在本文中，我们将为您详细介绍neuralink怎么读的相关知识，并且为您解答关于neuralink怎么读音的疑问，此外，我们还会提供一些关于1506.01186-CyclicalLearningRat

在本文中，我们将为您详细介绍neuralink怎么读的相关知识，并且为您解答关于neuralink怎么读音的疑问，此外，我们还会提供一些关于1506.01186-Cyclical Learning Rates for Training Neural Networks、2.8 亿美元融资进一步支持马斯克的脑机接口公司 Neuralink 的人体临床试验、Artificial Neurons and Single-Layer Neural Networks、Bilinear CNN与 Randomly Wired Neural Network的有用信息。

• neuralink怎么读（neuralink怎么读音）
• 1506.01186-Cyclical Learning Rates for Training Neural Networks
• 2.8 亿美元融资进一步支持马斯克的脑机接口公司 Neuralink 的人体临床试验
• Artificial Neurons and Single-Layer Neural Networks
• Bilinear CNN与 Randomly Wired Neural Network

neuralink怎么读（neuralink怎么读音）

有些用户对于neuralink这家公司的研究对象十分感兴趣，但却不清楚该怎么读这家公司的名字，这里就给大家介绍一下neuralink的读法，希望对您有所帮助。

neuralink怎么读

答：neuralink的谐音读法是“扭若林克“

其音标为：[ˈnʊrəllɪŋk]

意思是神经链接（美国神经科技和脑机接口公司）

neuralink扩展阅读：

1、Neuralink是一家美国科技公司，总部位于美国旧金山，

由埃隆·马斯克（Elon Musk）和其他八人于2016年7月联合创办。

2、该公司主要研发“神经织网（neural lace）”技术，致力于将人类大脑与计算机连接起来，

即支持人与机器直接交流，不再需要物理界面。

3、2021年7月29日，Neuralink完成了2.05亿美元C轮融资，领投的为迪拜风险投资公司Vy Capital，

谷歌风投、Founders

Fund、DFJ Growth等基金也出现在投资名单中。

4、Neuralink埃隆·马斯克成立的一家高科技公司，致力于通过制造微米等级的设备，将人脑与机器接口连接起来。

2016年7月，Neuralink在加州注册，是一家“医学研究”公司，马斯克提供公司运营资金。

5、Neuralink开发马斯克称之为“神经蕾丝”的技术，在人脑中植入细小的电极，

可能有朝一日能用来上传、下载人的思想。

1506.01186-Cyclical Learning Rates for Training Neural Networks

1506.01186-Cyclical Learning Rates for Training Neural Networks

论文中提出了一种循环调整学习率来训练模型的方式。

如下图：

通过循环的线性调整学习率，论文作者观察到的一种比较典型的曲线如下图：

图中，使用循环调整方式的模型，虽然训练中准确度有很大的波动，但是这种波动并不影像模型很快的收敛，并且以更快的速度收敛到了固定学习率或者学习率衰减方案中能达到的最高准确率。

这种方式需要设置的超参有三个， min bound，max bound ，step size。

step size一般选择epoch的step数的2-8倍。例如一个epoch里面有500步，则2000作为step size是不错的选择。

确定step size之后，使用一个简短的训练来确定 min-max bound，在一个训练里，以循环学习率的方式，线性的提升学习率，画出学习率与准确率关系的曲线。模型准确率最快增长的学习率适合用来做min bound，模型准确率增长减缓甚至直接出现负增长的地方，适合作max bound。如下图中所示：

模型一开始就出现了很快的准确率增长，所以选择0.001作为min bound，而0.005的位置出现了负增长，这里作为max bond。

另外论文提出了另外两种变种：

• triangular2; the same as the triangular policy except the learning rate difference is cut in half at the end of each cycle. This means the learning rate difference drops after each cycle.

