{"id":1840,"date":"2025-02-18T10:08:13","date_gmt":"2025-02-18T02:08:13","guid":{"rendered":"https:\/\/www.forillusion.com\/?p=1840"},"modified":"2025-02-18T10:16:01","modified_gmt":"2025-02-18T02:16:01","slug":"6-2-rnn","status":"publish","type":"post","link":"https:\/\/www.forillusion.com\/index.php\/6-2-rnn\/","title":{"rendered":"6.2 \u5faa\u73af\u795e\u7ecf\u7f51\u7edc"},"content":{"rendered":"\n<p><div class=\"has-toc have-toc\"><\/div><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4e0d\u542b\u9690\u85cf\u72b6\u6001\u7684\u795e\u7ecf\u7f51\u7edc<\/h2>\n\n\n\n<p>\u8003\u8651\u4e00\u4e2a\u542b\u5355\u9690\u85cf\u5c42\u7684\u591a\u5c42\u611f\u77e5\u673a\u3002\u7ed9\u5b9a\u6837\u672c\u6570\u4e3a$n$\u3001\u8f93\u5165\u4e2a\u6570\uff08\u7279\u5f81\u6570\u6216\u7279\u5f81\u5411\u91cf\u7ef4\u5ea6\uff09\u4e3a$d$\u7684\u5c0f\u6279\u91cf\u6570\u636e\u6837\u672c$\\boldsymbol{X} \\in \\mathbb{R}^{n \\times d}$\u3002\u8bbe\u9690\u85cf\u5c42\u7684\u6fc0\u6d3b\u51fd\u6570\u4e3a$\\phi$\uff0c\u90a3\u4e48\u9690\u85cf\u5c42\u7684\u8f93\u51fa$\\boldsymbol{H} \\in \\mathbb{R}^{n \\times h}$\u8ba1\u7b97\u4e3a<\/p>\n\n\n\n<p>$$<br>\\boldsymbol{H} = \\phi(\\boldsymbol{X} \\boldsymbol{W}_{xh} + \\boldsymbol{b}_h),<br>$$<\/p>\n\n\n\n<p>\u5176\u4e2d\u9690\u85cf\u5c42\u6743\u91cd\u53c2\u6570$\\boldsymbol{W}_{xh} \\in \\mathbb{R}^{d \\times h}$\uff0c\u9690\u85cf\u5c42\u504f\u5dee\u53c2\u6570 $\\boldsymbol{b}_h \\in \\mathbb{R}^{1 \\times h}$\uff0c$h$\u4e3a\u9690\u85cf\u5355\u5143\u4e2a\u6570\u3002\u628a\u9690\u85cf\u53d8\u91cf$\\boldsymbol{H}$\u4f5c\u4e3a\u8f93\u51fa\u5c42\u7684\u8f93\u5165\uff0c\u4e14\u8bbe\u8f93\u51fa\u4e2a\u6570\u4e3a$q$\uff08\u5982\u5206\u7c7b\u95ee\u9898\u4e2d\u7684\u7c7b\u522b\u6570\uff09\uff0c\u8f93\u51fa\u5c42\u7684\u8f93\u51fa\u4e3a<\/p>\n\n\n\n<p>$$<br>\\boldsymbol{O} = \\boldsymbol{H} \\boldsymbol{W}_{hq} + \\boldsymbol{b}_q,<br>$$<\/p>\n\n\n\n<p>\u5176\u4e2d\u8f93\u51fa\u53d8\u91cf$\\boldsymbol{O} \\in \\mathbb{R}^{n \\times q}$, \u8f93\u51fa\u5c42\u6743\u91cd\u53c2\u6570$\\boldsymbol{W}_{hq} \\in \\mathbb{R}^{h \\times q}$, \u8f93\u51fa\u5c42\u504f\u5dee\u53c2\u6570$\\boldsymbol{b}_q \\in \\mathbb{R}^{1 \\times q}$\u3002\u5982\u679c\u662f\u5206\u7c7b\u95ee\u9898\uff0c\u53ef\u4ee5\u4f7f\u7528$\\text{softmax}(\\boldsymbol{O})$\u6765\u8ba1\u7b97\u8f93\u51fa\u7c7b\u522b\u7684\u6982\u7387\u5206\u5e03\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u542b\u9690\u85cf\u72b6\u6001\u7684\u5faa\u73af\u795e\u7ecf\u7f51\u7edc<\/h2>\n\n\n\n<p>\u73b0\u5728\u8003\u8651\u8f93\u5165\u6570\u636e\u5b58\u5728\u65f6\u95f4\u76f8\u5173\u6027\u7684\u60c5\u51b5\u3002\u5047\u8bbe$\\boldsymbol{X}_t \\in \\mathbb{R}^{n \\times d}$\u662f\u5e8f\u5217\u4e2d\u65f6\u95f4\u6b65$t$\u7684\u5c0f\u6279\u91cf\u8f93\u5165\uff0c$\\boldsymbol{H}_t \\in \\mathbb{R}^{n \\times h}$\u662f\u8be5\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u53d8\u91cf\u3002\u4e0e\u591a\u5c42\u611f\u77e5\u673a\u4e0d\u540c\u7684\u662f\uff0c\u5efa\u7acb\u4e00\u4e2a\u9690\u85cf\u53d8\u91cf$\\boldsymbol{H}_{t-1}$\uff0c\u7528\u6765\u4fdd\u5b58\u4e0a\u4e00\u65f6\u95f4\u6b65\uff0c\u5e76\u5f15\u5165\u4e00\u4e2a\u65b0\u7684\u6743\u91cd\u53c2\u6570$\\boldsymbol{W}_{hh} \\in \\mathbb{R}^{h \\times h}$\uff0c\u5373\u9690\u85cf\u53d8\u91cf\u7684\u6743\u91cd\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u65f6\u95f4\u6b65$t$\u7684\u9690\u85cf\u53d8\u91cf\u7684\u8ba1\u7b97\u7531\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u8f93\u5165\u548c\u4e0a\u4e00\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u53d8\u91cf\u5171\u540c\u51b3\u5b9a\uff1a<\/p>\n\n\n\n<p>$$<br>\\boldsymbol{H}_t = \\phi(\\boldsymbol{X}_t \\boldsymbol{W}_{xh} + \\boldsymbol{H}_{t-1} \\boldsymbol{W}_{hh} + \\boldsymbol{b}_h).<br>$$<\/p>\n\n\n\n<p>\u4e0e\u591a\u5c42\u611f\u77e5\u673a\u76f8\u6bd4\uff0c\u5728\u8fd9\u91cc\u6dfb\u52a0\u4e86$\\boldsymbol{H}_{t-1} \\boldsymbol{W}_{hh}$\u4e00\u9879\u3002\u7531\u4e0a\u5f0f\u4e2d\u76f8\u90bb\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u53d8\u91cf$\\boldsymbol{H}_t$\u548c$\\boldsymbol{H}_{t-1}$\u4e4b\u95f4\u7684\u5173\u7cfb\u53ef\u77e5\uff0c\u8fd9\u91cc\u7684\u9690\u85cf\u53d8\u91cf\u80fd\u591f\u6355\u6349\u622a\u81f3\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u5e8f\u5217\u7684\u5386\u53f2\u4fe1\u606f\uff0c\u5c31\u50cf\u662f\u795e\u7ecf\u7f51\u7edc\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u72b6\u6001\u6216\u8bb0\u5fc6\u4e00\u6837\u3002\u56e0\u6b64\uff0c\u8be5\u9690\u85cf\u53d8\u91cf\u4e5f\u79f0\u4e3a\u9690\u85cf\u72b6\u6001\u3002\u7531\u4e8e\u9690\u85cf\u72b6\u6001\u5728\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u5b9a\u4e49\u4f7f\u7528\u4e86\u4e0a\u4e00\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u72b6\u6001\uff0c\u4e0a\u5f0f\u7684\u8ba1\u7b97\u662f\u5faa\u73af\u7684\u3002\u4f7f\u7528\u5faa\u73af\u8ba1\u7b97\u7684\u7f51\u7edc\u5373\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\uff08recurrent