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We consider a Johnson-N\'ed\'elec FEM-BEM coupling, which is a direct and non-symmetric coupling of finite and boundary element methods, in order to solve interface problems for the magnetostatic Maxwell's equations with the magnetic vector potential ansatz. In the FEM-domain, equations may be non-linear, whereas they are exclusively linear in the BEM-part to guarantee the existence of a fundamental solution. First, the weak problem is formulated in quotient spaces to avoid resolving to a saddle point problem. Second, we establish in this setting well-posedness of the arising problem using the framework of Lipschitz and strongly monotone operators as well as a stability result for a special type of non-linearity, which is typically considered in magnetostatic applications. Then, the discretization is performed in the isogeometric context, i.e., the same type of basis functions that are used for geometry design are considered as ansatz functions for the discrete setting. In particular, NURBS are employed for geometry considerations, and B-Splines, which can be understood as a special type of NURBS, for analysis purposes. In this context, we derive a priori estimates w.r.t. h-refinement, and point out to an interesting behavior of BEM, which consists in an amelioration of the convergence rates, when a functional of the solution is evaluated in the exterior BEM-domain. This improvement may lead to a doubling of the convergence rate under certain assumptions. Finally, we end the paper with a numerical example to illustrate the theoretical results, along with a conclusion and an outlook.

We propose and analyze a stochastic Newton algorithm for homogeneous distributed stochastic convex optimization, where each machine can calculate stochastic gradients of the same population objective, as well as stochastic Hessian-vector products (products of an independent unbiased estimator of the Hessian of the population objective with arbitrary vectors), with many such stochastic computations performed between rounds of communication. We show that our method can reduce the number, and frequency, of required communication rounds compared to existing methods without hurting performance, by proving convergence guarantees for quasi-self-concordant objectives (e.g., logistic regression), alongside empirical evidence.

Training neural networks with binary weights and activations is a challenging problem due to the lack of gradients and difficulty of optimization over discrete weights. Many successful experimental results have been achieved with empirical straight-through (ST) approaches, proposing a variety of ad-hoc rules for propagating gradients through non-differentiable activations and updating discrete weights. At the same time, ST methods can be truly derived as estimators in the stochastic binary network (SBN) model with Bernoulli weights. We advance these derivations to a more complete and systematic study. We analyze properties, estimation accuracy, obtain different forms of correct ST estimators for activations and weights, explain existing empirical approaches and their shortcomings, explain how latent weights arise from the mirror descent method when optimizing over probabilities. This allows to reintroduce ST methods, long known empirically, as sound approximations, apply them with clarity and develop further improvements.

Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size (HDLSS) setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work [1, 2] has proposed penalized PCA, indicating the feasibility of model selection in $L_2$- penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the intensive calculation of matrix inverse. In this paper, we propose a fast model selection method for penalized PCA, named Approximated Gradient Flow (AgFlow), which lowers the computation complexity through incorporating the implicit regularization effect introduced by (stochastic) gradient flow [3, 4] and obtains the complete solution path of $L_2$-penalized PCA under varying $L_2$-regularization. We perform extensive experiments on real-world datasets. AgFlow outperforms existing methods (Oja [5], Power [6], and Shamir [7] and the vanilla Ridge estimators) in terms of computation costs.

In this paper, we consider the asymptotical regularization with convex constraints for nonlinear ill-posed problems. The method allows to use non-smooth penalty terms, including the L1-like and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy. Under certain conditions we give convergence properties of the methods. Moreover, we propose Runge-Kutta type methods to discrete the initial value problems to construct new type iterative regularization methods.

Parameters of the covariance kernel of a Gaussian process model often need to be estimated from the data generated by an unknown Gaussian process. We consider fixed-domain asymptotics of the maximum likelihood estimator of the scale parameter under smoothness misspecification. If the covariance kernel of the data-generating process has smoothness $\nu_0$ but that of the model has smoothness $\nu \geq \nu_0$, we prove that the expectation of the maximum likelihood estimator is of the order $N^{2(\nu-\nu_0)/d}$ if the $N$ observation points are quasi-uniform in $[0, 1]^d$. This indicates that maximum likelihood estimation of the scale parameter alone is sufficient to guarantee the correct rate of decay of the conditional variance. We also discuss a connection the expected maximum likelihood estimator has to Driscoll's theorem on sample path properties of Gaussian processes. The proofs are based on reproducing kernel Hilbert space techniques and worst-case case rates for approximation in Sobolev spaces.

Recent development of Deep Reinforcement Learning (DRL) has demonstrated superior performance of neural networks in solving challenging problems with large or even continuous state spaces. One specific approach is to deploy neural networks to approximate value functions by minimising the Mean Squared Bellman Error (MSBE) function. Despite great successes of DRL, development of reliable and efficient numerical algorithms to minimise the MSBE is still of great scientific interest and practical demand. Such a challenge is partially due to the underlying optimisation problem being highly non-convex or using incomplete gradient information as done in Semi-Gradient algorithms. In this work, we analyse the MSBE from a smooth optimisation perspective and develop an efficient Approximate Newton's algorithm. First, we conduct a critical point analysis of the error function and provide technical insights on optimisation and design choices for neural networks. When the existence of global minima is assumed and the objective fulfils certain conditions, suboptimal local minima can be avoided when using over-parametrised neural networks. We construct a Gauss Newton Residual Gradient algorithm based on the analysis in two variations. The first variation applies to discrete state spaces and exact learning. We confirm theoretical properties of this algorithm such as being locally quadratically convergent to a global minimum numerically. The second employs sampling and can be used in the continuous setting. We demonstrate feasibility and generalisation capabilities of the proposed algorithm empirically using continuous control problems and provide a numerical verification of our critical point analysis. We outline the difficulties of combining Semi-Gradient approaches with Hessian information. To benefit from second-order information complete derivatives of the MSBE must be considered during training.

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.

The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, maximizing variance and preservation of pairwise relative distances. The derivation of their asymptotic correlation and numerical experiments tell that a projection usually cannot satisfy both objectives. In a standard classification problem we determine projections on the input data that balance them and compare subsequent results. Next, we extend our application of orthogonal projections to deep learning frameworks. We introduce new variational loss functions that enable integration of additional information via transformations and projections of the target data. In two supervised learning problems, clinical image segmentation and music information classification, the application of the proposed loss functions increase the accuracy.

Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.