  每次cycle之间才做一次lr decay

• exp range; the learning rate varies between the minimum and maximum boundaries and each boundary value declines by an exponential factor of gamme^iteration

  随着每个iteration都做一次lr decay

论文还尝试把循环学习率与各种优化算法都结合了一下，但是看结果，除了nestrov动量法，其他几种优化算法的效果都不好。

这是可以理解的，因为除了单纯的动量法，其他几种方法，都会针对学习率进行动态的调整。而循环学习率引入的学习率循环变化必然会破坏学习率动态调整的某些特性。所以结果就变得无法估计。

另外，针对学习率固定decay的效果与循环学习率的对比，论文作者单独设计了一个实验，循环学习率只循环一次，在学习率达到最小的时候，就固定住，或者做decay，不再上升，但是最后的结果，还是不如循环学习率的效果好。

这种循环学习率的方式，似乎是因为循环引入了更多的随机性？让梯度有了更多的机会往不同的方向探索？

论文中没有提出很好的理论解释。

2.8 亿美元融资进一步支持马斯克的脑机接口公司 Neuralink 的人体临床试验

马斯克创办的脑机接口初创公司 neuralink 最近完成了一轮新的融资，总额达到了2.8亿美元（约合人民币20.13亿元）。这次融资由彼得·蒂尔的创始人基金（founders fund）领投，几个月前，该公司还获得了美国监管机构的批准，开始进行首次人体临床试验

Neuralink 是一家致力于开发能够与人类大脑连接的芯片和设备的公司，其目标是让人类能够通过思想控制电子设备，以及治疗神经系统相关的疾病。该公司没有透露本轮融资的估值。据路透社今年 6 月报道，该公司在私下进行的股票交易中估值约为 50 亿美元。

在社交媒体平台X（原名Twitter）上，Neuralink公司发表了一篇文章，表达了对其未来发展的极大兴奋。去年，该公司的员工曾表示，由于马斯克的压力，要求尽快获得美国食品药品监督管理局（FDA）的批准，导致了动物实验的急于求成，不幸造成了一些不必要的动物死亡。这些动物实验所产生的数据将用于支持该公司申请进行人体临床试验

本站所有文章包含广告声明，其中包括对外跳转链接（如超链接、二维码、口令等形式），旨在传递更多信息，节省甄选时间，结果仅供参考

以上就是2.8 亿美元融资进一步支持马斯克的脑机接口公司 Neuralink 的人体临床试验的详细内容，更多请关注php中文网其它相关文章！

Artificial Neurons and Single-Layer Neural Networks

In [1]:

import sys
sys.path = [''/Users/sebastian/github/mlxtend/''] + sys.path

In [2]:

%load_ext watermark
%watermark -a ''Sebastian Raschka'' -v

Sebastian Raschka 

CPython 3.4.3
IPython 3.0.0

Artificial Neurons and Single-Layer Neural Networks

- How Machine Learning Algorithms Work Part 1

This article offers a brief glimpse of the history and basic concepts of machine learning. We will take a look at the first algorithmically described neural network and the gradient descent algorithm in context of adaptive linear neurons, which will not only introduce the principles of machine learning but also serve as the basis for modern multilayer neural networks in future articles.

Sections

• Introduction
• Artificial Neurons and the McCulloch-Pitts Model
• Frank Rosenblatt''s Perceptron
  • The Unit Step Function
  • The Perceptron Learning Rule
  • Implementing the Perceptron Rule in Python
  • Problems with Perceptrons
• Adaptive Linear Neurons and the Delta Rule
  • Gradient Descent
  • The Gradient Descent Rule in Action
  • Online Learning via Stochastic Gradient Descent
• What''s Next?
• References

Introduction

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Machine learning is one of the hottest and most exciting fields in the modern age of technology. Thanks to machine learning, we enjoy robust email spam filters, convenient text and voice recognition, reliable web search engines, challenging chess players, and, hopefully soon, safe and efficient self-driving cars.

Without any doubt, machine learning has become a big and popular field, and sometimes it may be challenging to see the (random) forest for the (decision) trees. Thus, I thought that it might be worthwhile to explore different machine learning algorithms in more detail by not only discussing the theory but also by implementing them step by step.

