{"id":99049,"date":"2026-09-01T21:25:46","date_gmt":"2026-09-01T13:25:46","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/99049.html"},"modified":"2026-09-01T21:25:46","modified_gmt":"2026-09-01T13:25:46","slug":"ai-09_%e7%bb%9f%e8%ae%a1%e5%ad%a6%e4%b9%a0%e7%90%86%e8%ae%ba%e7%9a%84%e5%b4%9b%e8%b5%b7%ef%bc%9asvm-%e4%b8%8e-vapnik","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/99049.html","title":{"rendered":"AI-09_\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u7684\u5d1b\u8d77\uff1aSVM \u4e0e Vapnik"},"content":{"rendered":"

\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u7684\u5d1b\u8d77：SVM \u4e0e Vapnik<\/h2>\n

\u5728\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u8005\u51ed\u501f\u76f4\u89c9\u548c\u5b9e\u9a8c\u63a8\u52a8AI\u524d\u8fdb\u65f6，\u4e00\u4f4d\u683c\u9c81\u5409\u4e9a\u6570\u5b66\u5bb6\u9009\u62e9\u4e86\u53e6\u4e00\u6761\u8def\u2014\u2014\u4e3a\u673a\u5668\u5b66\u4e60\u5efa\u7acb\u4e25\u683c\u7684\u6570\u5b66\u7406\u8bba，\u5e76\u4ece\u4e2d\u63a8\u5bfc\u51fa\u6700\u4f18\u7684\u5b66\u4e60\u7b97\u6cd5\u3002<\/p>\n

\u524d\u8a00<\/h3>\n

1990\u5e74\u4ee3\u4e2d\u671f，\u673a\u5668\u5b66\u4e60\u9886\u57df\u6b63\u7ecf\u5386\u4e00\u573a\u6df1\u523b\u7684\u5206\u88c2\u3002\u4e00\u65b9\u9762，\u795e\u7ecf\u7f51\u7edc\u57281980\u5e74\u4ee3\u672b\u7ecf\u5386\u4e86\u77ed\u6682\u7684\u590d\u5174\u540e，\u518d\u6b21\u9762\u4e34\u7406\u8bba\u6839\u57fa\u4e0d\u7a33\u7684\u8d28\u7591\u2014\u2014\u6ca1\u6709\u4eba\u80fd\u7cbe\u786e\u56de\u7b54"\u4e00\u4e2a\u4e09\u5c42\u7f51\u7edc\u9700\u8981\u591a\u5c11\u8bad\u7ec3\u6837\u672c\u624d\u80fd\u6cdb\u5316"\u8fd9\u6837\u7684\u57fa\u672c\u95ee\u9898\u3002\u53e6\u4e00\u65b9\u9762，Vladimir Vapnik \u548c Alexey Chervonenkis \u5728\u8fc7\u53bb\u4e8c\u5341\u4f59\u5e74\u95f4\u9ed8\u9ed8\u6784\u5efa\u7684**\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba（Statistical Learning Theory, SLT）**\u7ec8\u4e8e\u6210\u719f\u5230\u53ef\u4ee5\u4ea7\u751f\u5b9e\u7528\u7b97\u6cd5\u7684\u7a0b\u5ea6\u3002<\/p>\n

1995\u5e74，Vapnik \u548c Corinna Cortes \u53d1\u8868\u4e86\u652f\u6301\u5411\u91cf\u673a（Support Vector Machine, SVM）\u7684\u8bba\u6587，\u8fd9\u7bc7\u8bba\u6587\u4e0d\u4ec5\u63d0\u51fa\u4e86\u4e00\u4e2a\u65b0\u7684\u5206\u7c7b\u7b97\u6cd5，\u66f4\u5ba3\u544a\u4e86\u4e00\u79cd\u65b0\u7684\u673a\u5668\u5b66\u4e60\u8303\u5f0f：\u4ece\u6570\u5b66\u7406\u8bba\u51fa\u53d1\u63a8\u5bfc\u5b66\u4e60\u7b97\u6cd5，\u800c\u975e\u4f9d\u8d56\u76f4\u89c9\u548c\u542f\u53d1\u5f0f\u3002<\/p>\n

SVM \u5728\u968f\u540e\u7684\u5341\u5e74\u95f4\u51e0\u4e4e\u7edf\u6cbb\u4e86\u6574\u4e2a\u673a\u5668\u5b66\u4e60\u9886\u57df\u2014\u2014\u4ece\u6587\u672c\u5206\u7c7b\u5230\u751f\u7269\u4fe1\u606f\u5b66，\u4ece\u624b\u5199\u8bc6\u522b\u5230\u4eba\u8138\u68c0\u6d4b，\u51e0\u4e4e\u6240\u6709\u6709\u6807\u7b7e\u6570\u636e\u7684\u5206\u7c7b\u4efb\u52a1\u4e0a，SVM \u90fd\u662f\u9996\u9009\u65b9\u6cd5\u3002\u66f4\u91cd\u8981\u7684\u662f，Vapnik \u7684\u7406\u8bba\u6846\u67b6\u4e3a\u7406\u89e3"\u5b66\u4e60\u7684\u672c\u8d28"\u63d0\u4f9b\u4e86\u6df1\u523b\u7684\u6570\u5b66\u6d1e\u5bdf，\u8fd9\u4e9b\u6d1e\u5bdf\u81f3\u4eca\u4ecd\u5728\u5f71\u54cd\u6df1\u5ea6\u5b66\u4e60\u7684\u7406\u8bba\u7814\u7a76\u3002<\/p>\n

\n

\u4e00\u3001\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u7684\u6570\u5b66\u57fa\u7840<\/h3>\n

\"\u4e00\u3001\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u7684\u6570\u5b66\u57fa\u7840\"<\/p>\n

\"\u4e00\u3001\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u7684\u6570\u5b66\u57fa\u7840\"<\/p>\n

1.1 \u5b66\u4e60\u95ee\u9898\u7684\u6570\u5b66\u5f62\u5f0f\u5316<\/h4>\n

Vapnik \u7684\u51fa\u53d1\u70b9\u662f\u5c06\u673a\u5668\u5b66\u4e60\u95ee\u9898\u5f62\u5f0f\u5316\u4e3a\u4e00\u4e2a\u7edf\u8ba1\u4f30\u8ba1\u95ee\u9898\u3002\u7ed9\u5b9a\u8bad\u7ec3\u6570\u636e {(x1,y1),(x2,y2),\u2026,(xn,yn)}\\\\{(x_1, y_1), (x_2, y_2), \\\\ldots, (x_n, y_n)\\\\}<\/span><\/span>{(<\/span>x<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>y<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span>(<\/span>x<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>y<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>(<\/span>x<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>y<\/span><\/span>n<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)}<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d xi\u2208Rdx_i \\\\in \\\\mathbb{R}^d<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>R<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，yi\u2208{\u22121,+1}y_i \\\\in \\\\{-1, +1\\\\}<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2208<\/span><\/span><\/span><\/span>{<\/span>\u2212<\/span>1<\/span>,<\/span><\/span>+<\/span>1<\/span>}<\/span><\/span><\/span><\/span><\/span>，\u6211\u4eec\u5e0c\u671b\u627e\u5230\u4e00\u4e2a\u51fd\u6570 f(x)f(x)<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span>，\u4f7f\u5176\u5728\u672a\u89c1\u6570\u636e\u4e0a\u7684\u671f\u671b\u98ce\u9669（Expected Risk）\u6700\u5c0f：<\/p>\n

R[f]=\u222bL(y,f(x))\u2009dP(x,y)R[f] = \\\\int L(y, f(x)) \\\\, dP(x, y)<\/span><\/span>R<\/span>[<\/span>f<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>L<\/span>(<\/span>y<\/span>,<\/span><\/span>f<\/span>(<\/span>x<\/span>))<\/span><\/span>d<\/span>P<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d LL<\/span><\/span>L<\/span><\/span><\/span><\/span><\/span> \u662f\u635f\u5931\u51fd\u6570，P(x,y)P(x, y)<\/span><\/span>P<\/span>(<\/span>x<\/span>,<\/span><\/span>y<\/span>)<\/span><\/span><\/span><\/span><\/span> \u662f\u6570\u636e\u7684\u771f\u5b9e\u5206\u5e03\u2014\u2014\u800c\u8fd9\u4e2a\u5206\u5e03\u6211\u4eec\u4e0d\u77e5\u9053\u3002<\/p>\n

\u8fd9\u5c31\u662f\u5b66\u4e60\u95ee\u9898\u7684\u6838\u5fc3\u56f0\u96be：\u6211\u4eec\u53ea\u80fd\u770b\u5230\u6709\u9650\u7684\u8bad\u7ec3\u6837\u672c，\u5374\u9700\u8981\u5bf9\u6574\u4e2a\u5206\u5e03\u505a\u51fa\u63a8\u65ad\u3002<\/p>\n

1.2 \u7ecf\u9a8c\u98ce\u9669\u6700\u5c0f\u5316（ERM）\u7684\u9677\u9631<\/h4>\n

\u6700\u76f4\u89c2\u7684\u5b66\u4e60\u7b56\u7565\u662f\u7ecf\u9a8c\u98ce\u9669\u6700\u5c0f\u5316（Empirical Risk Minimization, ERM）：\u7528\u8bad\u7ec3\u96c6\u4e0a\u7684\u5e73\u5747\u635f\u5931\u6765\u8fd1\u4f3c\u671f\u671b\u98ce\u9669：<\/p>\n

Remp[f]=1n\u2211i=1nL(yi,f(xi))R_{emp}[f] = \\\\frac{1}{n} \\\\sum_{i=1}^{n} L(y_i, f(x_i))<\/span><\/span>R<\/span><\/span>e<\/span>m<\/span>p<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>f<\/span>]<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>L<\/span>(<\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>f<\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>))<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

ERM \u7684\u95ee\u9898\u662f：\u5982\u679c\u6a21\u578b\u592a\u590d\u6742，\u5b83\u53ef\u4ee5\u5728\u8bad\u7ec3\u96c6\u4e0a\u8fbe\u5230\u96f6\u8bef\u5dee，\u4f46\u5728\u65b0\u6570\u636e\u4e0a\u8868\u73b0\u5f88\u5dee\u2014\u2014\u8fd9\u5c31\u662f\u8fc7\u62df\u5408\u3002<\/p>\n

Vapnik \u7684\u5173\u952e\u95ee\u9898\u662f：\u5728\u4ec0\u4e48\u6761\u4ef6\u4e0b，\u7ecf\u9a8c\u98ce\u9669\u6700\u5c0f\u5316\u80fd\u591f\u4fdd\u8bc1\u671f\u671b\u98ce\u9669\u4e5f\u8db3\u591f\u5c0f？<\/p>\n

1.3 \u7ed3\u6784\u98ce\u9669\u6700\u5c0f\u5316（SRM）<\/h4>\n

Vapnik \u63d0\u51fa\u4e86**\u7ed3\u6784\u98ce\u9669\u6700\u5c0f\u5316（Structural Risk Minimization, SRM）**\u539f\u5219，\u5176\u6838\u5fc3\u601d\u60f3\u662f：\u5728\u7ecf\u9a8c\u98ce\u9669\u548c\u6a21\u578b\u590d\u6742\u5ea6\u4e4b\u95f4\u5bfb\u627e\u5e73\u8861\u3002<\/p>\n

