Research map / Learning theory and algorithms / Kernels and regression theory
Adaptivity of deep ReLU network for learning in Besov and mixed smooth Besov spaces: optimal rate and curse of dimensionality
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use deep ReLU networks instead of a linear method | Function approximation in Besov spaces · Besov space B^s_{p,q}([0,1]^d) | approximation error rate: \tilde{O}(N^{-\frac{s}{d} + (\frac{1}{p} - \frac{1}{r})_+}) -> \tilde{O}(N^{-\frac{s}{d}}) | Helped |
| Use deep ReLU networks instead of a linear estimator | Regression over Besov spaces · Besov space B^s_{p,q}([0,1]^d), especially p < 2 | estimation error rate: \tilde{O}(n^{-\frac{2s}{2s + d}}) vs. \Omega\big(n^{- \frac{2s - (2/(p \wedge 1) - 1)}{2s + 1 - (2/(p \wedge 1) - 1)}}\b | Helped |
| Assume mixed smoothness and use sparse-grid approximation | Deep ReLU network approximation · Mixed smooth Besov space | approximation error rate: \tilde{O}(N^{-\beta}) vs. the ordinary Besov rate \tilde{O}(N^{-\frac{s}{d}}) | Helped |
| Assume mixed smoothness rather than ordinary Besov smoothness | Deep ReLU network regression · Mixed smooth Besov space | squared L^2(P_X) estimation error: n^{- \frac{2s}{2s + 1}}\log(n)^{\frac{2(d-1)(u + s)}{1+2s}} \log(n)^2 vs. n^{-\frac{2s}{2s + d}}\log(n)^2 | Helped |
| Use nonlinear adaptive recovery rather than linear, non-adaptive recovery | Approximation in mixed smooth Besov spaces · Mixed smooth Besov space, p < r | approximation error | Helped |
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