To briefly summarize what machine learning is all about: "[Machine learning is the] field of study that gives computers the ability to learn without being explicitly programmed" (Arthur Samuel, 1959). Machine learning is about the development and use of algorithms that can recognize patterns in data in order to make decisions based on statistics, probability theory, combinatorics, and optimization.

The first article in this series will introduce perceptrons and the adaline (ADAptive LINear NEuron), which fall into the category of single-layer neural networks. The perceptron is not only the first algorithmically described learning algorithm [1], but it is also very intuitive, easy to implement, and a good entry point to the (re-discovered) modern state-of-the-art machine learning algorithms: Artificial neural networks (or "deep learning" if you like). As we will see later, the adaline is a consequent improvement of the perceptron algorithm and offers a good opportunity to learn about a popular optimization algorithm in machine learning: gradient descent.

Artificial Neurons and the McCulloch-Pitts Model

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The initial idea of the perceptron dates back to the work of Warren McCulloch and Walter Pitts in 1943 [2], who drew an analogy between biological neurons and simple logic gates with binary outputs. In more intuitive terms, neurons can be understood as the subunits of a neural network in a biological brain. Here, the signals of variable magnitudes arrive at the dendrites. Those input signals are then accumulated in the cell body of the neuron, and if the accumulated signal exceeds a certain threshold, a output signal is generated that which will be passed on by the axon.

Frank Rosenblatt''s Perceptron

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To continue with the story, a few years after McCulloch and Walter Pitt, Frank Rosenblatt published the first concept of the Perceptron learning rule [1]. The main idea was to define an algorithm in order to learn the values of the weights ww that are then multiplied with the input features in order to make a decision whether a neuron fires or not. In context of pattern classification, such an algorithm could be useful to determine if a sample belongs to one class or the other.

To put the perceptron algorithm into the broader context of machine learning: The perceptron belongs to the category of supervised learning algorithms, single-layer binary linear classifiers to be more specific. In brief, the task is to predict to which of two possible categories a certain data point belongs based on a set of input variables. In this article, I don''t want to discuss the concept of predictive modeling and classification in too much detail, but if you prefer more background information, please see my previous article "Introduction to supervised learning".

The Unit Step Function

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Before we dive deeper into the algorithm(s) for learning the weights of the artificial neuron, let us take a brief look at the basic notation. In the following sections, we will label the positive and negative class in our binary classification setting as "1" and "-1", respectively. Next, we define an activation function g(z)g(z) that takes a linear combination of the input values xx and weights ww as input (z=w1x1+⋯+wmxmz=w1x1+⋯+wmxm), and if g(z)g(z) is greater than a defined threshold θθ we predict 1 and -1 otherwise; in this case, this activation function gg is an alternative form of a simple "unit step function," which is sometimes also called "Heaviside step function."

(Please note that the unit step is classically defined as being equal to 0 if z<0z<0 and 1 for z≥0z≥0; nonetheless, we will refer to the following piece-wise linear function with -1 if z<θz<θ and 1 for z≥θz≥θ as unit step function for simplicity).

g(z)={1−1if z≥θotherwise.g(z)={1if z≥θ−1otherwise.

where

z=w1x1+⋯+wmxm=∑j=1mxjwj=wTxz=w1x1+⋯+wmxm=∑j=1mxjwj=wTx

ww is the feature vector, and xx is an mm-dimensional sample from the training dataset:

w=⎡⎣⎢⎢w1⋮wm⎤⎦⎥⎥x=⎡⎣⎢⎢x1⋮xm⎤⎦⎥⎥w=[w1⋮wm]x=[x1⋮xm]

In order to simplify the notation, we bring θθ to the left side of the equation and define w0=−θ and x0=1w0=−θ and x0=1

so that

g(z)={1−1if z≥0otherwise.(1)(1)g(z)={1if z≥0−1otherwise.

and

z=w0x0+w1x1+⋯+wmxm=∑j=0mxjwj=wTx.z=w0x0+w1x1+⋯+wmxm=∑j=0mxjwj=wTx.

