Orthogonally weighted $\ell_{2,1}$ regularization for rank-aware joint sparse recovery: algorithm and analysis

We propose and analyze an efficient algorithm for solving the joint sparse recovery problem using a new regularization-based method, named orthogonally weighted $\ell_{2,1}$ ($\mathit{ow}\ell_{2,1}$), which is specifically designed to take into account the rank of the solution matrix. This method has applications in feature extraction, matrix column selection, and dictionary learning, and it is distinct from commonly used $\ell_{2,1}$ regularization and other existing regularization-based approaches because it can exploit the full rank of the row-sparse solution matrix, a key feature in many applications. We provide a proof of the method's rank-awareness, establish the existence of solutions to the proposed optimization problem, and develop an efficient algorithm for solving it, whose convergence is analyzed. We also present numerical experiments to illustrate the theory and demonstrate the effectiveness of our method on real-life problems.

Point Cloud Data Simulation and Modelling with Aize Workspace

This work takes a look at data models often used in digital twins and presents preliminary results specifically from surface reconstruction and semantic segmentation models trained using simulated data. This work is expected to serve as a ground work for future endeavours in data contextualisation inside a digital twin.

Differentially Private Online-to-Batch for Smooth Losses

We develop a new reduction that converts any online convex optimization algorithm suffering $O(\sqrt{T})$ regret into an $\epsilon$-differentially private stochastic convex optimization algorithm with the optimal convergence rate $\tilde O(1/\sqrt{T} + \sqrt{d}/\epsilon T)$ on smooth losses in linear time, forming a direct analogy to the classical non-private "online-to-batch" conversion. By applying our techniques to more advanced adaptive online algorithms, we produce adaptive differentially private counterparts whose convergence rates depend on apriori unknown variances or parameter norms.

Momentum Aggregation for Private Non-convex ERM

We introduce new algorithms and convergence guarantees for privacy-preserving non-convex Empirical Risk Minimization (ERM) on smooth $d$-dimensional objectives. We develop an improved sensitivity analysis of stochastic gradient descent on smooth objectives that exploits the recurrence of examples in different epochs. By combining this new approach with recent analysis of momentum with private aggregation techniques, we provide an $(\epsilon,\delta)$-differential private algorithm that finds a gradient of norm $\tilde O\left(\frac{d^{1/3}}{(\epsilon N)^{2/3}}\right)$ in $O\left(\frac{N^{7/3}\epsilon^{4/3}}{d^{2/3}}\right)$ gradient evaluations, improving the previous best gradient bound of $\tilde O\left(\frac{d^{1/4}}{\sqrt{\epsilon N}}\right)$.

Model Calibration of the Liquid Mercury Spallation Target using Evolutionary Neural Networks and Sparse Polynomial Expansions

The mercury constitutive model predicting the strain and stress in the target vessel plays a central role in improving the lifetime prediction and future target designs of the mercury targets at the Spallation Neutron Source (SNS). We leverage the experiment strain data collected over multiple years to improve the mercury constitutive model through a combination of large-scale simulations of the target behavior and the use of machine learning tools for parameter estimation. We present two interdisciplinary approaches for surrogate-based model calibration of expensive simulations using evolutionary neural networks and sparse polynomial expansions. The experiments and results of the two methods show a very good agreement for the solid mechanics simulation of the mercury spallation target. The proposed methods are used to calibrate the tensile cutoff threshold, mercury density, and mercury speed of sound during intense proton pulse experiments. Using strain experimental data from the mercury target sensors, the newly calibrated simulations achieve 7\% average improvement on the signal prediction accuracy and 8\% reduction in mean absolute error compared to previously reported reference parameters, with some sensors experiencing up to 30\% improvement. The proposed calibrated simulations can significantly aid in fatigue analysis to estimate the mercury target lifetime and integrity, which reduces abrupt target failure and saves a tremendous amount of costs. However, an important conclusion from this work points out to a deficiency in the current constitutive model based on the equation of state in capturing the full physics of the spallation reaction. Given that some of the calibrated parameters that show a good agreement with the experimental data can be nonphysical mercury properties, we need a more advanced two-phase flow model to capture bubble dynamics and mercury cavitation.

Correcting Momentum with Second-order Information

We develop a new algorithm for non-convex stochastic optimization that finds an $\epsilon$-critical point in the optimal $O(\epsilon^{-3})$ stochastic gradient and hessian-vector product computations. Our algorithm uses Hessian-vector products to "correct" a bias term in the momentum of SGD with momentum. This leads to better gradient estimates in a manner analogous to variance reduction methods. In contrast to prior work, we do not require excessively large batch sizes (or indeed any restrictions at all on the batch size), and both our algorithm and its analysis are much simpler. We validate our results on a variety of large-scale deep learning benchmarks and architectures, where we see improvements over SGD and Adam.

