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scruel authored 2019-05-25 13:51 . images path in htmls
<!doctype html>
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<div id='write' class = 'is-node'><div class='md-toc' mdtype='toc'><p class="md-toc-content"><span class="md-toc-item md-toc-h1" data-ref="n2"><a class="md-toc-inner" href="#header-n2">1. 引言(Introduction)</a></span><span class="md-toc-item md-toc-h2" data-ref="n3"><a class="md-toc-inner" href="#header-n3">1.1 Welcome</a></span><span class="md-toc-item md-toc-h2" data-ref="n40"><a class="md-toc-inner" href="#header-n40">1.2 什么是机器学习(What is Machine Learning)</a></span><span class="md-toc-item md-toc-h2" data-ref="n79"><a class="md-toc-inner" href="#header-n79">1.3 监督学习(Supervised Learning)</a></span><span class="md-toc-item md-toc-h2" data-ref="n96"><a class="md-toc-inner" href="#header-n96">1.4 无监督学习(Unsupervised Learning)</a></span><span class="md-toc-item md-toc-h1" data-ref="n130"><a class="md-toc-inner" href="#header-n130">2 单变量线性回归(Linear Regression with One Variable)</a></span><span class="md-toc-item md-toc-h2" data-ref="n131"><a class="md-toc-inner" href="#header-n131">2.1 模型表示(Model Representation)</a></span><span class="md-toc-item md-toc-h2" data-ref="n165"><a class="md-toc-inner" href="#header-n165">2.2 代价函数(Cost Function)</a></span><span class="md-toc-item md-toc-h2" data-ref="n189"><a class="md-toc-inner" href="#header-n189">2.3 代价函数 - 直观理解1(Cost Function - Intuition I)</a></span><span class="md-toc-item md-toc-h2" data-ref="n203"><a class="md-toc-inner" href="#header-n203">2.4 代价函数 - 直观理解2(Cost Function - Intuition II)</a></span><span class="md-toc-item md-toc-h2" data-ref="n216"><a class="md-toc-inner" href="#header-n216">2.5 梯度下降(Gradient Descent)</a></span><span class="md-toc-item md-toc-h2" data-ref="n232"><a class="md-toc-inner" href="#header-n232">2.6 梯度下降直观理解(Gradient Descent Intuition)</a></span><span class="md-toc-item md-toc-h2" data-ref="n254"><a class="md-toc-inner" href="#header-n254">2.7 线性回归中的梯度下降(Gradient Descent For Linear Regression)</a></span><span class="md-toc-item md-toc-h1" data-ref="n279"><a class="md-toc-inner" href="#header-n279">3 Linear Algebra Review</a></span><span class="md-toc-item md-toc-h2" data-ref="n281"><a class="md-toc-inner" href="#header-n281">3.1 Matrices and Vectors</a></span><span class="md-toc-item md-toc-h2" data-ref="n286"><a class="md-toc-inner" href="#header-n286">3.2 Addition and Scalar Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n291"><a class="md-toc-inner" href="#header-n291">3.3 Matrix Vector Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n296"><a class="md-toc-inner" href="#header-n296">3.4 Matrix Matrix Multiplication</a></span><span class="md-toc-item md-toc-h2" data-ref="n301"><a class="md-toc-inner" href="#header-n301">3.5 Matrix Multiplication Properties</a></span><span class="md-toc-item md-toc-h2" data-ref="n306"><a class="md-toc-inner" href="#header-n306">3.6 Inverse and Transpose</a></span></p></div><h1><a name='header-n2' class='md-header-anchor '></a>1. 引言(Introduction)</h1><h2><a name='header-n3' class='md-header-anchor '></a>1.1 Welcome</h2><p>随着互联网数据不断累积,硬件不断升级迭代,在这个信息爆炸的时代,机器学习已被应用在各行各业中,可谓无处不在。</p><p>一些常见的机器学习的应用,例如:</p><ul><li>手写识别</li><li>垃圾邮件分类</li><li>搜索引擎</li><li>图像处理</li><li></li></ul><p>使用到机器学习的一些案例:</p><ul><li><p>数据挖掘</p><ul><li>网页点击流数据分析</li></ul></li><li><p>人工无法处理的工作(量大)</p><ul><li>手写识别</li><li>计算机视觉</li></ul></li><li><p>个人定制</p><ul><li>推荐系统</li></ul></li><li><p>研究大脑</p></li><li><p>……</p></li></ul><h2><a name='header-n40' class='md-header-anchor '></a>1.2 什么是机器学习(What is Machine Learning)</h2><ol start='' ><li>机器学习定义