The Perceptron Learning Rule

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It might sound like extreme case of a reductionist approach, but the idea behind this "thresholded" perceptron was to mimic how a single neuron in the brain works: It either "fires" or not. To summarize the main points from the previous section: A perceptron receives multiple input signals, and if the sum of the input signals exceed a certain threshold it either returns a signal or remains "silent" otherwise. What made this a "machine learning" algorithm was Frank Rosenblatt''s idea of the perceptron learning rule: The perceptron algorithm is about learning the weights for the input signals in order to draw linear decision boundary that allows us to discriminate between the two linearly separable classes +1 and -1.

Rosenblatt''s initial perceptron rule is fairly simple and can be summarized by the following steps:

1. Initialize the weights to 0 or small random numbers.
2. For each training sample x(i)x(i):
   1. Calculate the output value.
   2. Update the weights.

The output value is the class label predicted by the unit step function that we defined earlier (output =g(z)=g(z)) and the weight update can be written more formally as wj:=wj+Δwjwj:=wj+Δwj.

The value for updating the weights at each increment is calculated by the learning rule

Δwj=η(target(i)−output(i))x(i)jΔwj=η(target(i)−output(i))xj(i)

where ηη is the learning rate (a constant between 0.0 and 1.0), "target" is the true class label, and the "output" is the predicted class label.

It is important to note that all weights in the weight vector are being updated simultaneously. Concretely, for a 2-dimensional dataset, we would write the update as:

Δw0=η(target(i)−output(i))Δw0=η(target(i)−output(i))
Δw1=η(target(i)−output(i))x(i)1Δw1=η(target(i)−output(i))x1(i)
Δw2=η(target(i)−output(i))x(i)2Δw2=η(target(i)−output(i))x2(i)

Before we implement the perceptron rule in Python, let us make a simple thought experiment to illustrate how beautifully simple this learning rule really is. In the two scenarios where the perceptron predicts the class label correctly, the weights remain unchanged:

• Δwj=η(−1(i)−−1(i))x(i)j=0Δwj=η(−1(i)−−1(i))xj(i)=0
• Δwj=η(1(i)−1(i))x(i)j=0Δwj=η(1(i)−1(i))xj(i)=0

However, in case of a wrong prediction, the weights are being "pushed" towards the direction of the positive or negative target class, respectively:

• Δwj=η(1(i)−−1(i))x(i)j=η(2)x(i)jΔwj=η(1(i)−−1(i))xj(i)=η(2)xj(i)
• Δwj=η(−1(i)−1(i))x(i)j=η(−2)x(i)jΔwj=η(−1(i)−1(i))xj(i)=η(−2)xj(i)

It is important to note that the convergence of the perceptron is only guaranteed if the two classes are linearly separable. If the two classes can''t be separated by a linear decision boundary, we can set a maximum number of passes over the training dataset ("epochs") and/or a threshold for the number of tolerated misclassifications.

Implementing the Perceptron Rule in Python

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In this section, we will implement the simple perceptron learning rule in Python to classify flowers in the Iris dataset. Please note that I omitted some "safety checks" for clarity, for a more "robust" version please see the following code on GitHub.

In [3]:

import numpy as np

class Perceptron(object):
    
    def __init__(self, eta=0.01, epochs=50):
        self.eta = eta
        self.epochs = epochs

    def train(self, X, y):

        self.w_ = np.zeros(1 + X.shape[1])
        self.errors_ = []

        for _ in range(self.epochs):
            errors = 0
            for xi, target in zip(X, y):
                update = self.eta * (target - self.predict(xi))
                self.w_[1:] +=  update * xi
                self.w_[0] +=  update
                errors += int(update != 0.0)
            self.errors_.append(errors)
        return self

    def net_input(self, X):
        return np.dot(X, self.w_[1:]) + self.w_[0]

    def predict(self, X):
        return np.where(self.net_input(X) >= 0.0, 1, -1)

For the following example, we will load the Iris data set from the UCI Machine Learning Repository and only focus on the two flower species Setosa and Versicolor. Furthermore, we will only use the two features sepal length and petal length for visualization purposes.