\u8003\u8651\u4e00\u4e2a\u5d4c\u5957\u7684\u5047\u8bbe\u7a7a\u95f4\u5e8f\u5217：<\/p>\n

H1\u2282H2\u2282H3\u2282\u22ef\\\\mathcal{H}_1 \\\\subset \\\\mathcal{H}_2 \\\\subset \\\\mathcal{H}_3 \\\\subset \\\\cdots<\/span><\/span>H<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2282<\/span><\/span><\/span><\/span>H<\/span><\/span>2<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2282<\/span><\/span><\/span><\/span>H<\/span><\/span>3<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2282<\/span><\/span><\/span><\/span>\u22ef<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d Hk\\\\mathcal{H}_k<\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u590d\u6742\u5ea6\u968f kk<\/span><\/span>k<\/span><\/span><\/span><\/span><\/span> \u589e\u52a0\u3002\u5bf9\u4e8e\u6bcf\u4e2a Hk\\\\mathcal{H}_k<\/span><\/span>H<\/span><\/span>k<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u6211\u4eec\u6709\u6cdb\u5316\u8bef\u5dee\u4e0a\u754c：<\/p>\n

R[f]\u2264Remp[f]+\u03a6(hn)R[f] \\\\leq R_{emp}[f] + \\\\Phi\\\\left(\\\\frac{h}{n}\\\\right)<\/span><\/span>R<\/span>[<\/span>f<\/span>]<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>R<\/span><\/span>e<\/span>m<\/span>p<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>f<\/span>]<\/span><\/span>+<\/span><\/span><\/span><\/span>\u03a6<\/span><\/span>(<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>h<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d \u03a6\\\\Phi<\/span><\/span>\u03a6<\/span><\/span><\/span><\/span><\/span> \u662f\u590d\u6742\u5ea6\u60e9\u7f5a\u9879，hh<\/span><\/span>h<\/span><\/span><\/span><\/span><\/span> \u662f\u5047\u8bbe\u7a7a\u95f4\u7684"\u590d\u6742\u5ea6"（VC \u7ef4），nn<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span> \u662f\u6837\u672c\u6570\u3002<\/p>\n

SRM \u539f\u5219\u544a\u8bc9\u6211\u4eec：\u9009\u62e9\u4f7f\u7ecf\u9a8c\u98ce\u9669\u548c\u590d\u6742\u5ea6\u60e9\u7f5a\u4e4b\u548c\u6700\u5c0f\u7684\u5047\u8bbe\u7a7a\u95f4\u3002\u8fd9\u4e2a\u601d\u60f3\u76f4\u63a5\u5bfc\u81f4\u4e86 SVM \u7684\u6700\u5927\u95f4\u9694\u539f\u7406\u3002<\/p>\n

\n

\u4e8c\u3001VC \u7ef4\u4e0e\u6cdb\u5316\u7406\u8bba<\/h3>\n

\"\u4e8c\u3001VC<\/p>\n

2.1 VC \u7ef4\u7684\u5b9a\u4e49<\/h4>\n

**Vapnik-Chervonenkis \u7ef4（VC \u7ef4）**\u662f\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u4e2d\u6700\u6838\u5fc3\u7684\u6982\u5ff5\u4e4b\u4e00，\u5b83\u8861\u91cf\u4e00\u4e2a\u5047\u8bbe\u7a7a\u95f4\u7684"\u8868\u8fbe\u80fd\u529b"\u6216"\u590d\u6742\u5ea6"\u3002<\/p>\n

\u5b9a\u4e49：\u5bf9\u4e8e\u4e00\u4e2a\u5047\u8bbe\u7a7a\u95f4 H\\\\mathcal{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span>，\u5982\u679c\u5b58\u5728 hh<\/span><\/span>h<\/span><\/span><\/span><\/span><\/span> \u4e2a\u6837\u672c\u70b9，\u4f7f\u5f97 H\\\\mathcal{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u80fd\u591f\u5c06\u8fd9\u4e9b\u70b9\u7684\u6240\u6709 2h2^h<\/span><\/span>2<\/span><\/span>h<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u79cd\u53ef\u80fd\u6807\u7b7e\u7ec4\u5408\u90fd\u6b63\u786e\u5206\u7c7b（\u79f0\u4e3a"\u6253\u6563"，shattering），\u5219 H\\\\mathcal{H}<\/span><\/span>H<\/span><\/span><\/span><\/span><\/span> \u7684 VC \u7ef4\u81f3\u5c11\u4e3a hh<\/span><\/span>h<\/span><\/span><\/span><\/span><\/span>\u3002VC \u7ef4\u662f\u80fd\u88ab\u6253\u6563\u7684\u6700\u5927\u6837\u672c\u6570\u3002<\/p>\n

\u76f4\u89c2\u7406\u89e3：VC \u7ef4\u8d8a\u9ad8，\u6a21\u578b\u8d8a"\u7075\u6d3b"，\u80fd\u62df\u5408\u8d8a\u590d\u6742\u7684\u6a21\u5f0f，\u4f46\u4e5f\u8d8a\u5bb9\u6613\u8fc7\u62df\u5408\u3002<\/p>\n

2.2 \u7ecf\u5178\u4f8b\u5b50<\/h4>\n\n\u5047\u8bbe\u7a7a\u95f4VC \u7ef4\u76f4\u89c2\u7406\u89e3<\/tr>\n\n\n\n\n\n
1D \u9608\u503c\u5206\u7c7b\u5668<\/td>\n1<\/td>\n\u53ea\u80fd\u5b66\u4e60"\u5927\u4e8e\/\u5c0f\u4e8e\u67d0\u4e2a\u503c"<\/td>\n<\/tr>\n
2D \u7ebf\u6027\u5206\u7c7b\u5668<\/td>\n3<\/td>\n\u53ef\u4ee5\u6253\u6563\u4efb\u610f3\u4e2a\u4e0d\u5171\u7ebf\u7684\u70b9<\/td>\n<\/tr>\n
d\u7ef4\u7ebf\u6027\u5206\u7c7b\u5668<\/td>\nd+1<\/td>\n\u53c2\u6570\u6570\u91cf\u7684\u81ea\u7531\u5ea6<\/td>\n<\/tr>\n
\u6700\u8fd1\u90bb\u5206\u7c7b\u5668<\/td>\n\u221e<\/td>\n\u53ef\u4ee5\u6253\u6563\u4efb\u610f\u6570\u91cf\u7684\u70b9<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

2.3 VC \u6cdb\u5316\u754c<\/h4>\n

VC \u7ef4\u7406\u8bba\u7684\u6838\u5fc3\u6210\u679c\u662fVC \u6cdb\u5316\u754c：<\/p>\n

P(sup\u2061f\u2208H\u2223R[f]\u2212Remp[f]\u2223>\u03f5)\u22644exp\u2061(n\u03f52\/2\u2212(h(1+ln\u2061(2n\/h)))n)P\\\\left(\\\\sup_{f \\\\in \\\\mathcal{H}} |R[f] – R_{emp}[f]| > \\\\epsilon\\\\right) \\\\leq 4 \\\\exp\\\\left(n \\\\epsilon^2 \/ 2 – \\\\frac{(h(1 + \\\\ln(2n\/h)))}{n}\\\\right)<\/span><\/span>P<\/span><\/span>(<\/span><\/span><\/span>f<\/span>\u2208<\/span>H<\/span><\/span><\/span><\/span><\/span>sup<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2223<\/span>R<\/span>[<\/span>f<\/span>]<\/span><\/span>\u2212<\/span><\/span>R<\/span><\/span>e<\/span>m<\/span>p<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span>f<\/span>]<\/span>\u2223<\/span><\/span>><\/span><\/span>\u03f5<\/span>)<\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>4<\/span><\/span>exp<\/span><\/span>(<\/span><\/span>n<\/span>\u03f5<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\/2<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>h<\/span>(<\/span>1<\/span><\/span>+<\/span><\/span>ln<\/span>(<\/span>2<\/span>n<\/span>\/<\/span>h<\/span>)))<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u8fd9\u4e2a\u516c\u5f0f\u544a\u8bc9\u6211\u4eec\u51e0\u4e2a\u5173\u952e\u4e8b\u5b9e：<\/p>\n

  • \u6cdb\u5316\u8bef\u5dee\u4e0e VC \u7ef4\u6210\u6b63\u6bd4：\u6a21\u578b\u8d8a\u590d\u6742（VC \u7ef4\u8d8a\u9ad8），\u6cdb\u5316\u8bef\u5dee\u8d8a\u5927<\/li>\n
  • \u6cdb\u5316\u8bef\u5dee\u4e0e\u6837\u672c\u6570\u6210\u53cd\u6bd4：\u8bad\u7ec3\u6570\u636e\u8d8a\u591a，\u6cdb\u5316\u8d8a\u597d<\/li>\n
  • \u5b58\u5728\u6700\u4f18\u590d\u6742\u5ea6：\u592a\u7b80\u5355\u7684\u6a21\u578b\u6b20\u62df\u5408，\u592a\u590d\u6742\u7684\u6a21\u578b\u8fc7\u62df\u5408，\u5b58\u5728\u4e00\u4e2a"\u751c\u871c\u70b9"<\/li>\n

    \u53c2\u8003\u8bba\u6587：Vapnik, V. N., & Chervonenkis, A. Y. (1974). Theory of pattern recognition. Nauka, Moscow.<\/p>\n

    2.4 VC \u7ef4\u7684\u54f2\u5b66\u610f\u4e49<\/h4>\n

    VC \u7ef4\u7406\u8bba\u5bf9\u673a\u5668\u5b66\u4e60\u7684\u54f2\u5b66\u5f71\u54cd\u6df1\u8fdc：<\/p>\n

  • \u5b66\u4e60\u7684\u53ef\u884c\u6027：\u5b83\u7b2c\u4e00\u6b21\u7528\u6570\u5b66\u8bc1\u660e\u4e86"\u4ece\u6709\u9650\u6837\u672c\u5b66\u4e60"\u662f\u53ef\u80fd\u7684，\u53ea\u8981\u5047\u8bbe\u7a7a\u95f4\u7684\u590d\u6742\u5ea6（VC \u7ef4）\u4e0e\u6837\u672c\u6570\u6210\u9002\u5f53\u6bd4\u4f8b<\/li>\n
  • \u5965\u5361\u59c6\u5243\u5200\u7684\u6570\u5b66\u5316：\u7b80\u5355\u7684\u6a21\u578b\u66f4\u597d，\u4e0d\u662f\u56e0\u4e3a\u54f2\u5b66\u504f\u597d，\u800c\u662f\u56e0\u4e3a\u6570\u5b66\u4e0a\u53ef\u4ee5\u8bc1\u660e\u5176\u6cdb\u5316\u8bef\u5dee\u66f4\u5c0f<\/li>\n
  • \u6a21\u578b\u9009\u62e9\u7684\u7406\u8bba\u4f9d\u636e：\u4ea4\u53c9\u9a8c\u8bc1\u7b49\u7ecf\u9a8c\u65b9\u6cd5\u7ec8\u4e8e\u6709\u4e86\u7406\u8bba\u57fa\u7840<\/li>\n
    \n

