-
Notifications
You must be signed in to change notification settings - Fork 0
/
zf.tex
417 lines (380 loc) · 17.6 KB
/
zf.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
\documentclass[a4paper, 11pt, landscape]{article}
\usepackage{mathptmx} % more compact font
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{mathtools}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{paralist} % for compacter lists
\usepackage{hyperref} % for Todo's and similar things
\usepackage[left=4.5mm, right=4.5mm, top=4.5mm, bottom=6mm, landscape, nohead, nofoot]{geometry}
\usepackage[small,compact]{titlesec}
\usepackage[usenames,dvipsnames,svgnames,table]{xcolor}
\usepackage{xparse}
% compact text
\linespread{0.9}
\setlength{\parindent}{0pt}
% compact lists even more
\setdefaultleftmargin{0em}{0em}{0em}{0em}{0em}{0em}
% define expectation symbol
\DeclareMathOperator*{\E}{\mathbb{E}}
% compact sections
\titlespacing*{\section}{0pt}{0em}{0em}
\titlespacing*{\subsection}{0pt}{0em}{0em}
\titlespacing*{\subsubsection}{0pt}{0em}{0em}
% coloured section headings for easier read
\titleformat{name=\section}[block]
{\sffamily}
{}
{0pt}
{\colorsection}
\newcommand{\colorsection}[1]{%
\colorbox{red!10}{\parbox[t][0em]{\dimexpr\columnwidth-2\fboxsep}{\thesection\ #1}}}
\titleformat{name=\subsection}[block]
{\sffamily}
{}
{0pt}
{\subcolorsection}
\newcommand{\subcolorsection}[1]{%
\colorbox{orange!10}{\parbox[t][0em]{\dimexpr\columnwidth-2\fboxsep}{\thesubsection\ #1}}}
\titleformat{name=\subsubsection}[block]
{\sffamily}
{}
{0pt}
{\subsubcolorsection}
\newcommand{\subsubcolorsection}[1]{%
\colorbox{blue!10}{\parbox[t][0em]{\dimexpr\columnwidth-2\fboxsep}{\thesubsubsection\ #1}}}
% environment for multicols inside a list
\NewDocumentEnvironment{listcols}{O{2} O{0pt}}
{%
\bgroup %
\setlength{\multicolsep}{#2} %
\begin{multicols*}{#1} %
}
{%
\end{multicols*} %
\egroup %
}
% multicols lines & spacing
\setlength{\columnsep}{0.2cm}
\setlength{\columnseprule}{0.2pt}
% No page numbers
\pagenumbering{gobble}
% math helpers
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\begin{document}
\begin{multicols*}{3}
\section{Essentials}
\subsection{Adaptive Stepsize}
\begin{compactdesc}
\item[Line search:] Optimize step size at every step along gradient
\item[Bold driver:] Objective decrease $\rightarrow$ increase step size, objective increase $\rightarrow$ decrease step size
\end{compactdesc}
\subsection{Loss functions}
\begin{compactdesc}
\item[L1 loss:] $l_1(\mathbf{w}; x_i, y_i) = |y_i - \mathbf{w}^T x_i| $
\item[Lp loss:] $l_p(\mathbf{w}; x_i, y_i) = |y_i - \mathbf{w}^T x_i|^p$
\item[0/1 loss:] $l_{0/1}(\mathbf{w}; x_i, y_i) = \left\{
\begin{array}{lr}
0, \text{ if } sign(\mathbf{w}^Tx_i) = y_i\\
1, \text{ else}\\
\end{array}\right\}$
\item[Perceptron loss:] $l_\text{perc}(\mathbf{w}; x_i, y_i) = \left\{
\begin{array}{lr}
0, \text{ if } sign(\mathbf{w}^Tx_i) = y_i\\