这里主要有两种定义:</li></ol><ul><li><p>Arthur Samuel (1959). Machine Learning: Field of study that gives computers the ability to learn without being explicitly programmed.</p><p>这个定义有点不正式但提出的时间最早,来自于一个懂得计算机编程的下棋菜鸟。他编写了一个程序,但没有显式地编程每一步该怎么走,而是让计算机自己和自己对弈,并不断地计算布局的好坏,来判断什么情况下获胜的概率高,从而积累经验,好似学习,最后,这个计算机程序成为了一个比他自己还厉害的棋手。</p></li><li><p>Tom Mitchell (1998) Well-posed Learning Problem: A computer program is said to learn from experience E with respect to some <strong>task T</strong> and some <strong>performance measure P</strong>, if its performance on T, as measured by P, improves with <strong>experience E</strong>. </p><p>Tom Mitchell 的定义更为现代和正式。在过滤垃圾邮件这个例子中,电子邮件系统会根据用户对电子邮件的标记(是/不是垃圾邮件)不断学习,从而提升过滤垃圾邮件的准确率,定义中的三个字母分别代表:</p><ul><li>T(Task): 过滤垃圾邮件任务。</li><li>P(Performance): 电子邮件系统过滤垃圾邮件的准确率。</li><li>E(Experience): 用户对电子邮件的标记。</li></ul></li></ul><ol start='2' ><li><p>机器学习算法</p><p>主要有两种机器学习的算法分类</p><ol start='' ><li>监督学习</li><li>无监督学习</li></ol><p>两者的区别为<strong>是否需要人工参与数据结果的标注</strong>。这两部分的内容占比很大,并且很重要,掌握好了可以在以后的应用中节省大把大把的时间~</p><p>还有一些算法也属于机器学习领域,诸如:</p><ul><li>半监督学习: 介于监督学习于无监督学习之间</li><li>推荐算法: 没错,就是那些个买完某商品后还推荐同款的某购物网站所用的算法。</li><li>强化学习: 通过观察来学习如何做出动作,每个动作都会对环境有所影响,而环境的反馈又可以引导该学习算法。</li><li>迁移学习</li></ul></li></ol><h2><a name='header-n79' class='md-header-anchor '></a>1.3 监督学习(Supervised Learning)</h2><p>监督学习,即为教计算机如何去完成预测任务(有反馈),预先给一定数据量的输入<strong>和对应的结果</strong>即训练集,建模拟合,最后让计算机预测未知数据的结果。</p><p>监督学习一般有两种:</p><ol start='' ><li><p>回归问题(Regression)</p><p>回归问题即为预测一系列的<strong>连续值</strong></p><p>在房屋价格预测的例子中,给出了一系列的房屋面基数据,根据这些数据来预测任意面积的房屋价格。给出照片-年龄数据集,预测给定照片的年龄。</p><p><img src='images/20180105_194712.png' alt='' referrerPolicy='no-referrer' /></p></li><li><p>分类问题(Classification)</p><p>分类问题即为预测一系列的<strong>离散值</strong></p><p>即根据数据预测被预测对象属于哪个分类。</p><p>视频中举了癌症肿瘤这个例子,针对诊断结果,分别分类为良性或恶性。还例如垃圾邮件分类问题,也同样属于监督学习中的分类问题。</p><p><img src='images/20180105_194839.png' alt='' referrerPolicy='no-referrer' /></p></li></ol><p>视频中提到<strong>支持向量机</strong>这个算法,旨在解决当特征量很大的时候(特征即如癌症例子中的肿块大小,颜色,气味等各种特征),计算机内存一定会不够用的情况。<strong>支持向量机能让计算机处理无限多个特征。</strong></p><h2><a name='header-n96' class='md-header-anchor '></a>1.4 无监督学习(Unsupervised Learning)</h2><p>相对于监督学习,训练集不会有人为标注的结果(无反馈),我们<strong>不会给出</strong>结果或<strong>无法得知</strong>训练集的结果是什么样,而是单纯由计算机通过无监督学习算法自行分析,从而“得出结果”。计算机可能会把特定的数据集归为几个不同的簇,故叫做聚类算法。</p><p>无监督学习一般分为两种:</p><ol start='' ><li><p>聚类(Clustering)</p><ul><li>新闻聚合</li><li>DNA 个体聚类</li><li>天文数据分析</li><li>市场细分</li><li>社交网络分析</li></ul></li><li><p>非聚类(Non-clustering)</p><ul><li>鸡尾酒问题</li></ul></li></ol><p><strong>新闻聚合</strong></p><p>在例如谷歌新闻这样的网站中,每天后台都会收集成千上万的新闻,然后将这些新闻分组成一个个的新闻专题,这样一个又一个聚类,就是应用了无监督学习的结果。</p><p><strong>鸡尾酒问题</strong></p><p><img src='images/20180105_201639.png' alt='' referrerPolicy='no-referrer' /></p><p>在鸡尾酒会上,大家说话声音彼此重叠,几乎很难分辨出面前的人说了什么。我们很难对于这个问题进行数据标注,而这里的通过机器学习的无监督学习算法,就可以将说话者的声音同背景音乐分离出来,看视频,效果还不错呢~~。</p><p>嗯,这块是打打鸡血的,只需要一行代码就解决了问题,就是这么简单!当然,我没复现过 <sup>_</sup>……</p><p>神奇的一行代码:
