## 计算机代写|机器学习代写machine learning代考|CS446

2023年1月4日

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## 计算机代写|机器学习代写machine learning代考|Practical Course Material

In this section, a practical lecture on the (perhaps most) popular random Fourier feature approach, initially proposed to approximate the Gaussian kernel [Rahimi and Recht, 2008 , is discussed, the large-dimensional characterization of which is almost identity to that performed in Section 5.1.1, except for the major difference of employing two types of nonlinear activations (“sin” and “cos”) for random Fourier features. Both the training and test performance can be assessed, which, despite taking slightly more involved forms, (i) significantly differ from those of Gaussian kernel and (ii) also establish a double-descent-type test curve, as expected.

Practical Lecture Material 4 (Performance of large-dimensional random Fourier features, Liao et al. [2020]). As discussed in Remark 5.1, instead of the single-type nonlinearity setting in Figure $5.1$ thoroughly investigated in Section 5.1.1, from a random feature map and kernel approximation perspective, a mixture of nonlinearities such as “cos $+\sin$ ” in the case of random Fourier features [Rahimi and Recht, 2008] turns out to be a more natural choice. Specifically, for $\mathbf{W} \in \mathbb{R}^{N \times p}$ with independent standard Gaussian entries, the random Fourier features refer to the cascade of the random features from both ” $\cos$ ” and “sin” activations as
$$\boldsymbol{\Sigma}^{\boldsymbol{T}}=\left[\cos (\mathbf{W X})^{\top} \quad \sin (\mathbf{W X})^{\top}\right] \in \mathbb{R}^{n \times 2 N}$$
Check first that
$$\mathbb{E}{\mathbf{w}}\left[\cos \left(\mathbf{w}^{\top} \mathbf{x}_i\right) \cos \left(\mathbf{w}^{\top} \mathbf{x}_j\right)+\sin \left(\mathbf{w}^{\top} \mathbf{x}_i\right) \sin \left(\mathbf{w}^{\top} \mathbf{x}_j\right)\right]=\left[\mathbf{K}{\cos }\right]{i j}+\left[\mathbf{K}{\sin }\right]{i j}$$ so that by the strong law of large numbers, one has $$\frac{1}{N}\left[\Sigma^{\top} \Sigma\right]{i j} \stackrel{\text { a.s. }}{\longrightarrow}\left[\mathbf{K}{\mathrm{cos}}+\mathbf{K}{\sin }\right]{i j}=\left[\mathbf{K}{\text {Gauss }}\right]{i j}$$ as $N \rightarrow \infty$, for $\mathbf{K}{\mathrm{cos}}$ and $\mathbf{K}{\text {sin }}$ the limiting kernels of “cos” and “sin” non-linearities enlisted in Table 5.1, and $\mathbf{K}{\text {Gauss }}=\left{\exp \left(-\left|\mathbf{x}i-\mathbf{x}_j\right|^2 / 2\right)\right}{i, j=1}^n$ the Gaussian kernel. This justifies the use of random Fourier features, however only in the $N \gg n$ regime.
We move on to a large $n, p, N$ characterization of random Fourier features. Using the fact that $\mathbb{E}_{\mathbf{w}}\left[\cos \left(\mathbf{w}^{\top} \mathbf{x}_i\right) \sin \left(\mathbf{w}^{\top} \mathbf{x}_j\right)\right]=0$ for $\mathbf{w} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$ and that both $\cos (\cdot)$ and $\sin (\cdot)$ are 1-Lipschitz, show, with the help of Lemma 5.1 and similar to Theorem 5.1, that the random Fourier resolvent $\left(\frac{1}{n} \Sigma^{\top} \Sigma+\gamma \mathbf{I}_n\right)^{-1}$ admits the deterministic equivalent $$\mathbf{Q} \leftrightarrow \overline{\mathbf{Q}}, \quad \overline{\mathbf{Q}} \equiv\left(\frac{N}{n}\left(\frac{\mathbf{K}{\cos }}{1+\delta{\cos }}+\frac{\mathbf{K}{\sin }}{1+\delta{\sin }}\right)+\gamma \mathbf{I}n\right)^{-1}$$ for $\left(\delta{\cos }, \delta_{\sin }\right)$ the unique positive solution to
$$\delta_{\cos }=\frac{1}{n} \operatorname{tr} \mathbf{K}{\cos } \overline{\mathbf{Q}}, \quad \delta{\sin }=\frac{1}{n} \operatorname{tr} \mathbf{K}_{\sin } \overline{\mathbf{Q}} .$$

