## 计算机代写|机器学习代写machine learning代考|CS7641

2023年1月4日

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• (Generalized) Linear Models 广义线性模型
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## 计算机代写|机器学习代写machine learning代考|Generalized Linear Classifier

For simplicity of exposition, we consider the problem of classifying a binary “symmetric” Gaussian mixture of the form
$$\mathcal{C}1: \mathbf{x}_i \sim \mathcal{N}(-\boldsymbol{\mu}, \mathbf{C}), y_i=-1 \text { and } \mathcal{C}_2: \mathbf{x}_i \sim \mathcal{N}(+\boldsymbol{\mu}, \mathbf{C}), y_i=+1$$ each with a class prior probability of $1 / 2$, for some $\boldsymbol{\mu} \in \mathbb{R}^p$ and positive definite $\mathbf{C} \in \mathbb{R}^{p \times p}$. As in the previous chapters, we ensure that the classification problem is asymptotically nontrivial by specifying the following growth rate assumptions $$|\boldsymbol{\mu}|=O(1), \text { and } \max \left{|\mathbf{C}|,\left|\mathbf{C}^{-1}\right|\right}=O(1)$$ as $n, p \rightarrow \infty$ at the same pace. Note that the mixture model in (6.2) satisfies the logistic model in the sense that the conditional class probability is \begin{aligned} P(y \mid \mathbf{x}) & =\frac{P(y) P(\mathbf{x} \mid y)}{P(\mathbf{x})}=\frac{e^{-\frac{1}{2}(\mathbf{x}-y \boldsymbol{\mu}) \mathbf{C}^{-1}(\mathbf{x}-y \boldsymbol{\mu})}}{e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu}) \mathbf{C}^{-1}(\mathbf{x}-\boldsymbol{\mu})}+e^{-\frac{1}{2}(\mathbf{x}+\boldsymbol{\mu}) \mathbf{C}^{-1}(\mathbf{x}+\boldsymbol{\mu})}} \ & =\frac{1}{1+e^{-2 y \boldsymbol{\mu}^{\top} \mathbf{C}^{-1} \mathbf{x}}} \sigma \sigma\left(\boldsymbol{\beta}^{\top} y \mathbf{x}\right), \text { for } \quad \boldsymbol{\beta} \equiv 2 \mathbf{C}^{-1} \boldsymbol{\mu} \end{aligned}
with $\sigma(t)=\left(1+e^{-t}\right)^{-1}$ the logistic sigmoid function and the optimal Bayes solution $\beta*=2 \mathbf{C}^{-1} \mu$. By the symmetry of (6.2), it is convenient to use the shortcut notation $\tilde{\mathbf{x}}_i \equiv y_i \mathbf{x}_i$ so that $$\tilde{\mathbf{x}}i \sim \mathcal{N}(\boldsymbol{\mu}, \mathbf{C})$$ regardless of the class of $\mathbf{x}_i$. To investigate the large-dimensional asymptotics of the implicit classifier, which minimizes the empirical risk in (6.1), the main technical difficulty lies in the fact that $\boldsymbol{\beta}$, as the solution of a convex optimization problem, depends on all the random $\tilde{\mathbf{x}}_i \mathrm{~s}$ in a rather involved (and implicit) manner. Nonetheless, by canceling the loss function gradient with respect to $\beta$ in (6.1), we can still obtain the following seemingly simple equation satisfied by $\boldsymbol{\beta}$ : $$\gamma \boldsymbol{\beta}=\frac{1}{n} \sum{i=1}^n-L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_i\right) \tilde{\mathbf{x}}_i$$
where we assume the loss function $L$ is convex and at least three times continuously differentiable (making $\beta$ unique for $\gamma>0$ ). Of course, the technical difficulty remains: While $\boldsymbol{\beta}$ appears to be a linear combination of the independent $\tilde{\mathbf{x}}_i \mathrm{~s}$, the coefficients of the linear combination are themselves functions of $\boldsymbol{\beta}$, and thus functions of all $\tilde{\mathbf{x}}_j \mathrm{~s}$. Also, and possibly more fundamentally, unlike in Section $3.3$ on the robust estimators of scatter, where we already met similar fixed-point equations, the variables $\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_i$ will be seen not to converge (and thus cannot be “replaced” by a deterministic limit). This last remark crucially modifies the approach to study the large-dimensional statistics of $\boldsymbol{\beta}$. The next section introduces one of the natural angles of attack, based on a “leaveone-out” procedure. The method being somewhat intricate, we start by presenting the main intuitions and the basic developments to retrieve the system of equations, which (asymptotically) characterizes the statistical behavior of $\boldsymbol{\beta}$.