    \u4e09\u3001SVM \u7684\u6700\u5927\u95f4\u9694\u539f\u7406<\/h3>\n

    3.1 \u4ece VC \u7ef4\u5230\u6700\u5927\u95f4\u9694<\/h4>\n

    Vapnik \u7684\u5929\u624d\u5728\u4e8e：\u5c06 VC \u7ef4\u6700\u5c0f\u5316\u7684\u76ee\u6807\u8f6c\u5316\u4e3a\u4e00\u4e2a\u51e0\u4f55\u4f18\u5316\u95ee\u9898\u3002<\/p>\n

    \u5bf9\u4e8e\u7ebf\u6027\u5206\u7c7b\u5668 f(x)=w\u22c5x+bf(x) = w \\\\cdot x + b<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>w<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span><\/span>，\u5176 VC \u7ef4\u4e0e\u6743\u91cd\u5411\u91cf ww<\/span><\/span>w<\/span><\/span><\/span><\/span><\/span> \u7684\u8303\u6570\u6210\u6b63\u6bd4\u3002\u6700\u5c0f\u5316 VC \u7ef4\u7b49\u4ef7\u4e8e\u6700\u5c0f\u5316 \u2225w\u22252\\\\|w\\\\|^2<\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u3002\u540c\u65f6，\u4e3a\u4e86\u6b63\u786e\u5206\u7c7b\u8bad\u7ec3\u6570\u636e，\u6211\u4eec\u9700\u8981：<\/p>\n

    yi(w\u22c5xi+b)\u22651,\u2200iy_i(w \\\\cdot x_i + b) \\\\geq 1, \\\\quad \\\\forall i<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>w<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>)<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span><\/span>\u2200<\/span>i<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u8fd9\u4e24\u4e2a\u76ee\u6807\u7ed3\u5408，\u5c31\u5f97\u5230\u4e86\u786c\u95f4\u9694 SVM \u7684\u4f18\u5316\u95ee\u9898：<\/p>\n

    min\u2061w,b12\u2225w\u22252\\\\min_{w, b} \\\\frac{1}{2} \\\\|w\\\\|^2<\/span><\/span><\/span>w<\/span>,<\/span>b<\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \ns.t.yi(w\u22c5xi+b)\u22651,\u2200i=1,\u2026,n\\\\text{s.t.} \\\\quad y_i(w \\\\cdot x_i + b) \\\\geq 1, \\\\quad \\\\forall i = 1, \\\\ldots, n<\/span><\/span>s.t.<\/span><\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>w<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>)<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span><\/span>\u2200<\/span>i<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span>,<\/span><\/span>\u2026<\/span><\/span>,<\/span><\/span>n<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u51e0\u4f55\u89e3\u91ca：1\u2225w\u2225\\\\frac{1}{\\\\|w\\\\|}<\/span><\/span><\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u5206\u79bb\u8d85\u5e73\u9762\u5230\u6700\u8fd1\u6570\u636e\u70b9\u7684\u8ddd\u79bb（\u95f4\u9694）\u3002\u6700\u5c0f\u5316 \u2225w\u22252\\\\|w\\\\|^2<\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7b49\u4ef7\u4e8e\u6700\u5927\u5316\u95f4\u9694\u3002<\/p>\n

    3.2 \u8f6f\u95f4\u9694\u4e0e\u677e\u5f1b\u53d8\u91cf<\/h4>\n

    \u73b0\u5b9e\u4e16\u754c\u7684\u6570\u636e\u5f88\u5c11\u662f\u7ebf\u6027\u53ef\u5206\u7684\u3002Cortes \u548c Vapnik \u57281995\u5e74\u63d0\u51fa\u4e86\u8f6f\u95f4\u9694 SVM，\u5141\u8bb8\u90e8\u5206\u6837\u672c\u8fdd\u53cd\u95f4\u9694\u7ea6\u675f：<\/p>\n

    min\u2061w,b,\u03be12\u2225w\u22252+C\u2211i=1n\u03bei\\\\min_{w, b, \\\\xi} \\\\frac{1}{2} \\\\|w\\\\|^2 + C \\\\sum_{i=1}^{n} \\\\xi_i<\/span><\/span><\/span>w<\/span>,<\/span>b<\/span>,<\/span>\u03be<\/span><\/span><\/span><\/span><\/span>min<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>C<\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03be<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
    \ns.t.yi(w\u22c5xi+b)\u22651\u2212\u03bei,\u03bei\u22650\\\\text{s.t.} \\\\quad y_i(w \\\\cdot x_i + b) \\\\geq 1 – \\\\xi_i, \\\\quad \\\\xi_i \\\\geq 0<\/span><\/span>s.t.<\/span><\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>w<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>)<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>\u03be<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>\u03be<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2265<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

    \u5176\u4e2d \u03bei\\\\xi_i<\/span><\/span>\u03be<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u677e\u5f1b\u53d8\u91cf，\u5ea6\u91cf\u7b2c ii<\/span><\/span>i<\/span><\/span><\/span><\/span><\/span> \u4e2a\u6837\u672c\u8fdd\u53cd\u95f4\u9694\u7684\u7a0b\u5ea6；CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u662f\u6b63\u5219\u5316\u53c2\u6570，\u63a7\u5236\u95f4\u9694\u6700\u5927\u5316\u4e0e\u8bef\u5206\u7c7b\u60e9\u7f5a\u4e4b\u95f4\u7684\u6743\u8861\u3002<\/p>\n

      \n
    • CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u5f88\u5927：\u4e25\u683c\u8981\u6c42\u6bcf\u4e2a\u6837\u672c\u90fd\u6b63\u786e\u5206\u7c7b（\u53ef\u80fd\u8fc7\u62df\u5408）<\/li>\n
    • CC<\/span><\/span>C<\/span><\/span><\/span><\/span><\/span> \u5f88\u5c0f：\u5141\u8bb8\u66f4\u591a\u8bef\u5206\u7c7b，\u4f46\u95f4\u9694\u66f4\u5927（\u53ef\u80fd\u6b20\u62df\u5408）<\/li>\n<\/ul>\n

      3.3 \u5bf9\u5076\u95ee\u9898\u4e0e KKT \u6761\u4ef6<\/h4>\n

      \u901a\u8fc7\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6cd5，SVM \u7684\u539f\u59cb\u95ee\u9898\u53ef\u4ee5\u8f6c\u5316\u4e3a\u5bf9\u5076\u95ee\u9898：<\/p>\n

      max\u2061\u03b1\u2211i=1n\u03b1i\u221212\u2211i,j\u03b1i\u03b1jyiyjxi\u22c5xj\\\\max_{\\\\alpha} \\\\sum_{i=1}^{n} \\\\alpha_i – \\\\frac{1}{2} \\\\sum_{i,j} \\\\alpha_i \\\\alpha_j y_i y_j x_i \\\\cdot x_j<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span><\/span><\/span>max<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>=<\/span>1<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>i<\/span>,<\/span>j<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>
      \ns.t.0\u2264\u03b1i\u2264C,\u2211i\u03b1iyi=0\\\\text{s.t.} \\\\quad 0 \\\\leq \\\\alpha_i \\\\leq C, \\\\quad \\\\sum_{i} \\\\alpha_i y_i = 0<\/span><\/span>s.t.<\/span><\/span><\/span>0<\/span><\/span>\u2264<\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>C<\/span>,<\/span><\/span><\/span><\/span>i<\/span><\/span><\/span><\/span><\/span>\u2211<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

      \u5bf9\u5076\u95ee\u9898\u7684\u7f8e\u5999\u4e4b\u5904\u5728\u4e8e：<\/p>\n

    • \u53ea\u6d89\u53ca\u6570\u636e\u70b9\u4e4b\u95f4\u7684\u5185\u79ef xi\u22c5xjx_i \\\\cdot x_j<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>，\u8fd9\u4e3a\u6838\u65b9\u6cd5\u6253\u5f00\u4e86\u5927\u95e8<\/li>\n
    • \u5927\u591a\u6570 \u03b1i=0\\\\alpha_i = 0<\/span><\/span>\u03b1<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span>0<\/span><\/span><\/span><\/span><\/span>：\u53ea\u6709\u90a3\u4e9b\u6070\u597d\u5728\u95f4\u9694\u8fb9\u754c\u4e0a\u7684\u6837\u672c（\u652f\u6301\u5411\u91cf）\u624d\u5bf9\u51b3\u7b56\u51fd\u6570\u6709\u8d21\u732e<\/li>\n
    • \u89e3\u7684\u7a00\u758f\u6027\u4f7f\u5f97 SVM \u5728\u6d4b\u8bd5\u65f6\u975e\u5e38\u9ad8\u6548<\/li>\n

      3.4 \u652f\u6301\u5411\u91cf\u7684\u51e0\u4f55\u610f\u4e49<\/h4>\n

      \u652f\u6301\u5411\u91cf\u662f\u90a3\u4e9b\u6070\u597d\u4f4d\u4e8e\u95f4\u9694\u8fb9\u754c\u4e0a\u7684\u6570\u636e\u70b9，\u5373\u6ee1\u8db3 yi(w\u22c5xi+b)=1y_i(w \\\\cdot x_i + b) = 1<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>(<\/span>w<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>+<\/span><\/span><\/span><\/span>b<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u7684\u6837\u672c\u3002\u5b83\u4eec\u5177\u6709\u6df1\u523b\u7684\u51e0\u4f55\u548c\u7edf\u8ba1\u610f\u4e49：<\/p>\n

        \n
      • \u51b3\u5b9a\u6027：\u53ea\u6709\u652f\u6301\u5411\u91cf\u5f71\u54cd\u51b3\u7b56\u8fb9\u754c，\u79fb\u9664\u975e\u652f\u6301\u5411\u91cf\u4e0d\u4f1a\u6539\u53d8\u7ed3\u679c<\/li>\n
      • \u7a00\u758f\u6027：\u652f\u6301\u5411\u91cf\u7684\u6570\u91cf\u901a\u5e38\u8fdc\u5c0f\u4e8e\u8bad\u7ec3\u96c6\u5927\u5c0f<\/li>\n
      • \u9c81\u68d2\u6027：\u95f4\u9694\u6700\u5927\u5316\u4f7f\u51b3\u7b56\u8fb9\u754c\u5bf9\u6570\u636e\u6270\u52a8\u5177\u6709\u6700\u5927\u5bb9\u5fcd\u5ea6<\/li>\n<\/ul>\n

        \u53c2\u8003\u8bba\u6587：Cortes, C., & Vapnik, V. (1995). Support-vector networks. Machine Learning, 20(3), 273-297.<\/p>\n

