About Gradient Descent at a Saddle Point
This page uses f(x, y) = x squared minus y squared plus 0.1 y to the fourth, which has a saddle at the origin, and starts at (2.5, 0.05), very close to the ridge. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.
At a saddle the surface curves up in one direction (x) and down in another (y). The gradient along y is almost zero near the ridge, so a plain optimizer slides along x toward the saddle and then waits there. Each SGD step multiplies the small y offset by 1 + 2 lr, so a tiny offset takes many steps to grow into a real escape, and a smaller learning rate means a longer wait.
Once the offset has grown the optimizers slide down to the two valleys at y of about plus or minus 2.24, where f is -2.5. In high-dimensional neural network training saddle points are far more common than local minima, which is one reason noise and momentum are useful. Try the learning rate 0.01 and compare how long each optimizer waits.
How to use Gradient Descent at a Saddle Point
- Choose a function. The suggested learning rate for it is filled in.
- Change the learning rate if you like. Smaller is slower and safer, larger is faster until it blows up.
- Press Step to move every optimizer once, or Play to animate. "+25 steps" skips ahead.
- Tap the plot to move the starting point (the cross). The paths restart from there.
- Read the table for each optimizer's position, value and status: running, converged or diverged.
- Watch the log chart: a line that keeps falling is converging, and a line that climbs is diverging.
Features
- Seven functions: three curves and four 2D surfaces, including a narrow valley, Rosenbrock, Himmelblau and a saddle.
- SGD, Momentum, RMSprop and Adam, all from the same start with the same learning rate.
- Contour map for 2D functions and a curve for 1D, with a coloured path for each optimizer.
- Divergence detection that stops a run and says which optimizers blew up.
- A log-scale chart of the function value against step.
- Step, play and +25 steps controls, with slower playback if you prefer reduced motion.
- Tap the plot to choose the starting point.
Worked examples
Saddle from (2.5, 0.05): steps until f drops below -2.4 gives learning rate 0.1: SGD 24, Momentum 30, RMSprop 22, Adam 33
At a learning rate of 0.01 SGD needs 226 steps, while Momentum needs 43, because its velocity compounds.
Privacy: how it works
- Everything is computed in your browser tab. Nothing is sent to Ulkaa or anyone else.
Tips
- On the parabola, try a learning rate of 0.5: SGD jumps to the minimum in a single step because the step size matches the curvature exactly.
- On the double well, drag the start to the left side and see it settle in the other, shallower minimum.
- On Himmelblau, change the start point and see different optimizers choose different minima.
- If a path vanishes off the plot, check the status column: it probably diverged.
Frequently asked questions
Is it really free?
Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.
Do I need an account?
No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.
Does it run in my browser?
Yes. Your settings are processed by code running in your browser tab, on your own device.
Is my data uploaded?
No. Your settings are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.
What is a saddle point?
A point where the gradient is zero but the surface rises in some directions and falls in others, so it is neither a minimum nor a maximum.
Why do saddle points matter for deep learning?
In high dimensions most points with zero gradient are saddles, not minima. Optimizers must escape them to keep improving.
More Gradient Descent Visualizer pages
- Gradient Descent VisualizerCompare SGD, Momentum, RMSprop and Adam side by side, step by step, on curves and contour maps.
- Learning Rate Too High: Gradient Descent DivergenceSee the error grow instead of shrink when the learning rate crosses its limit.
- SGD Learning Rate VisualizerChange the learning rate and watch plain gradient descent speed up, oscillate or stall.
- Momentum vs SGDSee how a running velocity helps gradient descent in a long, narrow valley.
- Adam Optimizer VisualizerSee Adam adapt its step per direction on the curved Rosenbrock valley.
- RMSprop Optimizer VisualizerSee RMSprop normalise its steps on a surface with four minima.
- Gradient Descent and Local MinimaStart on one side of a double well and see which minimum each optimizer reaches.
- Gradient Descent on the Rosenbrock FunctionFollow gradient descent down the curved banana valley of the Rosenbrock function.
- Gradient Descent on a Bumpy LossSee how a bumpy loss traps plain gradient descent and how momentum helps.