Gradient

As explained in Ensemble Methods for Machine Learning

A gradient is the vector of partial derivatives of a scalar function with respect to its input variables.

A gradient collects the first derivatives of a function with respect to its variables. Intuitively, it measures the local slope of a surface and points toward steepest ascent. To minimize a function, move in the opposite direction. section 5.1.1

This turns optimization into a guided search. Rather than evaluating every point on a potentially expensive grid, use derivative information to choose a promising direction, then decide how far to move. For a loss function, descending the gradient seeks a model with lower error. section 5.1 section 5.1.1

A gradient-descent iteration proceeds as follows:

• Compute the negative gradient at the current position. section 5.1.1

Following the Branin surface

The two-variable Branin function provides a visual example. Starting at [-4, -5], gradient descent follows a descending path toward one of several local minima. Near the solution, steps become smaller and the path may zigzag; different starting points can reach different minima. section 5.1.1

Figure 5.2 The surface plot (left) and contour plot (right) of the Branin function. We can visually verify that this function has four minima, which are the centers of the elliptical regions in the contour plot.

Use the surface and contour plots to relate gradient direction to the geometry of local minima. section 5.1.1

How Ensemble Methods for Machine Learning covers it

How deep it goes

Explains first-order slope, descent direction, step selection, convergence, and a working implementation, while deferring second-order curvature.

Examples

Uses the Branin function, surface and contour plots, descent paths, and a NumPy/SciPy implementation.

What it assumes

Assumes functions, variables, and first derivatives, while grounding their optimization role through visual and procedural examples.

What you can do after

Builds a simple gradient-descent optimizer and interprets its path toward local minima.

How it connects

Links loss functions, gradient descent, gradient boosting, Newton’s method, and the Hessian matrix.

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Related concepts

• backpropagationproduced by · Backpropagation is a procedure for computing derivatives of an output or loss with respect to inputs and parameters by applying the chain rule backward through a computational graph.
• gradient descentused by · Gradient descent is an iterative optimization method that updates parameters opposite the gradient of an objective function to reduce its value.
• jax.gradproduced by · jax.grad is a JAX transformation that creates a function computing derivatives, typically gradients of a scalar-output function with respect to specified inputs.
• optimizationused by · Optimization is the systematic selection or adjustment of values, actions, or system designs to improve a specified objective, often subject to constraints.
• PyTorchused by · PyTorch is an open-source machine-learning framework for Python that provides tensor computation, automatic differentiation, and tools for building and training neural networks.
• tf.GradientTapeproduced by · tf.GradientTape is a TensorFlow context manager that records tensor operations so gradients can subsequently be computed automatically.

This explainer was generated by Manning's knowledge graph from the text of Ensemble Methods for Machine Learning. The section links point to the parts of the book it draws on.