        \n

        \u56db\u3001\u6838\u65b9\u6cd5\u7684\u5de7\u5999<\/h3>\n

        4.1 \u6838\u6280\u5de7\u7684\u6570\u5b66\u57fa\u7840<\/h4>\n

        SVM \u5bf9\u5076\u95ee\u9898\u4e2d\u53ea\u6d89\u53ca\u5185\u79ef xi\u22c5xjx_i \\\\cdot x_j<\/span><\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8fd9\u4e00\u4e8b\u5b9e，\u542f\u53d1\u4e86\u4e00\u4e2a\u6df1\u523b\u7684\u6d1e\u5bdf：\u5982\u679c\u6211\u4eec\u80fd\u7528\u67d0\u79cd\u65b9\u5f0f\u8ba1\u7b97\u9ad8\u7ef4（\u751a\u81f3\u65e0\u9650\u7ef4）\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef，\u5c31\u53ef\u4ee5\u5728\u4e0d\u663e\u5f0f\u8ba1\u7b97\u9ad8\u7ef4\u6620\u5c04\u7684\u60c5\u51b5\u4e0b，\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u8fdb\u884c\u7ebf\u6027\u5206\u7c7b\u3002<\/p>\n

        \u8fd9\u5c31\u662f\u6838\u6280\u5de7（Kernel Trick）：\u5b9a\u4e49\u6838\u51fd\u6570 K(xi,xj)=\u27e8\u03d5(xi),\u03d5(xj)\u27e9K(x_i, x_j) = \\\\langle \\\\phi(x_i), \\\\phi(x_j) \\\\rangle<\/span><\/span>K<\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>\u27e8<\/span>\u03d5<\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span>,<\/span><\/span>\u03d5<\/span>(<\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)\u27e9<\/span><\/span><\/span><\/span><\/span>，\u5176\u4e2d \u03d5:Rd\u2192RD\\\\phi: \\\\mathbb{R}^d \\\\to \\\\mathbb{R}^D<\/span><\/span>\u03d5<\/span><\/span>:<\/span><\/span><\/span><\/span>R<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span><\/span>R<\/span><\/span>D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u4ece\u8f93\u5165\u7a7a\u95f4\u5230\u7279\u5f81\u7a7a\u95f4\u7684\u6620\u5c04\u3002<\/p>\n

        \u5173\u952e\u5728\u4e8e：K(xi,xj)K(x_i, x_j)<\/span><\/span>K<\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>,<\/span><\/span>x<\/span><\/span>j<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span> \u53ef\u4ee5\u76f4\u63a5\u5728\u8f93\u5165\u7a7a\u95f4\u8ba1\u7b97，\u65e0\u9700\u77e5\u9053 \u03d5\\\\phi<\/span><\/span>\u03d5<\/span><\/span><\/span><\/span><\/span> \u7684\u5177\u4f53\u5f62\u5f0f\u3002<\/p>\n

        4.2 \u5e38\u7528\u6838\u51fd\u6570<\/h4>\n\n\u6838\u51fd\u6570\u516c\u5f0f\u9690\u5f0f\u6620\u5c04\u7ef4\u5ea6\u9002\u7528\u573a\u666f<\/tr>\n\n\n\n\n\n
        \u7ebf\u6027\u6838<\/td>\nK(x,z)=x\u22c5zK(x,z) = x \\\\cdot z<\/span><\/span>K<\/span>(<\/span>x<\/span>,<\/span><\/span>z<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>x<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>z<\/span><\/span><\/span><\/span><\/span><\/td>\ndd<\/span><\/span>d<\/span><\/span><\/span><\/span><\/span><\/td>\n\u7ebf\u6027\u53ef\u5206\u6570\u636e<\/td>\n<\/tr>\n
        \u591a\u9879\u5f0f\u6838<\/td>\nK(x,z)=(x\u22c5z+c)pK(x,z) = (x \\\\cdot z + c)^p<\/span><\/span>K<\/span>(<\/span>x<\/span>,<\/span><\/span>z<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>(<\/span>x<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>z<\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span>)<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n(d+pp)\\\\binom{d+p}{p}<\/span><\/span>(<\/span><\/span><\/span>p<\/span><\/span><\/span><\/span><\/span>d<\/span>+<\/span>p<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/td>\n\u4f4e\u9636\u7279\u5f81\u4ea4\u4e92<\/td>\n<\/tr>\n
        RBF（\u9ad8\u65af）\u6838<\/td>\nK(x,z)=exp\u2061(\u2212\u03b3\u2223x\u2212z\u22232)K(x,z) = \\\\exp(-\\\\gamma |x-z|^2)<\/span><\/span>K<\/span>(<\/span>x<\/span>,<\/span><\/span>z<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>exp<\/span>(<\/span>\u2212<\/span>\u03b3<\/span>\u2223<\/span>x<\/span><\/span>\u2212<\/span><\/span><\/span><\/span>z<\/span>\u2223<\/span><\/span>2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u221e\\\\infty<\/span><\/span>\u221e<\/span><\/span><\/span><\/span><\/span><\/td>\n\u901a\u7528，\u6700\u5e38\u7528<\/td>\n<\/tr>\n
        Sigmoid \u6838<\/td>\nK(x,z)=tanh\u2061(\u03bax\u22c5z+c)K(x,z) = \\\\tanh(\\\\kappa x \\\\cdot z + c)<\/span><\/span>K<\/span>(<\/span>x<\/span>,<\/span><\/span>z<\/span>)<\/span><\/span>=<\/span><\/span><\/span><\/span>tanh<\/span>(<\/span>\u03ba<\/span>x<\/span><\/span>\u22c5<\/span><\/span><\/span><\/span>z<\/span><\/span>+<\/span><\/span><\/span><\/span>c<\/span>)<\/span><\/span><\/span><\/span><\/span><\/td>\n\u2014<\/td>\n\u7c7b\u4f3c\u795e\u7ecf\u7f51\u7edc<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

        RBF \u6838\u7279\u522b\u503c\u5f97\u6ce8\u610f：\u5b83\u5bf9\u5e94\u7684\u7279\u5f81\u7a7a\u95f4\u662f\u65e0\u9650\u7ef4\u7684！\u8fd9\u610f\u5473\u7740\u7406\u8bba\u4e0a，RBF-SVM \u53ef\u4ee5\u62df\u5408\u4efb\u610f\u590d\u6742\u7684\u51b3\u7b56\u8fb9\u754c\u3002\u4f46 VC \u7ef4\u7406\u8bba\u544a\u8bc9\u6211\u4eec，\u901a\u8fc7\u6700\u5927\u5316\u95f4\u9694，SVM \u81ea\u52a8\u63a7\u5236\u4e86\u6709\u6548\u590d\u6742\u5ea6，\u907f\u514d\u4e86\u8fc7\u62df\u5408\u3002<\/p>\n

        4.3 \u6838\u65b9\u6cd5\u7684\u54f2\u5b66\u610f\u4e49<\/h4>\n

        \u6838\u65b9\u6cd5\u63ed\u793a\u4e86\u4e00\u4e2a\u6df1\u523b\u7684\u4e8b\u5b9e：\u5b66\u4e60\u7684\u672c\u8d28\u4e0d\u5728\u4e8e\u7279\u5f81\u7a7a\u95f4\u7684\u7ef4\u5ea6，\u800c\u5728\u4e8e\u6570\u636e\u5728\u8be5\u7a7a\u95f4\u4e2d\u7684\u51e0\u4f55\u7ed3\u6784\u3002<\/p>\n

          \n
        • \u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684\u7ebf\u6027\u5206\u7c7b = \u4f4e\u7ef4\u7a7a\u95f4\u4e2d\u7684\u975e\u7ebf\u6027\u5206\u7c7b<\/li>\n
        • \u6838\u51fd\u6570\u5b9a\u4e49\u4e86\u4e00\u79cd"\u76f8\u4f3c\u6027\u5ea6\u91cf"<\/li>\n
        • \u5b66\u4e60\u95ee\u9898\u53ef\u4ee5\u5b8c\u5168\u7528"\u76f8\u4f3c\u6027"\u7684\u8bed\u8a00\u6765\u63cf\u8ff0，\u65e0\u9700\u663e\u5f0f\u5b9a\u4e49\u7279\u5f81<\/li>\n<\/ul>\n

          \u8fd9\u4e2a\u601d\u60f3\u6df1\u523b\u5f71\u54cd\u4e86\u540e\u7eed\u7684\u7814\u7a76，\u5305\u62ec\u6838\u4e3b\u6210\u5206\u5206\u6790（Kernel PCA）\u3001\u9ad8\u65af\u8fc7\u7a0b（Gaussian Processes）\u3001\u4ee5\u53ca\u73b0\u4ee3Transformer \u4e2d\u7684\u6ce8\u610f\u529b\u673a\u5236（\u53ef\u4ee5\u89c6\u4e3a\u4e00\u79cd\u6570\u636e\u4f9d\u8d56\u7684\u6838\u51fd\u6570）\u3002<\/p>\n

          \n

          \u4e94\u3001\u4ee3\u7801\u5b9e\u73b0：SVM \u5206\u7c7b\u6f14\u793a<\/h3>\n

          5.1 \u4ece\u96f6\u5b9e\u73b0\u7b80\u5316\u7248 SVM<\/h4>\n

          \u4ee5\u4e0b\u4ee3\u7801\u4f7f\u7528\u68af\u5ea6\u4e0b\u964d\u5b9e\u73b0\u4e86\u7ebf\u6027 SVM \u7684\u8bad\u7ec3，\u5c55\u793a\u4e86\u6700\u5927\u95f4\u9694\u7684\u4f18\u5316\u8fc7\u7a0b：<\/p>\n

          import<\/span> numpy as<\/span> np<\/p>\n

          class<\/span> LinearSVM<\/span>:<\/span>
          \n """
          \n \u7ebf\u6027 SVM \u7684\u68af\u5ea6\u4e0b\u964d\u5b9e\u73b0<\/p>\n

          \u4f7f\u7528 hinge loss: L = max(0, 1 – y * (w\u00b7x + b))
          \n \u52a0\u4e0a L2 \u6b63\u5219\u5316: R = 0.5 * ||w||^2<\/p>\n

          \u603b\u635f\u5931: J = (1\/n) \u03a3 max(0, 1 – y_i * f(x_i)) + \u03bb * ||w||^2<\/p>\n

          \u6ce8\u610f：\u8fd9\u91cc\u7528\u68af\u5ea6\u4e0b\u964d\u6c42\u89e3，\u5b9e\u9645\u4e2d\u5e38\u7528 SMO \u7b97\u6cd5
          \n """<\/span><\/p>\n

          def<\/span> __init__<\/span>(<\/span>self,<\/span> learning_rate=<\/span>0.001<\/span>,<\/span> lambda_param=<\/span>0.01<\/span>,<\/span> n_iters=<\/span>1000<\/span>)<\/span>:<\/span>
          \n self.<\/span>lr =<\/span> learning_rate # \u5b66\u4e60\u7387<\/span>
          \n self.<\/span>lambda_param =<\/span> lambda_param # \u6b63\u5219\u5316\u5f3a\u5ea6（\u5bf9\u5e94 1\/C）<\/span>
          \n self.<\/span>n_iters =<\/span> n_iters # \u8fed\u4ee3\u6b21\u6570<\/span>
          \n self.<\/span>w =<\/span> None<\/span> # \u6743\u91cd\u5411\u91cf<\/span>
          \n self.<\/span>b =<\/span> None<\/span> # \u504f\u7f6e<\/span><\/p>\n

          def<\/span> fit<\/span>(<\/span>self,<\/span> X,<\/span> y)<\/span>:<\/span>
          \n """
          \n \u8bad\u7ec3 SVM<\/p>\n

