Repository navigation
Expand file tree
/
Copy pathdynamic-programming.html
More file actions
1460 lines (884 loc) · 51.7 KB
/
Copy pathdynamic-programming.html
File metadata and controls
1460 lines (884 loc) · 51.7 KB
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
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
<!DOCTYPE html>
<html class="theme-next mist use-motion" lang="zh-Hans">
<head><meta name="generator" content="Hexo 3.8.0">
<meta name="google-site-verification" content="zu-9nWphPjrzXV8v514mkHknIz4dNfHlib56-KNAu44">
<meta charset="UTF-8">
<meta http-equiv="X-UA-Compatible" content="IE=edge">
<meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1">
<meta name="theme-color" content="#222">
<script src="/lib/pace/pace.min.js?v=1.0.2"></script>
<link href="/lib/pace/pace-theme-flash.min.css?v=1.0.2" rel="stylesheet">
<meta http-equiv="Cache-Control" content="no-transform">
<meta http-equiv="Cache-Control" content="no-siteapp">
<script>
(function(i,s,o,g,r,a,m){i["DaoVoiceObject"]=r;i[r]=i[r]||function(){(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;a.charset="utf-8";m.parentNode.insertBefore(a,m)})(window,document,"script",('https:' == document.location.protocol ? 'https:' : 'http:') + "//widget.daovoice.io/widget/356f1943.js","daovoice")
daovoice('init', {
app_id: "356f1943"
});
daovoice('update');
</script>
<link href="/lib/fancybox/source/jquery.fancybox.css?v=2.1.5" rel="stylesheet" type="text/css">
<link href="/lib/font-awesome/css/font-awesome.min.css?v=4.6.2" rel="stylesheet" type="text/css">
<link href="/css/main.css?v=5.1.4" rel="stylesheet" type="text/css">
<link rel="apple-touch-icon" sizes="180x180" href="/images/apple-touch-icon.png?v=5.1.4">
<link rel="icon" type="image/png" sizes="32x32" href="/images/favicon-32x32png?v=5.1.4">
<link rel="icon" type="image/png" sizes="16x16" href="/images/favicon-16x16.png?v=5.1.4">
<link rel="mask-icon" href="/images/logo.svg?v=5.1.4" color="#222">
<meta name="keywords" content="rl,">
<link rel="alternate" href="/atom.xml" title="Keavnn'Blog" type="application/atom+xml">
<script>
(function(){
if(''){
if (prompt('请输入文章密码','') !== ''){
alert('密码错误！');
history.back();
}
}
})();
</script>
<meta name="description" content="本文介绍了强化学习问题中最简单基本的算法——动态规划（Dynamic Programming），介绍了贝尔曼方程在该算法中的应用。">
<meta name="keywords" content="rl">
<meta property="og:type" content="article">
<meta property="og:title" content="动态规划 Dynamic Programming">
<meta property="og:url" content="http://StepNeverStop.github.io/dynamic-programming.html">
<meta property="og:site_name" content="Keavnn'Blog">
<meta property="og:description" content="本文介绍了强化学习问题中最简单基本的算法——动态规划（Dynamic Programming），介绍了贝尔曼方程在该算法中的应用。">
<meta property="og:locale" content="zh-Hans">
<meta property="og:image" content="http://stepneverstop.github.io/dynamic-programming/gridworld.png">
<meta property="og:image" content="http://stepneverstop.github.io/dynamic-programming/iteration.png">
<meta property="og:image" content="http://stepneverstop.github.io/dynamic-programming/pi.png">
<meta property="og:image" content="http://stepneverstop.github.io/dynamic-programming/vi.png">
<meta property="og:image" content="http://stepneverstop.github.io/dynamic-programming/pivsvi.png">