## 计算机代写|机器学习代写machine learning代考|Intuitions and Main Results

In a way, the proof of the main results on the asymptotic characterization of $\beta$ is based on a similar leave-one-column-out perturbation approach used in the Bai-Silverstein method (for instance, when applied to the proof of the Marčenko-Pastur law, Theorem 2.4). Specifically here, we will compare the original $\boldsymbol{\beta}$, solution of (6.5), to $\boldsymbol{\beta}{-i}$, solution of a modified version of (6.5) in which the sum does not include the $i$ th datum $\tilde{\mathbf{x}}_i$. For $n$ large, $\boldsymbol{\beta}$ and $\boldsymbol{\beta}{-i}$ should be (asymptotically) close and, in particular, behave similarly when projected on deterministic vectors as well as on the $\tilde{\mathbf{x}}j$ s, except when $j=i$. When comparing $\boldsymbol{\beta}$ to $\boldsymbol{\beta}{-i}$, due to their asymptotic closeness, the nonlinear functions (here $L^{\prime}$ ) in (6.5) will be “linearized” by a Taylor expansion: This ultimately gives rise to a characterization of $\boldsymbol{\beta}$ involving only the sample mean and sample covariance of the $\tilde{\mathbf{x}}_i \mathrm{~s}$, and the first derivatives of $L$. This will, possibly surprisingly at first glance, allow us to fall back on classical sample covariance matrix characterizations as studied thoroughly in the book. We develop here the main ingredients and intuitive arguments of the approach, a complete and exhaustive proof being available in Mai and Liao [2019].

From (6.5), $\boldsymbol{\beta}$ can be viewed as a linear combination of all $\tilde{\mathbf{x}}_i \mathrm{~s}$, weighted by the coefficient $-L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_i\right)$. The idea is then to understand how each $\tilde{\mathbf{x}}_i$ affects the corresponding coefficient $-L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}i\right)$. To handle the complex dependence of $\boldsymbol{\beta}$ on all $\tilde{\mathbf{x}}_j \mathrm{~s}$, we create a “leave-one-out” version of $\boldsymbol{\beta}$, denoted $\boldsymbol{\beta}{-i}$, which is (i) asymptotically close to $\beta$ by removing the contribution of a single datum and (ii) independent of $\tilde{\mathbf{x}}i$, by solving (6.1) for all daata $\tilde{\mathbf{x}}_j$ different from $\tilde{\mathbf{x}}_i$, so thăt $$\gamma \boldsymbol{\beta}{-i}=\frac{1}{n} \sum_{j \neq i}-L^{\prime}\left(\boldsymbol{\beta}{-i}^{\top} \tilde{\mathbf{x}}_j\right) \tilde{\mathbf{x}}_j .$$ As a consequence, the difference $\gamma\left(\boldsymbol{\beta}-\boldsymbol{\beta}{-i}\right)$ satisfies
$$\gamma\left(\boldsymbol{\beta}-\boldsymbol{\beta}{-i}\right)=\frac{1}{n} \sum{j \neq i}\left(L^{\prime}\left(\boldsymbol{\beta}_{-i}^{\top} \tilde{\mathbf{x}}_j\right)-L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_j\right)\right) \tilde{\mathbf{x}}_j-\frac{1}{n} L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_i\right) \tilde{\mathbf{x}}_i$$

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Generalized Linear Classifier

$\mathcal{C} 1: \mathbf{x}_i \sim \mathcal{N}(-\boldsymbol{\mu}, \mathbf{C}), y_i=-1$ and $\mathcal{C}_2: \mathbf{x}_i \sim \mathcal{N}(+\boldsymbol{\mu}, \mathbf{C}), y_i$

$\mid$ 粗体符号 ${\backslash m u} \mid=O(1)$, \text ${$ and $} \backslash \max \backslash l$ ft ${|\backslash \operatorname{mathbf}{C}|, \backslash l$ eft $\mid \backslash \operatorname{math}$

$$P(y \mid \mathbf{x})=\frac{P(y) P(\mathbf{x} \mid y)}{P(\mathbf{x})}=\frac{e^{-\frac{1}{2}(\mathbf{x}-y \boldsymbol{\mu}) \mathbf{C}^{-1}(\mathbf{x}-y \boldsymbol{\mu})}}{e^{-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu}) \mathbf{C}^{-1}(\mathbf{x}-\boldsymbol{\mu})}+e^{-\frac{1}{2}(\mathbf{x}+\boldsymbol{\mu}) \mathbf{C}}}$$

$$\tilde{\mathbf{x}} i \sim \mathcal{N}(\boldsymbol{\mu}, \mathbf{C})$$

$$\gamma \boldsymbol{\beta}=\frac{1}{n} \sum i=1^n-L^{\prime}\left(\boldsymbol{\beta}^{\top} \tilde{\mathbf{x}}_i\right) \tilde{\mathbf{x}}_i$$

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## MATLAB代写

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