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Some continuous optimization methods can be connected to ordinary differential equations (ODEs) by taking continuous limits, and their convergence rates can be explained by the ODEs. However, since such ODEs can achieve any convergence rate by time scaling, the correspondence is not as straightforward as usually expected, and deriving new methods through ODEs is not quite direct. In this letter, we pay attention to stability restriction in discretizing ODEs and show that acceleration by time scaling always implies deceleration in discretization; they balance out so that we can define an attainable unique convergence rate which we call an "essential convergence rate".

We introduce a new type of Krasnoselskii's result. Using a simple differentiability condition, we relax the nonexpansive condition in Krasnoselskii's theorem. More clearly, we analyze the convergence of the sequence $x_{n+1}=\frac{x_n+g(x_n)}{2}$ based on some differentiability condition of $g$ and present some fixed point results. We introduce some iterative sequences that for any real differentiable function $g$ and any starting point $x_0\in \mathbb [a,b]$ converge monotonically to the nearest root of $g$ in $[a,b]$ that lay to the right or left side of $x_0$. Based on this approach, we present an efficient and novel method for finding the real roots of real functions. We prove that no root will be missed in our method. It is worth mentioning that our iterative method is free from the derivative evaluation which can be regarded as an advantage of this method in comparison with many other methods. Finally, we illustrate our results with some numerical examples.

We study sequential decision making problems aimed at maximizing the expected total reward while satisfying a constraint on the expected total utility. We employ the natural policy gradient method to solve the discounted infinite-horizon optimal control problem for Constrained Markov Decision Processes (constrained MDPs). Specifically, we propose a new Natural Policy Gradient Primal-Dual (NPG-PD) method that updates the primal variable via natural policy gradient ascent and the dual variable via projected sub-gradient descent. Although the underlying maximization involves a nonconcave objective function and a nonconvex constraint set, under the softmax policy parametrization we prove that our method achieves global convergence with sublinear rates regarding both the optimality gap and the constraint violation. Such convergence is independent of the size of the state-action space, i.e., it is~dimension-free. Furthermore, for log-linear and general smooth policy parametrizations, we establish sublinear convergence rates up to a function approximation error caused by restricted policy parametrization. We also provide convergence and finite-sample complexity guarantees for two sample-based NPG-PD algorithms. Finally, we use computational experiments to showcase the merits and the effectiveness of our approach.

In this paper, a new weighted average estimator (WAVE) is proposed to enhance the performance of the simple-averaging based distributed estimator, under a general loss with a high dimensional parameter. To obtain an efficient estimator, a weighted least-square ensemble framework plus an adaptive $L_1$ penalty is proposed, in which the local estimator is estimated via the adaptive-lasso and the weight is inversely proportional to the variance of local estimators. It can be proved that WAVE enjoys the same asymptotic properties as the global estimator and simultaneously spend a very low communication cost, only requiring the local worker to deliver two vectors to the master. Moreover, it is shown that WAVE is effective even when the samples across local workers have different mean and covariance. In particular, the asymptotic normality is established under such conditions, while other competitors may not own this property. The effectiveness of WAVE is further illustrated by an extensive numerical study and a real data analysis.

Blind super-resolution can be cast as a low rank matrix recovery problem by exploiting the inherent simplicity of the signal and the low dimensional structure of point spread functions. In this paper, we develop a simple yet efficient non-convex projected gradient descent method for this problem based on the low rank structure of the vectorized Hankel matrix associated with the target matrix. Theoretical analysis indicates that the proposed method exactly converges to the target matrix with a linear convergence rate under the similar conditions as convex approaches. Numerical results show that our approach is competitive with existing convex approaches in terms of recovery ability and efficiency.

We study stochastic gradient descent for solving conditional stochastic optimization problems, in which an objective to be minimized is given by a parametric nested expectation with an outer expectation taken with respect to one random variable and an inner conditional expectation with respect to the other random variable. The gradient of such a parametric nested expectation is again expressed as a nested expectation, which makes it hard for the standard nested Monte Carlo estimator to be unbiased. In this paper, we show under some conditions that a multilevel Monte Carlo gradient estimator is unbiased and has finite variance and finite expected computational cost, so that the standard theory from stochastic optimization for a parametric (non-nested) expectation directly applies. We also discuss a special case for which yet another unbiased gradient estimator with finite variance and cost can be constructed.

The synthetic control method has become a widely popular tool to estimate causal effects with observational data. Despite this, inference for synthetic control methods remains challenging. Often, inferential results rely on linear factor model data generating processes. In this paper, we characterize the conditions on the factor model primitives (the factor loadings) for which the statistical risk minimizers are synthetic controls (in the simplex). Then, we propose a Bayesian alternative to the synthetic control method that preserves the main features of the standard method and provides a new way of doing valid inference. We explore a Bernstein-von Mises style result to link our Bayesian inference to the frequentist inference. For linear factor model frameworks we show that a maximum likelihood estimator (MLE) of the synthetic control weights can consistently estimate the predictive function of the potential outcomes for the treated unit and that our Bayes estimator is asymptotically close to the MLE in the total variation sense. Through simulations, we show that there is convergence between the Bayes and frequentist approach even in sparse settings. Finally, we apply the method to re-visit the study of the economic costs of the German re-unification. The Bayesian synthetic control method is available in the bsynth R-package.

High-dimensional parabolic partial integro-differential equations (PIDEs) appear in many applications in insurance and finance. Existing numerical methods suffer from the curse of dimensionality or provide solutions only for a given space-time point. This gave rise to a growing literature on deep learning based methods for solving partial differential equations; results for integro-differential equations on the other hand are scarce. In this paper we consider an extension of the deep splitting scheme due to arXiv:1907.03452 and arXiv:2006.01496v3 to PIDEs. Our main contribution is an analysis of the approximation error which yields convergence rates in terms of the number of neurons for shallow neural networks. Moreover we discuss several test case studies to show the viability of our approach.

In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques.

We present a super-fast convergence approach to reconstructing the per-scene radiance field from a set of images that capture the scene with known poses. This task, which is often applied to novel view synthesis, is recently revolutionized by Neural Radiance Field (NeRF) for its state-of-the-art quality and flexibility. However, NeRF and its variants require a lengthy training time ranging from hours to days for a single scene. In contrast, our approach achieves NeRF-comparable quality and converges rapidly from scratch in less than 15 minutes with a single GPU. We adopt a representation consisting of a density voxel grid for scene geometry and a feature voxel grid with a shallow network for complex view-dependent appearance. Modeling with explicit and discretized volume representations is not new, but we propose two simple yet non-trivial techniques that contribute to fast convergence speed and high-quality output. First, we introduce the post-activation interpolation on voxel density, which is capable of producing sharp surfaces in lower grid resolution. Second, direct voxel density optimization is prone to suboptimal geometry solutions, so we robustify the optimization process by imposing several priors. Finally, evaluation on five inward-facing benchmarks shows that our method matches, if not surpasses, NeRF's quality, yet it only takes about 15 minutes to train from scratch for a new scene.