Locally optimal solution

As explained in Geometry for Programmers

A local minimum is a point whose function value is no greater than the values at all sufficiently nearby inputs.

A local minimum is a point where a function stops decreasing and begins increasing. Nearby inputs have no lower function value, even if the function falls lower elsewhere. A local minimum can therefore also be a global minimum, but it does not have to be. section 6.1.2

Finding a minimum on a cubic

For P(x) = x³ − x² − 2x + 1, the derivative is P′(x) = 3x² − 2x − 2. A local minimum occurs where the derivative changes from negative to positive, so finding derivative roots helps locate the curve’s bumps. A root alone is not enough: the derivative may touch the axis without changing sign, producing no extremum. section 6.1.2

Figure 6.5 A cubic polynomial (two bumps, three roots) and its derivative, which is a quadratic polynomial (two roots)

Compare the cubic’s two bumps with the roots of its quadratic derivative. section 6.1.2

How Geometry for Programmers covers it

How deep it goes

Defines the geometric turn, links it to derivative roots and sign changes, and notes that a derivative root alone may not produce an extremum.

Examples

Uses a cubic polynomial, its quadratic derivative, plotted bumps, and a derivative root that does not create an extremum.

What you can do after

Evaluate/understand local minima by interpreting derivative roots and checking whether the derivative changes from negative to positive.

How it connects

Connects local minima to derivatives, local maxima, global minima, and polynomial functions.

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Also explained in

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  Advanced Algorithms and Data StructuresA local minimum is a point whose function value is no greater than those of sufficiently nearby points, without necessarily being the smallest value over the entire domain.
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  Optimization Algorithms: AI techniques for design, planning, and control problemsA solution that is optimal relative to a neighborhood but may not be optimal over the entire feasible search space.
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  Data Without LabelsA local minimum is a point whose objective value is no greater than that of sufficiently nearby points, even if lower values exist elsewhere.
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  Grokking AI Algorithms, Second EditionA local minimum is a point whose objective value is no greater than that of nearby points, without necessarily being the global minimum.
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  Natural Language Processing in Action, Second EditionA local minimum is a point where an objective function has a value no greater than that of nearby points, without necessarily being the global minimum.
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  Deep Learning with R, Third EditionA local minimum is a point whose function value is no greater than the values at sufficiently nearby points.
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  Sutskever's ListA local minimum is a point whose objective value is no greater than that of nearby points, without necessarily being the global minimum.
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  Deep Learning with JAXA point where a function has a value no greater than that of nearby points.

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Related concepts

• global optimumcontrasts with · A global optimum is a feasible point or solution whose objective value is at least as favorable as that of every other point or solution in the domain.
• simulated annealingaddressed by · Simulated annealing is a stochastic optimization method that gradually reduces the probability of accepting worse solutions to escape local optima.
• K-means clusteringproduced by · K-means clustering partitions data into a specified number of groups by assigning observations to representative centers and iteratively updating those centers to reduce within-group distances.
• greedy algorithmused by · A greedy algorithm builds a solution incrementally by choosing the locally best available option at each step, without generally accounting for all future consequences or revising prior choices.
• momentumcontrasts with · Momentum is an optimization technique that incorporates a running history of gradients or parameter updates to smooth and accelerate movement during training.

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