机器学习基础：模型、数据与损失函数的融合

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"Machine Learning Basic Principles.pdf" 本教程深入探讨了机器学习的基础，它是一种让计算机从数据中学习并改进的技术。机器学习的核心思想在于利用数据来验证和优化假设，即构建模型。在这个过程中，首先设定一个关于现象的假设，然后收集相关的观察数据来验证这个假设，接着通过迭代优化来调整模型，以减少预测结果与实际观测值之间的差距，这个差距通常用损失函数来衡量。梯度下降法是机器学习中常用的一种优化算法，它用于更新模型的参数（权重向量），以最小化损失函数。这个过程不断进行，直到模型的预测性能达到最优。模型、损失函数和优化方法的选择是机器学习方法多样性的来源，不同的组合产生了各种各样的学习算法，如支持向量机、决策树、随机森林、神经网络等。在形式化机器学习问题时，主要涉及三个关键要素：假设空间（模型）、数据（观测）和损失函数。模型代表我们对数据生成过程的理解，数据是我们用来训练和测试模型的信息，而损失函数则量化了模型预测的错误程度。通过最小化损失，我们可以找到最佳的模型参数。本书是作者在阿尔托大学授课的课堂笔记基础上编写的，涵盖了从基础的监督学习到无监督学习，再到强化学习的各种机器学习方法。它不仅讨论了各种算法设计模式，还强调了实践中的问题，如数据预处理、过拟合与欠拟合的识别以及模型选择与验证。机器学习的应用已经广泛渗透到日常生活和各个科学领域，从推荐系统、图像识别到自然语言处理，无处不在。随着技术的发展，理解和掌握机器学习的基本原理变得越来越重要，这本书提供了一个良好的起点，帮助读者深入理解这个领域的核心概念。本教程的最新版本可以在指定链接中找到，供读者免费阅读和下载，以便进一步研究和学习。作者鼓励读者与社区分享反馈和建议，共同推进机器学习知识的传播和进步。

this is the true label value of the datapoint. By its very deﬁnition, this distribution q(ˆy) is

precisely the conditional probability distribution p(y|x) of the label value y, given the feature

value x of a randomly drawn datapoint z = (x, y) ∼ p(z).

Having an (estimate of) probability distribution p(z) for the observed datapoints not

only allows us to compute predictions but also to generate new datapoints. Indeed, we

can artiﬁcially augment the available data by randomly drawing new datapoints according

to the probability distribution p(z) (see Section 7.3). A recently popularized class of ML

methods that use probabilistic models to generate synthetic data is known as generative

adversarial networks [30].

p(z)

Figure 1.5: A scatterplot where each dot represents some day that is characterized by its

minimum daytime temperature x and its maximum daytime temperature y.

1.1.6 Artiﬁcial Intelligence

ML is instrumental for the design and analysis of artiﬁcial intelligence (AI). AI systems (hard

and software) interacts with their environment by taking certain actions. These actions

inﬂuence the environment as well as the state of the AI system. The behaviour of an AI

system is determined by how the perceptions made about the environment are used to form

the next action.

From an engineering point of view, AI aims at optimizing behaviour to maximize a long-

term return. The optimization of behaviour is based solely on the perceptions made by the

agent. Let us consider some application domains where AI systems can be used:

• a forest ﬁre management system: perceptions given by satellite images and local

observations using sensors or “crowd sensing” via some mobile application which allows

humans to notify about relevant events; actions amount to issuing warnings and bans

of open ﬁre; return is the reduction of number of forest ﬁres.

• a control unit for combustion engines: perceptions given by various measurements

such as temperature, fuel consistency; actions amount to varying fuel feed and timing

and the amount of recycled exhaust gas; return is measured in reduction of emissions.

• a severe weather warning service: perceptions given by weather radar; actions are

preventive measures taken by farmers or power grid operators; return is measured by

savings in damage costs (see https://www.munichre.com/)

• an automated beneﬁt application system for a social insurance institute (like “Kela”

in Finland): perceptions given by information about application and applicant; actions

are either to accept or to reject the application along with a justiﬁcation for the de-

cision; return is measured in reduction of processing time (applicants tend to prefer

getting decisions quickly)

• a personal diet assistant: perceived environment is the food preferences of the app

user and their health condition; actions amount to personalized suggestions for healthy

and tasty food; return is the increase in well-being or the reduction in public spending

for health-care.

• the cleaning robot Rumba (see Figure 1.6) perceives its environment using diﬀer-

ent sensors (distance sensors, on-board camera); actions amount to choosing diﬀerent

moving directions (“north”, “south”, “east”, “west”); return might be the amount of

cleaned ﬂoor area within a particular time period.

• personal health assistant: perceptions given by current health condition (blood

values, weight,. . . ), lifestyle (preferred food, exercise plan); actions amount to person-

alized suggestions for changing lifestyle habits (less meat, more jogging,. . . ); return is

measured via the level of well-being (or the reduction in public spending for health-

care).

• government-system for a community: perceived environment is constituted by cur-

rent economic and demographic indicators such as unemployment rate, budget deﬁcit,

age distribution,. . . ; actions involve the design of tax and employment laws, pub-

lic investment in infrastructure, organization of health-care system; return might be

determined by the gross domestic product, the budget deﬁcit or the gross national

happiness (cf. https://en.wikipedia.org/wiki/Gross_National_Happiness).