-y_i\mathbf{w}^Tx_i, \text{ else}\\
\end{array}\right\}\\ = max(0, -y_i \mathbf{w}^T x_i)$
\item[Cost sensitive perceptron:] $l_\text{cs}(\mathbf{w}; x_i, y_i) = c_y * l_p(\mathbf{w}; x_i, y_i)$
\item[Hinge loss:] $l_H(\mathbf{w}^T; x_i, y_i) = max(0, 1- y_i \mathbf{w}^Tx_i)$
\item[Logistic loss:] $l_\text{logistic}(\mathbf{w}^T, x_i, y_i) = log(1 + exp(-y_i\mathbf{w}^Tx_i))$
\end{compactdesc}
\subsection{Loss Function Derivatives}
\begin{compactdesc}
\item[Perceptron loss:] $\nabla_\mathbf{w} l_p = \left\{
\begin{array}{lr}
0, \text{ if } -y_i \mathbf{w}^T x_i < 0\\
-y_ix_i, \text{ else}\\
\end{array}\right\}\\$
\end{compactdesc}
\subsection{Distributions}
\begin{compactdesc}
\item[1D-Gaussian:] $P(X = x) = 1/\sqrt{2\pi\sigma^2}exp(-(x-\mu)^2/{2\sigma^2})$
\item[Bernoulli:] $Ber(y; x) = \left\{
\begin{array}{lr}
x, \text{ if } y= +1\\
1-x, \text{ if } y =-1\\
\end{array}
\right\}$
\end{compactdesc}
\subsection{Matrix Calculus}
\begin{compactdesc}
\item[Derivatives]
$\frac{\partial}{\partial x} \mathbf{A x} = \mathbf{A}^T$\\
$\frac{\partial}{\partial x} \mathbf{x}^T\mathbf{A} = \mathbf{A}$\\
$\frac{\partial}{\partial x} \mathbf{x}^T\mathbf{x} = 2\mathbf{x}$\\
$\frac{\partial}{\partial x} \mathbf{x}^T\mathbf{A x} = \mathbf{Ax} + \mathbf{A}^T\mathbf{x}$
\item[Ranks]
$rank(AB) \leq \min(rank(A), rank(B))$
\item[Diverse] $X$ psd $\Rightarrow u^TX u \geq 0$\\
$X$ pd $\Rightarrow u^TX u > 0$\\
$X$ psd and $Y$ pd $\Rightarrow X+Y$ pd\\
$X$ pd $\Rightarrow$ invertible
\end{compactdesc}
\subsection{Probabilistics}
\begin{compactdesc}
\item[Multiplication:] $P(A|B) = \frac{P(A, B)}{P(B)}$
\item[Bayes:] $P(A|B) = \frac{P(B|A) P(A)}{P(B)}$
\end{compactdesc}
\subsection{Gradient Descent}
\begin{compactdesc}
\item[Normal:] $\mathbf{w}_{t+1} = \mathbf{w}_t - \eta_t \nabla_\mathbf{w}\hat{R}(\mathbf{w}_t)$
\item[Stochastic:] $\mathbf{w}_{t+1} = \mathbf{w}_t - \eta_t \nabla_wl(\mathbf{w}_t, x', y')$ for random $(x', y') \in D$
\item[SGD L2:] $\mathbf{w}_{t+1} = \mathbf{w}_t(1-2\lambda\eta_t) - \eta_t \nabla_wl(\mathbf{w}_t, x', y')$
\end{compactdesc}
\subsection{Fundamental assumptions}
Optimal solution lies in span of data
\begin{compactdesc}
\item[Alternative Representation:] $\mathbf{w^*} = \sum_{i=1}^{n}(\alpha_iy_i)x_i$ for some $\alpha_{1:n}$
\end{compactdesc}
\section{Regression}
\subsection{Linear least sqares}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{w}) = \sum_{i=1}^{n}l_2(\mathbf{w}; x_i, y_i)$
\item[Closed Form:] $\mathbf{w^*} = (\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^Ty$ with $\mathbf{X} = (x_1, x_2, ..., x_n)^T$
\item[Gradient:] $\nabla_\mathbf{w} \hat{R}(\mathbf{w}) = -2 \sum_{i=1}^{n}(y_i - \mathbf{w}^Tx_i)x_i$
\item[Runtime:] Closed form $\varTheta(n*d^2 + d^3)$, Gradient descent $\varTheta(iter * n * d)$
\end{compactdesc}
\subsection{Polynomial features}
\begin{compactdesc}
\item[Aim:]Fit non-linear functions via linear regression.
