Research map / Learning theory and algorithms / Sample complexity
On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Represent smooth targets with an LP solution function | LP solution functions · C^2 functions on [0,1]^{n_x} | uniform approximation error: \|f-\pi\|_\infty\leq \epsilon; \mathcal{O}\Big(\left(\frac{n_x}{\epsilon}\right)^{\frac{n_x}{2}}\Big) constraints | Helped |
| Compose LP solution functions into a universal multilayer architecture | LP solution functions · F_{k,n_x} | uniform approximation error: depth of at most 2k; widest layer has at most \bigg( 2n_x+2+2\begin{pmatrix}n_x\\k+n_x\end{pmatrix}\bigg)N^{n_x} constra | Helped |
| Use a two-layer LP construction to represent multiplication | optimization-induced networks · Product function on (0,1]^2 | reconstruction error: with no error | Helped |
| Use exact LP bump functions to partition the input space | universal multilayer LP architecture · F_{k,n_x} | approximation complexity: a reduction at the order of 2^{n_x^2/k} | Helped |
| Bound LP/QP class complexity using constraint counts and condition numbers | LP/QP solution-function classes · bounded input space | log empirical L_1 covering number: \lesssim \frac{\kappa_\square^{\star 2}}{\epsilon^2} {\sum}_{0\leq i\leq {n_z-m_2}}\begin{pmatrix}m_1\\ i\end{pmatrix} | Helped |
| Assume definable objective and constraints to obtain finite Whitney stratification | generic optimization solution functions · Generic definable optimization | Whitney stratifiability | Helped |
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