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Level contour

As explained in Math and Architectures of Deep Learning

A curve in a two-dimensional domain along which a function has the same value everywhere.

A level contour is a curve in a two-dimensional domain where a function keeps the same value. For L(w₀,w₁)=√(w₀²+w₁²), the contours are concentric circles: w₀²+w₁²=1 gives L=1, while w₀²+w₁²=25 gives L=5. section 3.3.2

Moving along a contour follows its tangent and leaves the function value unchanged, so this is a direction of minimal change. The gradient is normal to the contour and points toward maximum change; moving across contours therefore changes the loss most rapidly. section 3.3.2

From contours to higher dimensions

For L(w₀,w₁,w₂)=w₀²+w₁²+w₂², constant-value sets become concentric spheres. For L(w₀,w₁,w₂)=w₀²+w₁², they become coaxial cylinders. Directions orthogonal to the gradient remain on one level surface, while gradient directions move between surfaces.

How Math and Architectures of Deep Learning covers it

How deep it goes
Explains definition, geometric meaning, gradient relation, and higher-dimensional analogues through circles, spheres, and cylinders.
Examples
Uses radial circles for √(w₀²+w₁²), spheres for a three-variable quadratic, and cylinders for a two-variable quadratic in three dimensions.
What you can do after
Evaluate loss geometry by identifying constant-value paths and reading gradient directions as directions of maximum change.
How it connects
Connects level contours with gradients, directional derivatives, level surfaces, level hypersurfaces, and loss minimization.
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This explainer was generated by Manning's knowledge graph from the text of Math and Architectures of Deep Learning. The section links point to the parts of the book it draws on.