## 计算机代写|机器学习代写machine learning代考|COMP3670

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|Results on ESN Asymptotics

Gathering the simplifications above, we consider here the model
$$\mathbf{s}{t+1}=\mathbf{W} \mathbf{s}_t+\mathbf{w}{\mathrm{in}} x_{t+1}+\eta \varepsilon_{t+1}$$
with the associated training and test errors
$$E_{\text {train }}=\frac{1}{T}\left|\mathbf{y}-\mathbf{S}^{\top} \boldsymbol{\beta}\right|^2, \quad E_{\text {test }}=\frac{1}{\hat{T}}\left|\hat{\mathbf{y}}-\hat{\mathbf{S}}^{\top} \boldsymbol{\beta}\right|^2$$
and $\beta \in \mathbb{R}^N$ such that
$$\beta= \begin{cases}\mathbf{S}\left(\mathbf{S}^{\top} \mathbf{S}\right)^{-1} \mathbf{y} & , N>T, \ \left(\mathbf{S S}^{\top}\right)^{-1} \mathbf{S} \mathbf{y} & , N<T .\end{cases}$$
For further simplicity of exposition, we particularly focus here on the training performance, which already conveys quite insightful results. The complete analyses of both train and test performances are available in Couillet et al. [2016b].

To investigate the large $N, T$ asymptotics of the training error $E_{\text {train }}$ defined in (5.29), first remark that, letting
$$\mathbf{Q}(\gamma) \equiv\left(\frac{1}{T} \mathbf{S S}^{\boldsymbol{\top}}+\gamma \mathbf{I}N\right)^{-1} \text { and } \tilde{\mathbf{Q}}(\gamma) \equiv\left(\frac{1}{T} \mathbf{S}^{\top} \mathbf{S}+\gamma \mathbf{I}_T\right)^{-1}$$ we have, irrespective of the sign of $N-T$, $$E{\text {train }}=\lim {\gamma \downarrow 0} \frac{\gamma}{T} \mathbf{y}^{\top} \tilde{\mathbf{Q}}(\gamma) \mathbf{y} .$$ The estimation of $E{\text {train }}$ thus reduces to the characterization of a quadratic form over the resolvent $\tilde{\mathbf{Q}}(\gamma)$ of $\frac{1}{T} \mathbf{S}^{\top} \mathbf{S}$, which is reminiscent of the (similar yet different) expression in (5.5) for feedforward networks.

The specific difficulty induced by $\mathbf{S}$ lies in the intricate dependence between its columns, as successive observations of a multivariate time series. In particular, in order to simplify the analysis and to avoid edge problems at time $t=0$, we assume (as is conventionally done in practice) that a sufficiently long “washout period” is performed preliminary to observing $x_0$, that is, the considered time series $x_0, \ldots, x_{T-1}$ is a finite time extraction of an infinite series $\ldots, x_{-1}, x_0, x_1, \ldots$; this discards the transition phase of the random network states $\mathbf{s}0, \ldots, \mathbf{s}{T-1}$.

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

In this chapter, by leveraging tools from the concentration of measure theory (see Section 2.7), we went beyond the simple Taylor expansion-based approach devised to understand kernel methods in Chapter 4 and were able to evaluate the largedimensional asymptotics of (single-hidden-layer) nonlinear neural network models

(Section 5.1) for real-world data. As we shall see later in Chapter 8, this “universal large-dimensional concentration” argument has an even more significant impact in practical applications and will be exploited to extend the current analyses and insights (on the choice of the kernel functions and the activation functions) to a much broader and more realistic setting.

The eigenspectra, or more generally the large-dimensional asymptotics of (random) neural network models have known a recent resurgence of interest. These investigations include the (limiting) spectral measure of the nonlinear Gram matrices [Pennington and Worah, 2017, Benigni and Péché, 2019] (similar to Section 5.1), as well as that of the input-output Jacobian matrices [Pennington et al., 2017, Pastur, 2020, Pastur and Slavin, 2020] (closely connected to the behavior of back propagation gradients) of a multilayer neural network with random Gaussian or orthogonal weights. These analyses are not limited to classical feedforward and fully connected networks but have been performed on networks with convolutional [Novak et al., 2019, Xiao et al., 2018], recurrent [Chen et al., 2018, Gilboa et al., 2019], and skipconnection structures [Ling and Qiu, 2019] (as in the case of the popular residual network architecture [He et al., 2016]), to name a few. Since random (Gaussian) initializations are widely used in training such deep networks [Glorot and Bengio, 2010, He et al., 2015], these works shed a new light on the “landscape” of deep neural networks at the initialization point as well as on the impact of nonlinear activations.

