{"resolved":true,"match_type":"uuid","source_type":"original_book","book_id":"cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","title":"智慧的疆界 从图灵机到人工智能","author":null,"edition":null,"publisher":null,"year":null,"isbn":"9787111778004","requested_by":"printed_page","book":{"source_type":"original_book","book_id":"cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","id":"cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","uuid":"cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","title":"智慧的疆界 从图灵机到人工智能","normalized_title":"智慧的疆界从图灵机到人工智能","author":null,"edition":null,"publisher":null,"year":null,"isbn":"9787111778004","isbns":["9787111778004","9787111695714"],"parsed":true,"ai_readable":true,"document_type":"book","page_count":456,"navigation_url":"https://workshop.closeclaw.top/ai/book/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","title_url":"https://workshop.closeclaw.top/ai/title/%E6%99%BA%E6%85%A7%E7%9A%84%E7%96%86%E7%95%8C%20%E4%BB%8E%E5%9B%BE%E7%81%B5%E6%9C%BA%E5%88%B0%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD","fixed_page_url_template":"https://workshop.closeclaw.top/ai/book/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb/page/{printed_page}","path_api_url_template":"https://workshop.closeclaw.top/api/ai/book/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb/page/{printed_page}","read_url":"https://workshop.closeclaw.top/api/ai/read/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb","page_url":"https://workshop.closeclaw.top/api/ai/books/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb/page","search_url":"https://workshop.closeclaw.top/api/ai/books/cf76bc49-7e9b-4f42-bee8-3050a9fb9feb/search"},"page":{"printed_page":251,"pdf_page":266,"physical_index":266,"physical_index_base":0,"printed_page_status":"recognized","page_sha256":"db046d1cd3bb234fd8570127c88d7afb239e1584b6d1df02773e688a681eec0d"},"chapter":"内容简介","content":"那么，对于这三个神经元而言，它们之间的变换关系就是函数嵌套关系，微积分中称这种嵌套函数为“复合函数”，记作 $( f \\circ g ) ( x )$ ，如以下式子所示：\n\n$$\ny = f ( u ) , \\ u = \\ g ( x ) , \\ \\not \\equiv \\ y = \\ f ( g ( x ) ) \\ \\not \\equiv \\not \\equiv \\not \\equiv \\not \\langle \\not { F } ( f \\circ g ) ( x )\n$$\n\n对于这种复合函数的求导，通常使用微积分中非常经典而且重要的“链式法则”（Chaining Rule）来处理，这条法则的含义是：“由两个函数组成的复合函数，其导数等于内部函数代入外部函数值之导数乘以内部函数的导数。”具体应用到上述讨论的场景中，经过 $f ( )$ 和 $g ( \\ u )$ 两层函数变换后，x 对最终输出y施加的影响，可以通过导数 $\\cdot \\frac { \\partial y } { \\partial x }$ 体现出来。根据链式法则， $\\frac { \\partial y } { \\partial x }$ 的计算方法为\n\n$$\n1 \\sharp \\vec { \\chi } \\ : ; \\ : \\varkappa : \\ : y = f ( u ) , u = g ( x )\n$$\n\n$$\n\\mathbb { H } \\mathbb { 3 } \\textrm {  : } \\frac { \\partial \\boldsymbol { y } } { \\partial \\boldsymbol { x } } = \\frac { \\partial \\boldsymbol { y } } { \\partial \\boldsymbol { u } } \\cdot \\frac { \\partial \\boldsymbol { u } } { \\partial \\boldsymbol { x } }\n$$\n\n基于这个场景的计算过程是最简单的，只涉及三个神经元，而更贴近实际的场景是在全连接前馈神经网络里，输出层的任意一个神经元y 都会受到其前一个隐层所有神经元的共同影响，而每一个隐层神经元 $u _ { i }$ 也都受到所有输入层神经元 $x _ { i }$ 的输入值的共同影响，那对于单独任一个输入值 x 求导数，其计算过程如下所示：\n\n假设： $y = f ( u _ { 1 } , u _ { 2 } , \\cdots , u _ { k } ) , ~ u _ { i } = { g } _ { i } ( x ) , ~ i = ~ 1 , \\cdots , k$\n\n$$\n\\mathbb { H } \\mathbb { 3 } \\ ; \\ \\underline { { \\partial } } \\underline { { y } } = \\ \\sum _ { i = 1 } ^ { k } \\frac { \\partial { y } } { \\partial { u } _ { i } } \\ \\bullet \\ \\frac { \\partial { u } _ { i } } { \\partial { x } }\n$$\n\n第 7 章 深度学习时代\n\n251","previous_page":250,"next_page":252}