          \u68af\u5ea6\u8ba1\u7b97：
          \n \u5bf9\u4e8e\u6b63\u786e\u5206\u7c7b\u4e14\u5728\u95f4\u9694\u5916\u7684\u6837\u672c（y*f(x) >= 1）：
          \n \u2202L\/\u2202w = \u03bb * w（\u4ec5\u6b63\u5219\u5316\u9879）
          \n \u5bf9\u4e8e\u8bef\u5206\u7c7b\u6216\u5728\u95f4\u9694\u5185\u7684\u6837\u672c（y*f(x) < 1）：
          \n \u2202L\/\u2202w = \u03bb * w – y_i * x_i（hinge loss + \u6b63\u5219\u5316）
          \n """<\/span>
          \n n_samples,<\/span> n_features =<\/span> X.<\/span>shape<\/p>\n

          # \u5c06\u6807\u7b7e\u8f6c\u6362\u4e3a {-1, +1}<\/span>
          \n y_ =<\/span> np.<\/span>where(<\/span>y <=<\/span> 0<\/span>,<\/span> –<\/span>1<\/span>,<\/span> 1<\/span>)<\/span><\/p>\n

          # \u521d\u59cb\u5316\u53c2\u6570<\/span>
          \n self.<\/span>w =<\/span> np.<\/span>zeros(<\/span>n_features)<\/span>
          \n self.<\/span>b =<\/span> 0<\/span><\/p>\n

          # \u8bb0\u5f55\u8bad\u7ec3\u8fc7\u7a0b<\/span>
          \n losses =<\/span> [<\/span>]<\/span><\/p>\n

          for<\/span> epoch in<\/span> range<\/span>(<\/span>self.<\/span>n_iters)<\/span>:<\/span>
          \n for<\/span> i in<\/span> range<\/span>(<\/span>n_samples)<\/span>:<\/span>
          \n # \u8ba1\u7b97\u95f4\u9694\u6761\u4ef6：y_i * (w \u00b7 x_i + b)<\/span>
          \n condition =<\/span> y_[<\/span>i]<\/span> *<\/span> (<\/span>np.<\/span>dot(<\/span>X[<\/span>i]<\/span>,<\/span> self.<\/span>w)<\/span> +<\/span> self.<\/span>b)<\/span><\/p>\n

          if<\/span> condition >=<\/span> 1<\/span>:<\/span>
          \n # \u6b63\u786e\u5206\u7c7b\u4e14\u5728\u95f4\u9694\u5916：\u4ec5\u66f4\u65b0\u6b63\u5219\u5316<\/span>
          \n self.<\/span>w -=<\/span> self.<\/span>lr *<\/span> (<\/span>2<\/span> *<\/span> self.<\/span>lambda_param *<\/span> self.<\/span>w)<\/span>
          \n else<\/span>:<\/span>
          \n # \u8bef\u5206\u7c7b\u6216\u5728\u95f4\u9694\u5185：hinge loss \u68af\u5ea6 + \u6b63\u5219\u5316<\/span>
          \n self.<\/span>w -=<\/span> self.<\/span>lr *<\/span> (<\/span>2<\/span> *<\/span> self.<\/span>lambda_param *<\/span> self.<\/span>w –<\/span> y_[<\/span>i]<\/span> *<\/span> X[<\/span>i]<\/span>)<\/span>
          \n self.<\/span>b -=<\/span> self.<\/span>lr *<\/span> y_[<\/span>i]<\/span><\/p>\n

          # \u8ba1\u7b97\u603b\u635f\u5931<\/span>
          \n distances =<\/span> 1<\/span> –<\/span> y_ *<\/span> (<\/span>X @ self.<\/span>w +<\/span> self.<\/span>b)<\/span>
          \n hinge_loss =<\/span> np.<\/span>sum<\/span>(<\/span>np.<\/span>maximum(<\/span>0<\/span>,<\/span> distances)<\/span>)<\/span> \/<\/span> n_samples
          \n reg_loss =<\/span> self.<\/span>lambda_param *<\/span> np.<\/span>dot(<\/span>self.<\/span>w,<\/span> self.<\/span>w)<\/span>
          \n total_loss =<\/span> hinge_loss +<\/span> reg_loss
          \n losses.<\/span>append(<\/span>total_loss)<\/span><\/p>\n

          if<\/span> epoch %<\/span> 200<\/span> ==<\/span> 0<\/span>:<\/span>
          \n print<\/span>(<\/span>f"Epoch <\/span>{<\/span>epoch:<\/span>4d<\/span>}<\/span><\/span> | Loss: <\/span>{<\/span>total_loss:<\/span>.4f<\/span>}<\/span><\/span> | "<\/span><\/span>
          \n f"Hinge: <\/span>{<\/span>hinge_loss:<\/span>.4f<\/span>}<\/span><\/span> | Reg: <\/span>{<\/span>reg_loss:<\/span>.4f<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

          # \u7edf\u8ba1\u652f\u6301\u5411\u91cf<\/span>
          \n distances =<\/span> y_ *<\/span> (<\/span>X @ self.<\/span>w +<\/span> self.<\/span>b)<\/span>
          \n self.<\/span>support_vectors =<\/span> X[<\/span>distances <=<\/span> 1.0<\/span> +<\/span> 1e-7<\/span>]<\/span>
          \n print<\/span>(<\/span>f"\\\\n\u652f\u6301\u5411\u91cf\u6570\u91cf: <\/span>{<\/span>len<\/span>(<\/span>self.<\/span>support_vectors)<\/span>}<\/span><\/span> \/ <\/span>{<\/span>n_samples}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

          return<\/span> losses<\/p>\n

          def<\/span> predict<\/span>(<\/span>self,<\/span> X)<\/span>:<\/span>
          \n """\u9884\u6d4b：sign(w \u00b7 x + b)"""<\/span>
          \n return<\/span> np.<\/span>sign(<\/span>X @ self.<\/span>w +<\/span> self.<\/span>b)<\/span><\/p>\n

          # === \u6f14\u793a：\u5728\u4e8c\u7ef4\u6570\u636e\u4e0a\u8bad\u7ec3 SVM ===<\/span>
          \nnp.<\/span>random.<\/span>seed(<\/span>42<\/span>)<\/span><\/p>\n

          # \u751f\u6210\u7ebf\u6027\u53ef\u5206\u7684\u4e8c\u5206\u7c7b\u6570\u636e<\/span>
          \nn_samples =<\/span> 100<\/span>
          \n# \u7c7b\u522b 1：\u4e2d\u5fc3\u5728 (2, 2)<\/span>
          \nX1 =<\/span> np.<\/span>random.<\/span>randn(<\/span>n_samples \/\/<\/span> 2<\/span>,<\/span> 2<\/span>)<\/span> *<\/span> 0.8<\/span> +<\/span> np.<\/span>array(<\/span>[<\/span>2<\/span>,<\/span> 2<\/span>]<\/span>)<\/span>
          \n# \u7c7b\u522b -1：\u4e2d\u5fc3\u5728 (-2, -2)<\/span>
          \nX2 =<\/span> np.<\/span>random.<\/span>randn(<\/span>n_samples \/\/<\/span> 2<\/span>,<\/span> 2<\/span>)<\/span> *<\/span> 0.8<\/span> +<\/span> np.<\/span>array(<\/span>[<\/span>–<\/span>2<\/span>,<\/span> –<\/span>2<\/span>]<\/span>)<\/span><\/p>\n

          X =<\/span> np.<\/span>vstack(<\/span>[<\/span>X1,<\/span> X2]<\/span>)<\/span>
          \ny =<\/span> np.<\/span>array(<\/span>[<\/span>1<\/span>]<\/span> *<\/span> (<\/span>n_samples \/\/<\/span> 2<\/span>)<\/span> +<\/span> [<\/span>–<\/span>1<\/span>]<\/span> *<\/span> (<\/span>n_samples \/\/<\/span> 2<\/span>)<\/span>)<\/span><\/p>\n

          # \u8bad\u7ec3 SVM<\/span>
          \nsvm =<\/span> LinearSVM(<\/span>learning_rate=<\/span>0.0005<\/span>,<\/span> lambda_param=<\/span>0.001<\/span>,<\/span> n_iters=<\/span>1000<\/span>)<\/span>
          \nlosses =<\/span> svm.<\/span>fit(<\/span>X,<\/span> y)<\/span><\/p>\n

          # \u8bc4\u4f30<\/span>
          \npredictions =<\/span> svm.<\/span>predict(<\/span>X)<\/span>
          \naccuracy =<\/span> np.<\/span>mean(<\/span>predictions ==<\/span> y)<\/span>
          \nprint<\/span>(<\/span>f"\\\\n\u8bad\u7ec3\u7cbe\u5ea6: <\/span>{<\/span>accuracy:<\/span>.2%<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
          \nprint<\/span>(<\/span>f"\u6743\u91cd\u5411\u91cf w: [<\/span>{<\/span>svm.<\/span>w[<\/span>0<\/span>]<\/span>:<\/span>.4f<\/span>}<\/span><\/span>, <\/span>{<\/span>svm.<\/span>w[<\/span>1<\/span>]<\/span>:<\/span>.4f<\/span>}<\/span><\/span>]"<\/span><\/span>)<\/span>
          \nprint<\/span>(<\/span>f"\u504f\u7f6e b: <\/span>{<\/span>svm.<\/span>b:<\/span>.4f<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
          \nprint<\/span>(<\/span>f"\u95f4\u9694\u5bbd\u5ea6: <\/span>{<\/span>2<\/span> \/<\/span> np.<\/span>linalg.<\/span>norm(<\/span>svm.<\/span>w)<\/span>:<\/span>.4f<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

