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Here's one possible solution: $$\nabla f(x) = \left[ \frac{\partial f(x)}{\partial x_1} \cdots \frac{\partial f(x)}{\partial x_n}\right]^T$$ The coordinate descent algorithm is exploring all the coordinate axes, so you have an estimate of $\hat{\nabla} f_i(x_k)=\frac{\partial f(x^k)}{\partial x_i}$. In particular, when $\Big| \hat{\nabla} f_i(x_k) \Big| < ...


VC-dimension has multiple multiclass extensions: pseudo dimension, Natarajan dimension, graph dimension. See here for example: There are also extensions to continuous-valued functions (fat-shattering dimension):

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