Posted on Categories:Convex optimization, 凸优化, 数学代写

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

Another variant of incremental gradient is the incremental aggregated gradient method, which has the form
$$x_{k+1}=P_X\left(x_k-\alpha_k \sum_{\ell=0}^{m-1} \nabla f_{i_{k-\ell}}\left(x_{k-\ell}\right)\right),$$
where $f_{i_k}$ is the new component function selected for iteration $k$. Here, the component indexes $i_k$ may either be selected in a cyclic order $\left[i_k=\right.$ $(k$ modulo $m)+1]$, or according to some randomization scheme, consistently with Eq. (2.31). Also for $k<m$, the summation should go up to $\ell=k$, and $\alpha$ should be replaced by a corresponding larger value, such as $\alpha_k=m \alpha /(k+1)$. This method, first proposed in [BHG08], computes the gradient incrementally, one component per iteration, but in place of the single component gradient, it uses an approximation to the total cost gradient $\nabla f\left(x_k\right)$, which is the sum of the component gradients computed in the past $m$ iterations.

There is analytical and experimental evidence that by aggregating the component gradients one may be able to attain a faster asymptotic convergence rate, by ameliorating the effect of approximating the full gradient with component gradients; see the original paper [BHG08], which provides an analysis for quadratic problems, the paper [SLB13], which provides a more general convergence and convergence rate analysis, and extensive computational results, and the papers [Mai13], [Mai14], [DCD14], which describe related methods. The expectation of faster convergence should be tempered, however, because in order for the effect of aggregating the component gradients to fully manifest itself, at least one pass (and possibly quite a few more) through the components must be made, which may be too long if $m$ is very large.

## 数学代写|凸优化代写Convex Optimization代考|Incremental Gradient Method with Momentum

There is an incremental version of the gradient method with momentum or heavy ball method, discussed in Section 2.1.1 [cf. Eq. (2.12)]. It is given by
$$x_{k+1}=x_k-\alpha_k \nabla f_{i_k}\left(x_k\right)+\beta_k\left(x_k-x_{k-1}\right),$$
where $f_{i_k}$ is the component function selected for iteration $k, \beta_k$ is a scalar in $[0,1)$, and we define $x_{-1}=x_0$; see e.g., [MaS94], [Tse98]. As noted earlier, special nonincremental methods with similarities to the one above have optimal iteration complexity properties under certain conditions; cf. Section 6.2. However, there have been no proposals of incremental versions of these optimal complexity methods.

The heavy ball method $(2.36)$ is related with the aggregated gradient method $(2.35)$ when $\beta_k \approx 1$. In particular, when $\alpha_k \equiv \alpha$ and $\beta_k \equiv \beta$, the sequence generated by Eq. (2.36) satisfies
$$x_{k+1}=x_k-\alpha \sum_{\ell=0}^k \beta^{\ell} \nabla f_{i_{k-\ell}}\left(x_{k-\ell}\right)$$

[both iterations (2.35) and (2.37) involve different types of diminishing dependence on past gradient components]. Thus, the heavy ball iteration (2.36) provides an approximate implementation of the incremental aggregated gradient method (2.35), while it does not have the memory storage issue of the latter.

A further way to intertwine the ideas of the aggregated gradient method (2.35) and the heavy ball method (2.36) for the unconstrained case $\left(X=\Re^n\right)$ is to form an infinite sequence of components
$$f_1, f_2, \ldots, f_m, f_1, f_2, \ldots, f_m, f_1, f_2, \ldots,$$
and group together blocks of successive components into batches. One way to implement this idea is to add $p$ preceding gradients (with $1<p<m$ ) to the current component gradient in iteration (2.36), thus iterating according to
$$x_{k+1}=x_k-\alpha_k \sum_{\ell=0}^p \nabla f_{i_{k-\ell}}\left(x_{k-\ell}\right)+\beta_k\left(x_k-x_{k-1}\right)$$

## 凸优化代写

$$x_{k+1}=P_X\left(x_k-\alpha_k \sum_{\ell=0}^{m-1} \nabla f_{i_{k-\ell}}\left(x_{k-\ell}\right)\right),$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。