neural network\uff09\u3002<\/p>\n\n\n\n<p>\u5728\u65f6\u95f4\u6b65$t$\uff0c\u8f93\u51fa\u5c42\u7684\u8f93\u51fa\u548c\u591a\u5c42\u611f\u77e5\u673a\u4e2d\u7684\u8ba1\u7b97\u7c7b\u4f3c\uff1a<\/p>\n\n\n\n<p>$$<br>\\boldsymbol{O}_t = \\boldsymbol{H}_t \\boldsymbol{W}_{hq} + \\boldsymbol{b}_q.<br>$$<\/p>\n\n\n\n<p>\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u53c2\u6570\u5305\u62ec\u9690\u85cf\u5c42\u7684\u6743\u91cd$\\boldsymbol{W}_{xh} \\in \\mathbb{R}^{d \\times h}$\u3001$\\boldsymbol{W}_{hh} \\in \\mathbb{R}^{h \\times h}$\u548c\u504f\u5dee $\\boldsymbol{b}_h \\in \\mathbb{R}^{1 \\times h}$\uff0c\u4ee5\u53ca\u8f93\u51fa\u5c42\u7684\u6743\u91cd$\\boldsymbol{W}_{hq} \\in \\mathbb{R}^{h \\times q}$\u548c\u504f\u5dee$\\boldsymbol{b}_q \\in \\mathbb{R}^{1 \\times q}$\u3002\u503c\u5f97\u4e00\u63d0\u7684\u662f\uff0c\u5373\u4fbf\u5728\u4e0d\u540c\u65f6\u95f4\u6b65\uff0c\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u4e5f\u59cb\u7ec8\u4f7f\u7528\u8fd9\u4e9b\u6a21\u578b\u53c2\u6570\u3002\u56e0\u6b64\uff0c\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u53c2\u6570\u7684\u6570\u91cf\u4e0d\u968f\u65f6\u95f4\u6b65\u7684\u589e\u52a0\u800c\u589e\u957f\u3002<\/p>\n\n\n\n<p>\u4e0b\u56fe\u5c55\u793a\u4e86\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u57283\u4e2a\u76f8\u90bb\u65f6\u95f4\u6b65\u7684\u8ba1\u7b97\u903b\u8f91\u3002\u5728\u65f6\u95f4\u6b65$t$\uff0c\u9690\u85cf\u72b6\u6001\u7684\u8ba1\u7b97\u53ef\u4ee5\u770b\u6210\u662f\u5c06\u8f93\u5165$\\boldsymbol{X}_t$\u548c\u524d\u4e00\u65f6\u95f4\u6b65\u9690\u85cf\u72b6\u6001$\\boldsymbol{H}_{t-1}$\u8fde\u7ed3\u540e\u8f93\u5165\u4e00\u4e2a\u6fc0\u6d3b\u51fd\u6570\u4e3a$\\phi$\u7684\u5168\u8fde\u63a5\u5c42\u3002\u8be5\u5168\u8fde\u63a5\u5c42\u7684\u8f93\u51fa\u5c31\u662f\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u9690\u85cf\u72b6\u6001$\\boldsymbol{H}_t$\uff0c\u4e14\u6a21\u578b\u53c2\u6570\u4e3a$\\boldsymbol{W}_{xh}$\u4e0e$\\boldsymbol{W}_{hh}$\u7684\u8fde\u7ed3\uff0c\u504f\u5dee\u4e3a$\\boldsymbol{b}_h$\u3002\u5f53\u524d\u65f6\u95f4\u6b65$t$\u7684\u9690\u85cf\u72b6\u6001$\\boldsymbol{H}_t$\u5c06\u53c2\u4e0e\u4e0b\u4e00\u4e2a\u65f6\u95f4\u6b65$t+1$\u7684\u9690\u85cf\u72b6\u6001$\\boldsymbol{H}_{t+1}$\u7684\u8ba1\u7b97\uff0c\u5e76\u8f93\u5165\u5230\u5f53\u524d\u65f6\u95f4\u6b65\u7684\u5168\u8fde\u63a5\u8f93\u51fa\u5c42\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/6.2_rnn.