Gradient Descent Method

Recall from Previous Lectures:

• Necessary conditions for optimality (KKT conditions)

• Sufficient conditions using Hessian matrix

• Classification of critical points

Focus of this Topic:

• How do we find these optimal points algorithmically?

• Gradient descent: the foundational iterative method

Historical Context:

• Cauchy (1847): First systematic use of steepest descent

• Foundation for modern machine learning algorithms

• Still the workhorse for large-scale optimization

Key Applications:

• Machine Learning: Training neural networks

• Statistics: Maximum likelihood estimation

• Engineering: Parameter identification

• Finance: Portfolio optimization

Key Insights:

• Gradient \(\nabla f(x)\) is perpendicular to level curves

• Points in direction of steepest increase

• Negative gradient gives steepest decrease

First-Order Taylor Expansion:

To minimize locally, choose \(d = -\alpha \nabla f(x)\)

• At point \(x_k\), the gradient \(\nabla f(x_k)\) points in the direction of steepest ascent

• To minimize \(f\), we move in the opposite direction: \(-\nabla f(x_k)\)

• This is the direction of steepest descent

Key Components:

• Direction: \(d_k = -\nabla f(x_k)\) (steepest descent direction)

• Step size: \(\alpha_k\) (how far to move)

• Stopping criterion: \(\|\nabla f(x_k)\| \leq \epsilon\)

Gradient Norm: \(\|\nabla f(x_k)\| \leq \epsilon_g\)

• Most common and theoretically justified

• Measures proximity to stationary point

Function Value Change: \(|f(x_k) - f(x_{k-1})| \leq \epsilon_f\)

• Practical when function evaluation is expensive

• May terminate prematurely in flat regions

Step Size: \(\|x_k - x_{k-1}\| \leq \epsilon_x\)

• Useful when solution accuracy is more important than optimality

• Can be misleading with small step sizes

Combined Criteria: Often use multiple conditions simultaneously

• Fixed step size: \(\alpha_k = \alpha\) (constant)

• Exact line search: \(\alpha_k = \arg\min_{\alpha > 0} f(x_k - \alpha \nabla f(x_k))\)

• Inexact line search: Armijo rule, Wolfe conditions

• Adaptive methods: AdaGrad, RMSprop, Adam (for ML)

Idea: Start with a large step size and reduce until sufficient decrease condition is met.

Backtracking Algorithm:

1. Start with \(\alpha = \alpha_0\) (e.g., \(\alpha_0 = 1\))

2. While Armijo condition fails: \(\alpha \leftarrow \rho \alpha\) (e.g., \(\rho = 0.5\))

3. Accept current \(\alpha\) as \(\alpha_k\)

Advantages: Guarantees sufficient decrease, adaptive to local function behavior

More sophisticated line search conditions:

Typical values: \(c_1 = 10^{-4}\), \(c_2 = 0.9\) (for Newton-type methods) or \(c_2 = 0.1\) (for conjugate gradient)

Benefits:

• Ensures both sufficient decrease and sufficient progress

• Prevents steps that are too small

• Theoretical guarantees for many optimization algorithms

Standard Assumptions:

1. \(f: \mathbb{R}^n \to \mathbb{R}\) is continuously differentiable

2. \(f\) is bounded below: \(f(x) \geq f_{\text{inf}} > -\infty\) for all \(x\)

3. \(\nabla f\) is Lipschitz continuous with constant \(L > 0\):

   \[\|\nabla f(x) - \nabla f(y)\| \leq L \|x - y\|, \quad \forall x, y \in \mathbb{R}^n\]

Proof idea: Use Taylor expansion and Lipschitz condition to bound \(f(x_{k+1})\).

Application to Gradient Descent: Setting \(y = x_k - \alpha_k \nabla f(x_k)\):

If \(\alpha_k < 2/L\), then \(f(x_{k+1}) < f(x_k)\) (strict decrease).