<meta property="og:updated_time" content="2019-05-13T14:27:05.707Z">
<meta name="twitter:card" content="summary">
<meta name="twitter:title" content="动态规划 Dynamic Programming">
<meta name="twitter:description" content="本文介绍了强化学习问题中最简单基本的算法——动态规划（Dynamic Programming），介绍了贝尔曼方程在该算法中的应用。">
<meta name="twitter:image" content="http://stepneverstop.github.io/dynamic-programming/gridworld.png">
<script type="text/javascript" id="hexo.configurations">
var NexT = window.NexT || {};
var CONFIG = {
root: '/',
scheme: 'Mist',
version: '5.1.4',
sidebar: {"position":"left","display":"post","offset":12,"b2t":false,"scrollpercent":true,"onmobile":true},
fancybox: true,
tabs: true,
motion: {"enable":true,"async":true,"transition":{"post_block":"fadeIn","post_header":"slideDownIn","post_body":"slideDownIn","coll_header":"slideLeftIn","sidebar":"slideUpIn"}},
duoshuo: {
userId: '0',
author: '博主'
},
algolia: {
applicationID: '',
apiKey: '',
indexName: '',
hits: {"per_page":10},
labels: {"input_placeholder":"Search for Posts","hits_empty":"We didn't find any results for the search: ${query}","hits_stats":"${hits} results found in ${time} ms"}
}
};
</script>
<link rel="canonical" href="http://StepNeverStop.github.io/dynamic-programming.html">
<title>动态规划 Dynamic Programming | Keavnn'Blog</title>
</head>
<body itemscope="" itemtype="http://schema.org/WebPage" lang="zh-Hans">
<div class="container sidebar-position-left page-post-detail">
<div class="headband"></div>
<a href="https://github.com/StepNeverStop" class="github-corner" aria-label="View source on GitHub" rel="external nofollow" target="_blank"><svg width="80" height="80" viewbox="0 0 250 250" style="fill:#151513; color:#fff; position: absolute; top: 0; border: 0; right: 0;" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"/><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"/><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"/></svg></a><style>.github-corner:hover .octo-arm{animation:octocat-wave 560ms ease-in-out}@keyframes octocat-wave{0%,100%{transform:rotate(0)}20%,60%{transform:rotate(-25deg)}40%,80%{transform:rotate(10deg)}}@media (max-width:500px){.github-corner:hover .octo-arm{animation:none}.github-corner .octo-arm{animation:octocat-wave 560ms ease-in-out}}</style>
<header id="header" class="header" itemscope="" itemtype="http://schema.org/WPHeader">
<div class="header-inner"><div class="site-brand-wrapper">
<div class="site-meta ">
<div class="custom-logo-site-title">
<a href="/" class="brand" rel="start">
<span class="logo-line-before"><i></i></span>
<span class="site-title">Keavnn'Blog</span>
<span class="logo-line-after"><i></i></span>
</a>
</div>
<h1 class="site-subtitle" itemprop="description">If it is to be, it is up to me.</h1>
</div>
<div class="site-nav-toggle">
<button>
<span class="btn-bar"></span>
<span class="btn-bar"></span>
<span class="btn-bar"></span>
</button>
</div>
</div>
<nav class="site-nav">
<ul id="menu" class="menu">
<li class="menu-item menu-item-home">
<a href="/" rel="section">
<i class="menu-item-icon fa fa-fw fa-home"></i> <br>
首页
</a>
</li>
<li class="menu-item menu-item-about">
<a href="/about/" rel="section">