In [4]:

import pandas as pd
df = pd.read_csv(''https://archive.ics.uci.edu/ml/machine-learning-databases/iris/iris.data'', header=None)

# setosa and versicolor
y = df.iloc[0:100, 4].values
y = np.where(y == ''Iris-setosa'', -1, 1)

# sepal length and petal length
X = df.iloc[0:100, [0,2]].values

In [5]:

%matplotlib inline
import matplotlib.pyplot as plt
from mlxtend.plotting import plot_decision_regions

ppn = Perceptron(epochs=10, eta=0.1)

ppn.train(X, y)
print(''Weights: %s'' % ppn.w_)
plot_decision_regions(X, y, clf=ppn)
plt.title(''Perceptron'')
plt.xlabel(''sepal length [cm]'')
plt.ylabel(''petal length [cm]'')
plt.show()

plt.plot(range(1, len(ppn.errors_)+1), ppn.errors_, marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''Misclassifications'')
plt.show()

Weights: [-0.4  -0.68  1.82]

As we can see, the perceptron converges after the 6th iteration and separates the two flower classes perfectly.

Problems with Perceptrons

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Although the perceptron classified the two Iris flower classes perfectly, convergence is one of the biggest problems of the perceptron. Frank Rosenblatt proofed mathematically that the perceptron learning rule converges if the two classes can be separated by linear hyperplane, but problems arise if the classes cannot be separated perfectly by a linear classifier. To demonstrate this issue, we will use two different classes and features from the Iris dataset.

In [6]:

# versicolor and virginica
y2 = df.iloc[50:150, 4].values
y2 = np.where(y2 == ''Iris-virginica'', -1, 1)

# sepal width and petal width
X2 = df.iloc[50:150, [1,3]].values

ppn = Perceptron(epochs=25, eta=0.01)
ppn.train(X2, y2)

plot_decision_regions(X2, y2, clf=ppn)
plt.show()

plt.plot(range(1, len(ppn.errors_)+1), ppn.errors_, marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''Misclassifications'')
plt.show()

In [7]:

print(''Total number of misclassifications: %d of 100'' % (y2 != ppn.predict(X2)).sum())

Total number of misclassifications: 43 of 100

Even at a lower training rate, the perceptron failed to find a good decision boundary since one or more samples will always be misclassified in every epoch so that the learning rule never stops updating the weights.

It may seem paradoxical in this context that another shortcoming of the perceptron algorithm is that it stops updating the weights as soon as all samples are classified correctly. Our intuition tells us that a decision boundary with a large margin between the classes (as indicated by the dashed line in the figure below) likely has a better generalization error than the decision boundary of the perceptron. But large-margin classifiers such as Support Vector Machines are a topic for another time.

Adaptive Linear Neurons and the Delta Rule

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The perceptron surely was very popular at the time of its discovery, however, it only took a few years until Bernard Widrow and his doctoral student Tedd Hoff proposed the idea of the Adaptive Linear Neuron (adaline) [3].

In contrast to the perceptron rule, the delta rule of the adaline (also known as Widrow-Hoff" rule or Adaline rule) updates the weights based on a linear activation function rather than a unit step function; here, this linear activation function g(z)g(z) is just the identity function of the net input g(wTx)=wTxg(wTx)=wTx. In the next section, we will see why this linear activation is an improvement over the perceptron update and where the name "delta rule" comes from.

Gradient Descent

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Being a continuous function, one of the biggest advantages of the linear activation function over the unit step function is that it is differentiable. This property allows us to define a cost function J(w)J(w) that we can minimize in order to update our weights. In the case of the linear activation function, we can define the cost function J(w)J(w) as the sum of squared errors (SSE), which is similar to the cost function that is minimized in ordinary least squares (OLS) linear regression.