AdaDGS: An adaptive black-box optimization method with a nonlocal directional Gaussian smoothing gradient

The local gradient points to the direction of the steepest slope in an infinitesimal neighborhood. An optimizer guided by the local gradient is often trapped in local optima when the loss landscape is multi-modal. A directional Gaussian smoothing (DGS) approach was recently proposed in (Zhang et al., 2020) and used to define a truly nonlocal gradient, referred to as the DGS gradient, for high-dimensional black-box optimization. Promising results show that replacing the traditional local gradient with the DGS gradient can significantly improve the performance of gradient-based methods in optimizing highly multi-modal loss functions. However, the optimal performance of the DGS gradient may rely on fine tuning of two important hyper-parameters, i.e., the smoothing radius and the learning rate. In this paper, we present a simple, yet ingenious and efficient adaptive approach for optimization with the DGS gradient, which removes the need of hyper-parameter fine tuning. Since the DGS gradient generally points to a good search direction, we perform a line search along the DGS direction to determine the step size at each iteration. The learned step size in turn will inform us of the scale of function landscape in the surrounding area, based on which we adjust the smoothing radius accordingly for the next iteration. We present experimental results on high-dimensional benchmark functions, an airfoil design problem and a game content generation problem. The AdaDGS method has shown superior performance over several the state-of-the-art black-box optimization methods.

Analysis of The Ratio of $\ell_1$ and $\ell_2$ Norms in Compressed Sensing

We first propose a novel criterion that guarantees that an $s$-sparse signal is the local minimizer of the $\ell_1/\ell_2$ objective; our criterion is interpretable and useful in practice. We also give the first uniform recovery condition using a geometric characterization of the null space of the measurement matrix, and show that this condition is easily satisfied for a class of random matrices. We also present analysis on the stability of the procedure when noise pollutes data. Numerical experiments are provided that compare $\ell_1/\ell_2$ with some other popular non-convex methods in compressed sensing. Finally, we propose a novel initialization approach to accelerate the numerical optimization procedure. We call this initialization approach \emph{support selection}, and we demonstrate that it empirically improves the performance of existing $\ell_1/\ell_2$ algorithms.

Accelerating Reinforcement Learning with a Directional-Gaussian-Smoothing Evolution Strategy

Evolution strategy (ES) has been shown great promise in many challenging reinforcement learning (RL) tasks, rivaling other state-of-the-art deep RL methods. Yet, there are two limitations in the current ES practice that may hinder its otherwise further capabilities. First, most current methods rely on Monte Carlo type gradient estimators to suggest search direction, where the policy parameter is, in general, randomly sampled. Due to the low accuracy of such estimators, the RL training may suffer from slow convergence and require more iterations to reach optimal solution. Secondly, the landscape of reward functions can be deceptive and contains many local maxima, causing ES algorithms to prematurely converge and be unable to explore other parts of the parameter space with potentially greater rewards. In this work, we employ a Directional Gaussian Smoothing Evolutionary Strategy (DGS-ES) to accelerate RL training, which is well-suited to address these two challenges with its ability to i) provide gradient estimates with high accuracy, and ii) find nonlocal search direction which lays stress on large-scale variation of the reward function and disregards local fluctuation. Through several benchmark RL tasks demonstrated herein, we show that DGS-ES is highly scalable, possesses superior wall-clock time, and achieves competitive reward scores to other popular policy gradient and ES approaches.

A Scalable Evolution Strategy with Directional Gaussian Smoothing for Blackbox Optimization

We developed a new scalable evolution strategy with directional Gaussian smoothing (DGS-ES) for high-dimensional blackbox optimization. Standard ES methods have been proved to suffer from the curse of dimensionality, due to the random directional search and low accuracy of Monte Carlo estimation. The key idea of this work is to develop Gaussian smoothing approach which only averages the original objective function along $d$ orthogonal directions. In this way, the partial derivatives of the smoothed function along those directions can be represented by one-dimensional integrals, instead of $d$-dimensional integrals in the standard ES methods. As such, the averaged partial derivatives can be approximated using the Gauss-Hermite quadrature rule, as opposed to MC, which significantly improves the accuracy of the averaged gradients. Moreover, the smoothing technique reduces the barrier of local minima, such that global minima become easier to achieve. We provide three sets of examples to demonstrate the performance of our method, including benchmark functions for global optimization, and a rocket shell design problem.