<i class="menu-item-icon fa fa-fw fa-user"></i> <br>
关于
</a>
</li>
<li class="menu-item menu-item-tags">
<a href="/tags/" rel="section">
<i class="menu-item-icon fa fa-fw fa-tags"></i> <br>
标签
</a>
</li>
<li class="menu-item menu-item-categories">
<a href="/categories/" rel="section">
<i class="menu-item-icon fa fa-fw fa-th"></i> <br>
分类
</a>
</li>
<li class="menu-item menu-item-archives">
<a href="/archives/" rel="section">
<i class="menu-item-icon fa fa-fw fa-archive"></i> <br>
归档
</a>
</li>
<li class="menu-item menu-item-search">
<a href="javascript:;" class="popup-trigger">
<i class="menu-item-icon fa fa-search fa-fw"></i> <br>
搜索
</a>
</li>
</ul>
<div class="site-search">
<div class="popup search-popup local-search-popup">
<div class="local-search-header clearfix">
<span class="search-icon">
<i class="fa fa-search"></i>
</span>
<span class="popup-btn-close">
<i class="fa fa-times-circle"></i>
</span>
<div class="local-search-input-wrapper">
<input autocomplete="off" placeholder="搜索..." spellcheck="false" type="text" id="local-search-input">
</div>
</div>
<div id="local-search-result"></div>
</div>
</div>
</nav>
</div>
</header>
<main id="main" class="main">
<div class="main-inner">
<div class="content-wrap">
<div id="content" class="content">
<div id="posts" class="posts-expand">
<article class="post post-type-normal" itemscope="" itemtype="http://schema.org/Article">
<div class="post-block">
<link itemprop="mainEntityOfPage" href="http://StepNeverStop.github.io/dynamic-programming.html">
<span hidden itemprop="author" itemscope="" itemtype="http://schema.org/Person">
<meta itemprop="name" content="Keavnn">
<meta itemprop="description" content="">
<meta itemprop="image" content="/images/Kicon.jpg">
</span>
<span hidden itemprop="publisher" itemscope="" itemtype="http://schema.org/Organization">
<meta itemprop="name" content="Keavnn'Blog">
</span>
<header class="post-header">
<h2 class="post-title" itemprop="name headline">动态规划 Dynamic Programming</h2>
<div class="post-meta">
<span class="post-time">
<span class="post-meta-item-icon">
<i class="fa fa-calendar-o"></i>
</span>
<span class="post-meta-item-text">发表于</span>
<time title="创建于" itemprop="dateCreated datePublished" datetime="2019-05-12T20:16:22+08:00">
2019-05-12
</time>
<span class="post-meta-divider">|</span>
<span class="post-meta-item-icon">
<i class="fa fa-calendar-check-o"></i>
</span>
<span class="post-meta-item-text">更新于:</span>
<time title="更新于" itemprop="dateModified" datetime="2019-05-13T22:27:05+08:00">
2019-05-13
</time>
</span>
<span class="post-category">
<span class="post-meta-divider">|</span>
<span class="post-meta-item-icon">
<i class="fa fa-folder-o"></i>
</span>
<span class="post-meta-item-text">分类于</span>
<span itemprop="about" itemscope="" itemtype="http://schema.org/Thing">
<a href="/categories/ReinforcementLearning/" itemprop="url" rel="index">
<span itemprop="name">ReinforcementLearning</span>
</a>
</span>
</span>
<div class="post-wordcount">
<span class="post-meta-item-icon">
<i class="fa fa-file-word-o"></i>
</span>
<span class="post-meta-item-text">字数统计:</span>
<span title="字数统计">
2.5k
</span>
<span class="post-meta-divider">|</span>
