Subgradient generalizes the notion of gradient/derivative and quantifies the rate of change of non-differentiable/non-smooth function ([1], section 8.1 in [2]). For a real-valued function, the *subgradients* of the function at a point x_{0} are all values of c such that:

Unlike derivative of smooth function, which is a singleton value, the subgradients form a set (e.g. an interval of points). The set of all subgradients is called the *subdifferential*.

Subdifferential always exists if f(x) is a convex function with a convex domain X, and vice versa (proposition 1.1 in [3]):

The generalized subgradient descent scheme is given as:

**References**

[1] Wikipedia, Subgradient

[2] J. F. Bonnan, J. C. Gilbert, C. Lemaréchal, C. A. Sagastizábal. Numerical Optimization: Theoretical and Practical Aspects. Second Edition, Springer, 2006

[3] S. Bubeck. Theory of Convex Optimization for Machine Learning. Lecture Notes, 2014

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