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/6.2_rnn.png\" alt=\"\"\/><\/figure><\/noscript>\n\n\n\n<p>\u9690\u85cf\u72b6\u6001\u4e2d$\\boldsymbol{X}_t \\boldsymbol{W}_{xh} + \\boldsymbol{H}_{t-1} \\boldsymbol{W}_{hh}$\u7684\u8ba1\u7b97\u7b49\u4ef7\u4e8e$\\boldsymbol{X}_t$\u4e0e$\\boldsymbol{H}_{t-1}$\u8fde\u7ed3\u540e\u7684\u77e9\u9635\u4e58\u4ee5$\\boldsymbol{W}_{xh}$\u4e0e$\\boldsymbol{W}_{hh}$\u8fde\u7ed3\u540e\u7684\u77e9\u9635\u3002\u9996\u5148\uff0c\u6784\u9020\u77e9\u9635<code>X<\/code>\u3001<code>W_xh<\/code>\u3001<code>H<\/code>\u548c<code>W_hh<\/code>\uff0c\u5b83\u4eec\u7684\u5f62\u72b6\u5206\u522b\u4e3a(3, 1)\u3001(1, 4)\u3001(3, 4)\u548c(4, 4)\u3002\u5c06<code>X<\/code>\u4e0e<code>W_xh<\/code>\u3001<code>H<\/code>\u4e0e<code>W_hh<\/code>\u5206\u522b\u76f8\u4e58\uff0c\u518d\u628a\u4e24\u4e2a\u4e58\u6cd5\u8fd0\u7b97\u7684\u7ed3\u679c\u76f8\u52a0\uff0c\u5f97\u5230\u5f62\u72b6\u4e3a(3, 4)\u7684\u77e9\u9635\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>import torch\n\nX, W\\_xh = torch.randn(3, 1), torch.randn(1, 4)\nH, W\\_hh = torch.randn(3, 4), torch.randn(4, 4)\nprint(torch.matmul(X, W_xh) + torch.matmul(H, W_hh))<\/code><\/pre>\n\n\n\n<p>\u8f93\u51fa\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tensor(&#91;&#91; 5.2633, -3.2288,  0.6037, -1.3321],\n        &#91; 9.4012, -6.7830,  1.0630, -0.1809],\n        &#91; 7.0355, -2.2361,  0.7469, -3.4667]])<\/code><\/pre>\n\n\n\n<p>\u5c06\u77e9\u9635<code>X<\/code>\u548c<code>H<\/code>\u6309\u5217\uff08\u7ef4\u5ea61\uff09\u8fde\u7ed3\uff0c\u8fde\u7ed3\u540e\u7684\u77e9\u9635\u5f62\u72b6\u4e3a(3, 5)\u3002\u53ef\u89c1\uff0c\u8fde\u7ed3\u540e\u77e9\u9635\u5728\u7ef4\u5ea61\u7684\u957f\u5ea6\u4e3a\u77e9\u9635<code>X<\/code>\u548c<code>H<\/code>\u5728\u7ef4\u5ea61\u7684\u957f\u5ea6\u4e4b\u548c\uff08$1+4$\uff09\u3002\u7136\u540e\uff0c\u5c06\u77e9\u9635<code>W_xh<\/code>\u548c<code>W_hh<\/code>\u6309\u884c\uff08\u7ef4\u5ea60\uff09\u8fde\u7ed3\uff0c\u8fde\u7ed3\u540e\u7684\u77e9\u9635\u5f62\u72b6\u4e3a(5, 4)\u3002\u6700\u540e\u5c06\u4e24\u4e2a\u8fde\u7ed3\u540e\u7684\u77e9\u9635\u76f8\u4e58\uff0c\u5f97\u5230\u4e0e\u4e0a\u9762\u4ee3\u7801\u8f93\u51fa\u76f8\u540c\u7684\u5f62\u72b6\u4e3a(3, 