<span class="post-meta-item-icon">
<i class="fa fa-clock-o"></i>
</span>
<span class="post-meta-item-text">阅读时长 ≈</span>
<span title="阅读时长">
9
</span>
</div>
</div>
</header>
<div class="post-body" itemprop="articleBody">
<p>本文介绍了强化学习问题中最简单基本的算法——动态规划（Dynamic Programming），介绍了贝尔曼方程在该算法中的应用。</p>
<a id="more"></a>
<h1 id="DP的基本概念"><a href="#DP的基本概念" class="headerlink" title="DP的基本概念"></a>DP的基本概念</h1><blockquote>
<p>动态规划(dynamic programming)是运筹学的一个分支，是求解决策过程(decision process)最优化的数学方法。20世纪50年代初美国数学家R.E.Bellman等人在研究多阶段决策过程(multistep decision process)的优化问题时，提出了著名的最优化原理(principle of optimality)，把多阶段过程转化为一系列单阶段问题，利用各阶段之间的关系，逐个求解，创立了解决这类过程优化问题的新方法——动态规划。1957年出版了他的名著《Dynamic Programming》，这是该领域的第一本著作。——<a href="[https://baike.baidu.com/item/%E5%8A%A8%E6%80%81%E8%A7%84%E5%88%92/529408?fr=aladdin](https://baike.baidu.com/item/动态规划/529408?fr=aladdin">百度百科</a>)</p>
</blockquote>
<p>动态规划-DP算法指的不是单一一个算法，而是<strong>一系列可以在给定满足MDP的完全可知环境模型中计算出最优策略的算法</strong>。</p>
<p>DP的特点：</p>
<ul>
<li>Model-Based</li>
<li>Value-Based</li>
<li>Off-Policy(这个比较牵强，因为DP不涉及采样、预测，完全靠planning)</li>
</ul>
<p>DP具有很重要的理论基础作用，但是在现在的强化学习问题中，DP并不常使用，主要原因有二：</p>
<ul>
<li>需要完全可知的模型，状态空间、动作空间离散，状态转移、奖励函数可知且确定</li>
<li>计算量很大(每次更新都需要完全规划所有可能性)</li>
</ul>
<p>在一些表格型的问题中，如完全可知的迷宫，可以使用DP，但是要解决人类现实世界极其复杂的问题、任务，DP可能就有些力不从心啦。</p>
<p>其实，所有的强化学习算法都可以被认为是在<strong>不完全可知的环境</strong>中使用<strong>少量计算</strong>得到如DP效果一样的策略（最优策略）。</p>
<h1 id="算法"><a href="#算法" class="headerlink" title="算法"></a>算法</h1><p>先回顾一下之前提到的贝尔曼方程。</p>
<p>贝尔曼期望方程：</p>
<script type="math/tex; mode=display">
v_{\pi}(s) =\sum_{a}\pi(a\mid s)\sum_{s',r}p(s',r \mid s,a)\left[r+\gamma v_{\pi}(s')\right]</script><script type="math/tex; mode=display">
q_{\pi}(s,a) =\sum_{s',r}p(s',r \mid s,a)\left[r+\gamma v_{\pi}(s')\right]</script><p>贝尔曼最优方程：</p>
<script type="math/tex; mode=display">
v_{*}(s) =\max_{a}\sum_{s',r}p(s',r\mid s,a)\left[r+\gamma v_{*}(s')\right]</script><script type="math/tex; mode=display">
q_{*}(s,a) =\sum_{s',r}p(s',r\mid s,a)\left[r+\gamma v_{*}(s')\mid s,a\right]</script><p>为什么要再次提到贝尔曼方程呢？因为动态规划算法中的策略迭代、值迭代就是将贝尔曼方程给结合了起来。</p>
<p>回顾一下在<a href="./价值与贝尔曼方程.html">价值与贝尔曼方差</a>最后的例子中迭代计算$V(S)$和$Q(S,A)$，遍历所有$Q(S,A)$以计算$V(S)$，再遍历所有$V(S)$以计算$Q(S,A)$。在一个简单的场景下循环迭代至收敛就需要很大的计算量，如果在复杂场景中(还是完全可知的)迭代计算可想有多费时费力，动态规划DP下的Policy Iteration和Value Iteration减轻了计算的负担，同时又不影响收敛性。</p>
<p>引用《Reinfocement Learning : An Introduction》中的一个网格世界的例子：</p>
<p><img src="./dynamic-programming/gridworld.png" alt=""></p>
<p>在这个例子中，有1-14个非终态以及两个终态(左上角、右下角)，动作空间为上下左右四种，在边缘位置的状态，例如$S=1$，可选的动作只有左右下三种，且等概率选择每种动作，每进行一次移动，就给予-1的奖励值。智能体需要尽快的到达网格世界的出口-终态，以获得尽量少的负奖励(即累计奖励最大)。</p>
<p>如果按照<strong>先遍历所有$Q(S,A)$以计算$V(S)$，再遍历所有$V(S)$以计算$Q(S,A)$</strong>的方式计算，值函数的表格将会如下图所示：</p>
<p><img src="./dynamic-programming/iteration.png" alt=""></p>
<p>左边展示的是进行$k$次迭代，使值函数表格可以收敛，右边表示在每次迭代中，取$a=argmax_{a}q(s,a)$的策略。</p>
<p>可以发现，这样迭代$V(S)$至收敛有两个浪费算力的地方：</p>
<ol>
<li>选择动作的概率完全按照环境设置，导致计算状态$s$的值函数时，最差的动作$a$所带来的影响也被计算在其中，但其实真正执行的时候，永远不会执行该动作。</li>
<li>$k=3$与$k=10$时的策略表示一样，也就意味着，不必等到$V(S)$迭代至完全收敛就有可能可以获得最优策略，那么后续迭代完全没有用处，造成了资源浪费。</li>
</ol>
<p>策略迭代、值迭代的思想都是贪心策略，但策略迭代针对问题1通过<strong>剪裁可选动作</strong>的方式进行了优化，值迭代针对问题2通过<strong>取最大动作值函数</strong>的方式进行了优化。</p>