Figure 1.6: A cleaning robot chooses actions (moving directions) to maximize a long-term

reward measured by the amount of cleaned ﬂoor area per day.

ML methods are used on diﬀerent levels within AI systems. On a low-level, ML methods

help to extract the relevant information from raw data. AI systems use ML methods to

classify images into diﬀerent categories. The AI system subsequently only needs to process

the category of the image instead of its raw digital form.

ML methods are also used for higher-level tasks of an AI system. To behave optimally,

an AI system or agent is required to learn a good hypothesis about how its behaviour aﬀects

its environment. We can think of optimal behaviour as a consequent choice of actions that

are predicted as optimal according to a hypothesis which could be obtained by ML methods.

What sets AI methods apart from other ML methods is that they must compute predic-

tions in real-time while collecting data and choosing the next action. Consider an AI system

that steers a toy car. In any given state (or point of time), the resulting prediction inﬂuences

immediately the features of the following datapoints.

Consider datapoints that represent diﬀerent states of a toy car. For such datapoints we

could deﬁne their labels as the optimal steering angle for these states. However, it might

be very challenging to obtain accurate label values for any of these datapoints. Instead, we

could evaluate the usefulness of a particular steering angle only in an indirect fashion by

using a reward signal. For the toy car example, we might obtain a reward from a distance

sensor that indicates if the car reduces the distance to some goal or target location.

1.2 Flavours of Machine Learning

The main component of any form of ML are collections of data points which can carry

information about some phenomenon. An individual data point is characterized by various

properties. We ﬁnd it convenient to divide the properties of data points into two groups:

features and labels (see Chapter 2.1). Features are properties that we measure or compute

easily in an automated fashion. Labels are properties that cannot be measured easily and

often represent some higher-level fact whose discovery often requires human experts.

ML aims at predicting (approximating or guessing) the label of a datapoint based solely

on the features of this datapoint. Formally, the prediction is obtained as the function value

of a hypothesis function (or map) which takes the features of a data point as its argument.

Since any ML method must be implemented with ﬁnite computational resources, it can only

consider a subset of all possible hypothesis maps. This subset is referred to as the hypothesis

space or model underlying a ML method. Based on how ML methods assess the quality of

diﬀerent hypothesis maps we distinguish three main ﬂavours of ML: supervised, unsupervised

and reinforcement learning.

Supervised Learning. The main focus of this book is on supervised ML methods.

These methods use a training set that consists of labeled data points. We refer to a data

point as labeled if its label value is known. Such labeled data points can be obtained by

some human annotating data points with their label values. There are even marketplaces

for renting human labelling workforce [73]. Supervised ML searches for a hypothesis that

can imitate the human annotator and allows to predict the label solely from the features of

a data point.

Figure 1.7 illustrates the basic principle of supervised methods. These methods learn

a hypothesis with minimum discrepancy between its predictions and the true labels of the

datapoint in the training set. Loosely speaking, supervised ML ﬁts a curve (the graph of the

predictor map) to labeled datapoints in a training set. For the actual implementing of this

curve ﬁtting we need a loss function that quantiﬁes the ﬁtting error. Supervised ML method

diﬀer in their choice for a loss function to measure the discrepancy between predicted label

and true label of a data point.

While the principle behind supervised ML sounds trivial, the challenge of modern ML

applications is the sheer amount of datapoints and their complexity. ML methods must

process billions of datapoints with each single datapoint characterized by a potentially vast

number of features. Consider datapoints representing social network users, whose features

include all media that has been posted (videos, images, text). Besides the size and complexity

of datasets, another challenge for modern ML methods is that they must be able to ﬁt

highly non-linear predictor maps. Deep learning methods address this challenge by using a

computationally convenient representation of non-linear maps via artiﬁcial neural networks

[29].

(2)

, y

(2)

)

(1)

, y

(1)

)

feature x

predictor h(x)

label y

Figure 1.7: Supervised ML methods ﬁt a curve to (a huge number of) datapoints.

Unsupervised Learning. Some ML methods do not require any labeled datapoint and

are referred to as unsupervised. Loosely speaking, these methods only use the intrinsic

structure of data points to learn a good hypothesis. They do not need a teacher or human

expert which provides labels for the datapoints in a training set. Chapters 8 and 9 discuss

two large families of unsupervised methods, clustering and feature learning methods.

Clustering methods group data points into few subsets such that data points within the

same subset or cluster are more similar with each other than with data points outside the

cluster. Feature learning methods determine numeric features such that data points can be

processed eﬃciently using these features. Two important applications of feature learning are

dimensionality reduction and data visualization.

Reinforcement Learning. ML methods use a loss function to evaluate and compare

diﬀerent hypotheses. The loss function assigns a (typically non-negative) loss value to a

given pair of a data point and a hypothesis. Loosely speaking, ML is the search for a

hypothesis that incurs minimum loss for any datapoint. Reinforcement learning (RL) studies

applications where data point are generated sequentially. We could think of data points

forming a time series, each time instant a new data point comes in. Based on the most recent

data point coming in, RL methods determine a new candidate for an optimal hypothesis.

What sets RL methods apart from supervised and unsupervised methods is that they

must learn a hypothesis in an online fashion. In each time instant, providing a new data

point, RL methods must deliver a guess or estimate for the optimal hypothesis and apply it

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