-
Notifications
You must be signed in to change notification settings - Fork 101
/
Copy pathregularization.html
196 lines (146 loc) · 4.15 KB
/
regularization.html
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
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<!-- saved from url=(0014)about:internet -->
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title></title>
<base target="_blank"/>
<style type="text/css">
body, td {
font-family: sans-serif;
background-color: white;
font-size: 12px;
margin: 8px;
}
tt, code, pre {
font-family: 'DejaVu Sans Mono', 'Droid Sans Mono', 'Lucida Console', Consolas, Monaco, monospace;
}
h1 {
font-size:2.2em;
}
h2 {
font-size:1.8em;
}
h3 {
font-size:1.4em;
}
h4 {
font-size:1.0em;
}
h5 {
font-size:0.9em;
}
h6 {
font-size:0.8em;
}
a:visited {
color: rgb(50%, 0%, 50%);
}
pre {
margin-top: 0;
max-width: 95%;
border: 1px solid #ccc;
}
pre code {
display: block; padding: 0.5em;
}
code.r {
background-color: #F8F8F8;
}
table, td, th {
border: none;
}
blockquote {
color:#666666;
margin:0;
padding-left: 1em;
border-left: 0.5em #EEE solid;
}
hr {
height: 0px;
border-bottom: none;
border-top-width: thin;
border-top-style: dotted;
border-top-color: #999999;
}
@media print {
* {
background: transparent !important;
color: black !important;
filter:none !important;
-ms-filter: none !important;
}
body {
font-size:12pt;
max-width:100%;
}
a, a:visited {
text-decoration: underline;
}
hr {
visibility: hidden;
page-break-before: always;
}
pre, blockquote {
padding-right: 1em;
page-break-inside: avoid;
}
tr, img {
page-break-inside: avoid;
}
img {
max-width: 100% !important;
}
@page :left {
margin: 15mm 20mm 15mm 10mm;
}
@page :right {
margin: 15mm 10mm 15mm 20mm;
}
p, h2, h3 {
orphans: 3; widows: 3;
}
h2, h3 {
page-break-after: avoid;
}
}
</style>
<!-- MathJax scripts -->
<script type="text/javascript" src="https://c328740.ssl.cf1.rackcdn.com/mathjax/2.0-latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
</head>
<body>
<p>Every statistical model performs the following decomposition:</p>
<ul>
<li>data = signal + noise</li>
</ul>
<p>Every statistical model can be fit to data by attempting to minimize the amount of the data that gets labelled as noise. Minimizing prediction error is valuable, but many models in ML also benefit from <em>regularization</em>, which involves penalizing the model for being complex in addition to penalizing the model for making errors:</p>
<ul>
<li>Unregularized model: Fit model by minimizing prediction error penalty</li>
<li>Regularized model: Fit model by minimizing a combination of a prediction error penalty and a complexity penalty</li>
</ul>
<p>There are many reasons to believe that regularization is valuable, but the best reason of all is empirical: anyone who works with data long enough discovers that the model that fits our training data best is often not the model that predicts held-out test data best.</p>
<p>In practice, regularization is often indistinguishable from replacing MLE's with MAP estimates after including a prior. To see this, consider the problem of maximizing a likelihood function:</p>
<p>\[<br/>
\theta_{MLE} = \arg \max_{\theta} P(D | \theta)<br/>
\]</p>
<p>If one instead attempts to maximize the posterior probability of the data, the solution becomes:</p>
<p>\[<br/>
\theta_{MAP} = \arg \max_{\theta} \frac{P(D | \theta) P(\theta)}{P(D)} = \arg \max_{\theta} \frac{1}{Z} P(D | \theta) P(\theta)<br/>
\]</p>
<p>Because maxima are not shifted by monotonic transformations, we can take logs and remove constants to get:</p>
<p>\[<br/>
\theta_{MAP} = \arg \max_{\theta} \log(P(D | \theta) P(\theta))<br/>
\]</p>
<p>Using the basic properties of logs, this implies that:</p>
<p>\[<br/>
\theta_{MAP} = \arg \max_{\theta} \log(P(D | \theta)) + \log(P(\theta))<br/>
\]</p>
<p>This implies that MAP estimates are penalized MLE's. Indeed, many common regularization schemes can be described using priors over \(\theta\):</p>
<ul>
<li>Ridge</li>
<li>LASSO</li>
</ul>
</body>
</html>