Taylor Expansion
A mathematical tool for locally approximating a function with polynomials. The first-order expansion corresponds to the linear approximation used in gradient descent, while the second-order expansion introduces curvature and corresponds to second-order optimization methods such as Newton's method.
Timeline (last 90 days)
The basic idea of the first-order analysis is to Taylor-expand the effect of weight perturbation on model output and keep only the first-order linear term: the output change is approximately the inner product of the perturbation vector and the gradient of the output with respect to the weights.
First-order Taylor expansion corresponds to the linear approximation of gradient descent, while second-order expansion introduces curvature information and corresponds to second-order optimization methods such as Newton's method.
Taylor expansion is an analysis tool rather than a training algorithm, and "training networks with Taylor expansion" is a confusion of terms.
The second-order Taylor expansion approximates exp(x) as 1 + x + x²/2, where (q^T k)² can be written in the form (q⊙q)^T (k⊙k) to maintain linear time complexity.
All Facts (4)
The basic idea of the first-order analysis is to Taylor-expand the effect of weight perturbation on model output and keep only the first-order linear term: the output change is approximately the inner product of the perturbation vector and the gradient of the output with respect to the weights.
50%UnverifiedFirst-order Taylor expansion corresponds to the linear approximation of gradient descent, while second-order expansion introduces curvature information and corresponds to second-order optimization methods such as Newton's method.
50%UnverifiedTaylor expansion is an analysis tool rather than a training algorithm, and "training networks with Taylor expansion" is a confusion of terms.
50%UnverifiedThe second-order Taylor expansion approximates exp(x) as 1 + x + x²/2, where (q^T k)² can be written in the form (q⊙q)^T (k⊙k) to maintain linear time complexity.
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