<h2 id="策略迭代-Policy-Iteration"><a href="#策略迭代-Policy-Iteration" class="headerlink" title="策略迭代 Policy Iteration"></a>策略迭代 Policy Iteration</h2><p>动作是通过策略产生的，因此势必需要对初始策略（GridWorld中的完全随机策略）进行替换，以达到更改动作选取概率的目的。</p>
<p>既然要在不同的阶段更改动作选择的概率，那么要进行多次策略的更改，随之而来的问题就是：</p>
<ol>
<li>值函数迭代更新到什么情况下时，开始更新策略</li>
<li>如何更新策略？</li>
</ol>
<p>针对第一个问题，我们使用<strong>策略评估Policy Evaluation</strong>的方式来解决，针对第二个问题，我们使用<strong>策略提升Policy Improvation</strong>的方式来解决，最终当新策略与旧策略相同时，终止迭代，得到最优策略。</p>
<script type="math/tex; mode=display">
\pi_{0} \xrightarrow{E} v_{\pi_{0}} \xrightarrow{I} \pi_{1} \xrightarrow{E} v_{\pi_{1}} \xrightarrow{I} \pi_{2} \xrightarrow{E} ... \xrightarrow{I} \pi_{\ast} \xrightarrow{E} v_{\pi_{\ast}}</script><h3 id="伪代码"><a href="#伪代码" class="headerlink" title="伪代码"></a>伪代码</h3><p><img src="./dynamic-programming/pi.png" alt=""></p>
<h3 id="策略评估-Policy-Evaluation"><a href="#策略评估-Policy-Evaluation" class="headerlink" title="策略评估 Policy Evaluation"></a>策略评估 Policy Evaluation</h3><p>问题：值函数迭代更新到什么情况下时，开始更新策略</p>
<p>答：设置更新幅度阈值$\theta$，当sweep(横扫)一遍状态空间计算$V(S)$时，与上次更新时的$V(S)$相比较，如果最大的更新幅度小于阈值$\theta$，即$\Delta \lt \theta$，则认为策略评估已经完成，开始进行策略更新</p>
<p>特点：使用了贝尔曼期望方程-$v_{\pi}(s)$</p>
<p><em>注：策略评估指的不是评估一个策略的好坏，而是在当前策略下评估所有的状态值，使状态值表格近似收敛。</em></p>
<h3 id="策略提升-Policy-Improvement"><a href="#策略提升-Policy-Improvement" class="headerlink" title="策略提升 Policy Improvement"></a>策略提升 Policy Improvement</h3><p>问题：如何更新策略？</p>
<p>答：根据策略评估步骤得到的值函数$V(S)$，计算$Q(S,A)$表格，选取每个状态下使动作值函数最大的动作作为新的动作集，每个动作的选择概率相同，接着进行策略评估。</p>
<p>特点：使用了贝尔曼期望方程-$q_{\pi}(s,a)$</p>
<h2 id="值迭代-Value-Iteration"><a href="#值迭代-Value-Iteration" class="headerlink" title="值迭代 Value Iteration"></a>值迭代 Value Iteration</h2><p>策略迭代的一个缺点是在得到最优策略$\pi_{\ast}$之前，需要多次更新策略，每次更新策略都会引起可选取动作的改变，这会引起在更新完策略后的前几次策略评估中值函数偏差比较大，导致在策略评估过程中需要花费大量的迭代来减小更新幅度$\Delta$，因此需要多次sweep(横扫)$V(S)$来迭代计算。</p>
<p>值迭代的思想是：<strong>能不能通过早停的方式，在不更改策略的情况下，直接一次得到最优策略？</strong></p>
<p>值迭代运用了策略评估、策略迭代的思想，并将它们融合在一起，即不更新策略，直接选择动作值函数$q(s,a)$最大的动作作为状态值$v(s)$，直接迭代出<strong>近似最优</strong>（早停，$\Delta \lt \theta$即可）状态价值函数$V_{\ast}(S)$，使用贪心策略进而得到最优策略$\pi_{\ast}$。</p>
<p>特点：使用了贝尔曼最优方程-$v_{\ast}(s)，q_{\ast}(s,a)$。</p>
<h3 id="伪代码-1"><a href="#伪代码-1" class="headerlink" title="伪代码"></a>伪代码</h3><p><img src="./dynamic-programming/vi.png" alt=""></p>
<h2 id="PI与VI的比较"><a href="#PI与VI的比较" class="headerlink" title="PI与VI的比较"></a>PI与VI的比较</h2><p><img src="./dynamic-programming/pivsvi.png" alt=""></p>
<p>相同点：</p>
<ul>
<li>在$0 \leq \gamma \lt 1$，有限MDPs环境中，两种方式都可以收敛到最优策略$\pi_{\ast}$</li>
<li>都使用了贝尔曼方程进行状态值函数的迭代</li>
</ul>
<p>不同点：</p>
<ol>
<li>收敛方式<ul>
<li>策略迭代PI包括策略评估Policy Evaluation和策略提升Policy Improvement，这两部循环迭代至策略收敛</li>
<li>值迭代VI包括找到最优状态值函数和一步提取策略，这两步不需要循环迭代，而是根据最优值函数直接得到最优策略</li>
</ul>
</li>
<li>动作改变方式<ul>
<li>策略迭代PI获得新策略$\pi_{new}$后，<strong>更改每个状态的可选动作集</strong>，多次横扫（遍历）$V(S)$</li>
<li>值迭代VI过程中不产生策略，不更改每个状态的可选动作集，但是<strong>只取每个状态下动作值函数最大的动作作为状态值</strong>，一次横扫（遍历）$V(S)$。（这里需要解释一下，虽然循环是多次遍历，但是因为max操作，每次遍历每个状态所选取的动作不一定一样，虽然策略一直是随机策略，没有产生新策略，但是计算过程没有遍历到所有动作，可以隐含的看作是一个新策略，因此每次遍历时这个“隐策略”都会改变，所以称为一次遍历。）</li>
</ul>
</li>
<li>计算方式<ul>
<li>策略迭代PI使用贝尔曼期望方程</li>
<li>值迭代VI使用贝尔曼最优方程</li>
</ul>
</li>
<li>Policy方式<ul>
<li>策略迭代PI是On-Policy</li>
<li>值迭代VI是Off-Policy</li>
</ul>
</li>
<li>稳定性检查<ul>
<li>策略迭代PI中更新策略时进行了策略稳定性检查，判断是否收敛</li>