4)\u7684\u77e9\u9635\u3002<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>print(torch.matmul(torch.cat((X, H), dim=1), torch.cat((W_xh, W_hh), dim=0)))<\/code><\/pre>\n\n\n\n<p>\u8f93\u51fa\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-code\"><code>tensor(&#91;&#91; 5.2633, -3.2288,  0.6037, -1.3321],\n        &#91; 9.4012, -6.7830,  1.0630, -0.1809],\n        &#91; 7.0355, -2.2361,  0.7469, -3.4667]])<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">\u5e94\u7528\uff1a\u57fa\u4e8e\u5b57\u7b26\u7ea7\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u8bed\u8a00\u6a21\u578b<\/h2>\n\n\n\n<p>\u8bbe\u5c0f\u6279\u91cf\u4e2d\u6837\u672c\u6570\u4e3a1\uff0c\u6587\u672c\u5e8f\u5217\u4e3a\u201c\u60f3\u201d\u201c\u8981\u201d\u201c\u6709\u201d\u201c\u76f4\u201d\u201c\u5347\u201d\u201c\u673a\u201d\u3002\u4e0b\u56fe\u6f14\u793a\u4e86\u5982\u4f55\u4f7f\u7528\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u57fa\u4e8e\u5f53\u524d\u548c\u8fc7\u53bb\u7684\u5b57\u7b26\u6765\u9884\u6d4b\u4e0b\u4e00\u4e2a\u5b57\u7b26\u3002\u5728\u8bad\u7ec3\u65f6\uff0c\u5bf9\u6bcf\u4e2a\u65f6\u95f4\u6b65\u7684\u8f93\u51fa\u5c42\u8f93\u51fa\u4f7f\u7528softmax\u8fd0\u7b97\uff0c\u7136\u540e\u4f7f\u7528\u4ea4\u53c9\u71b5\u635f\u5931\u51fd\u6570\u6765\u8ba1\u7b97\u5b83\u4e0e\u6807\u7b7e\u7684\u8bef\u5dee\u3002\u5728\u56fe\u4e2d\uff0c\u7531\u4e8e\u9690\u85cf\u5c42\u4e2d\u9690\u85cf\u72b6\u6001\u7684\u5faa\u73af\u8ba1\u7b97\uff0c\u65f6\u95f4\u6b653\u7684\u8f93\u51fa$\\boldsymbol{O}_3$\u53d6\u51b3\u4e8e\u6587\u672c\u5e8f\u5217\u201c\u60f3\u201d\u201c\u8981\u201d\u201c\u6709\u201d\u3002 \u7531\u4e8e\u8bad\u7ec3\u6570\u636e\u4e2d\u8be5\u5e8f\u5217\u7684\u4e0b\u4e00\u4e2a\u8bcd\u4e3a\u201c\u76f4\u201d\uff0c\u65f6\u95f4\u6b653\u7684\u635f\u5931\u5c06\u53d6\u51b3\u4e8e\u8be5\u65f6\u95f4\u6b65\u57fa\u4e8e\u5e8f\u5217\u201c\u60f3\u201d\u201c\u8981\u201d\u201c\u6709\u201d\u751f\u6210\u4e0b\u4e00\u4e2a\u8bcd\u7684\u6982\u7387\u5206\u5e03\u4e0e\u8be5\u65f6\u95f4\u6b65\u7684\u6807\u7b7e\u201c\u76f4\u201d\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image\"><img decoding=\"async\"   class=\"lazyload\" data-src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/6.2_rnn-train.png\" src=\"https:\/\/cdn.forillusion.com\/moezx\/img\/svg\/loader\/trans.ajax-spinner-preloader.svg\" onerror=\"imgError(this)\"  alt=\"\"\/><\/figure >\n<noscript><img decoding=\"async\" src=\"https:\/\/cos.forillusion.top\/wp-content\/uploads\/2025\/02\/6.2_rnn-train.png\" 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