<code>[W,s,v] = svd((repmat(sum(x.*x,1),size(x,1),1).*x)*x&#39;);</code></p><p><strong>编程语言建议</strong></p><p>在机器学习刚开始时,<strong>推荐使用 Octave 类的工程计算编程软件</strong>,因为在 C++ 或 Java 等编程语言中,编写对应的代码需要用到复杂的库以及要写大量的冗余代码,比较耗费时间,建议可以在学习过后再考虑使用其他语言来构建系统。
另外,在做<strong>原型搭建</strong>的时候也应该先考虑使用类似于 Octave 这种便于计算的编程软件,当其已经可以工作后,才将模型移植到其他的高级编程语言中。</p><blockquote><p>注:Octave 与 MATLAB 语法相近,由于 MATLAB 为商业软件,课程中使用开源且免费的 Octave。</p><p>机器学习领域发展迅速,现在也可使用 Tensorflow 等开源机器学习框架编写机器学习代码,这些框架十分友好,易于编写及应用。</p></blockquote><h1><a name='header-n130' class='md-header-anchor '></a>2 单变量线性回归(Linear Regression with One Variable)</h1><h2><a name='header-n131' class='md-header-anchor '></a>2.1 模型表示(Model Representation)</h2><ol start='' ><li>房价预测训练集</li></ol><figure><table><thead><tr><th>Size in <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="5.334ex" height="2.811ex" viewBox="0 -906.7 2296.6 1210.2" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E1-MJMATHI-66" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 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d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></path><path stroke-width="0" id="E1-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E1-MJMATHI-66" x="0" y="0"></use><use xlink:href="#E1-MJMATHI-65" x="550" y="0"></use><use xlink:href="#E1-MJMATHI-65" x="1016" y="0"></use><g transform="translate(1482,0)"><use xlink:href="#E1-MJMATHI-74" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E1-MJMAIN-32" x="510" y="513"></use></g></g></svg></span><script type="math/tex">feet^2</script> (<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script>)</th><th>Price ($) in 1000&#39;s(<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" 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type="math/tex">y</script>)</th></tr></thead><tbody><tr><td>2104</td><td>460</td></tr><tr><td>1416</td><td>232</td></tr><tr><td>1534</td><td>315</td></tr><tr><td>852</td><td>178</td></tr><tr><td>...</td><td>...</td></tr></tbody></table></figure><p>房价预测训练集中,同时给出了输入 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script> 和输出结果 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.154ex" height="1.877ex" viewBox="0 -504.6 497 808.1" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E27-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script>,即给出了人为标注的<strong>”正确结果“</strong>,且预测的量是连续的,属于监督学习中的回归问题。</p><ol start='2' ><li><strong>问题解决模型</strong></li></ol><p><img src='images/20180105_212048.png' alt='' referrerPolicy='no-referrer' /></p><p>其中 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: 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代表结果函数,也称为<strong>假设(hypothesis)</strong> 。假设函数根据输入(房屋的面积),给出预测结果输出(房屋的价格),即是一个 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="7.364ex" height="1.994ex" viewBox="0 -755.9 3170.6 858.4" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E7-MJMATHI-58" d="M42 0H40Q26 0 26 11Q26 15 29 27Q33 41 36 43T55 46Q141 49 190 98Q200 108 306 224T411 342Q302 620 297 625Q288 636 234 637H206Q200 643 200 645T202 664Q206 677 212 683H226Q260 681 347 681Q380 681 408 681T453 682T473 682Q490 682 490 671Q490 670 488 658Q484 643 481 640T465 637Q434 634 411 620L488 426L541 485Q646 598 646 610Q646 628 622 635Q617 635 609 637Q594 637 594 648Q594 650 596 664Q600 677 606 683H618Q619 