<li>值迭代VI获得新策略$\pi$时没有进行策略稳定性检查</li>
</ul>
</li>
</ol>
<p>至于策略迭代PI与值迭代VI的收敛速度，<strong>通常情况</strong>下，PI的迭代次数更少，VI的运行时间更少。</p>
<blockquote>
<p><a href="https://stackoverflow.com/a/42493295/11483803" rel="external nofollow" target="_blank">What is the difference between value iteration and policy iteration?</a></p>
<p><a href="http://incompleteideas.net/book/RLbook2018.pdf" rel="external nofollow" target="_blank">《Reinforcement Learning : An Introduction 2nd Edition》p77</a></p>
</blockquote>
</div>
<div>
<div>
<div style="text-align:center;color: #ccc;font-size:14px;">-------------本文结束<i class="fa fa-heart"></i>感谢您的阅读-------------</div>
</div>
</div>
<div>
<div class="my_post_copyright">
<script src="//cdn.bootcss.com/clipboard.js/1.5.10/clipboard.min.js"></script>
<!-- JS库 sweetalert 可修改路径 -->
<script src="https://cdn.bootcss.com/jquery/2.0.0/jquery.min.js"></script>
<script src="https://unpkg.com/sweetalert/dist/sweetalert.min.js"></script>
<p><span>本文标题:</span><a href="/dynamic-programming.html">动态规划 Dynamic Programming</a></p>
<p><span>文章作者:</span><a href="/" title="访问 Keavnn 的个人博客">Keavnn</a></p>
<p><span>发布时间:</span>2019年05月12日 - 20:05</p>
<p><span>最后更新:</span>2019年05月13日 - 22:05</p>
<p><span>原始链接:</span><a href="/dynamic-programming.html" title="动态规划 Dynamic Programming">http://StepNeverStop.github.io/dynamic-programming.html</a>
<span class="copy-path" title="点击复制文章链接"><i class="fa fa-clipboard" data-clipboard-text="http://StepNeverStop.github.io/dynamic-programming.html" aria-label="复制成功！"></i></span>
</p>
<p><span>许可协议:</span><i class="fa fa-creative-commons"></i> <a rel="external nofollow" href="https://creativecommons.org/licenses/by-nc-sa/4.0/" target="_blank" title="Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)">署名-非商业性使用-相同方式共享 4.0 国际</a> 转载请保留原文链接及作者。</p>
</div>
<script>
var clipboard = new Clipboard('.fa-clipboard');
$(".fa-clipboard").click(function(){
clipboard.on('success', function(){
swal({
title: "",
text: '复制成功',
icon: "success",
showConfirmButton: true
});
});
});
</script>
</div>
<div>
<div style="padding: 10px 0; margin: 20px auto; width: 90%; text-align: center;">
<div>如果您获得了帮助，也可以资助一下小的啦~</div>
<button id="rewardButton" disable="enable" onclick="var qr = document.getElementById('QR'); if (qr.style.display === 'none') {qr.style.display='block';} else {qr.style.display='none'}">
<span>打赏啦</span>
</button>
<div id="QR" style="display: none;">
<div id="wechat" style="display: inline-block">
<img id="wechat_qr" src="/images/wechatpay.jpg" alt="Keavnn 微信">
<p>微信</p>
</div>
<div id="alipay" style="display: inline-block">
<img id="alipay_qr" src="/images/alipay.jpg" alt="Keavnn 支付宝">
<p>支付宝</p>
</div>
</div>
</div>
</div>
<footer class="post-footer">
<div class="post-tags">
<a href="/tags/rl/" rel="tag"> <i class="fa fa-tag"></i> rl</a>
</div>
<div class="post-nav">
<div class="post-nav-next post-nav-item">
<a href="/rl-classification.html" rel="next" title="强化学习的类别">
<i class="fa fa-chevron-left"></i> 强化学习的类别
</a>
</div>
<span class="post-nav-divider"></span>
<div class="post-nav-prev post-nav-item">
<a href="/mc-td.html" rel="prev" title="Monte Carlo and Temporal-Difference">
Monte Carlo and Temporal-Difference <i class="fa fa-chevron-right"></i>