\item[Solution:]Use non-linear transformation of data
\item[Ojbective Function:] $\hat{R}(\mathbf{w}) = \sum_{i=1}^{n}(y_i - f(x))^2$
\item[Transformation:]$f(x) = \sum_{j=1}^{d}w_i\phi_i(x)$
\item[Polynomial features:]$x \rightarrow \phi(x)$
\end{compactdesc}
\subsection{Ridge Regression}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{w}) = \sum_{i=1}^{n}l_2(\mathbf{w}, x_i, y_i) + \lambda\|\mathbf{w}\|_2^2$
\item[Closed Form:] $\mathbf{w^*} = (\mathbf{X}^T\mathbf{X} + \lambda\mathbf{I})^{-1}\mathbf{X}^Ty$
\item[Gradient:] $\nabla_\mathbf{w} \hat{R}(\mathbf{w}) = -2 \sum_{i=1}^{n}(y_i - \mathbf{w}^Tx_i)x_i + 2\lambda\mathbf{w}$
\item[Notes:] Scale of features matter. All features should be zero mean and unit variance\\ Use L1 regularizer to get LASSO for better feature selection
\end{compactdesc}
\section{Classification}
\subsection{Nearest Neighbor}
\begin{compactdesc}
\item[Idea:] Use k closest neighbors to vote on new point x's class
\item[Prediction:] $\hat{y} = sign(\sum_{i:x_i \in KNN(x)}^{}y_i)$ (binary case)
\end{compactdesc}
\subsection{Perceptron algorithm}
Stochastic gradient descent on perceptron loss
\subsection{SVM}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{w}) = \sum_{i=1}^{n}l_H(\mathbf{w}, x_i, y_i) + \lambda\|\mathbf{w}\|_2^2$
\item[Gradient:]???
\item[Notes:] Use L1 regularizer for better feature selection (L1 SVM)
\end{compactdesc}
\subsection{Multiclass classification}
\begin{compactdesc}
\item[1vAll:] Train 1 classifier for each class. Choose classifier with biggest confidence.
\item[1vAll prediction:] $\hat{y} = \argmax_{i=1:c}f_i(x)$ where $f_i(x)$ is the classifier for class $i$
\item[1vAll Notes:] Normalize weights $\hat{\mathbf{w_i}}=\mathbf{w_i^*}/\|\mathbf{w_i^*}\|$ for determining confidence.
\item[1v1:] Train 1 classifier for each class pair. ($c(c-1)/2$)
\item[1v1 prediction:] Use voting to determine class
\end{compactdesc}
\section{Kernels}
\subsection{Notation}
\begin{compactdesc}
\item[Kernel Function:] $k(x_i, x_j) = \phi(x_i)^T\phi(x_j)$
\item[Gram Matrix:] $\mathbf{K} = \left(
\begin{array}{ccc}
k(x_1, x_1), ..., k(x_1, x_n)\\
..., ..., ...\\
k(x_n, x_1), ..., k(x_n, x_n)
\end{array}\right)$
\item[Kernelitem:] $k_i = [y_1k(x_i, x_1), y_2k(x_i, x_2), ..., y_nk(x_i, x_n)]$
\end{compactdesc}
\subsection{General}
\begin{compactdesc}
\item[Properties:] inner product, symmetric, positive semidefinite
\item[K. engineering:] $k_1 + k_2$, $k_1 * k_2$, $c * k_1 \text{ for } c > 0$, $f(k_1)$ for $f(x)$ polynomial or exp
\item[Monomials of deg. m] $k(x, x') = (x^Tx')^m$
\item[Monomials up to deg. m] $k(x, x') = (1 + x^Tx)^m$
\end{compactdesc}
\subsection{Kernelized Linear Ridge Regression}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{\alpha}) = \|\mathbf{\alpha}^T\mathbf{K} - y\|_2^2 + \lambda \mathbf{\alpha}^T\mathbf{K}\mathbf{\alpha}$
\item[Closed Form:] $\mathbf{\alpha^*} = (\mathbf{K} + \lambda\mathbf{I})^{-1}y$
\item[Prediction:] $\hat{y} = \sum_{i=1}^{d}\alpha^*_ik(x_i, x)$
\end{compactdesc}
\subsection{Kernelized Perceptron}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{\alpha}) = \sum_{i=1}^{n}\max(0, - y_i \mathbf{\alpha}^Tk_i)$\\