          \u4ee3\u7801\u8bf4\u660e：\u8fd9\u4e2a\u5b9e\u73b0\u4f7f\u7528\u968f\u673a\u68af\u5ea6\u4e0b\u964d（SGD）\u6c42\u89e3 SVM \u7684\u4f18\u5316\u95ee\u9898\u3002\u5173\u952e\u7ec6\u8282\u5305\u62ec：(1) hinge loss \u7684\u68af\u5ea6\u5728 yif(xi)\u22651y_i f(x_i) \\\\geq 1<\/span><\/span>y<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span><\/span>i<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>)<\/span><\/span>\u2265<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u65f6\u4e3a\u96f6（\u6b63\u786e\u5206\u7c7b\u7684\u6837\u672c\u4e0d\u8d21\u732e\u68af\u5ea6），\u8fd9\u81ea\u7136\u5b9e\u73b0\u4e86\u652f\u6301\u5411\u91cf\u7684\u9009\u62e9\u6027；(2) \u6b63\u5219\u5316\u53c2\u6570 \u03bb\\\\lambda<\/span><\/span>\u03bb<\/span><\/span><\/span><\/span><\/span> \u63a7\u5236\u95f4\u9694\u5927\u5c0f\u4e0e\u8bad\u7ec3\u8bef\u5dee\u7684\u6743\u8861；(3) \u95f4\u9694\u5bbd\u5ea6 2\u2225w\u2225\\\\frac{2}{\\\\|w\\\\|}<\/span><\/span><\/span><\/span>\u2225<\/span>w<\/span>\u2225<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>2<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f SVM \u6cdb\u5316\u80fd\u529b\u7684\u76f4\u63a5\u5ea6\u91cf\u3002<\/p>\n

          5.2 \u6838 SVM \u7684\u5b9e\u73b0<\/h4>\n

          \u4ee5\u4e0b\u4ee3\u7801\u5b9e\u73b0\u4e86\u5e26 RBF \u6838\u7684 SVM，\u5c55\u793a\u4e86\u6838\u6280\u5de7\u5982\u4f55\u5c06\u7ebf\u6027\u5206\u7c7b\u5668\u6269\u5c55\u5230\u975e\u7ebf\u6027\u51b3\u7b56\u8fb9\u754c：<\/p>\n

          import<\/span> numpy as<\/span> np<\/p>\n

          class<\/span> KernelSVM<\/span>:<\/span>
          \n """
          \n \u4f7f\u7528\u6838\u6280\u5de7\u7684 SVM（\u7b80\u5316\u7248 SMO \u7b97\u6cd5）<\/p>\n

          \u6838\u51fd\u6570\u5c06\u6570\u636e\u9690\u5f0f\u6620\u5c04\u5230\u9ad8\u7ef4\u7a7a\u95f4，
          \n \u4f7f\u5f97\u5728\u539f\u59cb\u7a7a\u95f4\u4e2d\u7ebf\u6027\u4e0d\u53ef\u5206\u7684\u6570\u636e\u53d8\u5f97\u7ebf\u6027\u53ef\u5206\u3002
          \n """<\/span><\/p>\n

          def<\/span> __init__<\/span>(<\/span>self,<\/span> kernel=<\/span>'rbf'<\/span>,<\/span> C=<\/span>1.0<\/span>,<\/span> gamma=<\/span>1.0<\/span>,<\/span> max_iter=<\/span>1000<\/span>)<\/span>:<\/span>
          \n self.<\/span>C =<\/span> C # \u6b63\u5219\u5316\u53c2\u6570<\/span>
          \n self.<\/span>gamma =<\/span> gamma # RBF \u6838\u7684\u5e26\u5bbd\u53c2\u6570<\/span>
          \n self.<\/span>max_iter =<\/span> max_iter
          \n self.<\/span>kernel =<\/span> kernel
          \n self.<\/span>alpha =<\/span> None<\/span> # \u62c9\u683c\u6717\u65e5\u4e58\u5b50<\/span>
          \n self.<\/span>b =<\/span> 0<\/span> # \u504f\u7f6e<\/span>
          \n self.<\/span>support_vectors =<\/span> None<\/span>
          \n self.<\/span>support_labels =<\/span> None<\/span>
          \n self.<\/span>support_alphas =<\/span> None<\/span><\/p>\n

          def<\/span> _kernel_function<\/span>(<\/span>self,<\/span> x1,<\/span> x2)<\/span>:<\/span>
          \n """
          \n \u6838\u51fd\u6570\u8ba1\u7b97<\/p>\n

          RBF \u6838: K(x1, x2) = exp(-\u03b3 ||x1 – x2||\u00b2)<\/p>\n

          \u8fd9\u4e2a\u51fd\u6570\u7684\u7cbe\u5999\u4e4b\u5904\u5728\u4e8e：
          \n \u5b83\u8ba1\u7b97\u7684\u662f\u4e24\u4e2a\u6837\u672c\u5728\u65e0\u9650\u7ef4\u7279\u5f81\u7a7a\u95f4\u4e2d\u7684\u5185\u79ef，
          \n \u4f46\u53ea\u9700\u8981\u5728\u539f\u59cb\u7a7a\u95f4\u4e2d\u8fdb\u884c\u7b80\u5355\u7684\u8ddd\u79bb\u8ba1\u7b97\u3002
          \n """<\/span>
          \n if<\/span> self.<\/span>kernel ==<\/span> 'linear'<\/span>:<\/span>
          \n return<\/span> np.<\/span>dot(<\/span>x1,<\/span> x2)<\/span>
          \n elif<\/span> self.<\/span>kernel ==<\/span> 'rbf'<\/span>:<\/span>
          \n # ||x1 – x2||\u00b2 = ||x1||\u00b2 + ||x2||\u00b2 – 2 * x1\u00b7x2<\/span>
          \n sq_dist =<\/span> np.<\/span>sum<\/span>(<\/span>x1**<\/span>2<\/span>)<\/span> +<\/span> np.<\/span>sum<\/span>(<\/span>x2**<\/span>2<\/span>)<\/span> –<\/span> 2<\/span> *<\/span> np.<\/span>dot(<\/span>x1,<\/span> x2)<\/span>
          \n return<\/span> np.<\/span>exp(<\/span>–<\/span>self.<\/span>gamma *<\/span> sq_dist)<\/span><\/p>\n

          def<\/span> _compute_kernel_matrix<\/span>(<\/span>self,<\/span> X)<\/span>:<\/span>
          \n """\u9884\u8ba1\u7b97\u6838\u77e9\u9635 K[i,j] = K(x_i, x_j)"""<\/span>
          \n n =<\/span> len<\/span>(<\/span>X)<\/span>
          \n K =<\/span> np.<\/span>zeros(<\/span>(<\/span>n,<\/span> n)<\/span>)<\/span>
          \n for<\/span> i in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
          \n for<\/span> j in<\/span> range<\/span>(<\/span>i,<\/span> n)<\/span>:<\/span>
          \n K[<\/span>i,<\/span> j]<\/span> =<\/span> self.<\/span>_kernel_function(<\/span>X[<\/span>i]<\/span>,<\/span> X[<\/span>j]<\/span>)<\/span>
          \n K[<\/span>j,<\/span> i]<\/span> =<\/span> K[<\/span>i,<\/span> j]<\/span> # \u6838\u77e9\u9635\u662f\u5bf9\u79f0\u7684<\/span>
          \n return<\/span> K<\/p>\n

          def<\/span> fit<\/span>(<\/span>self,<\/span> X,<\/span> y)<\/span>:<\/span>
          \n """
          \n \u7b80\u5316\u7684 SMO（Sequential Minimal Optimization）\u8bad\u7ec3<\/p>\n

          \u6838\u5fc3\u601d\u60f3：\u6bcf\u6b21\u9009\u62e9\u4e24\u4e2a alpha \u8fdb\u884c\u4f18\u5316，
          \n \u4fdd\u6301 KKT \u6761\u4ef6\u7684\u6ee1\u8db3，\u9010\u6b65\u903c\u8fd1\u6700\u4f18\u89e3\u3002
          \n """<\/span>
          \n n =<\/span> len<\/span>(<\/span>X)<\/span>
          \n y_ =<\/span> np.<\/span>where(<\/span>y <=<\/span> 0<\/span>,<\/span> –<\/span>1<\/span>,<\/span> 1<\/span>)<\/span>.<\/span>astype(<\/span>float<\/span>)<\/span><\/p>\n

          # \u9884\u8ba1\u7b97\u6838\u77e9\u9635（\u907f\u514d\u91cd\u590d\u8ba1\u7b97）<\/span>
          \n K =<\/span> self.<\/span>_compute_kernel_matrix(<\/span>X)<\/span><\/p>\n

          # \u521d\u59cb\u5316 alpha \u4e3a\u96f6<\/span>
          \n self.<\/span>alpha =<\/span> np.<\/span>zeros(<\/span>n)<\/span>
          \n self.<\/span>b =<\/span> 0<\/span><\/p>\n

          for<\/span> iteration in<\/span> range<\/span>(<\/span>self.<\/span>max_iter)<\/span>:<\/span>
          \n alpha_prev =<\/span> self.<\/span>alpha.<\/span>copy(<\/span>)<\/span><\/p>\n

          for<\/span> i in<\/span> range<\/span>(<\/span>n)<\/span>:<\/span>
          \n # \u8ba1\u7b97\u7b2c i \u4e2a\u6837\u672c\u7684\u9884\u6d4b\u503c<\/span>
          \n # f(x_i) = \u03a3 \u03b1_j y_j K(x_j, x_i) + b<\/span>
          \n f_i =<\/span> np.<\/span>sum<\/span>(<\/span>self.<\/span>alpha *<\/span> y_ *<\/span> K[<\/span>i,<\/span> :<\/span>]<\/span>)<\/span> +<\/span> self.<\/span>b<\/p>\n

          # \u68c0\u67e5 KKT \u6761\u4ef6\u662f\u5426\u6ee1\u8db3<\/span>
          \n # KKT: \u03b1_i = 0 \u4e14 y_i f(x_i) >= 1（\u6b63\u786e\u5206\u7c7b\u5728\u5916\u4fa7）<\/span>
          \n # 0 < \u03b1_i < C \u4e14 y_i f(x_i) = 1（\u5728\u95f4\u9694\u4e0a）<\/span>
          \n # \u03b1_i = C \u4e14 y_i f(x_i) <= 1（\u5728\u95f4\u9694\u5185\u6216\u8bef\u5206\u7c7b）<\/span>
          \n E_i =<\/span> f_i –<\/span> y_[<\/span>i]<\/span> # \u9884\u6d4b\u8bef\u5dee<\/span><\/p>\n

          # \u5982\u679c\u8fdd\u53cd KKT \u6761\u4ef6，\u5c1d\u8bd5\u66f4\u65b0<\/span>
          \n if<\/span> (<\/span>y_[<\/span>i]<\/span> *<\/span> E_i <<\/span> –<\/span>0.01<\/span> and<\/span> self.<\/span>alpha[<\/span>i]<\/span> <<\/span> self.<\/span>C)<\/span> or<\/span> \\\\
          \n (<\/span>y_[<\/span>i]<\/span> *<\/span> E_i ><\/span> 0.01<\/span> and<\/span> self.<\/span>alpha[<\/span>i]<\/span> ><\/span> 0<\/span>)<\/span>:<\/span><\/p>\n

          # \u968f\u673a\u9009\u62e9\u53e6\u4e00\u4e2a\u6837\u672c j<\/span>
          \n j =<\/span> i
          \n while<\/span> j ==<\/span> i:<\/span>
          \n j =<\/span> np.<\/span>random.<\/span>randint(<\/span>0<\/span>,<\/span> n)<\/span><\/p>\n

          f_j =<\/span> np.<\/span>sum<\/span>(<\/span>self.<\/span>alpha *<\/span> y_ *<\/span> K[<\/span>j,<\/span> :<\/span>]<\/span>)<\/span> +<\/span> self.<\/span>b
          \n E_j =<\/span> f_j –<\/span> y_[<\/span>j]<\/span><\/p>\n