</a>
</div>
</div>
</footer>
</div>
</article>
<div class="post-spread">
<!-- Go to www.addthis.com/dashboard to customize your tools -->
<div class="addthis_inline_share_toolbox">
<script type="text/javascript" src="//s7.addthis.com/js/300/addthis_widget.js#pubid=ra-5cefbfc88c13b0e7" async="async"></script>
</div>
</div>
</div>
</div>
<div class="comments" id="comments">
<div id="lv-container" data-id="city" data-uid="MTAyMC80MTk0NS8xODQ5MQ=="></div>
</div>
</div>
<div class="sidebar-toggle">
<div class="sidebar-toggle-line-wrap">
<span class="sidebar-toggle-line sidebar-toggle-line-first"></span>
<span class="sidebar-toggle-line sidebar-toggle-line-middle"></span>
<span class="sidebar-toggle-line sidebar-toggle-line-last"></span>
</div>
</div>
<aside id="sidebar" class="sidebar">
<div id="sidebar-dimmer"></div>
<div class="sidebar-inner">
<ul class="sidebar-nav motion-element">
<li class="sidebar-nav-toc sidebar-nav-active" data-target="post-toc-wrap">
文章目录
</li>
<li class="sidebar-nav-overview" data-target="site-overview-wrap">
站点概览
</li>
</ul>
<section class="site-overview-wrap sidebar-panel">
<div class="site-overview">
<div class="site-author motion-element" itemprop="author" itemscope="" itemtype="http://schema.org/Person">
<img class="site-author-image" itemprop="image" src="/images/Kicon.jpg" alt="Keavnn">
<p class="site-author-name" itemprop="name">Keavnn</p>
<p class="site-description motion-element" itemprop="description">If it is to be, it is up to me.</p>
</div>
<nav class="site-state motion-element">
<div class="site-state-item site-state-posts">
<a href="/archives/">
<span class="site-state-item-count">51</span>
<span class="site-state-item-name">日志</span>
</a>
</div>
<div class="site-state-item site-state-categories">
<a href="/categories/index.html">
<span class="site-state-item-count">11</span>
<span class="site-state-item-name">分类</span>
</a>
</div>
<div class="site-state-item site-state-tags">
<a href="/tags/index.html">
<span class="site-state-item-count">26</span>
<span class="site-state-item-name">标签</span>
</a>
</div>
</nav>
<div class="feed-link motion-element">
<a href="/atom.xml" rel="alternate">
<i class="fa fa-rss"></i>
RSS
</a>
</div>
<div class="links-of-author motion-element">
<span class="links-of-author-item">
<a href="https://github.com/StepNeverStop" target="_blank" title="GitHub" rel="external nofollow">
<i class="fa fa-fw fa-github"></i>GitHub</a>
</span>
<span class="links-of-author-item">
<a href="mailto:oooooooooooooooocoo@vip.qq.com" target="_blank" title="E-Mail" rel="external nofollow">
<i class="fa fa-fw fa-envelope"></i>E-Mail</a>
</span>
</div>
<div class="cc-license motion-element" itemprop="license">
<a href="https://creativecommons.org/licenses/by-nc-sa/4.0/" class="cc-opacity" target="_blank" rel="external nofollow">
<img src="/images/cc-by-nc-sa.svg" alt="Creative Commons">
</a>
</div>
<div class="links-of-blogroll motion-element links-of-blogroll-inline">
<div class="links-of-blogroll-title">
<i class="fa fa-fw fa-link"></i>
推荐阅读
</div>
<ul class="links-of-blogroll-list">
<li class="links-of-blogroll-item">
<a href="https://bluefisher.github.io" title="Fisher Chang" target="_blank" rel="external nofollow">Fisher Chang</a>
