斯坦福统计学习理论课程CS229T/STAT231：理论与应用探讨

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CS229T/STAT231: Statistical Learning Theory (Winter 2016) 是斯坦福大学的一门高级课程，旨在深入探讨统计学习理论在机器学习中的核心概念和应用。该课程由Percy Liang主讲，更新时间为2018年12月25日，随着课程的进行，讲座笔记会不断更新。课程大纲涵盖了多个关键主题，旨在帮助学生理解机器学习算法的理论基础： 1. 概述（Lecture 1）：课程首先介绍课程的目的，强调理解机器学习算法的工作原理，包括它们为何有效，以及如何将学习的概念形式化。此外，还讨论了诸如均匀收敛性这样的概念，这是评估学习算法性能的关键指标。 2. 渐近分析（Lecture 1-4）：这部分深入探讨了估计问题，如高斯均值估计、多项式估计，以及涉及概率模型如指数家族的方法。课程还介绍了最大熵原理，它在构建复杂模型时起到指导作用，以及方法论上的考虑，如隐变量模型的矩法和线性回归。 3. 在线学习（Lecture 1）：在快速变化的数据环境中，课程讲解了在线学习策略，这种学习方式允许模型在接收到新数据时动态调整。 4. 均匀收敛（Lecture 5）：此部分关注理论框架中的一个重要概念，即学习过程中的收敛性，特别是当假设空间有限或可数时，如何确保模型在所有可能的数据分布下都能表现良好。 5. 核方法（Lecture 1）：这是一种重要的非线性特征转换技术，使得机器学习算法能够在高维空间中有效地工作，即使原始数据是非线性可分的。 6. 固定设计下的线性回归与正规化（Lecture 3-4）：课程详细解释了固定设计情况下的线性回归模型，以及引入正则化技术（如L2范数惩罚）以防止过拟合的重要性。通过这些章节，学生不仅能够掌握统计学习的基本理论，还能学会如何利用这些理论来设计和优化机器学习算法。课程内容具有理论深度，但又注重实际应用，为理解和实施现代机器学习提供了坚实的数学基础。

[begin lecture 2] (2)

2.4 Exponential families (Lecture 2)

• So far, we’ve analyzed the Gaussian and the multinomial. Now let us delve into

exponential families, a much more general family of distributions which include the

Gaussian, multinomial, gamma, beta, etc., but not things like the uniform distribution

or mixture models.

• Exponential families are also the basis of undirected graphical models. You probably

ﬁrst saw exponential families in a univariate continuous setting, but we will introduce

them here in the multivariable discrete setting in order to build some diﬀerent intuitions

(and to simply the math).

• Deﬁnition 1 (exponential family)

– Let X be a discrete set.

∗ Example: X = {−1, +1}

– Let φ : X → R

be a feature function.

∗ Example (unary and binary factors):

φ(x) = [x

, x

] ∈ R

. (19)

– Deﬁne a family of distributions P, where each p

∈ P assigns each x ∈ X a

probability that depends on x only through φ(x):

def

= {p

: θ ∈ R

}, p

(x)

def

= exp{θ · φ(x) − A(θ)}, (20)

where the log-partition function

A(θ)

def

= log

x∈X

exp{θ · φ(x)} (21)

ensures the distribution is normalized.

– Example: if θ = [0, 0, 0, 0, 0], then p

is the uniform distribution.

– Example: if θ = [0, 0, 0, log 2, 0] (which favors x

and x

having the same sign),

then p

assigns probability:

What do we do with the RHS? The delta method allows us to transfer asymp-

totic normality results on one quantity (ˆµ in this case) to a non-linear transfor-

mation of another quantity (f(ˆµ) in this case). The asymptotic variance is just

multiplied by a linearization of the non-linear transformation. Speciﬁcally:

√

n(f(ˆµ) − f(µ

∗

))

−→ N(0, ∇f(µ

∗

)

Cov

(φ(x))∇f(µ

∗

)). (33)

Conveniently, ∇f(µ) = ∇

A(µ)

−1

= Cov

(φ(x))

−1

for θ = f(µ), so we get that:

√

θ − θ

∗

)

−→ N(0, Cov

(φ(x))

−1

). (34)

– Notice that the asymptotic variance is the inverse of the covariance of the features

φ(x). Intuitively, if the features vary more, we can estimate the parameters better

(smaller asymptotic variance).

– The parameter error k

θ − θk

can be computed exactly the same way as for

multinomial mean estimation, so we will skip this.

• Method of moments

– We have introduced

θ as a maximum likelihood estimator. But we can also

interpret it as a method of moments estimator. If we rewrite the optimality

conditions for

θ using the fact that ∇A(θ) = E

[φ(x)], we see that

[φ(x)] = ˆµ. (35)

In other words, the empirical moments ˆµ are identical to those deﬁned by the

model p

– The method of moments principle is: Choose model parameters to make the

moments under the model match moments of the data.

– The maximum likelihood principle is: Choose model parameters to maximize the

likelihood of the entire data.

– The method of moments dates back to Karl Pearson from 1894, which predates

the full development of maximum likelihood by Fisher in the 1920s. Since then,

maximum likelihood has been the dominant paradigm for parameter estimation,

mainly due to its statistical eﬃciency and naturalness. There is more ad-hocness

in the method of moments, but it is exactly this ad-hocness that allows us to get

computationally eﬃcient algorithms at the price of reduced statistical eﬃciency,

as we’ll see later for latent-variable models.

2.5 Maximum entropy principle (Lecture 2)

• Let us scrap exponential families for now and embrace the method of moments per-

spective for a moment. Can we develop an estimator from ﬁrst principles?

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斯坦福统计学习理论课程CS229T/STAT231：理论与应用探讨

Statistical Learning Theory

斯坦福大学机器学习CS229笔记

Statistical Learning Theory Lecture Notes (Stanford CS229t)

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