$\hat{R}(\mathbf{\alpha}) = \sum_{i=1}^{n}\max(0, - y_i\sum_{j}\alpha_jy_jk(x_j, x_i))$\\
\item[Prediction:] $\hat{y} = sign(\sum_{i=1}^{d}\alpha^*_iy_ik(x_i, x))$
\item[Optimize:] $\text{if } \hat{y} \neq y_i \text{ set } \alpha_i = \alpha_i + \eta_t$
\end{compactdesc}
\subsection{Kernelized SVM}
\begin{compactdesc}
\item[Objective Function:] $\hat{R}(\mathbf{\alpha}) = \sum_{i=1}^{n}\max(0, 1- y_i \mathbf{\alpha}^Tk_i) + \lambda\mathbf{\alpha}^T\mathbf{K}\mathbf{\alpha}$
\item[Prediction:] $\hat{y} = sign(\sum_{i=1}^{d}\alpha^*_iy_ik(x_i, x))$
\item[Optimize:] $\text{if } \hat{y} \neq y_i \text{ set } \alpha_i = \alpha_i + \eta_t$
\end{compactdesc}
\section{Artificial Neural Networks}
\begin{compactdesc}
\item[Transfer Function:] $\sum_{j}^{}w_j\varphi(\theta_j^Tx)$, $w_j \hat{=}$ hidden-to-output weight, $\theta_j^T \hat{=}$ input-to-hidden weight
\item[ReLU Act. Func::] $\varphi(z) = \max(z,0)$
\end{compactdesc}
\subsection{Propagation Algorithms}
??
\section{Practical Issues}
\subsection{Feature Selection}
\begin{compactdesc}
\item[Greedy Forward:] Start with no features. Always choose best feature to add next, until no improvement.
\item[Greedy Backward:] Start with all features. Always choose best feature to remove next, until no improvement.
\end{compactdesc}
\subsection{Imbalanced Data}
\begin{compactdesc}
\item[Subsampling] Remove samples from majority class until balanced
\item[Upsampling] Repeat samples from minority class until balanced
\item[Cost sensitive loss functions:] See cost sensitive perceptron loss
\end{compactdesc}
\subsection{Performance Metrics}
\begin{compactdesc}
\item[Accuracy:] $\frac{TP + TN}{TP + TN + FP + FN}$
\item[Precision:] $\frac{TP}{TP + FP}$
\item[Recall:] $\frac{TP}{TP + FN}$
\item[F1-Score:] $\frac{2TP}{2TP + FP + FN}$
\item[What we want:] Good F1-Score
\end{compactdesc}
\section{Clustering}
\subsection{k-means}
\begin{compactdesc}
\item[1:] Initialize cluster centers at random
\item[2:] Assign each point to closest center\\ $z_i \leftarrow \argmin_{j \in 1:k} \|x_i-\mu_j^{t-1}\|_2^2$
\item[3:] Update centers as mean of assigned points\\ $\mu_j^t \leftarrow 1/n_j \sum_{i:z_i=j}^{} x_i$
\item[Runtime] $\varTheta(iter * n * k * d)$
\end{compactdesc}
\subsection{k-means++ (Initialization)}
\begin{compactdesc}
\item[1:] Start with random datapoint as center
\item[2:] Pick $\mu_j = x_i$ randomly s.t.\\ $P(\mu_j = x_i) = 1/Z \min_{l \in 1:j-1}\|x_i - \mu_l\|_2^2$
\end{compactdesc}
\section{Probabilistic Modelling}
\subsection{Bayes Optimal Predictor}
\begin{compactdesc}
\item[Assumption:] $(x_i, y_i) \sim P(X, Y)$ i.i.d
\item[Minimize:] $\int P(x, y) l(y;h(x))dx dy = \E_{x,y}[l(y;h(x))]$ by finding best $h(x)$
\item[LS Solution:] $h^*(x) = \E[Y | X=x] = \int P(Y|X=x)y dy$
\item[Application:] Estimate $P(Y | X=x)$ to predict label
\end{compactdesc}
\subsection{Maximum Likelihood}
\begin{compactdesc}
\item[Idea:] Estimate parameters of model such that the likelihood of the labels is maximized
\item[1:] $\theta^* = \argmax_\theta \hat{P}(y_1, ..., y_n | x_1, ..., x_n, \theta)$\\
$\Rightarrow\theta^* = \argmin_\theta - \sum_{i}^{} \log \hat{P}(y_i |x_i, \theta)$
\item[2:] Set derivative to zero, get $\theta^*$
