Gradients

When Gradients Are Unavailable

Gradient descent is normally described using the gradient of a function. That requirement becomes restrictive when the function is not differentiable at some points. Subgradients extend the descent idea by allowing the method to use a vector that satisfies a supporting inequality at the current point instead of requiring an ordinary gradient.

The Supporting Slope

For a convex function, a subgradient at w is a vector that supplies a lower-supporting linear slope at that point. Geometrically, the associated linear expression touches the function at the reference point and remains below the function elsewhere in its domain. This lower-supporting behavior is the central idea behind a subgradient.

Testing a Candidate Vector

The supporting-slope picture becomes precise through the subgradient inequality. Begin with the reference point w and a candidate vector v. Form the linear expression f(w) + vᵀ(u − w). Then compare it with f(u) for every u in the function's domain. If the expression is no greater than f(u) everywhere, v has the required subgradient property at w.

f(w) + vᵀ(u − w) ≤ f(u) for every u in the domain

Gradient and Subdifferential

A gradient is tied to differentiability. A subgradient is selected through the supporting inequality, so it can be used in the broader setting that includes nondifferentiable functions. For a convex function, the subdifferential set ∂f(w) contains all subgradients of f at w.

Choosing One Useful Member

The subdifferential set may contain all valid subgradients, but many practical uses do not require calculating every member. The practical goal is often to construct one element of ∂f(w). If f is differentiable at w, that construction is immediate: use the gradient ∇f(w).

Pointwise maximum functions are another situation for which the source material introduces a construction method. The important practical role of that method is to produce a subgradient of the maximum function. The objective is not necessarily to list the entire subdifferential set; obtaining one member can be enough for many uses.

Mistakes in Subgradient Reasoning

• Treating a subgradient as automatically identical to a gradient.

  A gradient is tied to differentiability, while a subgradient is identified through the supporting inequality and can extend the discussion to nondifferentiable functions.

  Fix: Check the subgradient condition. If f is differentiable at w, the subdifferential is the singleton containing ∇f(w); otherwise, use the supporting inequality to identify a valid vector.

• Checking the inequality at only one point u.

  The subgradient inequality must hold for every u in the domain.

  Fix: Treat the comparison as a domain-wide test.

• Assuming that finding every subgradient is always necessary.

  For many practical uses, constructing one member of the subdifferential set is sufficient.

  Fix: Focus first on obtaining one valid subgradient.

• Ignoring the reference point.

  The subgradient property is defined at a particular point w through f(w) and the displacement u − w.

  Fix: Identify w before forming and checking the supporting expression.

Practice Check

Key Takeaways

1. A subgradient is a vector that supplies a lower-supporting linear slope at a point.
2. The subgradient inequality tests whether a candidate vector remains below the function everywhere in the required sense.
3. The subdifferential set ∂f(w) contains all subgradients at w.
4. When f is differentiable at w, ∂f(w) is the singleton containing ∇f(w).
5. For many practical uses, constructing one subgradient is sufficient, including through a suitable construction method for pointwise maximum functions.