Aku Kammonen’s research while affiliated with King Abdullah University of Science and Technology and other places

This paper presents an enhanced adaptive random Fourier features (ARFF) training algorithm for shallow neural networks, building upon the work introduced in "Adaptive Random Fourier Features with Metropolis Sampling", Kammonen et al., Foundations of Data Science, 2(3):309--332, 2020. This improved method uses a particle filter type resampling technique to stabilize the training process and reduce sensitivity to parameter choices. With resampling, the Metropolis test may also be omitted, reducing the number of hyperparameters and reducing the computational cost per iteration, compared to ARFF. We present comprehensive numerical experiments demonstrating the efficacy of our proposed algorithm in function regression tasks, both as a standalone method and as a pre-training step before gradient-based optimization, here Adam. Furthermore, we apply our algorithm to a simple image regression problem, showcasing its utility in sampling frequencies for the random Fourier features (RFF) layer of coordinate-based multilayer perceptrons (MLPs). In this context, we use the proposed algorithm to sample the parameters of the RFF layer in an automated manner.

We present experimental results highlighting two key differences resulting from the choice of training algorithm for two-layer neural networks. The spectral bias of neural networks is well known, while the spectral bias dependence on the choice of training algorithm is less studied. Our experiments demonstrate that an adaptive random Fourier features algorithm (ARFF) can yield a spectral bias closer to zero compared to the stochastic gradient descent optimizer (SGD). Additionally, we train two identically structured classifiers, employing SGD and ARFF, to the same accuracy levels and empirically assess their robustness against adversarial noise attacks.

Estimates of the generalization error are proved for a residual neural network with L random Fourier features layers $\bar z_{\ell +1}=\bar z_\ell + \textrm {Re}\sum _{k=1}^K\bar b_{\ell k}\,e^{\textrm {i}\omega _{\ell k}\bar z_\ell }+ \textrm {Re}\sum _{k=1}^K\bar c_{\ell k}\,e^{\textrm {i}\omega ^{\prime}_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega _{\ell k},\omega ^{\prime}_{\ell k})$ of the random Fourier features $e^{\textrm {i}\omega _{\ell k}\bar z_\ell }$ and $e^{\textrm {i}\omega ^{\prime}_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values f(x). The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1({\mathbb {R}}^d)}}/{(KL)}$ of the generalization error for random Fourier features, with one hidden layer and the same total number of nodes KL, in the case of the $L^\infty$-norm of f is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network. Promising performance of the proposed new algorithm is demonstrated in computational experiments.

Estimates of the generalization error are proved for a residual neural network with L random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \text{Re}\sum_{k=1}^K\bar b_{\ell k}e^{{\rm i}\omega_{\ell k}\bar z_\ell}+ \text{Re}\sum_{k=1}^K\bar c_{\ell k}e^{{\rm i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{{\rm i}\omega_{\ell k}\bar z_\ell}$ and $e^{{\rm i}\omega'_{\ell k}\cdot x}$ is derived. The derivation is based on the corresponding generalization error to approximate function values f(x). The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(LK)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes LK, in the case the $L^\infty$-norm of f is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network that shows promising results.

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.

On page 2744 in [1], it is stated that the nonlinear eigenvalue problem (3.8).

It is known that ab initio molecular dynamics based on the electron ground state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature. The proof uses the semiclassical Weyl law to show that canonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The result covers observables that depend on time-correlations. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's law shows that the error estimate holds for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position and the observables are in the diagonal form with respect to the electron eigenstates.

Ab initio molecular dynamics based on the electron ground state eigenvalue can be used to approximate quantum observables in the canonical ensemble when the temperature is low compared to the first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for all temperatures. The proof uses the semi-classical Weyl law to show that canonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The result includes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's law shows that the error estimate holds for observable and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei position.