683 643 683T697 681T738 680Q828 680 837 683H845Q852 676 852 672Q850 647 840 637H824Q790 636 763 628T722 611T698 593L687 584Q687 585 592 480L505 384Q505 383 536 304T601 142T638 56Q648 47 699 46Q734 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373T411 410T454 461Q546 568 561 587T577 618Q577 634 545 637Q528 637 528 647Q528 649 530 661Q533 676 535 679T549 683Q551 683 578 682T657 680Q684 680 713 681T746 682Q763 682 763 673Q763 669 760 657T755 643Q753 637 734 637Q662 632 617 587Q608 578 477 424L348 273L322 169Q295 62 295 57Q295 46 363 46Q379 46 384 45T390 35Q390 33 388 23Q384 6 382 4T366 1Q361 1 324 1T232 2Q170 2 138 2T102 1Q84 1 84 9Q84 14 87 24Q88 27 89 30T90 35T91 39T93 42T96 44T101 45T107 45T116 46T129 46Q168 47 180 50T198 63Q201 68 227 171L252 274L129 623Q128 624 127 625T125 627T122 629T118 631T113 633T105 634T96 635T83 636T66 637Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E7-MJMATHI-58" x="0" y="0"></use><use xlink:href="#E7-MJMAIN-2192" x="1129" y="0"></use><use xlink:href="#E7-MJMATHI-59" x="2407" y="0"></use></g></svg></span><script type="math/tex">X\to Y</script> 的映射。</p><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 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629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path stroke-width="0" id="E73-MJMAIN-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path stroke-width="0" id="E73-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E73-MJMATHI-68" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMATHI-3B8" x="814" y="-218"></use><use xlink:href="#E73-MJMAIN-28" x="1007" y="0"></use><use xlink:href="#E73-MJMATHI-78" x="1396" y="0"></use><use xlink:href="#E73-MJMAIN-29" x="1968" y="0"></use><use xlink:href="#E73-MJMAIN-3D" x="2635" y="0"></use><g transform="translate(3691,0)"><use xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMAIN-30" x="663" y="-213"></use></g><use xlink:href="#E73-MJMAIN-2B" x="4835" y="0"></use><g transform="translate(5836,0)"><use xlink:href="#E73-MJMATHI-3B8" x="0" y="0"></use><use transform="scale(0.707)" xlink:href="#E73-MJMAIN-31" x="663" y="-213"></use></g><use xlink:href="#E73-MJMATHI-78" x="6758" y="0"></use></g></svg></span><script type="math/tex">h_\theta(x)=\theta_0+\theta_1x</script>,为解决房价问题的一种可行表达式。</p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path 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100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script> 为参数,<span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script> 的变化才决定了输出结果,不同以往,这里的 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.329ex" height="1.41ex" viewBox="0 -504.6 572 607.1" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E12-MJMATHI-78" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E12-MJMATHI-78" x="0" y="0"></use></g></svg></span><script type="math/tex">x</script> 被我们<strong>视作已知</strong>(不论是数据集还是预测时的输入),所以怎样解得 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.089ex" height="2.11ex" viewBox="0 -806.1 469 908.7" role="img" focusable="false" style="vertical-align: -0.238ex;"><defs><path stroke-width="0" id="E70-MJMATHI-3B8" d="M35 200Q35 302 74 415T180 610T319 704Q320 704 327 704T339 705Q393 701 423 656Q462 596 462 495Q462 380 417 261T302 66T168 -10H161Q125 -10 99 10T60 63T41 130T35 200ZM383 566Q383 668 330 668Q294 668 260 623T204 521T170 421T157 371Q206 370 254 370L351 371Q352 372 359 404T375 484T383 566ZM113 132Q113 26 166 26Q181 26 198 36T239 74T287 161T335 307L340 324H145Q145 321 136 286T120 208T113 132Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E70-MJMATHI-3B8" x="0" y="0"></use></g></svg></span><script type="math/tex">\theta</script> 以更好地拟合数据,成了求解该问题的最终问题。