J(w)=12∑i(target(i)−output(i))2output(i)∈RJ(w)=12∑i(target(i)−output(i))2output(i)∈R

(The fraction 1212 is just used for convenience to derive the gradient as we will see in the next paragraphs.)

In order to minimize the SSE cost function, we will use gradient descent, a simple yet useful optimization algorithm that is often used in machine learning to find the local minimum of linear systems.

Before we get to the fun part (calculus), let us consider a convex cost function for one single weight. As illustrated in the figure below, we can describe the principle behind gradient descent as "climbing down a hill" until a local or global minimum is reached. At each step, we take a step into the opposite direction of the gradient, and the step size is determined by the value of the learning rate as well as the slope of the gradient.

Now, as promised, onto the fun part -- deriving the Adaline learning rule. As mentioned above, each update is updated by taking a step into the opposite direction of the gradient Δw=−η∇J(w)Δw=−η∇J(w), thus, we have to compute the partial derivative of the cost function for each weight in the weight vector: Δwj=−η∂J∂wjΔwj=−η∂J∂wj.

The partial derivative of the SSE cost function for a particular weight can be calculated as follows:

∂J∂wj=∂∂wj12∑i(t(i)−o(i))2=12∑i∂∂wj(t(i)−o(i))2=12∑i2(t(i)−o(i))∂∂wj(t(i)−o(i))=∑i(t(i)−o(i))∂∂wj(t(i)−∑jwjx(i)j)=∑i(t(i)−o(i))(−x(i)j)(2)(2)∂J∂wj=∂∂wj12∑i(t(i)−o(i))2=12∑i∂∂wj(t(i)−o(i))2=12∑i2(t(i)−o(i))∂∂wj(t(i)−o(i))=∑i(t(i)−o(i))∂∂wj(t(i)−∑jwjxj(i))=∑i(t(i)−o(i))(−xj(i))

(t = target, o = output)

And if we plug the results back into the learning rule, we get

Δwj=−η∂J∂wj=−η∑i(t(i)−o(i))(−x(i)j)=η∑i(t(i)−o(i))x(i)jΔwj=−η∂J∂wj=−η∑i(t(i)−o(i))(−xj(i))=η∑i(t(i)−o(i))xj(i),

Eventually, we can apply a simultaneous weight update similar to the perceptron rule:

w:=w+Δww:=w+Δw.

Although, the learning rule above looks identical to the perceptron rule, we shall note the two main differences:

1. Here, the output "o" is a real number and not a class label as in the perceptron learning rule.
2. The weight update is calculated based on all samples in the training set (instead of updating the weights incrementally after each sample), which is why this approach is also called "batch" gradient descent.

The Gradient Descent Rule in Action

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Now, it''s time to implement the gradient descent rule in Python.

In [8]:

import numpy as np

class AdalineGD(object):
    
    def __init__(self, eta=0.01, epochs=50): 
        self.eta = eta
        self.epochs = epochs

    def train(self, X, y):

        self.w_ = np.zeros(1 + X.shape[1])
        self.cost_ = []

        for i in range(self.epochs):
            output = self.net_input(X)
            errors = (y - output)
            self.w_[1:] += self.eta * X.T.dot(errors)
            self.w_[0] += self.eta * errors.sum()
            cost = (errors**2).sum() / 2.0
            self.cost_.append(cost)
        return self

    def net_input(self, X):
        return np.dot(X, self.w_[1:]) + self.w_[0]

    def activation(self, X):
        return self.net_input(X)

    def predict(self, X):
        return np.where(self.activation(X) >= 0.0, 1, -1)

In practice, it often requires some experimentation to find a good learning rate for optimal convergence, thus, we will start by plotting the cost for two different learning rates.