</li>
</ul>
</div>
</div>
</section>
<!--noindex-->
<section class="post-toc-wrap motion-element sidebar-panel sidebar-panel-active">
<div class="post-toc">
<div class="post-toc-content"><ol class="nav"><li class="nav-item nav-level-1"><a class="nav-link" href="#DP的基本概念"><span class="nav-number">1.</span> <span class="nav-text">DP的基本概念</span></a></li><li class="nav-item nav-level-1"><a class="nav-link" href="#算法"><span class="nav-number">2.</span> <span class="nav-text">算法</span></a><ol class="nav-child"><li class="nav-item nav-level-2"><a class="nav-link" href="#策略迭代-Policy-Iteration"><span class="nav-number">2.1.</span> <span class="nav-text">策略迭代 Policy Iteration</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#伪代码"><span class="nav-number">2.1.1.</span> <span class="nav-text">伪代码</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#策略评估-Policy-Evaluation"><span class="nav-number">2.1.2.</span> <span class="nav-text">策略评估 Policy Evaluation</span></a></li><li class="nav-item nav-level-3"><a class="nav-link" href="#策略提升-Policy-Improvement"><span class="nav-number">2.1.3.</span> <span class="nav-text">策略提升 Policy Improvement</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#值迭代-Value-Iteration"><span class="nav-number">2.2.</span> <span class="nav-text">值迭代 Value Iteration</span></a><ol class="nav-child"><li class="nav-item nav-level-3"><a class="nav-link" href="#伪代码-1"><span class="nav-number">2.2.1.</span> <span class="nav-text">伪代码</span></a></li></ol></li><li class="nav-item nav-level-2"><a class="nav-link" href="#PI与VI的比较"><span class="nav-number">2.3.</span> <span class="nav-text">PI与VI的比较</span></a></li></ol></li></ol></div>
</div>
</section>
<!--/noindex-->
</div>
</aside>
</div>
</main>
<footer id="footer" class="footer">
<div class="footer-inner">
<script async src="https://busuanzi.ibruce.info/busuanzi/2.3/busuanzi.pure.mini.js"></script>
<div class="copyright">© <span itemprop="copyrightYear">2020</span>
<span class="with-love">
<i class="fa fa-heart"></i>
</span>
<span class="author" itemprop="copyrightHolder">Keavnn</span>
<span class="post-meta-divider">|</span>
<span class="post-meta-item-icon">
<i class="fa fa-area-chart"></i>
</span>
<span class="post-meta-item-text">Site words total count:</span>
<span title="Site words total count">80.3k</span>
</div>
<div class="powered-by">
<i class="fa fa-user-md"></i><span id="busuanzi_container_site_pv">
本站总访问量<span id="busuanzi_value_site_pv"></span>次
</span>
</div>
<!-- <div class="theme-info">
<div class="powered-by"></div>
<span class="post-count">博客全站共80.3k字</span>
</div> -->
</div>
</footer>
<div class="back-to-top">
<i class="fa fa-arrow-up"></i>
<span id="scrollpercent"><span>0</span>%</span>
</div>
</div>
<script type="text/javascript">
if (Object.prototype.toString.call(window.Promise) !== '[object Function]') {
window.Promise = null;
}
</script>
<script type="text/javascript" src="/lib/jquery/index.js?v=2.1.3"></script>
<script type="text/javascript" src="/lib/fastclick/lib/fastclick.min.js?v=1.0.6"></script>
<script type="text/javascript" src="/lib/jquery_lazyload/jquery.lazyload.js?v=1.9.7"></script>