Adaptive random Fourier features with Metropolis sampling

Aku Kammonen; Jonas Kiessling; Petr Plecháč; Mattias Sandberg; Anders Szepessy

doi:10.3934/fods.2020014

Article Contents

2020, Volume 2, Issue 3: 309-332. Doi: 10.3934/fods.2020014

This issue Previous Article Probabilistic learning on manifolds Next Article Posterior contraction rates for non-parametric state and drift estimation

Adaptive random Fourier features with Metropolis sampling

1.
Department of Mathematics, Royal Institute of Technology, 100 44, Stockholm, Sweden
2.
H-Ai AB, Grevgatan 23,114 53, Stockholm, Sweden
3.
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

^* Corresponding author: Aku Kammonen
^* Corresponding author: Aku Kammonen

Received: July 2020

Revised: August 2020

Early access: September 2020

Published: September 2020

The research was supported by Swedish Research Council grant 2019-03725. The work of P. P. was supported in part by the ARO Grant W911NF-19-1-0243

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Abstract

The supervised learning problem to determine a neural network approximation $ \mathbb{R}^d\ni x\mapsto\sum_{k = 1}^K\hat\beta_k e^{{{\mathrm{i}}}\omega_k\cdot x} $ with one hidden layer is studied as a random Fourier features algorithm. The Fourier features, i.e., the frequencies $ \omega_k\in\mathbb{R}^d $, are sampled using an adaptive Metropolis sampler. The Metropolis test accepts proposal frequencies $ \omega_k' $, having corresponding amplitudes $ \hat\beta_k' $, with the probability $ \min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^{\gamma}\big\} $, for a certain positive parameter $ {\gamma} $, determined by minimizing the approximation error for given computational work. This adaptive, non-parametric stochastic method leads asymptotically, as $ K\to\infty $, to equidistributed amplitudes $ |\hat\beta_k| $, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods. Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is tested both on synthetic data and a real-world high-dimensional benchmark.

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Mathematics Subject Classification: Primary: 65D15; Secondary: 65D40, 65C05.

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Figure 1. Case 1: Graph of the target function $ f $ with sampled data set points $ (x_n,y_n) $ marked (red on-line). The inset shows $ |\hat f| $ of its Fourier transform and the detail of its behaviour at the origin

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Figure 2. Case 1: The same data are shown in all figures with each of the six different experiments highlighted (blue on-line)

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Figure 3. Case 2: The figure illustrates the generalization error with respect to $ K $ for a target function in dimension $ d = 5 $

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Figure 4. Case 3: The figure illustrates the generalization error over time when the approximate problem solution is computed using Algorithm 1, Algorithm 2 and stochastic gradient descent

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Figure 5. Case 4: Dependence on $ K $ of the misclassification percentage in the MNIST

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Figure 6. The generalization error as a function of the standard deviation $ \sigma_\omega $ of $ p({\omega}) $

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Table 2. Summary of numerical experiments together with their corresponding parameter choices

	Regression
Case	Case 1: a regularised step function					Case 2: a high dimensional function
Purpose	Find $p$ such that the constant $\mathbb{E}_\omega[\frac{\|\hat f(\omega)\|^2}{(2\pi)^{d}p^2(\omega)}]$ does not become large					Study the ability of Alg. 1 to find a high dimensional dimensional function
Target $f(x)$	$\text{Si}\left(\frac{x}{a}\right)e^{-\frac{x^2}{2}}$ $a = 10^{-3}$					$\text{Si}\left(\frac{x_1}{a}\right)e^{-\frac{\|x\|^2}{2}}$ $a = 10^{-1}$
$d$	$1$					$5$
$K$	$2^i, \, i = 1, 2, ..., 11$					$2^i, \, i = 1, 2, ..., 10$
Experiment	Exp. 1	Exp. 2	Exp. 3	Exp. 4	Exp. 5	Exp. 1	Exp. 4
Method	Alg. 1	Alg. 2	RFF$^{1}$	SGM	Alg. 1 tabular	Alg. 1	SGM
$N$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$
$\tilde{N}$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$
$γ$	$3d-2$	$3d-2$			$3d-2$	$3d-2$
$\lambda$	$0.1$	$0.1$	$0.1$	$0$	$0.1$	$0.1$	0
$M$	$10^3$	$5000$	N/A	$10^7$	$10^4$	$2.5\times 10^3$	$10^7$
$\bar{M}$	$10$	$10$	$10$	$10$	$10$	$10$	$10$
$\delta$	$2.4^2/d$	$0.1$			$2.4^2/d$	$\frac{2.4^2}{10d}$
$\Delta t$				$\mathtt{1.5e-4}$			$\mathtt{3.0e-4}$
$t_0$		$M/10$
$ω_{\text{max}}$		$\infty$
$m$	$10$	$50$			$100$	$25$

	Regression			Classification
Case	Case 3: anisotropic Gaussian function			Case 4: The MNIST data set
Purpose	Computational complexity comparison			Study the ability of Alg. 1 to work with non synthetic data
Target $f(x)$	$e^{-(32 x_1)^2/2}e^{-(32^{-1} x_2)^2/2}$
$d$	$2$			$784$
$K$	$256$			$2^i, \, i = 1, 2, ..., 13$
Experiment	Exp. 1	Exp. 2	Exp. 4	Exp. 1	Exp. 3
Method	Alg. 1	Alg. 2	SGM	Alg. 1	RFF$^{2}$
$N$	$10^4$	$10^4$	$3\times 10^7$	$6\times 10^4$	$6 \times 10^4$
$\tilde{N}$	$10^4$	$10^4$	$10^4$	$10^4$	$10^4$
$γ$	$3d-2$	$3d-2$		$3d-2$
$\lambda$	$0.1$	$0.1$	$0$	$0.1$	$0.1$
$M$	$10^4$	$10^4$	$3\times 10^7$	$10^2$
$\bar{M}$	$1$	$1$	$1$	$1$	$1$
$\delta$	$0.5$	$0.1$		$0.1$
$\Delta t$			$\mathtt{1.5e-3}$
$t_0$		$M/10$
$ω_{\text{max}}$		$\infty$
$m$	$100$	$100$		$M+1$
$ ^1 $ – Random Fourier features with frequencies sampled from the fixed distribution $ \mathcal{N}(0,1) $ $ ^2 $ – Random Fourier features with frequencies sampled from the fixed distribution $ \mathcal{N}(0,1) $, or $ \mathcal{N}(0,0.1^2) $

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Table 1. Case 4: The table shows the percentage of misclassified digits in the MNIST test data set for different values of $ K $. Comparison between adaptively computed distribution of frequencies $ \omega_k $ and sampling a fixed (normal) distribution

K	Fixed	Fixed	Adaptive
	$ \omega\sim\mathcal{N}(0, 1) $	$ \omega\sim\mathcal{N}(0, 0.1^2) $
2	89.97%	81.65%	80.03%
4	89.2%	70.65%	65.25%
8	88.98%	55.35%	53.44%
16	88.69%	46.77%	36.42%
32	88.9%	30.43%	23.52%
64	88.59%	19.73%	16.98%
128	88.7%	13.6%	11.13%
256	88.09%	10.12%	7.99%
512	88.01%	8.16%	5.93%
1024	87.34%	6.29%	4.57%
2048	86.5%	4.94%	3.5%
4096	85.21%	3.76%	2.74%
8192	83.98%	3.16%	1.98%

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