Learn AI /How Models Improve /Local and global minimum
Local and global minimum
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- How Models Improve
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Also called: local minimum, global minimum, local minima.
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In five seconds
What it is
A local minimum is the lowest loss in a nearby region, while a global minimum is the lowest loss anywhere in the whole set of possible model settings.
In plain words
A local minimum is the best place your training method found nearby. A global minimum is the best place anywhere. Training often aims downhill using only local clues, so it may stop at a nearby valley instead of the deepest valley overall.
An everyday example
Amina and Diego are checking a water pump site before sunrise, when thick fog hides the wider landscape. They walk downhill from where they stand, using only the ground under their boots, and stop when every step nearby would go upward. They have found the bottom of the small hollow they are in. But once the fog clears, they can see a deeper basin on the other side of the site that would have been a better place if they had known the whole area.
A loss landscape for a real model is not a simple physical hill. It usually has very many dimensions, not just a ground surface you can walk on. Also, training methods do not literally inspect the whole landscape after the fog clears, so they usually cannot confirm a true global minimum in large systems.
Remember thisLocal means lowest nearby; global means lowest overall, and training methods that follow local slope often cannot tell the difference by themselves.
Understand it
Understand it
Why it matters
This term matters because a lower training loss does not automatically mean the best possible model was found. If you hear that a system has 'converged', you should ask what it converged to: a good enough low-loss solution or a poor local minimum. That affects decisions about whether to restart training, change hyperparameters, compare multiple runs, or judge the model on validation performance instead of training loss alone. For non-specialists, the key point is that optimisation can get stuck in a nearby good spot, so claims about model quality need evidence from evaluation, not just one training curve.
Local minimum versus global minimum on a loss landscape
Help the reader explain why training can stop at a nearby low point without reaching the lowest point overall.
01
Loss as height
Treat vertical height as loss: high means more wrong, low means less wrong.
Draw a labelled y-axis 'loss' and a smooth wavy landscape in both panels. Use darker shading lower down.
02
Local view
In the left panel, mark one valley and highlight only a small neighbourhood around it. Within that neighbourhood, it is the lowest point.
Circle a valley in the left panel with a dashed oval labelled 'nearby region'. Add a marker labelled 'local minimum'.
03
Whole landscape
In the right panel, mark the deepest valley anywhere on the landscape. That point is lower than all other shown points.
Draw several valleys in the right panel. Make one clearly deepest and label it 'global minimum'.
04
Why training may stop early
An optimiser that follows local slope can move into a nearby valley and stop there, even if a deeper valley exists elsewhere.
Add arrows descending into the local minimum from a starting point. In the right panel, place another deeper valley separated by an uphill section.
Show two side-by-side loss landscapes. The left panel marks a valley that is lower than its neighbours but not the lowest on the page. The right panel marks the deepest valley across the whole page. Read left to right: first compare the scope of the word local, then the scope of the word global, then note how gradient-based training only follows nearby slope.
Hover or tap each label to highlight the relevant region: nearby region for local minimum, entire landscape for global minimum, and descent arrows for the optimiser path.
Describe this without the diagram
The diagram compares two views of a loss landscape, where height represents error. In the first view, a point in a valley is labelled local minimum because it is lower than the points close to it inside a highlighted nearby region. In the second view, a deeper valley elsewhere is labelled global minimum because it is the lowest point across the full landscape. Arrows show that an optimiser can follow downhill slope into the local minimum and stop there without ever reaching the deeper valley.
With reduced motion: A static two-panel comparison replaces any highlighting. The left panel shows a valley marked as lowest only within a dashed nearby region. The right panel shows a deeper valley marked as lowest across the whole landscape. Fixed arrows indicate a downhill path into the local minimum.
During training, a system updates model settings and calculates how wrong each setting is using a loss function. You can imagine the possible settings laid out on a landscape where height means loss: lower points mean less error.
A local minimum is a point whose loss is no greater than the loss at points close to it, but it may not be lowest elsewhere. A global minimum has loss no greater than any point across the entire landscape. Several points can tie as global minima. The plural of local minimum is local minima.
Optimisation methods such as gradient descent look at the slope around the current point and step downhill. That means they use local information. If the landscape has many valleys, a method can settle into one valley that is good nearby but not best overall.
What goes in are the current model settings, training data, and a rule for measuring loss. What happens is repeated updates that try to reduce loss. What comes out is a new set of settings, often near a minimum. In practice, for large neural networks, people often care less about proving they reached a global minimum and more about reaching a low-loss solution that generalises well to new data.
How “Walking around hills in fog” maps onto the real thing
| In the story | In the AI system |
|---|---|
| small hollow where nearby steps all go upward | local minimum |
| deeper basin elsewhere on the site | global minimum |
| fog limiting what they can see | optimisation using only local slope information |
| walking downhill step by step | gradient-based updates reducing loss |
A worked example
A tiny model has one adjustable setting, x. Its loss at several tested values is measured during training.
Starting from: Tested values: x=-3 gives loss 9, x=-1 gives loss 2, x=0 gives loss 3, x=2 gives loss 1, x=4 gives loss 5.
- Compare nearby values around x=-1: losses at x=-3, -1, and 0 are 9, 2, and 3, so x=-1 is lower than its neighbours.