          # \u4fdd\u5b58\u65e7\u7684 alpha \u503c<\/span>
          \n alpha_i_old =<\/span> self.<\/span>alpha[<\/span>i]<\/span>
          \n alpha_j_old =<\/span> self.<\/span>alpha[<\/span>j]<\/span><\/p>\n

          # \u8ba1\u7b97 alpha_j \u7684\u4e0a\u4e0b\u754c<\/span>
          \n if<\/span> y_[<\/span>i]<\/span> !=<\/span> y_[<\/span>j]<\/span>:<\/span>
          \n L =<\/span> max<\/span>(<\/span>0<\/span>,<\/span> self.<\/span>alpha[<\/span>j]<\/span> –<\/span> self.<\/span>alpha[<\/span>i]<\/span>)<\/span>
          \n H =<\/span> min<\/span>(<\/span>self.<\/span>C,<\/span> self.<\/span>C +<\/span> self.<\/span>alpha[<\/span>j]<\/span> –<\/span> self.<\/span>alpha[<\/span>i]<\/span>)<\/span>
          \n else<\/span>:<\/span>
          \n L =<\/span> max<\/span>(<\/span>0<\/span>,<\/span> self.<\/span>alpha[<\/span>i]<\/span> +<\/span> self.<\/span>alpha[<\/span>j]<\/span> –<\/span> self.<\/span>C)<\/span>
          \n H =<\/span> min<\/span>(<\/span>self.<\/span>C,<\/span> self.<\/span>alpha[<\/span>i]<\/span> +<\/span> self.<\/span>alpha[<\/span>j]<\/span>)<\/span><\/p>\n

          if<\/span> L ==<\/span> H:<\/span>
          \n continue<\/span><\/p>\n

          # \u8ba1\u7b97\u4e8c\u9636\u5bfc\u6570（\u6838\u5316\u7248\u672c）<\/span>
          \n eta =<\/span> 2<\/span> *<\/span> K[<\/span>i,<\/span> j]<\/span> –<\/span> K[<\/span>i,<\/span> i]<\/span> –<\/span> K[<\/span>j,<\/span> j]<\/span>
          \n if<\/span> eta >=<\/span> 0<\/span>:<\/span>
          \n continue<\/span><\/p>\n

          # \u66f4\u65b0 alpha_j<\/span>
          \n self.<\/span>alpha[<\/span>j]<\/span> -=<\/span> y_[<\/span>j]<\/span> *<\/span> (<\/span>E_i –<\/span> E_j)<\/span> \/<\/span> eta
          \n self.<\/span>alpha[<\/span>j]<\/span> =<\/span> np.<\/span>clip(<\/span>self.<\/span>alpha[<\/span>j]<\/span>,<\/span> L,<\/span> H)<\/span><\/p>\n

          # \u66f4\u65b0 alpha_i（\u4fdd\u6301\u7ea6\u675f \u03a3 \u03b1_i y_i = 0）<\/span>
          \n self.<\/span>alpha[<\/span>i]<\/span> +=<\/span> y_[<\/span>i]<\/span> *<\/span> y_[<\/span>j]<\/span> *<\/span> (<\/span>alpha_j_old –<\/span> self.<\/span>alpha[<\/span>j]<\/span>)<\/span><\/p>\n

          # \u66f4\u65b0\u504f\u7f6e b<\/span>
          \n b1 =<\/span> self.<\/span>b –<\/span> E_i –<\/span> y_[<\/span>i]<\/span> *<\/span> (<\/span>self.<\/span>alpha[<\/span>i]<\/span> –<\/span> alpha_i_old)<\/span> *<\/span> K[<\/span>i,<\/span> j]<\/span> \\\\
          \n –<\/span> y_[<\/span>j]<\/span> *<\/span> (<\/span>self.<\/span>alpha[<\/span>j]<\/span> –<\/span> alpha_j_old)<\/span> *<\/span> K[<\/span>i,<\/span> j]<\/span>
          \n b2 =<\/span> self.<\/span>b –<\/span> E_j –<\/span> y_[<\/span>i]<\/span> *<\/span> (<\/span>self.<\/span>alpha[<\/span>i]<\/span> –<\/span> alpha_i_old)<\/span> *<\/span> K[<\/span>i,<\/span> j]<\/span> \\\\
          \n –<\/span> y_[<\/span>j]<\/span> *<\/span> (<\/span>self.<\/span>alpha[<\/span>j]<\/span> –<\/span> alpha_j_old)<\/span> *<\/span> K[<\/span>j,<\/span> j]<\/span><\/p>\n

          if<\/span> 0<\/span> <<\/span> self.<\/span>alpha[<\/span>i]<\/span> <<\/span> self.<\/span>C:<\/span>
          \n self.<\/span>b =<\/span> b1
          \n elif<\/span> 0<\/span> <<\/span> self.<\/span>alpha[<\/span>j]<\/span> <<\/span> self.<\/span>C:<\/span>
          \n self.<\/span>b =<\/span> b2
          \n else<\/span>:<\/span>
          \n self.<\/span>b =<\/span> (<\/span>b1 +<\/span> b2)<\/span> \/<\/span> 2<\/span><\/p>\n

          # \u68c0\u67e5\u6536\u655b<\/span>
          \n diff =<\/span> np.<\/span>linalg.<\/span>norm(<\/span>self.<\/span>alpha –<\/span> alpha_prev)<\/span>
          \n if<\/span> iteration %<\/span> 100<\/span> ==<\/span> 0<\/span>:<\/span>
          \n print<\/span>(<\/span>f"Iter <\/span>{<\/span>iteration:<\/span>4d<\/span>}<\/span><\/span> | Alpha change: <\/span>{<\/span>diff:<\/span>.6f<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span>
          \n if<\/span> diff <<\/span> 1e-5<\/span>:<\/span>
          \n print<\/span>(<\/span>f"\u6536\u655b\u4e8e\u7b2c <\/span>{<\/span>iteration}<\/span><\/span> \u6b21\u8fed\u4ee3"<\/span><\/span>)<\/span>
          \n break<\/span><\/p>\n

          # \u63d0\u53d6\u652f\u6301\u5411\u91cf（alpha > 0 \u7684\u6837\u672c）<\/span>
          \n sv_mask =<\/span> self.<\/span>alpha ><\/span> 1e-7<\/span>
          \n self.<\/span>support_vectors =<\/span> X[<\/span>sv_mask]<\/span>
          \n self.<\/span>support_labels =<\/span> y_[<\/span>sv_mask]<\/span>
          \n self.<\/span>support_alphas =<\/span> self.<\/span>alpha[<\/span>sv_mask]<\/span>
          \n print<\/span>(<\/span>f"\u652f\u6301\u5411\u91cf\u6570\u91cf: <\/span>{<\/span>len<\/span>(<\/span>self.<\/span>support_vectors)<\/span>}<\/span><\/span> \/ <\/span>{<\/span>n}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

          def<\/span> predict<\/span>(<\/span>self,<\/span> X)<\/span>:<\/span>
          \n """\u9884\u6d4b：f(x) = \u03a3 \u03b1_i y_i K(x_i, x) + b"""<\/span>
          \n predictions =<\/span> [<\/span>]<\/span>
          \n for<\/span> x in<\/span> X:<\/span>
          \n f =<\/span> sum<\/span>(<\/span>a *<\/span> y *<\/span> self.<\/span>_kernel_function(<\/span>sv,<\/span> x)<\/span>
          \n for<\/span> a,<\/span> y,<\/span> sv in<\/span> zip<\/span>(<\/span>self.<\/span>support_alphas,<\/span>
          \n self.<\/span>support_labels,<\/span>
          \n self.<\/span>support_vectors)<\/span>)<\/span>
          \n predictions.<\/span>append(<\/span>np.<\/span>sign(<\/span>f +<\/span> self.<\/span>b)<\/span>)<\/span>
          \n return<\/span> np.<\/span>array(<\/span>predictions)<\/span><\/p>\n

          # === \u6f14\u793a：\u7528 RBF \u6838 SVM \u5206\u7c7b\u975e\u7ebf\u6027\u6570\u636e ===<\/span>
          \nnp.<\/span>random.<\/span>seed(<\/span>42<\/span>)<\/span><\/p>\n

          # \u751f\u6210 XOR \u95ee\u9898（\u7ebf\u6027\u4e0d\u53ef\u5206）<\/span>
          \nn =<\/span> 100<\/span>
          \nX_xor =<\/span> np.<\/span>random.<\/span>randn(<\/span>n,<\/span> 2<\/span>)<\/span>
          \ny_xor =<\/span> np.<\/span>sign(<\/span>X_xor[<\/span>:<\/span>,<\/span> 0<\/span>]<\/span> *<\/span> X_xor[<\/span>:<\/span>,<\/span> 1<\/span>]<\/span>)<\/span> # XOR \u903b\u8f91<\/span>
          \n# \u6dfb\u52a0\u4e00\u4e9b\u566a\u58f0<\/span>
          \ny_xor =<\/span> np.<\/span>where(<\/span>np.<\/span>random.<\/span>rand(<\/span>n)<\/span> ><\/span> 0.9<\/span>,<\/span> –<\/span>y_xor,<\/span> y_xor)<\/span><\/p>\n

          # \u8bad\u7ec3 RBF \u6838 SVM<\/span>
          \nkernel_svm =<\/span> KernelSVM(<\/span>kernel=<\/span>'rbf'<\/span>,<\/span> C=<\/span>10.0<\/span>,<\/span> gamma=<\/span>1.0<\/span>,<\/span> max_iter=<\/span>500<\/span>)<\/span>
          \nkernel_svm.<\/span>fit(<\/span>X_xor,<\/span> y_xor)<\/span><\/p>\n

          # \u8bc4\u4f30<\/span>
          \npredictions =<\/span> kernel_svm.<\/span>predict(<\/span>X_xor)<\/span>
          \naccuracy =<\/span> np.<\/span>mean(<\/span>predictions ==<\/span> y_xor)<\/span>
          \nprint<\/span>(<\/span>f"\\\\nRBF \u6838 SVM \u5728 XOR \u6570\u636e\u4e0a\u7684\u7cbe\u5ea6: <\/span>{<\/span>accuracy:<\/span>.2%<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