In [9]:

ada = AdalineGD(epochs=10, eta=0.01).train(X, y)
plt.plot(range(1, len(ada.cost_)+1), np.log10(ada.cost_), marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''log(Sum-squared-error)'')
plt.title(''Adaline - Learning rate 0.01'')
plt.show()

ada = AdalineGD(epochs=10, eta=0.0001).train(X, y)
plt.plot(range(1, len(ada.cost_)+1), ada.cost_, marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''Sum-squared-error'')
plt.title(''Adaline - Learning rate 0.0001'')
plt.show()

The two plots above nicely emphasize the importance of plotting learning curves by illustrating two most common problems with gradient descent:

1. If the learning rate is too large, gradient descent will overshoot the minima and diverge.
2. If the learning rate is too small, the algorithm will require too many epochs to converge and can become trapped in local minima more easily.

Gradient descent is also a good example why feature scaling is important for many machine learning algorithms. It is not only easier to find an appropriate learning rate if the features are on the same scale, but it also often leads to faster convergence and can prevent the weights from becoming too small (numerical stability).

A common way of feature scaling is standardization

xj,std=xj−μjσjxj,std=xj−μjσj

where μjμj is the sample mean of the feature xjxj and σjσj the standard deviation, respectively. After standardization, the features will have unit variance and are centered around mean zero.

In [10]:

# standardize features
X_std = np.copy(X)
X_std[:,0] = (X[:,0] - X[:,0].mean()) / X[:,0].std()
X_std[:,1] = (X[:,1] - X[:,1].mean()) / X[:,1].std()

In [11]:

%matplotlib inline
import matplotlib.pyplot as plt
from mlxtend.evaluate import plot_decision_regions

ada = AdalineGD(epochs=15, eta=0.01)

ada.train(X_std, y)
plot_decision_regions(X_std, y, clf=ada)
plt.title(''Adaline - Gradient Descent'')
plt.xlabel(''sepal length [standardized]'')
plt.ylabel(''petal length [standardized]'')
plt.show()

plt.plot(range(1, len( ada.cost_)+1), ada.cost_, marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''Sum-squared-error'')
plt.show()

Online Learning via Stochastic Gradient Descent

[back to top]

The previous section was all about "batch" gradient descent learning. The "batch" updates refers to the fact that the cost function is minimized based on the complete training data set. If we think back to the perceptron rule, we remember that it performed the weight update incrementally after each individual training sample. This approach is also called "online" learning, and in fact, this is also how Adaline was first described by Bernard Widrow et al. [3]

The process of incrementally updating the weights is also called "stochastic" gradient descent since it approximates the minimization of the cost function. Although the stochastic gradient descent approach might sound inferior to gradient descent due its "stochastic" nature and the "approximated" direction (gradient), it can have certain advantages in practice. Often, stochastic gradient descent converges much faster than gradient descent since the updates are applied immediately after each training sample; stochastic gradient descent is computationally more efficient, especially for very large datasets. Another advantage of online learning is that the classifier can be immediately updated as new training data arrives, e.g., in web applications, and old training data can be discarded if storage is an issue. In large-scale machine learning systems, it is also common practice to use so-called "mini-batches", a compromise with smoother convergence than stochastic gradient descent.

In the interests of completeness let us also implement the stochastic gradient descent Adaline and confirm that it converges on the linearly separable iris dataset.

In [14]:

import numpy as np

class AdalineSGD(object):
    
    def __init__(self, eta=0.01, epochs=50):
        self.eta = eta
        self.epochs = epochs

    def train(self, X, y, reinitialize_weights=True):

        if reinitialize_weights:
            self.w_ = np.zeros(1 + X.shape[1])
        self.cost_ = []

        for i in range(self.epochs):
            for xi, target in zip(X, y):
                output = self.net_input(xi)
                error = (target - output)
                self.w_[1:] += self.eta * xi.dot(error)
                self.w_[0] += self.eta * error
            
            cost = ((y - self.activation(X))**2).sum() / 2.0
            self.cost_.append(cost)
        return self

    def net_input(self, X):
        return np.dot(X, self.w_[1:]) + self.w_[0]

    def activation(self, X):
        return self.net_input(X)

    def predict(self, X):
        return np.where(self.activation(X) >= 0.0, 1, -1)

One more advice before we let the adaline learn via stochastic gradient descent is to shuffle the training dataset to iterate over the training samples in random order.

