Gradient Descent Path Plot Calculator

Optimisation Results

Results appear here after each calculation.

Contour Path

Three-Dimensional Surface

Loss Convergence

Gradient, Learning Rate, and Distance

Path Animation

Speed

Comparison Summary

Run	Final loss	Iterations	Final gradient	Distance	Time	Status

Iteration Details

Run	Iteration	X	Y	Loss	Gradient X	Gradient Y	Gradient Norm	Learning Rate	Step	Improvement

Objective Function

Choose a surface or define your own two-variable objective.

Objective function

Function scale

Regularisation coefficient

Starting Point and Search Bounds

Initial X

Initial Y

Random starting points

Adds extra runs inside the selected bounds.

Minimum X

Maximum X

Minimum Y

Maximum Y

Additional starting points

Keep updates inside bounds

Optimiser and Learning Settings

Primary optimiser

Initial learning rate

Learning-rate schedule

Minimum learning rate

Decay rate

Decay step interval

Momentum coefficient

Adam beta one

Adam beta two

Epsilon

Weight decay

Mini-batch size

Gradient noise

Applied to stochastic and mini-batch modes.

Gradient clipping threshold

Use zero to disable clipping.

Gradient calculation

Finite-difference step

Normalise gradients

Stop on divergence

Iterations and Stopping Rules

Maximum iterations

Gradient tolerance

Loss-change tolerance

Parameter-change tolerance

Early-stopping patience

Minimum improvement

Divergence loss limit

Known optimum X

Known optimum Y

Enable early stopping

Stop near known optimum

Comparison Mode

Overlay optimisers, learning rates, and starting points.

Optimisers to compare

Use Control or Command for multiple choices.

Learning rates

Maximum comparison runs

Plot and Display Options

Plot title

X-axis title

Y-axis title

Loss-axis title

Decimal precision

Contour resolution

Contour levels

Path marker size 6

Path line width 3

Surface opacity 0.78

Show grid

Show legend

Show direction arrows

Show iteration labels

Show contour labels

Scientific notation

Colour markers by loss

Formula Used

θ(t + 1) = θ(t) − α(t)∇J(θ(t))

The parameter vector moves opposite the local gradient. The learning rate controls each update distance. Optimiser rules can modify that direction and scale.

Adam update

m(t) = β₁m(t−1) + (1−β₁)g(t)
v(t) = β₂v(t−1) + (1−β₂)g(t)²
θ(t+1) = θ(t) − α·m̂(t)/(√v̂(t)+ε)

How to Use

Select an objective function.
Choose a starting point and bounds.
Set an optimiser and learning rate.
Adjust stopping and display controls.
Add comparison settings when needed.
Press Calculate and Plot.
Inspect paths, diagnostics, and iteration details.
Export the required report or dataset.

Example Settings

Objective	Start	Optimiser	Learning rate	Expected behaviour
Quadratic	(2.5, -1.5)	Momentum	0.08	Fast movement toward one minimum.
Rosenbrock	(-1.4, 1.6)	Adam	0.002	Curved progress through a narrow valley.
Himmelblau	(-3, -3)	RMSProp	0.01	Convergence toward one local minimum.
Linear MSE	(0, 0)	Batch gradient descent	0.03	Slope and intercept approach fitted values.

Interpretation and Common Diagnostics

A smooth falling loss usually indicates stable optimisation. Repeated crossings often indicate an excessive learning rate. Very slow movement can indicate undersized updates.

Plateaus can occur near flat regions or saddle points. Exploding gradients create very large steps. Clipping can limit those unstable updates.

Different starting points may reach different local minima. Optimiser comparisons reveal sensitivity to update rules. Always inspect both loss and path geometry.

Frequently Asked Questions

What does the path represent?

It shows every parameter position visited during optimisation. Each segment represents one update.

Why can the path oscillate?

The learning rate may be too large. Momentum can also preserve excessive movement.

Why does gradient descent stop early?

A tolerance, patience limit, optimum distance, or divergence rule may have triggered.

What is a contour line?

It connects locations with equal objective values. Tight spacing indicates steeper surface regions.

Which optimiser should I choose?

Start with Adam for difficult surfaces. Compare it with simpler methods before deciding.

What does gradient clipping do?

It limits gradient magnitude before updating parameters. This can reduce unstable jumps.

Can custom functions be used?

Yes. Enter a supported expression using x and y. Numerical derivatives provide a fallback.

Why use several starting points?

Non-convex functions can contain multiple minima. Multiple starts expose different convergence outcomes.

What is the gradient norm?

It measures gradient magnitude. Small values often indicate a stationary region.