</p><p>单变量,即只有一个特征(如例子中房屋的面积这个特征)。</p><h2><a name='header-n165' class='md-header-anchor '></a>2.2 代价函数(Cost Function)</h2><blockquote><p>李航《统计学习方法》一书中,损失函数与代价函数两者为<strong>同一概念</strong>,未作细分区别,全书没有和《深度学习》一书一样混用,而是统一使用<strong>损失函数</strong>来指代这类类似概念。</p><p>吴恩达(Andrew Ng)老师在其公开课中对两者做了细分。<strong>如果要听他的课做作业,不细分这两个概念是会被打小手扣分的</strong>!这也可能是因为老师发现了业内混用的乱象,想要治一治吧。</p><p><strong>损失函数</strong>(Loss/Error Function): 计算<strong>单个</strong>样本的误差。<a href='https://www.coursera.org/learn/neural-networks-deep-learning/lecture/yWaRd/logistic-regression-cost-function'>link</a></p><p><strong>代价函数</strong>(Cost Function): 计算整个训练集<strong>所有损失函数之和的平均值</strong></p><p>&nbsp;</p><p>综合考虑,本笔记对两者概念进行细分,若有所谬误,欢迎指正。</p><p><a href='https://www.zhihu.com/question/52398145/answer/298003145'>机器学习中的目标函数、损失函数、代价函数有什么区别?- 知乎</a></p></blockquote><p>&nbsp;</p><p>我们的目的在于求解预测结果 <span class="MathJax_SVG" tabindex="-1" style="font-size: 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y="674"></use></g></g></g></g></svg></span><script type="math/tex">J(\theta_0,\theta_1)=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(\hat{y}_{i}-y_{i} \right)^2=\dfrac{1}{2m}\displaystyle\sum_{i=1}^m\left(h_\theta(x_{i})-y_{i}\right)^2</script> </p><blockquote><p><span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.301ex" height="2.461ex" viewBox="0 -755.9 560.2 1059.4" role="img" focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E26-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 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focusable="false" style="vertical-align: -0.705ex;"><defs><path stroke-width="0" id="E27-MJMATHI-79" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><use xlink:href="#E27-MJMATHI-79" x="0" y="0"></use></g></svg></span><script type="math/tex">y</script> 的预测值</p><p>系数 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.657ex" height="3.278ex" viewBox="0 -956.9 713.6 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E29-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path stroke-width="0" id="E29-MJMAIN-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="matrix(1 0 0 -1 0 0)"><g transform="translate(120,0)"><rect stroke="none" width="473" height="60" x="0" y="220"></rect><use transform="scale(0.707)" xlink:href="#E29-MJMAIN-31" x="84" y="571"></use><use transform="scale(0.707)" xlink:href="#E29-MJMAIN-32" x="84" y="-531"></use></g></g></svg></span><script type="math/tex">\frac{1}{2}</script> 存在与否都不会影响结果,这里是为了在应用梯度下降时便于求解,平方的导数会抵消掉 <span class="MathJax_SVG" tabindex="-1" style="font-size: 100%; display: inline-block;"><svg xmlns:xlink="http://www.w3.org/1999/xlink" width="1.657ex" height="3.278ex" viewBox="0 -956.9 713.6 1411.3" role="img" focusable="false" style="vertical-align: -1.055ex;"><defs><path stroke-width="0" id="E29-MJMAIN-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 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