The investigations on randomly weighted neural networks are of even greater interest to the neural tangent kernel recently introduced by Jacot et al. [2018] as an approximation of extremely “wide” layers: While the weight matrices after training are no longer random, the underlying neural tangent kernel is determined only by the random initialization and remains unchanged during the whole training procedure in the “infinite-neurons” limit. As such, the eigenspectral assessments of the neural tangent kernel for randomly weighted deep networks go beyond the initial stage of training and lead to much richer results on, for example, the learning dynamics of networks [Fan and Wang, 2020, Adlam and Pennington, 2020] – at least in this neural tangent limit where the network widths are much larger than both the number of training data $n$ and their dimension $p$. Nonetheless, most of these works are concerned with random noise-like input data (e.g., i.i.d. Gaussian data or almost orthonormal data [Fan and Wang, 2020, Adlam et al., 2019]) with no information structure, and thus fail to provide sharp qualitative predictions on real-world datasets.

# 机器学习代考

## 计算机代写|机器学习代写machine learning代考|Results on ESN Asymptotics

$$\mathbf{s} t+1=\mathbf{W} \mathbf{s}t+\mathbf{w i n} x{t+1}+\eta \varepsilon_{t+1}$$

$$E_{\text {train }}=\frac{1}{T}\left|\mathbf{y}-\mathbf{S}^{\top} \boldsymbol{\beta}\right|^2, \quad E_{\text {test }}=\frac{1}{\hat{T}}\left|\hat{\mathbf{y}}-\hat{\mathbf{S}}^{\top} \boldsymbol{\beta}\right|^2$$

$$\beta=\left{\mathbf{S}\left(\mathbf{S}^{\top} \mathbf{S}\right)^{-1} \mathbf{y} \quad, N>T,\left(\mathbf{S S}^{\top}\right)^{-1} \mathbf{S y} \quad, N<T .\right.$$

$\mathbf{Q}(\gamma) \equiv\left(\frac{1}{T} \mathbf{S S}^{\top}+\gamma \mathbf{I} N\right)^{-1}$ and $\tilde{\mathbf{Q}}(\gamma) \equiv\left(\frac{1}{T} \mathbf{S}^{\top} \mathbf{S}+\gamma \mathbf{I}T\right)^{-1}$ 我们有，不管标志 $N-T$ ， $$E \text { train }=\lim \gamma \downarrow 0 \frac{\gamma}{T} \mathbf{y}^{\top} \tilde{\mathbf{Q}}(\gamma) \mathbf{y}$$ 的估计 $E$ train 因此减少到二次形式的特征 $\tilde{\mathbf{Q}}(\gamma)$ 的 $\frac{1}{T} \mathbf{S}^{\top} \mathbf{S}$ ，这让人 想起 (5.5) 中用于前馈网络的 (相似但不同的) 表达式。 具体难度由 $\mathbf{S}$ 在于其列之间错综复杂的依赖性，作为多元时间序列的 连续观察。特别是，为了简化分析和避免时间边缘问题 $t=0$ ，我们假 设 (正如实践中通常所做的那样) 在观察之前执行了足够长的 “冲洗期” $x_0$ ，即考虑的时间序列 $x_0, \ldots, x{T-1}$ 是无限序列的有限时间提取 $\ldots, x_{-1}, x_0, x_1, \ldots ;$ 这去弃了随机网络状态的过渡阶段 $\mathbf{s} 0, \ldots, \mathbf{s} T-1$

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

（第 5.1 节）用于真实世界的数据。正如我们将在第 8 章后面看到的那样，这种“普遍的大维集中”论点在实际应用中具有更重要的影响，并将被用来扩展当前的分析和见解（关于核函数和激活函数的选择） ) 到更广泛和更现实的环境。

Jacot 等人最近引入的神经正切核对随机加权神经网络的研究更加有趣。[2018] 作为极“宽”层的近似：虽然训练后的权重矩阵不再随机，但底层神经正切核仅由随机初始化确定，并且在“无限神经元”的整个训练过程中保持不变“ 限制。因此，随机加权深度网络的神经正切核的特征谱评估超出了训练的初始阶段，并导致了更丰富的结果，例如网络的学习动态 [Fan and Wang, 2020, Adlam and Pennington, 2020] – 至少在这个神经切线限制中，网络宽度远大于训练数据的数量n和他们的维度p. 尽管如此，这些工作大多涉及没有信息结构的随机类噪声输入数据（例如，独立同分布高斯数据或几乎正交数据 [Fan and Wang, 2020, Adlam et al., 2019]），因此无法提供对真实世界数据集的清晰定性预测。

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