For time-dependent problems with high-contrast multiscale coefficients, the time step size for explicit methods is affected by the magnitude of the coefficient parameter. With a suitable construction of multiscale space, one can achieve a stable temporal splitting scheme where the time step size is independent of the contrast. Consider the parabolic equation with heterogeneous diffusion parameter, the flow rates vary significantly in different regions due to the high-contrast features of the diffusivity. In this work, we aim to introduce a multirate partially explicit splitting scheme to achieve efficient simulation with the desired accuracy. We first design multiscale subspaces to handle flow with different speed. For the fast flow, we obtain a low-dimensional subspace with respect to the high-diffusive component and adopt an implicit time discretization scheme. The other multiscale subspace will take care of the slow flow, and the corresponding degrees of freedom are treated explicitly. Then a multirate time stepping is introduced for the two parts. The stability of the multirate methods is analyzed for the partially explicit scheme. Moreover, we derive local error estimators corresponding to the two components of the solutions and provide an upper bound of the errors. An adaptive local temporal refinement framework is then proposed to achieve higher computational efficiency. Several numerical tests are presented to demonstrate the performance of the proposed method.
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In this paper, we present a novel maximum entropy formulation of the Differential Dynamic Programming algorithm and derive two variants using unimodal and multimodal value functions parameterizations. By combining the maximum entropy Bellman equations with a particular approximation of the cost function, we are able to obtain a new formulation of Differential Dynamic Programming which is able to escape from local minima via exploration with a multimodal policy. To demonstrate the efficacy of the proposed algorithm, we provide experimental results using four systems on tasks that are represented by cost functions with multiple local minima and compare them against vanilla Differential Dynamic Programming. Furthermore, we discuss connections with previous work on the linearly solvable stochastic control framework and its extensions in relation to compositionality.
This paper studies the problem of federated learning (FL) in the absence of a trustworthy server/clients. In this setting, each client needs to ensure the privacy of its own data without relying on the server or other clients. We study local differential privacy (LDP) and provide tight upper and lower bounds that establish the minimax optimal rates (up to logarithms) for LDP convex/strongly convex federated stochastic optimization. Our rates match the optimal statistical rates in certain practical parameter regimes ("privacy for free"). Second, we develop a novel time-varying noisy SGD algorithm, leading to the first non-trivial LDP risk bounds for FL with non-i.i.d. clients. Third, we consider the special case where each client's loss function is empirical and develop an accelerated LDP FL algorithm to improve communication complexity compared to existing works. We also provide matching lower bounds, establishing the optimality of our algorithm for convex/strongly convex settings. Fourth, with a secure shuffler to anonymize client reports (but without a trusted server), our algorithm attains the optimal central DP rates for stochastic convex/strongly convex optimization, thereby achieving optimality in the local and central models simultaneously. Our upper bounds quantify the role of network communication reliability in performance.
A new explicit stochastic scheme of order 1 is proposed for solving commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. The proposed method is a semi-tamed version of Milstein scheme to solve SDEs with the drift coefficient consisting of non-Lipschitz continuous term and globally Lipschitz continuous term. It is easily implementable and achieves higher strong convergence order. A stability criterion for this method is derived, which shows that the stability condition of the numerical methods and that of the solved equations keep uniform. Compared with some widely used numerical schemes, the proposed method has better performance in inheriting the mean square stability of the exact solution of SDEs. Numerical experiments are given to illustrate the obtained convergence and stability properties.
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The construction of the basis functions is based on the local snapshot space that takes all possible flows on the interface between coarse cells into account. In order to reduce the size of the snapshot space and obtain the multiscale approximation, we solve a local spectral problem to identify dominant modes in the snapshot space. We present a convergence analysis of the presented multiscale method. Numerical results are presented for two-dimensional problems in three testing geometries along with the errors associated to different numbers of the multiscale basis functions used for the velocity field. Numerical investigations are conducted for problems with homogeneous and heterogeneous properties respectively.
The dispersion error is often the dominant error for computed solutions of wave propagation problems with high-frequency components. In this paper, we define and give explicit examples of $\alpha$-dispersion-relation-preserving schemes. These are dual-pair finite-difference schemes for systems of hyperbolic partial differential equations which preserve the dispersion-relation of the continuous problem uniformly to an $\alpha \%$-error tolerance. We give a general framework to design provably stable finite difference operators that preserve the dispersion relation for hyperbolic systems such as the elastic wave equation. The operators we derive here can resolve the highest frequency ($\pi$-mode) present on any equidistant grid at a tolerance of $5\%$ error. This significantly improves on the current standard that have a tolerance of $100 \%$ error.
Adaptive gradient methods, such as Adam, have achieved tremendous success in machine learning. Scaling gradients by square roots of the running averages of squared past gradients, such methods are able to attain rapid training of modern deep neural networks. Nevertheless, they are observed to generalize worse than stochastic gradient descent (SGD) and tend to be trapped in local minima at an early stage during training. Intriguingly, we discover that substituting the gradient in the second moment estimation term with the momentumized version in Adam can well solve the issues. The intuition is that gradient with momentum contains more accurate directional information and therefore its second moment estimation is a better choice for scaling than that of the raw gradient. Thereby we propose AdaMomentum as a new optimizer reaching the goal of training fast while generalizing better. We further develop a theory to back up the improvement in optimization and generalization and provide convergence guarantees under both convex and nonconvex settings. Extensive experiments on a wide range of tasks and models demonstrate that AdaMomentum exhibits state-of-the-art performance consistently.
We provide a control-theoretic perspective on optimal tensor algorithms for minimizing a convex function in a finite-dimensional Euclidean space. Given a function $\Phi: \mathbb{R}^d \rightarrow \mathbb{R}$ that is convex and twice continuously differentiable, we study a closed-loop control system that is governed by the operators $\nabla \Phi$ and $\nabla^2 \Phi$ together with a feedback control law $\lambda(\cdot)$ satisfying the algebraic equation $(\lambda(t))^p\|\nabla\Phi(x(t))\|^{p-1} = \theta$ for some $\theta \in (0, 1)$. Our first contribution is to prove the existence and uniqueness of a local solution to this system via the Banach fixed-point theorem. We present a simple yet nontrivial Lyapunov function that allows us to establish the existence and uniqueness of a global solution under certain regularity conditions and analyze the convergence properties of trajectories. The rate of convergence is $O(1/t^{(3p+1)/2})$ in terms of objective function gap and $O(1/t^{3p})$ in terms of squared gradient norm. Our second contribution is to provide two algorithmic frameworks obtained from discretization of our continuous-time system, one of which generalizes the large-step A-HPE framework and the other of which leads to a new optimal $p$-th order tensor algorithm. While our discrete-time analysis can be seen as a simplification and generalization of~\citet{Monteiro-2013-Accelerated}, it is largely motivated by the aforementioned continuous-time analysis, demonstrating the fundamental role that the feedback control plays in optimal acceleration and the clear advantage that the continuous-time perspective brings to algorithmic design. A highlight of our analysis is that we show that all of the $p$-th order optimal tensor algorithms that we discuss minimize the squared gradient norm at a rate of $O(k^{-3p})$, which complements the recent analysis.
We introduce kernel thinning, a new procedure for compressing a distribution $\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\mathbf{k}$ and $\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\mathbb{P}$ into a $\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. With high probability, the maximum discrepancy in integration error is $\mathcal{O}_d(n^{-\frac{1}{2}}\sqrt{\log n})$ for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} \sqrt{(\log n)^{d+1}\log\log n})$ for sub-exponential $\mathbb{P}$ on $\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-\frac14})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning, in dimensions $d=2$ through $100$.
Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
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