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# 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Convexity

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## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Convexity

Convex functions define the main class of functions which are somehow “simple” to optimize, in the sense that all minimizers are global minimizers, and that there are often efficient methods to find these minimizers (at least for smooth convex functions). A convex function is such that for any pair of point $(x, y) \in\left(\mathbb{R}^p\right)^2$,
$$\forall t \in[0,1], \quad f((1-t) x+t y) \leqslant(1-t) f(x)+t f(y)$$
which means that the function is below its secant (and actually also above its tangent when this is well defined), see Fig. 4. If $x^{\star}$ is a local minimizer of a convex $f$, then $x^{\star}$ is a global minimizer, i.e. $x^{\star} \in$ argmin $f$. Convex function are very convenient because they are stable under lots of transformation. In particular, if $f, g$ are convex and $a, b$ are positive, $a f+b g$ is convex (the set of convex function is itself an infinite dimensional convex cone!) and so is $\max (f, g)$. If $g: \mathbb{R}^q \rightarrow \mathbb{R}$ is convex and $B \in \mathbb{R}^{q \times p}, b \in \mathbb{R}^q$ then $f(x)=g(B x+b)$ is convex. This shows immediately that the square loss appearing in (3) is convex, since $|\cdot|^2 / 2$ is convex (as a sum of squares). Also, similarly, if $\ell$ and hence $L$ is convex, then the classification loss function (4) is itself convex.

Strict convexity. When $f$ is convex, one can strengthen the condition (5) and impose that the inequality is strict for $t \in] 0,1[$ (see Fig. 4, right), i.e.
$$\forall t \in] 0,1[, \quad f((1-t) x+t y)<(1-t) f(x)+t f(y) .$$
In this case, if a minimum $x^{\star}$ exists, then it is unique. Indeed, if $x_1^{\star} \neq x_2^{\star}$ were two different minimizer, one would have by strict convexity $f\left(\frac{x_1^{\star}+x_2^{\star}}{2}\right)<f\left(x_1^{\star}\right)$ which is impossible.
Example 2 (Least squares). For the quadratic loss function $f(x)=\frac{1}{2}|A x-y|^2$, strict convexity is equivalent to $\operatorname{ker}(A)={0}$. Indeed, we see later that its second derivative is $\partial^2 f(x)=A^{\top} A$ and that strict convexity is implied by the eigenvalues of $A^{\top} A$ being strictly positive. The eigenvalues of $A^{\top} A$ being positive, it is equivalent to $\operatorname{ker}\left(A^{\top} A\right)={0}$ (no vanishing eigenvalue), and $A^{\top} A z=0$ implies $\left\langle A^{\top} A z, z\right\rangle=|A z|^2=0$ i.e. $z \in \operatorname{ker}(A)$

## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代考|Convex Sets

A set $\Omega \subset \mathbb{R}^p$ is said to be convex if for any $(x, y) \in \Omega^2,(1-t) x+t y \in \Omega$ for $t \in[0,1]$. The connexion between convex function and convex sets is that a function $f$ is convex if and only if its epigraph $\operatorname{epi}(f) \stackrel{\text { def. }}{=}\left{(x, t) \in \mathbb{R}^{p+1} ; t \geqslant f(x)\right}$ is a convex set.
Remark 1 (Convexity of the set of minimizers). In general, minimizers $x^{\star}$ might be non-unique, as shown on Figure 3. When $f$ is convex, the set argmin $(f)$ of minimizers is itself a convex set. Indeed, if $x_1^{\star}$ and $x_2^{\star}$ are minimizers, so that in particular $f\left(x_1^{\star}\right)=f\left(x_2^{\star}\right)=\min (f)$, then $f\left((1-t) x_1^{\star}+t x_2^{\star}\right) \leqslant(1-t) f\left(x_1^{\star}\right)+t f\left(x_2^{\star}\right)=$ $f\left(x_1^{\star}\right)=\min (f)$, so that $(1-t) x_1^{\star}+t x_2^{\star}$ is itself a minimizer. Figure 5 shows convex and non-convex sets.

## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代 考|Convexity

$$\forall t \in[0,1], \quad f((1-t) x+t y) \leqslant(1-t) f(x)+t f(y)$$

$$\forall t \in] 0,1[, \quad f((1-t) x+t y)<(1-t) f(x)+t f(y) .$$

## 数学代写|机器学习中的优化理论代写Optimization for Machine Learning代写|Convex Sets

$f\left((1-t) x_1^{\star}+t x_2^{\star}\right) \leqslant(1-t) f\left(x_1^{\star}\right)+t f\left(x_2^{\star}\right)=f\left(x_1^{\star}\right)=\min (f)$ ，以便 $(1-t) x_1^{\star}+t x_2^{\star}$ 本身就是一个最小化器。图 5

## 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。