- Compare nearby values around x=2: losses at x=0, 2, and 4 are 3, 1, and 5, so x=2 is also lower than its neighbours.
- Now compare all tested points together.
- The smallest loss overall is 1 at x=2, so x=2 is the global minimum among these tested values, while x=-1 is only a local minimum.
Result: Among the five tested values, x=-1 and x=2 have lower loss than their adjacent tested values, while x=2 has the lowest tested loss. These samples do not establish minima for every possible x in the range.
In a United Nations workplace
At Headquarters, the analyst responsible for security review is drafting a note for the file before close of business on a completed classification pilot. The approved internal model was tuned on synthetic incident summaries. Its training logs show one run settling where nearby parameter changes increased loss, while a run from a different starting point reached a lower loss basin.
The first run may have reached a local minimum, not the global minimum. Because gradient-based training follows local slope information, the review should compare multiple starting points, validation performance and output quality. Even the best pilot run is only the best observed result, not proof that the global minimum was found.
Before you try thisDo not upload operational details, identifiers, access-control records or vulnerability information to a tool that is not approved for such material. Use synthetic or properly authorised data in the approved environment, apply access controls and retain human review. Do not describe the lowest observed loss as the global minimum unless that claim can actually be established.
Where you may hear it
- A post-pilot technical review at Headquarters
- A vendor demonstration to an information and communications technology governance board
- A model-validation meeting before a progress report is cleared
- A research briefing on why separate training runs produce different results
Avoid confusion
Avoid confusion
Often confused with
Convergence
Convergence means training updates are settling towards a stable, lower-loss solution instead of continuing to wander or blow up.
Optimisation
The process of adjusting a model’s internal settings to reduce loss, plus the algorithms and rules used to choose each adjustment.
Gradient descent
A method for reducing a model’s loss by repeatedly adjusting its internal settings in the direction opposite the loss gradient.
Common misconceptions
Common misconception: If training reaches a minimum, the model has found the best possible solution.
Correction: A minimum only tells you something about the area you reached. If it is local, there may be better settings elsewhere. In practice, many training methods use nearby slope information, so they cannot guarantee the absolute best point across the whole landscape.
Common misconception: A global minimum always means the model will perform best on new data.
Correction: The global minimum usually refers to the chosen loss on the training objective, not automatically to real-world performance. A model can fit the training data extremely well and still generalise poorly, which is why validation data and regularisation matter.
Common misconception: Every low point that looks flat is a local minimum.
Correction: Not every flat or nearly flat region is a minimum. Some flat areas are plateaus, and some points with zero slope can be saddle points, where one direction goes up and another goes down. So low gradient alone does not prove a local minimum.
Quick check
Which statement is the best description of the difference?
Option A is correct because the difference is about scope: nearby region versus entire search space. Zero gradient alone does not guarantee a minimum, and training stopping does not prove the absolute lowest point was found. Validation performance is related but not identical to the training-loss landscape.
Go deeper
Go deeper
The simple version
A local minimum is the best place your training method found nearby. A global minimum is the best place anywhere. Training often aims downhill using only local clues, so it may stop at a nearby valley instead of the deepest valley overall.
The technical version
Let L(theta) be the loss as a function of model parameters theta. A point theta* is a local minimum if there exists a neighbourhood around theta* such that L(theta*) is less than or equal to L(theta) for all theta in that neighbourhood. It is a global minimum if L(theta*) is less than or equal to L(theta) for all theta in the full domain. In differentiable optimisation, gradient-based methods update parameters using local derivative information, often computed by backpropagation. Because the objective in deep learning is typically non-convex, there may be many local minima and saddle points. The reached solution depends on initialisation, learning rate, optimiser choice, noise from mini-batches, and other hyperparameters. Also, the optimisation target is the training objective, so even a global minimum of training loss does not guarantee the best validation or test performance.
Reviewed external resources
EncyclopaediaGlobal minimum corporate tax rate (opens in a new tab)Wikipedia
This Wikipedia page is about a policy term in international taxation that uses the same words, so it helps readers notice that 'global minimum' can mean very different things outside machine learning.
Intermediatelink checked 2026-07-31
CourseReducing Loss (opens in a new tab)Google
This course material explains how training reduces loss over time, which suits readers who want a practical introduction to optimisation without heavy maths.
Intermediatelink checked 2026-07-31
VideoNeural networks, explained visually (opens in a new tab)3Blue1Brown
This video gives an intuitive visual picture of neural networks and training, which suits readers who learn well from geometry and motion before formal definitions.
Intermediatelink checked 2026-07-31
VideoVideo explanations: Local and global minimum (opens in a new tab)YouTube search
This is a YouTube search results page, not a single vetted lesson, so it helps readers compare different teaching styles but requires care about accuracy and level.
search results, not one pagelink checked 2026-07-31
DiscussionCommunity discussion: Local and global minimum (opens in a new tab)Reddit search
This is a Reddit search results page, not a single authoritative source, so it can show common questions and confusions but should be used cautiously for technical correctness.
search results, not one pagelink checked 2026-07-31
Where this came from
Drafted with a language model through an offline pipeline, checked against this site's schema, and every external link above fetched to see whether it answers. Treat it as a starting point and check anything that matters against the primary sources. How this site is built and what that means.
The one thing to remember
Local means lowest nearby; global means lowest overall, and training methods that follow local slope often cannot tell the difference by themselves.