## 计算机代写|机器学习代写machine learning代考|Large-Dimensional Convex Optimization

Unlike the kernel methods discussed in Section 4 or the simple neural network models of Section 5, where the objects under study (kernel matrices and random feature ridge regressors) assume an explicit form, many other machine learning algorithms are the solutions of optimization problems having in general no closed-form formulation. A first example is the popular logistic regression method, where one aims to find (say in a binary classification setting) an optimal (generalized) linear classifier $\boldsymbol{\beta} \in \mathbb{R}^p$ by minimizing the logistic loss $\frac{1}{n} \sum_{i=1}^n \log \left(1+e^{-y_i \boldsymbol{\beta}^{\top} \mathbf{x}i}\right)$ over a training set $\left{\left(\mathbf{x}_i, y_i\right)\right}{i=1}^n$ with labels $y_i \in{-1,+1}{ }^1$ More generally, by choosing other loss functions beyond the logistic loss, logistic regression can be viewed as a special case of the empirical risk minimization [Vapnik, 1992] problem
$$\underset{\boldsymbol{\beta} \in \mathbb{R}^p}{\arg \min } \frac{1}{n} \sum_{i=1}^n L\left(y_i \boldsymbol{\beta}^{\top} \mathbf{x}_i\right)+\frac{\gamma}{2}|\boldsymbol{\beta}|^2$$
for some convex loss $L: \mathbb{R} \rightarrow \mathbb{R}^{+}$and regularization factor $\gamma \geq 0$. With the logistic loss $L(t)=\log \left(1+e^{-t}\right)$ one gets the logistic regression, while the least-squares classifier (or ridge regressor) can be obtained with the squared loss $L(t)=(t-1)^2$. Other popular choices of $L(\cdot)$ include the exponential loss $L(t)=e^{-t}$ widely used in boosting algorithms [Schapire, 1999] and the hinge loss $L(t)=\max (0,1-t)$ in the case of support vector machines (SVMs) [Rosasco et al., 2004]. Figure $6.1$ illustrates these different losses.

Except for the least-squares solution where $L(t)=(t-1)^2$, the minimization of (ridge-regularized) a generic loss $L$ generally leads to a classifier $\boldsymbol{\beta}$ that only takes an implicit form. It is thus not clear how the resulting $\boldsymbol{\beta}$ depends on the data $\mathbf{X}$ and labels $\mathbf{y}$, making its (large-dimensional) statistical behavior more challenging to investigate. The technical challenge posed by implicit optimization problems appears not only in the analysis of logistic regression, but also in most machine learning algorithms of daily use, starting with the popular deep learning schemes. It is therefore of crucial chapters to assess the performance of nonexplicit optimization-based learning methods. In this chapter, we focus on the quite generic empirical risk minimization example of (6.1) and evaluate the large-dimensional behavior of the resulting classifier $\boldsymbol{\beta}$. Technically, a major emphasis will be cast on the “leave-one-out” approach, which aims to “decouple” the intricate statistical dependencies induced by the optimization scheme into the statistical learning algorithms. Other approaches to overcome this technical difficulty of “intrinsic dependence” will be discussed in Section $6.3$ at the end of this chapter.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Practical Course Material

$$\mathbf{\Sigma}^{\boldsymbol{T}}=\left[\cos (\mathbf{W X})^{\top} \quad \sin (\mathbf{W} \mathbf{X})^{\top}\right] \in \mathbb{R}^{n \times 2 N}$$

$\mathbb{E} \mathbf{w}\left[\cos \left(\mathbf{w}^{\top} \mathbf{x}i\right) \cos \left(\mathbf{w}^{\top} \mathbf{x}_j\right)+\sin \left(\mathbf{w}^{\top} \mathbf{x}_i\right) \sin \left(\mathbf{w}^{\top} \mathbf{x}_j\right)\right]=[\mathbf{K} \cos ]$ 所以根据强大的大数定律，一个人有 $$\frac{1}{N}\left[\Sigma^{\top} \Sigma\right] i j \stackrel{\text { a.s. }}{\longrightarrow}[\mathbf{K} \cos +\mathbf{K} \sin ] i j=[\mathbf{K G a u s s}] i j$$ 作为 $N \rightarrow \infty$ ，为了 $\mathbf{K} \cos$ 和 $\mathbf{K} \sin$ 表 $5.1$ 中列出的 “cos”和“ $\sin$ “非线 性的限制核，以及 高斯内核。这证明了使用随机傅里叶特征是合理的，但仅在 $N \gg n$ 政权。 我们转向一个大 $n, p, N$ 表征随机傅立叶特征。使用的事实是 $\mathbb{E}{\mathbf{w}}\left[\cos \left(\mathbf{w}^{\top} \mathbf{x}i\right) \sin \left(\mathbf{w}^{\top} \mathbf{x}_j\right)\right]=0$ 为了 $\mathbf{w} \sim \mathcal{N}\left(\mathbf{0}, \mathbf{I}_p\right)$ 并且两者 $\cos (\cdot)$ 和 $\sin (\cdot)$ 是 1-Lipschitz，在引理 $5.1$ 的帮助下并类似于定理 5.1，表明随机傅里叶分解 $\left(\frac{1}{n} \Sigma^{\top} \Sigma+\gamma \mathbf{I}_n\right)^{-1}$ 冸认确定性等价物 $\mathbf{Q} \leftrightarrow \overline{\mathbf{Q}}, \quad \overline{\mathbf{Q}} \equiv\left(\frac{N}{n}\left(\frac{\mathbf{K} \cos }{1+\delta \cos }+\frac{\mathbf{K} \sin }{1+\delta \sin }\right)+\gamma \mathbf{I} n\right)^{-1}$ 为了 $\left(\delta \cos , \delta{\sin }\right)$ 的唯一正解
$$\delta_{\cos }=\frac{1}{n} \operatorname{tr} \mathbf{K} \cos \overline{\mathbf{Q}}, \quad \delta \sin =\frac{1}{n} \operatorname{tr} \mathbf{K}_{\sin } \overline{\mathbf{Q}} .$$

## 计算机代写|机器学习代写machine learning代考|Large-Dimensional Convex Optimization

$$\underset{\boldsymbol{\beta} \in \mathbb{R}^p}{\arg \min } \frac{1}{n} \sum_{i=1}^n L\left(y_i \boldsymbol{\beta}^{\top} \mathbf{x}_i\right)+\frac{\gamma}{2}|\boldsymbol{\beta}|^2$$

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