Lecture 1 notes

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machine learning

发布于：2021年9月15日

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Lecture 1: Introduction

(1). Examples of Machine Learning Applications

• Learning associations
• Supervised learning
  • Classification
  • Regression
• Unsupervised learning
• Reinforcement learning

(2). Illustrative Example: Polynomial Curve Fitting

• Regression problem:

  • input variable: x
  • target variable: t
  • To find a curve that fit these points

• Polynomial function for fitting data:

  $$y(x, \textbf{w}) = w_0+w_1x+w_2x^2+\cdots+w_Mx^M=\sum_{i=0}^Mw_jx^j$$

  The function for x is nonlinear but for $\textbf{w}$ is linear.

• Error function (loss):

  $$E(\textbf{w})=\frac{1}{2}\sum_{n=1}^N[y(x_n,\textbf{w}-t_n)]^2$$

  Our aim is to minimize the loss function to fit the curve. But sometimes we will get overfitting which we can detect by calculating the RMS error.

• Regularization:

  We can rewrite the loss function as:

  $$\tilde{E}(\textbf{w})=\frac{1}{2}\sum_{n=1}^N[y(x_n,\textbf{w}-t_n)]^2+\frac{\lambda}{2}||\textbf{w}||^2$$

  In order to restrict the parameters we get by optimize the loss function to get rid of overfitting.

  As $\lambda$ goes bigger, the RMS error goes down but when $\lambda$ is big enough, the RMS error goes up.

(3). Something about classification

• Training set $\chi$: from where the model learn the informations

• Class $\mathcal{C}$: the expected best fitted range of a specific class of data.

• Hypothesis Class $\mathcal{H}$: The learning algorithm finds a hypothesis $h\in\mathcal{H}$ to approximate the class $\mathcal{C}$.

• Empirical Error: $\mathcal{C}(x)$ is usually unknown, so we use empirical error to test how well $h(x)$ matches $\mathcal{C}(x)$.

  $$E(h|\chi) = \sum_{\mathcal{l}=1}^N\mathbf{1}_{[h(\mathbf{x}^{\mathcal{(l)}})\neq \mathbf{y}^\mathcal{(l)}]}$$

• Version Space:

  • Most specific hypothesis $\mathcal{S}$: tightest rectangle in $\mathcal{H}$ that includes all positive example but no negative example.
  • Most general hypothesis $\mathcal{G}$: largest rectangle in $\mathcal{H}$ that includes all positive example but no negative example.
  • Version space: the set in $\mathcal{H}$ between $\mathcal{S}$ and $\mathcal{G}$.

• Vapnik-Chervonenkis (VC) Dimension:

  N points can lay out $2^N$ ways as +/-. So the definition of VC dimension of a hypothesis $\mathcal{H}$ is: the maximum number of points that can be shattered by $\mathcal{H}$.

  In other words: if all the $2^N$ ways can find a $h\in \mathcal{H}$ that can separate the these $N$ points by +/- correctly, then the maximum value of $N$ is the VC dimension of $\mathcal{H}$.

  An example:

  

  

• Multiple Classes: A K-class classification problem can be considered as K 2-classes problems.

(4). Something about regression

(5). Dimensions of Supervised Learning Algorithm

• Model:

  $$g(\mathbf{x}|\theta)$$

• Loss function:

  $$E(\theta|\chi)=\sum_{\mathcal{l}}L(\mathbf{y}^l,g(\mathbf{x}^l|\theta))$$

• Optimization procedure/algorithm:

  $$\theta^{*}=\mathop{\arg\min}\limits_{\theta}E(\theta|\chi)$$

更新于：2021年9月15日