We consider structured optimisation problems defined in terms of the sum of a smooth and convex function, and a proper, l.s.c., convex (typically non-smooth) one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to their intrinsic space-variant properties, such spaces can be naturally used as solution space and combined with space-variant functionals for the solution of ill-posed inverse problems. For this purpose, we propose and analyse two instances (primal and dual) of proximal gradient algorithms in $L_{p(\cdot)}(\Omega)$, where the proximal step, rather than depending on the natural (non-separable) $L_{p(\cdot)}(\Omega)$ norm, is defined in terms of its modular function, which, thanks to its separability, allows for the efficient computation of algorithmic iterates. Convergence in function values is proved for both algorithms, with convergence rates depending on problem/space smoothness. To show the effectiveness of the proposed modelling, some numerical tests highlighting the flexibility of the space $L_{p(\cdot)}(\Omega)$ are shown for exemplar deconvolution and mixed noise removal problems. Finally, a numerical verification on the convergence speed and computational costs of both algorithms in comparison with analogous ones defined in standard $L_{p}(\Omega)$ spaces is presented.
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We present a midpoint policy iteration algorithm to solve linear quadratic optimal control problems in both model-based and model-free settings. The algorithm is a variation of Newton's method, and we show that in the model-based setting it achieves cubic convergence, which is superior to standard policy iteration and policy gradient algorithms that achieve quadratic and linear convergence, respectively. We also demonstrate that the algorithm can be approximately implemented without knowledge of the dynamics model by using least-squares estimates of the state-action value function from trajectory data, from which policy improvements can be obtained. With sufficient trajectory data, the policy iterates converge cubically to approximately optimal policies, and this occurs with the same available sample budget as the approximate standard policy iteration. Numerical experiments demonstrate effectiveness of the proposed algorithms.
This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.
A fundamental computational problem is to find a shortest non-zero vector in Euclidean lattices, a problem known as the Shortest Vector Problem (SVP). This problem is believed to be hard even on quantum computers and thus plays a pivotal role in post-quantum cryptography. In this work we explore how (efficiently) Noisy Intermediate Scale Quantum (NISQ) devices may be used to solve SVP. Specifically, we map the problem to that of finding the ground state of a suitable Hamiltonian. In particular, (i) we establish new bounds for lattice enumeration, this allows us to obtain new bounds (resp.~estimates) for the number of qubits required per dimension for any lattices (resp.~random q-ary lattices) to solve SVP; (ii) we exclude the zero vector from the optimization space by proposing (a) a different classical optimisation loop or alternatively (b) a new mapping to the Hamiltonian. These improvements allow us to solve SVP in dimension up to 28 in a quantum emulation, significantly more than what was previously achieved, even for special cases. Finally, we extrapolate the size of NISQ devices that is required to be able to solve instances of lattices that are hard even for the best classical algorithms and find that with approximately $10^3$ noisy qubits such instances can be tackled.
The study of generalising the central difference for integer order Laplacian to fractional order is discussed in this paper. Analysis shows that, in contrary to the conclusion of a previous study, difference stencils evaluated through fast Fourier transform prevents the convergence of the solution of fractional Laplacian. We propose a composite quadrature rule in order to efficiently evaluate the stencil coefficients with the required convergence rate in order to guarantee convergence of the solution. Furthermore, we propose the use of generalised higher order lattice Boltzmann method to generate stencils which can approximate fractional Laplacian with higher order convergence speed and error isotropy. We also review the formulation of the lattice Boltzmann method and discuss the explicit sparse solution formulated using Smolyak's algorithm, as well as the method for the evaluation of the Hermite polynomials for efficient generation of the higher order stencils. Numerical experiments are carried out to verify the error analysis and formulations.
We propose an efficient numerical method for computing natural gradient descent directions with respect to a generic metric in the state space. Our technique relies on representing the natural gradient direction as a solution to a standard least-squares problem. Hence, instead of calculating, storing, or inverting the information matrix directly, we apply efficient methods from numerical linear algebra to solve this least-squares problem. We treat both scenarios where the derivative of the state variable with respect to the parameter is either explicitly known or implicitly given through constraints. We apply the QR decomposition to solve the least-squares problem in the former case and utilize the adjoint-state method to compute the natural gradient descent direction in the latter case. As a result, we can reliably compute several natural gradient descents, including the Wasserstein natural gradient, for a large-scale parameter space with thousands of dimensions, which was believed to be out of reach. Finally, our numerical results shed light on the qualitative differences among the standard gradient descent method and various natural gradient descent methods based on different metric spaces in large-scale nonconvex optimization problems.
We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for the case of the 2-sphere, including an explicit example for Gaussian measures with positive densities.
In the Strip Packing problem (SP), we are given a vertical half-strip $[0,W]\times[0,\infty)$ and a set of $n$ axis-aligned rectangles of width at most $W$. The goal is to find a non-overlapping packing of all rectangles into the strip such that the height of the packing is minimized. A well-studied and frequently used practical constraint is to allow only those packings that are guillotine separable, i.e., every rectangle in the packing can be obtained by recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine cuts) that do not intersect any item of the solution. In this paper, we study approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e., the Strip Packing problem where we require additionally that the packing needs to be guillotine separable. This problem generalizes the classical Bin Packing problem and also makespan minimization on identical machines, and thus it is already strongly NP-hard. Moreover, due to a reduction from the Partition problem, it is NP-hard to obtain a polynomial-time $(3/2-\varepsilon)$-approximation algorithm for GSP for any $\varepsilon>0$ (exactly as Strip Packing). We provide a matching polynomial time $(3/2+\varepsilon)$-approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time $(1+\varepsilon)$-approximation algorithm for GSP. This is surprising as it is NP-hard to obtain a $(5/4-\varepsilon)$-approximation algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.
Stochastic approximation algorithms are iterative procedures which are used to approximate a target value in an environment where the target is unknown and direct observations are corrupted by noise. These algorithms are useful, for instance, for root-finding and function minimization when the target function or model is not directly known. Originally introduced in a 1951 paper by Robbins and Monro, the field of Stochastic approximation has grown enormously and has come to influence application domains from adaptive signal processing to artificial intelligence. As an example, the Stochastic Gradient Descent algorithm which is ubiquitous in various subdomains of Machine Learning is based on stochastic approximation theory. In this paper, we give a formal proof (in the Coq proof assistant) of a general convergence theorem due to Aryeh Dvoretzky, which implies the convergence of important classical methods such as the Robbins-Monro and the Kiefer-Wolfowitz algorithms. In the process, we build a comprehensive Coq library of measure-theoretic probability theory and stochastic processes.
Assume that we observe i.i.d.~points lying close to some unknown $d$-dimensional $\mathcal{C}^k$ submanifold $M$ in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses ($L_p$, Hellinger, total variation, etc.), we focus on the Wasserstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends on $d$ and the regularity $s\leq k-1$ of the underlying density, but not on the ambient dimension. In particular, we show that the estimator is minimax and matches previous rates in the literature in the case where the manifold $M$ is a $d$-dimensional cube. The related problem of the estimation of the volume measure of $M$ for the Wasserstein loss is also considered, for which a minimax estimator is exhibited.
In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.
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