Implications:

• \(\|\nabla f(x_k)\| \to 0\) as \(k \to \infty\) (subsequence convergence)

• Convergence rate: \(O(1/k)\) (sublinear)

• To achieve \(\|\nabla f(x_k)\| \leq \epsilon\), need \(O(1/\epsilon^2)\) iterations

Note: This is a worst-case bound; actual performance often much better.

Condition Number: \(\kappa = L/\mu\) determines convergence rate.

• Theorem: For strongly convex function, condition number \(\kappa = L/\mu\)

• Convergence rate: \(\rho = \frac{\kappa - 1}{\kappa + 1}\)

Per iteration cost:

• Gradient computation: \(O(n)\) for general functions

• Vector operations: \(O(n)\)

• Line search (if used): depends on function evaluation cost

Memory requirements:

• Current point: \(x_k \in \mathbb{R}^n\)

• Current gradient: \(\nabla f(x_k) \in \mathbb{R}^n\)

• Total: \(O(n)\) memory

Advantages:

• Simple implementation

• Low memory requirements

• Guaranteed descent (with proper step size)

• Scales well to high dimensions

Numerical Issues:

• Gradient computation accuracy

• Finite difference approximations

• Round-off errors in step size selection

Practical Tips:

• Monitor both gradient norm and function value

• Use relative tolerances: \(\|\nabla f(x_k)\| \leq \epsilon_{\text{rel}} \cdot \|\nabla f(x_0)\|\)

• Restart if progress stalls

• Consider scaling/preconditioning for better conditioning

Debugging:

• Verify gradient with finite differences

• Check descent property: \(f(x_{k+1}) < f(x_k)\)

• Plot convergence history

1. Slow convergence: Only \(O(1/k)\) rate in general

2. Condition number dependence: Poor performance when \(\kappa = L/\mu\) is large

3. No curvature information: Ignores second-order information

4. Sensitive to scaling: Performance depends on coordinate system

5. Local minima: Can get stuck in non-convex problems

Momentum Methods:

• Heavy ball method: \(x_{k+1} = x_k - \alpha \nabla f(x_k) + \beta(x_k - x_{k-1})\)

• Nesterov acceleration: Look-ahead gradient evaluation

Adaptive Step Sizes:

• AdaGrad: Accumulate squared gradients

• RMSprop: Exponential moving average of squared gradients

• Adam: Combine momentum and adaptive step sizes

Stochastic Variants:

• Stochastic Gradient Descent (SGD)

• Mini-batch methods

• Variance reduction techniques (SVRG, SAGA)

Coming in next lectures:

• Newton’s Method: Uses Hessian information

  \[x_{k+1} = x_k - [\nabla^2 f(x_k)]^{-1} \nabla f(x_k)\]

• Quasi-Newton Methods: Approximate Hessian (BFGS, L-BFGS)

• Conjugate Gradient: Exploits quadratic structure

• Trust Region Methods: Adaptive step size selection

• Consider \(f(x) = \frac{1}{2} x^T A x - b^T x\) where \(A \succ 0\).

• Gradient: \(\nabla f(x) = Ax - b\)

• Optimal point: \(x^* = A^{-1} b\)

• Gradient descent:

  \[x_{k+1} = x_k - \alpha (Ax_k - b) = (I - \alpha A) x_k + \alpha b\]

• Error analysis: \(e_k = x_k - x^*\) satisfies \(e_{k+1} = (I - \alpha A) e_k\)

• Convergence: Linear with rate \(\rho = \max\{|1 - \alpha \lambda_{\min}|, |1 - \alpha \lambda_{\max}|\}\)

• Optimal step size: \(\alpha^* = \frac{2}{\lambda_{\min} + \lambda_{\max}}\) gives \(\rho^* = \frac{\kappa - 1}{\kappa + 1}\)

Problem: Given data \((x_i, y_i)_{i=1}^m\) where \(y_i \in \{0,1\}\), minimize:

Gradient:

Properties:

• \(f\) is convex (with regularization \(\lambda > 0\))

• \(\nabla f\) is Lipschitz continuous

• Gradient descent converges to global optimum

• Widely used in machine learning

Problem: \(\min_x \frac{1}{2}\|Ax - b\|^2 + \frac{\lambda}{2}\|x\|^2\)

Gradient: \(\nabla f(x) = A^T(Ax - b) + \lambda x = (A^TA + \lambda I)x - A^Tb\)

Analysis:

• Condition number: \(\kappa = \frac{\sigma_1^2 + \lambda}{\sigma_n^2 + \lambda}\) where \(\sigma_1 \geq \cdots \geq \sigma_n \geq 0\)

• Regularization improves conditioning

• For \(\lambda > 0\), always strongly convex

Practical considerations:

• Can solve directly: \(x^* = (A^TA + \lambda I)^{-1} A^Tb\)

• Gradient descent useful for very large problems

• Iterative methods can be more efficient than direct solve

Good Cases:

• Large-scale problems (high dimensional)

• When Hessian is expensive or unavailable

• Smooth, well-conditioned problems

• As a warm-start for other methods

• Online/streaming optimization

Poor Cases:

• Ill-conditioned problems

• When high accuracy is needed quickly

• Small problems where direct methods work

• Highly non-convex functions with many local minima

Alternatives to Consider:

• Newton’s method (if Hessian available)

• Quasi-Newton methods (L-BFGS)

• Conjugate gradient (for quadratic problems)

Step Size Selection:

• Start with \(\alpha = 1\) and use backtracking

• For quadratic functions: estimate condition number and use \(\alpha = 2/(L + \mu)\)

• Monitor convergence: reduce if oscillating, increase if too slow

• Consider adaptive methods for complex problems

Stopping Criteria:

• Primary: \(\|\nabla f(x_k)\| \leq \epsilon_{abs} + \epsilon_{rel} \|\nabla f(x_0)\|\)

• Secondary: Maximum iterations (prevent infinite loops)

• Optional: Function value change or step size criteria

Implementation Tips:

• Scale variables to similar ranges

• Use good initial guess when possible

• Monitor and plot convergence history

Common Issues:

• Divergence: Step size too large, check \(\alpha < 2/L\)

• Slow convergence: Problem is ill-conditioned, consider preconditioning

• Oscillation: Step size too large or poor scaling

• Premature termination: Tolerance too loose or wrong stopping criterion

Debugging Strategies:

• Verify gradient implementation with finite differences

• Check that function decreases at each iteration

• Plot function value and gradient norm vs iteration

• Test on simple quadratic functions first

• Use line search to automatically adjust step size

Performance Monitoring:

• Track: \(f(x_k)\), \(\|\nabla f(x_k)\|\), \(\|x_k - x_{k-1}\|\)

• Look for: monotonic decrease, exponential convergence patterns

Gradient Descent Fundamentals:

• Simple, intuitive algorithm: move opposite to gradient direction

• Guaranteed convergence under mild conditions

• Sublinear convergence rate: \(O(1/k)\) in general case

Critical Factors:

• Step size selection determines success/failure

• Problem conditioning (\(\kappa = L/\mu\)) affects convergence speed

• Proper scaling and initialization matter significantly

Practical Impact:

• Foundation of modern machine learning optimization

• Building block for more sophisticated methods

• Still widely used for large-scale problems

Gradient Descent as Template:

Unifying Principle: All methods choose direction \(d_k\) and step size \(\alpha_k\) to ensure convergence while optimizing computational efficiency.

Next Topics:

• Newton’s Method: Second-order information for faster convergence

• Quasi-Newton Methods: Approximating Hessian efficiently

• Constrained Optimization: Handling equality/inequality constraints

• Stochastic Methods: Large-scale and noisy optimization

Advanced Topics:

• Non-convex optimization landscapes

• Parallel and distributed optimization

• Online and streaming optimization

• Robust optimization under uncertainty