\end{compactdesc}
\subsection{Maximum a Posteriori}
\begin{compactdesc}
\item[Idea:] Introduce assumption on distribution of parameters
\item[1:] $\argmax_w P(w|x_{1:n}, y_{1:n}) = \argmax_w \frac{P(w|x_{1:n})P(y_{1:n}|x_{1:n}, w)}{P(y_{1:n}|x_{1:n})}$\\
$\Rightarrow \argmin_w -\log P(w|x_{1:n}) - \log P(Y_{1:n}|x_{1:n}, w) + \log P(y_{1:n}|x_{1:n})$\\
$\Rightarrow \argmin_w -\log P(w) - \log P(Y_{1:n}|x_{1:n}, w) + \log P(y_{1:n}|x_{1:n})$ (indep.)\\
$\Rightarrow \argmin_w -\log P(w) - MLE + \log P(y_{1:n}|x_{1:n})$\\
$\Rightarrow \argmin_w -\log P(w) - MLE$ (irrelevant, indep. of w)\\
$\Rightarrow \argmin_w -\log \prod P(w_j) - MLE$ (iid)
\item[2:] Set derivative to zero
\end{compactdesc}
\subsection{Logistic Regression}
\begin{compactdesc}
\item[Link Function:] $\sigma(\mathbf{w}'Tx) = \frac{1}{1 + exp(-\mathbf{w}^Tx)}$
\item[Noise:] Assume Bernoulli noise
\item[Distribution:] $P(y|x,\mathbf{w}) = Ber(y; \sigma(\mathbf{w}^Tx)$\\
$\Rightarrow P(y|x,\mathbf{w}) = \frac{1}{1 + exp(-y\mathbf{w}^Tx)}$
\item[Idea.] Estimate above distribution using MLE
\item[Gradient:] $y\mathbf{x}\hat{P}(Y=-y|\mathbf{w}, x)$
\end{compactdesc}
\subsection{Bayesian Decision Theory}
\begin{compactdesc}
\item[Idea:] Assign cost to actions and minimize cost
\item[Given:] $P(y|x)$, Actions $A$, Cost function $y \times A \rightarrow \mathbb{R}$
\item[Minimize:] $a^* = \argmin_{a \in A} \E_y[C(y,a)|x]$\\
$\Rightarrow a^* = \argmin_{a \in A} \sum_{y}^{}{P(y|x)C(y,a)}$ (discrete)\\
$\Rightarrow a^* = \argmin_{a \in A} \int_{y}^{}{P(y|x)C(y,a)}dy$ (cont.)\\
\end{compactdesc}
\subsection{Uncertainity Sampling}
\begin{compactdesc}
\item[Idea:] Classify most uncertain points first
\item[1:] Estimate $\hat{P}(y_i|x_i)$ given $D$
\item[2:] Pick most uncertain data point
\item[3:] Classify point and set $D \leftarrow D \cup {(x_i, y_i)}$
\item[4:] Restart at \textbf{1}
\end{compactdesc}
\section{Generative Modeling}
\begin{compactdesc}
\item[1:] Estimate prior $P(y)$
\item[2:] Estimate conditional $P(x|y)$
\item[3:] Then: $P(y|x) = \frac{1}{P(x)} P(y) P(x|y) $ and\\
$P(x,y) = P(x|y)P(x)$ with\\
$p(x) = \sum_y P(y)P(x|y)$\\
\item[Prediction:] $\hat{y} = \argmax_y P(y|x)$
\end{compactdesc}
\subsection{Naive Bayes Classifier}
\begin{compactdesc}
\item[Class label:] $P(Y=y) = p_y$ (categorical)\\
$\Rightarrow p_y = \frac{Count(Y=y)}{n}$
\item[Features:] $P(X_1, ... ,X_n|Y) = \prod_{i=1}^{d}P(X_i|Y)$ (independent)\\
$\Rightarrow$ Use MLE to estimate
\item[Gauss NBC:] $P(X_i|y) = \mathcal{N}(X_i | \mu_{y,i}, \sigma_{y,i}^2)$\\
$\mu_{y,i} = \frac{1}{Count(Y=y)} \sum_{j:y_j=y}x_{j,i}$\\
$\sigma_{y,i}^2=\frac{1}{Count(Y=y)} \sum_{j:y_j=y}^{}(x_{j,i} - \mu_{y,i})^2$
\end{compactdesc}
\subsection{Gaussian Bayes Classifier}
\begin{compactdesc}
\item[Class label:] $P(Y=y) = p_y$ (categorical)\\
$\Rightarrow p_y = \frac{Count(Y=y)}{n}$
\item[Features:] $P(x|y) = \mathcal{N}(x, \mu_y, \Sigma_y)$ (multivariate)
\item[Estimates:] $\mu_y = \frac{1}{Count(Y=y)}\sum_{i:y_i=y}x_i$\\
$\Sigma_y = \frac{1}{Count(Y=y)} \sum_{i:y_i=y}((x_i - \mu_y)(x_i - \mu_y)^T$
\end{compactdesc}
\subsection{Outlier Detection}
If $P(x) < \mathcal{T} \hat{=}$ Threshold, throw away point.