We shall note that the "standard" stochastic gradient descent algorithm uses sampling "with replacement," which means that at each iteration, a training sample is chosen randomly from the entire training set. In contrast, sampling "without replacement," which means that each training sample is evaluated exactly once in every epoch, is not only easier to implement but also shows a better performance in empirical comparisons. A more detailed discussion about this topic can be found in Benjamin Recht and Christopher Re''s paper Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences [4].

In [15]:

ada = AdalineSGD(epochs=15, eta=0.01)

# shuffle data
np.random.seed(123)
idx = np.random.permutation(len(y))
X_shuffled, y_shuffled =  X_std[idx], y[idx]

# train and adaline and plot decision regions
ada.train(X_shuffled, y_shuffled)
plot_decision_regions(X_shuffled, y_shuffled, clf=ada)
plt.title(''Adaline - Gradient Descent'')
plt.xlabel(''sepal length [standardized]'')
plt.ylabel(''petal length [standardized]'')
plt.show()

plt.plot(range(1, len(ada.cost_)+1), ada.cost_, marker=''o'')
plt.xlabel(''Iterations'')
plt.ylabel(''Sum-squared-error'')
plt.show()

What''s Next?

[back to top]

Although we covered many different topics during this article, we just scratched the surface of artificial neurons.

In later articles, we will take a look at different approaches to dynamically adjust the learning rate, the concepts of "One-vs-All" and "One-vs-One" for multi-class classification, regularization to overcome overfitting by introducing additional information, dealing with nonlinear problems and multilayer neural networks, different activation functions for artificial neurons, and related concepts such as logistic regression and support vector machines.

References

[back to top]

[1] F. Rosenblatt. The perceptron, a perceiving and recognizing automaton Project Para. Cornell Aeronautical Laboratory, 1957.

[2] W. S. McCulloch and W. Pitts. A logical calculus of the ideas immanent in nervous activity. The bulletin of mathematical biophysics, 5(4):115–133, 1943.

[3] B. Widrow et al. Adaptive ”Adaline” neuron using chemical ”memistors”. Number Technical Report 1553-2. Stanford Electron. Labs., Stanford, CA, October 1960.

[4] B. Recht and C. R ́e. Beneath the valley of the noncommutative arithmetic-geometric mean inequality: conjectures, case-studies, and consequences. arXiv preprint arXiv:1202.4184, 2012.

Bilinear CNN与 Randomly Wired Neural Network

最近主要学习了两篇论文以及相关的代码。

1、Bilinear CNN

这篇论文主要是在细粒度分类上应用的，在全连接层之前，在所有的卷积计算完成之后，进行的Bilinear计算，关键的代码如下：

def forward(self, X):
        """Forward pass of the network.

        Args:
            X, torch.autograd.Variable of shape N*3*448*448.

        Returns:
            Score, torch.autograd.Variable of shape N*200.
        """
        N = X.size()[0]
        assert X.size() == (N, 3, 448, 448)
        X = self.features(X)
        assert X.size() == (N, 512, 28, 28)
        X = X.view(N, 512, 28**2)
        X = torch.bmm(X, torch.transpose(X, 1, 2)) / (28**2)  # Bilinear
        assert X.size() == (N, 512, 512)
        X = X.view(N, 512**2)
        X = torch.sqrt(X + 1e-5)
        X = torch.nn.functional.normalize(X)
        X = self.fc(X)
        assert X.size() == (N, 200)
        return X

这个地方和判断卷积核提取的特征的相似度判别，求取的相似矩阵有很大的相似之处。

2、 Randomly Wired Neural Network

将随机图应用到神经网络的构建之中，讲道理，构建的网络其实和DenseNet这种有很大的相似性。具体的过程，寄到笔记本上了。

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