Gradient Descent

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Article
Topic area	Optimization
Difficulty	Introductory

Gradient descent is a first-order iterative optimisation algorithm for finding a local minimum of a differentiable function. It is the foundation of nearly all modern machine-learning training procedures, from simple linear regression to billion-parameter deep neural networks.

Intuition

Imagine standing on a mountainside in thick fog. You cannot see the valley floor, but you can feel the slope beneath your feet. The most natural strategy is to take a step in the steepest downhill direction, then reassess. Gradient descent formalises precisely this idea: at each step, the algorithm computes the direction of steepest increase of the function (the gradient) and moves in the opposite direction.

The size of each step is controlled by a scalar called the learning rate (often denoted $ \eta $). A large learning rate covers ground quickly but risks overshooting the minimum; a small learning rate converges more reliably but may take prohibitively many steps.

Mathematical formulation

Given a differentiable objective function $ f:\mathbb{R}^n \to \mathbb{R} $, gradient descent generates a sequence of iterates by the update rule:

$ \theta_{t+1} = \theta_t - \eta \, \nabla f(\theta_t) $

where $ \nabla f(\theta_t) $ is the gradient vector evaluated at the current point $ \theta_t $ and $ \eta > 0 $ is the learning rate.

In the one-dimensional case this simplifies to:

$ \theta_{t+1} = \theta_t - \eta \, f'(\theta_t) $

The gradient $ \nabla f $ points in the direction of steepest ascent, so subtracting it moves the iterate downhill.

Batch, stochastic, and mini-batch variants

When the objective has the form of an average over data points,

$ f(\theta) = \frac{1}{N}\sum_{i=1}^{N} \ell(\theta;\, x_i, y_i) $

three common strategies differ in how much data is used to estimate the gradient:

Variant	Gradient computed over	Per-step cost	Gradient noise
Batch (full) gradient descent	All $ N $ samples	High	None
Stochastic gradient descent (SGD)	1 random sample	Low	High
Mini-batch gradient descent	$ B $ random samples ($ 1 < B < N $)	Medium	Medium

Full batch gradient descent computes the exact gradient and therefore follows a smooth trajectory toward the minimum. Stochastic gradient descent uses a single sample to estimate the gradient, drastically reducing computation per step at the cost of a noisier trajectory. Mini-batch gradient descent strikes a balance and is the most common choice in practice, with typical batch sizes between 32 and 512.

Convergence

Convex functions

For a convex function with Lipschitz-continuous gradients (constant $ L $), gradient descent with a fixed learning rate $ \eta \leq 1/L $ converges at a rate of $ O(1/t) $. If the function is additionally strongly convex with parameter $ \mu > 0 $, convergence accelerates to a linear (exponential) rate:

$ f(\theta_t) - f(\theta^*) \leq \left(1 - \frac{\mu}{L}\right)^t \bigl(f(\theta_0) - f(\theta^*)\bigr) $

The ratio $ \kappa = L / \mu $ is called the condition number and governs how quickly the algorithm converges. Ill-conditioned problems (large $ \kappa $) converge slowly.

Non-convex functions

Most deep-learning objectives are non-convex. In this setting gradient descent is only guaranteed to converge to a stationary point (where $ \nabla f = 0 $), which could be a local minimum, saddle point, or even a local maximum. In practice, saddle points are more problematic than local minima in high-dimensional spaces.

Learning rate selection

Choosing the learning rate is one of the most important practical decisions:

• Too large — the iterates oscillate or diverge.
• Too small — convergence is unacceptably slow.
• Learning rate schedules — many practitioners start with a larger rate and reduce it over time (step decay, exponential decay, cosine annealing).
• Line search — classical numerical methods choose $ \eta $ at each step to satisfy conditions such as the Wolfe or Armijo conditions, though this is rare in deep learning.

A common heuristic is to try several values on a logarithmic scale (e.g. $ 10^{-1}, 10^{-2}, 10^{-3} $) and pick the one that reduces the loss fastest without instability.

Extensions and improvements

Several important modifications address limitations of vanilla gradient descent:

• Momentum — accumulates a velocity vector from past gradients, helping to accelerate convergence in ravine-like landscapes.
• Nesterov accelerated gradient — a momentum variant that evaluates the gradient at a look-ahead position, yielding better theoretical convergence rates.
• Adaptive methods (Adagrad, RMSProp, Adam) — maintain per-parameter learning rates that adapt based on the history of gradients.
• Second-order methods — algorithms like Newton's method and L-BFGS use curvature information (the Hessian or its approximation) for faster convergence, but are often too expensive for large-scale problems.

Practical tips

• Feature scaling — normalising input features so they have similar ranges dramatically improves convergence, because the loss surface becomes more isotropic.
• Gradient clipping — capping the norm of the gradient prevents excessively large updates.
• Random initialisation — starting from a reasonable random initialisation (e.g. Xavier or He initialisation for neural networks) avoids symmetry-breaking issues.
• Monitoring the loss curve — plotting the training loss over iterations is the simplest diagnostic: a smoothly decreasing curve indicates healthy training; oscillations suggest the learning rate is too high.

Applications

Gradient descent and its variants are used throughout science and engineering:

• Training machine-learning models (linear models, neural networks, support vector machines)
• Signal processing and control systems
• Inverse problems in physics and imaging
• Operations research and logistics optimisation
• Economics and game-theoretic equilibrium computation

See also

• Stochastic Gradient Descent
• Backpropagation
• Loss Functions
• Neural Networks
• Overfitting and Regularization

References

• Cauchy, A. (1847). "Méthode générale pour la résolution des systèmes d'équations simultanées". Comptes Rendus de l'Académie des Sciences.
• Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press.
• Ruder, S. (2016). "An overview of gradient descent optimization algorithms". arXiv:1609.04747.
• Goodfellow, I., Bengio, Y. and Courville, A. (2016). Deep Learning, Chapter 8. MIT Press.