\section{Mixture Models for Clustering}
\begin{compactdesc}
\item[Idea:] Model each cluster j as $P(x|\theta_j)$
\item[Assumption iid:] $P(D|\mathbf{\theta}) = \prod_{i=1}^{n}\sum_{j=1}^{k}{w_jP(x_i|\theta_j)}$\\
Minimization difficult! $\Rightarrow$ Soft/Hard EM
\end{compactdesc}
\subsection{Hard EM}
\begin{compactdesc}
\item[for t=1, ...]
\item[1:] $z_i^t = \argmax_z P(z|x_i, \theta^{t-1}) $\\
$ \Rightarrow z_i^t = \argmax_z P(z|\theta^{t-1})P(x_i|z,\theta^{t-1}) $
\item[2:] $\theta^t = \argmax_\theta P(D^t|\theta)$
\end{compactdesc}
\subsection{Soft EM}
\begin{compactdesc}
\item[for t=1, ...]
\item[E-Step:] $y_j^t(x) = P(Z=j | x, \Sigma, \mu, w)$\\
$\Rightarrow y_j^t(x) = \frac{w_jP(x|\Sigma_j, \mu_j)}{\sum_l w_lP(x|\Sigma_l, \mu_l)}$
\item[M-Step:] $\theta^t = \argmax_\theta Q(\theta;\theta^{t-1})$\\
$Q(\theta;\theta^{t-1}) = \E_{z_{1:n}}[\log P(x_{1:n}, z_{1:n}| \theta)|x_{1:n}, \theta^{t-1}]$\\
$\Rightarrow \theta^t = \argmax_\theta \sum_{i=1}^{n} \sum_{z_i=1}^{k} y_{z_i}(x_i) \log P(x_i, z_i | \theta)$
\item[M-Step Gaussian:] $w_j^t \leftarrow \frac{1}{n} \sum_{i=1}^{n}y_j^t(x_i)$\\
$\mu_j^t \leftarrow \frac{\sum_{i=1}^{n} y_j^t(x_i)*x_i}{\sum_{i=1}^{n}y_j^t(x_i)}$\\
$\Sigma_j^t \leftarrow \frac{\sum_{i=1}^{n}y_j^t(x_i)(x_i-\mu_j^t)(x_i-\mu_j^t)^T}{\sum_{i=1}^{n}y_j^t(x_i)}$
\end{compactdesc}
\section{Markov Chains / Markov Model}
\begin{compactdesc}
\item[Markov Assumption:] $\forall t: P(Y_t | Y_1, ..., Y_{t-1}) = P(Y_t|Y_{t-1})$
\item[Stationary Assumption:] $\forall t,y,y': P(Y_{t+1} = y | Y_t = y') = P(Y_t = y | Y_{t-1} = y')$
\item[Markov Chain:] $p^t = [p_1^t, p_2^t, ..., p_c^t]$\\
$T_{y',y} = P(Y_{t+1} = y | Y_t=y') = \theta_{y|y'}$\\
$\Rightarrow p^{t+1} = p^t*T$
\item[MLE Estimation:] $\hat{p_y} = \frac{Count(Y_1=y)}{m}$\\
$\hat{\theta}_{y|y'} = \frac{Count(Y_t = y, Y_{t-1} = y')}{Count(Y_{t-1} = y')}$
\end{compactdesc}
\raggedcolumns
\end{multicols*}
\end{document}