计算机代写|数据分析信号处理和机器学习中的矩阵方法代写Matrix Methods In Data Analysis, Signal Processing, And Machine Learning代考|CSC2321

计算机代写|数据分析信号处理和机器学习中的矩阵方法代写Matrix Methods In Data Analysis, Signal Processing, And Machine Learning代考|Stochastic Gradient Descent

We can obtain better results for more restrictive classes of functions. Recall that we can only set a small step because the gradient changes as we change the solution. A function is smooth if the change in the gradient is bounded by how far the solution changes. More precisely, a function is $\beta$ smooth if for any two points $x, y$, we have
$$|\nabla f(x)-\nabla f(y)| \leq \beta|x-y|$$
Note that this implies for any $x, y$
$$f(y) \leq f(x)+\langle\nabla f(x), y-x\rangle+\frac{\beta}{2}|y-x|^2$$
Thus, we can always bound the objective using a quadratic function. In order to find the next location for the solution, we minimize the quadratic approximation at the current point. The solution turns out to be $\eta_t=1 / \beta$. With this choice, we have
$$f\left(x^{(t)}\right) \leq f\left(x^{(t-1)}\right)-\frac{1}{2 \beta} \mid \nabla f\left(\left.x^{(t-1)}\right|^2\right.$$
We analyze this algorithm using the potential Missing \left or extra \right. The change in the potential is
Missing \left or extra \right }

计算机代写|数据分析信号处理和机器学习中的矩阵方法代写Matrix Methods In Data Analysis, Signal Processing, And Machine Learning代考|Two Notes on Proof

We now extend our results to the constrained case $\min {x \in S} f(x)$ for a convex set $S$. In the unconstrained case, we take the quadratic approximation of the function at the current solution and the next solution is the minimizer of the quadratic approximation. Notice that this step is still meaningful when we have constraints. Thus our algorithm is $$x^{(t)} \leftarrow \underset{z \in S}{\operatorname{argmin}} f\left(x^{(t-1)}\right)+\left\langle\nabla f\left(x^{(t-1)}\right), z-x^{(t-1)}\right\rangle+\frac{\beta}{2}\left|z-x^{(t-1)}\right|^2$$ Another idea is to move in the direction of the gradient, which might take us out of the feasible region, and then project back to the feasible region. In other words, our algorithm is $$y^{(t)} \leftarrow x^{(t-1)}-\eta_t \nabla f\left(x^{(t-1)}\right) \quad x^{(t)} \leftarrow \underset{x \in S}{\operatorname{argmin}}\left|y^{(t)}-x\right|$$ It turns out that for step size $\eta_t=1 / \beta$, these two algorithms are identical. In order to analyze this algorithm, we need a property of the projection operation. Lemma 6.1. Given a convex set $S$, let $a \in S$ and $b^{\prime} \in \mathbb{R}^n$. Let $b=\operatorname{argmin}{x \in S} \frac{1}{2}\left|x-b^{\prime}\right|^2$. Then $\left\langle a-b, b-b^{\prime}\right\rangle \geq 0$ and therefore, $|a-b|^2 \leq\left|a-b^{\prime}\right|^2$.

Proof. The lemma follows from the optimality of $b$. The gradient of $\frac{1}{2}\left|x-b^{\prime}\right|^2$ at $x=b$ is $b-b^{\prime}$. Because of the optimality of $b$, we have $\left\langle a-b, b-b^{\prime}\right\rangle \geq 0$.
Using this property, we obtain Missing \left or extra \right and we can observe that the rest of the original proof goes through in the constrained setting.

机器学习中的矩阵方法代考

计算机代写|数据分析信号处理和机器学习中的矩阵方法代写Matrix Methods In Data Analysis, Signal Processing, And Machine Learning代 考|Smooth functions

|∇f(x)−∇f(y)|≤β|x−y|

f(y)≤f(x)+⟨∇f(x),y−x⟩+β2|y−x|2

f(x(t))≤f(x(t−1))−12β∣∇f(x(t−1)|2

$$|\nabla f(x)-\nabla f(y)| \leq \beta|x-y|$$

$$f(y) \leq f(x)+\langle\nabla f(x), y-x\rangle+\frac{\beta}{2}|y-x|^2$$

$$f\left(x^{(t)}\right) \leq f\left(x^{(t-1)}\right)-\frac{1}{2 \beta} \mid \nabla f\left(\left.x^{(t-1)}\right|^2\right.$$

计算机代写|数据分析信号处理和机器学习中的矩阵方法代写Matrix Methods In Data Analysis, Signal Processing, And Machine Learning代 考|Constrained optimization

$$x^{(t)} \leftarrow \underset{z \in S}{\operatorname{argmin}} f\left(x^{(t-1)}\right)+\left\langle\nabla f\left(x^{(t-1)}\right), z-x^{(t-1)}\right\rangle+\frac{\beta}{2}\left|z-x^{(t-1)}\right|^2$$

$$y^{(t)} \leftarrow x^{(t-1)}-\eta_t \nabla f\left(x^{(t-1)}\right) \quad x^{(t)} \leftarrow \underset{x \in S}{\operatorname{argmin}}\left|y^{(t)}-x\right|$$

myassignments-help数学代考价格说明

1、客户需提供物理代考的网址，相关账户，以及课程名称，Textbook等相关资料~客服会根据作业数量和持续时间给您定价~使收费透明，让您清楚的知道您的钱花在什么地方。

2、数学代写一般每篇报价约为600—1000rmb，费用根据持续时间、周作业量、成绩要求有所浮动(持续时间越长约便宜、周作业量越多约贵、成绩要求越高越贵)，报价后价格觉得合适，可以先付一周的款，我们帮你试做，满意后再继续，遇到Fail全额退款。

3、myassignments-help公司所有MATH作业代写服务支持付半款，全款，周付款，周付款一方面方便大家查阅自己的分数，一方面也方便大家资金周转，注意:每周固定周一时先预付下周的定金，不付定金不予继续做。物理代写一次性付清打9.5折。

Math作业代写、数学代写常见问题

myassignments-help擅长领域包含但不是全部:

myassignments-help服务请添加我们官网的客服或者微信/QQ，我们的服务覆盖：Assignment代写、Business商科代写、CS代考、Economics经济学代写、Essay代写、Finance金融代写、Math数学代写、report代写、R语言代考、Statistics统计学代写、物理代考、作业代写、加拿大代考、加拿大统计代写、北美代写、北美作业代写、北美统计代考、商科Essay代写、商科代考、数学代考、数学代写、数学作业代写、physics作业代写、物理代写、数据分析代写、新西兰代写、澳洲Essay代写、澳洲代写、澳洲作业代写、澳洲统计代写、澳洲金融代写、留学生课业指导、经济代写、统计代写、统计作业代写、美国Essay代写、美国代考、美国数学代写、美国统计代写、英国Essay代写、英国代考、英国作业代写、英国数学代写、英国统计代写、英国金融代写、论文代写、金融代考、金融作业代写。

Scroll to Top