge slower but are more computationally efficient)
• Variants of SGD include momentum (accelerates convergence by considering previous gradients), Adagrad (adapts the learning rate for each parameter based on historical gradients), and Adam (combines momentum and adaptive learning rates)

Convergence of stochastic algorithms

• Convergence analysis studies the behavior of the optimization algorithm as the number of iterations increases, aiming to establish convergence guarantees and rates
• Convex objective functions and constraints simplify convergence analysis, and SGD converges to the global optimum for convex problems (linear regression)
• For non-convex problems (deep neural networks), convergence to a global optimum is not guaranteed, and SGD may converge to a local optimum or a saddle point
• Convergence rates:
  • Sublinear convergence: $O(1/\sqrt{k})$ for general convex problems
  • Linear convergence: $O(1/k)$ for strongly convex problems, where $k$ denotes the number of iterations
• Factors affecting convergence include the learning rate schedule (constant vs. adaptive), mini-batch size (larger batches may slow convergence), and noise in the gradients (stochastic vs. batch gradients)

Comparison of stochastic optimization techniques

• Stochastic gradient descent (SGD) is simple, widely used, and suitable for large-scale problems but may require careful tuning of learning rate and mini-batch size
• Stochastic average gradient (SAG) maintains a running average of the gradients, leading to faster convergence than SGD for strongly convex problems but has higher memory requirements due to storing gradients
• Stochastic variance reduced gradient (SVRG) reduces the variance of the gradients by using a full gradient computation periodically, resulting in faster convergence than SGD and SAG for strongly convex problems but incurs additional computational overhead for the full gradient computation
• Stochastic dual coordinate ascent (SDCA) operates in the dual space and is suitable for problems with sparse data (text classification), achieving linear convergence for strongly convex problems
• Trade-offs between stochastic optimization techniques include convergence speed vs. computational complexity, memory requirements (storing gradients), and sensitivity to hyperparameters (learning rate, mini-batch size)