          \u4ee3\u7801\u8bf4\u660e：\u8fd9\u4e2a\u5b9e\u73b0\u5c55\u793a\u4e86\u6838 SVM \u7684\u6838\u5fc3\u673a\u5236：(1) \u6838\u77e9\u9635\u7684\u9884\u8ba1\u7b97\u907f\u514d\u4e86\u91cd\u590d\u7684\u9ad8\u7ef4\u6620\u5c04\u8ba1\u7b97；(2) RBF \u6838\u5c06\u6570\u636e\u9690\u5f0f\u6620\u5c04\u5230\u65e0\u9650\u7ef4\u7a7a\u95f4，\u4f7f\u5f97 XOR \u8fd9\u6837\u7684\u975e\u7ebf\u6027\u95ee\u9898\u53d8\u5f97\u7ebf\u6027\u53ef\u5206；(3) SMO \u7b97\u6cd5\u901a\u8fc7\u6bcf\u6b21\u4f18\u5316\u4e24\u4e2a\u62c9\u683c\u6717\u65e5\u4e58\u5b50\u6765\u4fdd\u6301 KKT \u6761\u4ef6，\u8fd9\u662f Platt \u57281998\u5e74\u63d0\u51fa\u7684\u9ad8\u6548\u6c42\u89e3\u65b9\u6cd5\u7684\u7b80\u5316\u7248\u672c\u3002\u652f\u6301\u5411\u91cf\u7684\u7a00\u758f\u6027（\u901a\u5e38\u53ea\u6709\u8bad\u7ec3\u6837\u672c\u7684\u4e00\u5c0f\u90e8\u5206）\u4f7f\u5f97 SVM \u5728\u6d4b\u8bd5\u65f6\u975e\u5e38\u9ad8\u6548\u3002<\/p>\n

          \n

          \u516d\u3001SVM \u65f6\u4ee3\u7684\u5386\u53f2\u5730\u4f4d<\/h3>\n

          6.1 SVM \u7684\u7edf\u6cbb\u65f6\u671f（1995-2012）<\/h4>\n

          \u4ece1995\u5e74\u52302012\u5e74\u6df1\u5ea6\u5b66\u4e60\u5d1b\u8d77\u4e4b\u524d，SVM \u51e0\u4e4e\u7edf\u6cbb\u4e86\u6574\u4e2a\u673a\u5668\u5b66\u4e60\u9886\u57df\u3002\u5176\u6210\u529f\u7684\u539f\u56e0\u5305\u62ec：<\/p>\n

        • \u7406\u8bba\u5b8c\u5907：\u6709 VC \u7ef4\u7406\u8bba\u548c SRM \u539f\u5219\u4f5c\u4e3a\u575a\u5b9e\u7684\u6570\u5b66\u57fa\u7840<\/li>\n
        • \u5168\u5c40\u6700\u4f18：\u51f8\u4f18\u5316\u95ee\u9898\u4fdd\u8bc1\u627e\u5230\u5168\u5c40\u6700\u4f18\u89e3（\u4e0d\u50cf\u795e\u7ecf\u7f51\u7edc\u53ef\u80fd\u9677\u5165\u5c40\u90e8\u6700\u5c0f\u503c）<\/li>\n
        • \u6838\u6280\u5de7：\u4f18\u96c5\u5730\u5904\u7406\u975e\u7ebf\u6027\u95ee\u9898<\/li>\n
        • \u7a00\u758f\u89e3：\u53ea\u6709\u652f\u6301\u5411\u91cf\u5f71\u54cd\u51b3\u7b56，\u8ba1\u7b97\u6548\u7387\u9ad8<\/li>\n
        • \u6cdb\u5316\u4fdd\u8bc1：\u95f4\u9694\u6700\u5927\u5316\u63d0\u4f9b\u4e86\u660e\u786e\u7684\u6cdb\u5316\u7406\u8bba<\/li>\n

          6.2 SVM \u7684\u7ecf\u5178\u5e94\u7528<\/h4>\n\n\u5e94\u7528\u9886\u57df\u5177\u4f53\u4efb\u52a1\u4ee3\u8868\u6027\u5de5\u4f5c<\/tr>\n\n\n\n\n\n\n
          \u6587\u672c\u5206\u7c7b<\/td>\n\u5783\u573e\u90ae\u4ef6\u8fc7\u6ee4\u3001\u60c5\u611f\u5206\u6790<\/td>\nJoachims 1998<\/td>\n<\/tr>\n
          \u751f\u7269\u4fe1\u606f\u5b66<\/td>\n\u86cb\u767d\u8d28\u5206\u7c7b\u3001\u57fa\u56e0\u8868\u8fbe\u5206\u6790<\/td>\n\u7b49<\/td>\n<\/tr>\n
          \u8ba1\u7b97\u673a\u89c6\u89c9<\/td>\n\u4eba\u8138\u68c0\u6d4b\u3001\u7269\u4f53\u8bc6\u522b<\/td>\n\u4e0e HOG \u7279\u5f81\u7ed3\u5408<\/td>\n<\/tr>\n
          \u624b\u5199\u8bc6\u522b<\/td>\nMNIST \u5206\u7c7b<\/td>\n\u4e0e LeNet \u7ade\u4e89<\/td>\n<\/tr>\n
          \u81ea\u7136\u8bed\u8a00\u5904\u7406<\/td>\n\u8bcd\u6027\u6807\u6ce8\u3001\u547d\u540d\u5b9e\u4f53\u8bc6\u522b<\/td>\n\u7b49<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          6.3 SVM \u88ab\u6df1\u5ea6\u5b66\u4e60\u8d85\u8d8a<\/h4>\n

          2012\u5e74，AlexNet \u5728 ImageNet \u4e0a\u7684\u7a81\u7834\u6807\u5fd7\u7740 SVM \u7edf\u6cbb\u5730\u4f4d\u7684\u7ec8\u7ed3\u3002\u6df1\u5ea6\u5b66\u4e60\u80dc\u51fa\u7684\u539f\u56e0：<\/p>\n

        • \u7279\u5f81\u5b66\u4e60：\u6df1\u5ea6\u7f51\u7edc\u81ea\u52a8\u5b66\u4e60\u7279\u5f81，\u4e0d\u9700\u8981\u624b\u5de5\u8bbe\u8ba1（\u5982 HOG\u3001SIFT）<\/li>\n
        • \u89c4\u6a21\u6269\u5c55：\u6df1\u5ea6\u7f51\u7edc\u7684\u6027\u80fd\u968f\u6570\u636e\u548c\u8ba1\u7b97\u91cf\u6301\u7eed\u63d0\u5347，\u800c SVM \u7684\u6027\u80fd\u8d8b\u4e8e\u9971\u548c<\/li>\n
        • \u7aef\u5230\u7aef\u8bad\u7ec3：\u6df1\u5ea6\u5b66\u4e60\u53ef\u4ee5\u5c06\u7279\u5f81\u63d0\u53d6\u548c\u5206\u7c7b\u7edf\u4e00\u5230\u4e00\u4e2a\u6846\u67b6\u4e2d<\/li>\n
        • GPU \u52a0\u901f：\u77e9\u9635\u8fd0\u7b97\u5929\u7136\u9002\u5408 GPU \u5e76\u884c<\/li>\n

          \u4f46 SVM \u7684\u7406\u8bba\u9057\u4ea7\u2014\u2014\u95f4\u9694\u6700\u5927\u5316\u3001\u6838\u65b9\u6cd5\u3001VC \u7ef4\u7406\u8bba\u2014\u2014\u4ecd\u7136\u6df1\u523b\u5f71\u54cd\u7740\u6df1\u5ea6\u5b66\u4e60\u7684\u7406\u8bba\u7814\u7a76\u3002<\/p>\n

          \u53c2\u8003\u8bba\u6587：<\/p>\n

            \n
          • Sch\u00f6lkopf, B., & Smola, A. J. (2002). Learning with Kernels. MIT Press.<\/li>\n
          • Platt, J. (1998). Sequential minimal optimization: A fast algorithm for training support vector machines. Microsoft Research Technical Report.<\/li>\n<\/ul>\n
            \n

            \u603b\u7ed3<\/h3>\n

            Vapnik \u7684\u7edf\u8ba1\u5b66\u4e60\u7406\u8bba\u548c SVM \u7684\u5386\u53f2\u610f\u4e49，\u8fdc\u8fdc\u8d85\u51fa\u4e86\u4e00\u4e2a\u5206\u7c7b\u7b97\u6cd5\u7684\u8303\u7574\u3002\u5b83\u4ee3\u8868\u4e86\u673a\u5668\u5b66\u4e60\u7814\u7a76\u7684\u4e00\u6b21\u8303\u5f0f\u8f6c\u53d8：\u4ece\u4f9d\u8d56\u76f4\u89c9\u548c\u5b9e\u9a8c，\u8f6c\u5411\u57fa\u4e8e\u4e25\u683c\u6570\u5b66\u7406\u8bba\u7684\u7b97\u6cd5\u8bbe\u8ba1\u3002<\/p>\n

            \u56de\u987e\u8fd9\u6bb5\u5386\u53f2，\u6211\u4eec\u53ef\u4ee5\u83b7\u5f97\u51e0\u4e2a\u6df1\u523b\u7684\u542f\u793a：<\/p>\n

          • \n

            \u7406\u8bba\u7684\u4ef7\u503c：\u5728\u6df1\u5ea6\u5b66\u4e60\u65f6\u4ee3，\u201c\u5148\u5b9e\u9a8c\u540e\u89e3\u91ca"\u4f3c\u4e4e\u6210\u4e86\u4e3b\u6d41\u3002\u4f46 SVM \u7684\u6210\u529f\u544a\u8bc9\u6211\u4eec，\u624e\u5b9e\u7684\u7406\u8bba\u57fa\u7840\u53ef\u4ee5\u6307\u5bfc\u7b97\u6cd5\u8bbe\u8ba1，\u63d0\u4f9b\u6cdb\u5316\u4fdd\u8bc1，\u5e76\u5e2e\u52a9\u6211\u4eec\u7406\u89e3"\u4e3a\u4ec0\u4e48\u201d\u3002<\/p>\n<\/li>\n

          • \n

            \u7b80\u5355\u5047\u8bbe\u7684\u529b\u91cf：SVM \u7684\u6838\u5fc3\u5047\u8bbe\u2014\u2014\u95f4\u9694\u6700\u5927\u5316\u2014\u2014\u6781\u5176\u7b80\u5355，\u5374\u4ea7\u751f\u4e86\u6df1\u8fdc\u7684\u5f71\u54cd\u3002\u8fd9\u63d0\u9192\u6211\u4eec，\u5728\u673a\u5668\u5b66\u4e60\u4e2d，\u6b63\u786e\u7684\u5f52\u7eb3\u504f\u7f6e\u6bd4\u590d\u6742\u7684\u6a21\u578b\u7ed3\u6784\u66f4\u91cd\u8981\u3002<\/p>\n<\/li>\n

          • \n

            \u6838\u65b9\u6cd5\u7684\u9057\u4ea7：\u867d\u7136 SVM \u672c\u8eab\u53ef\u80fd\u4e0d\u518d\u662f\u9996\u9009\u65b9\u6cd5，\u4f46\u6838\u65b9\u6cd5\u7684\u601d\u60f3\u2014\u2014\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u5bfb\u627e\u7ebf\u6027\u7ed3\u6784\u2014\u2014\u5df2\u7ecf\u6df1\u6df1\u878d\u5165\u4e86\u73b0\u4ee3\u673a\u5668\u5b66\u4e60\u7684\u65b9\u65b9\u9762\u9762\u3002\u4ece Transformer \u7684\u6ce8\u610f\u529b\u673a\u5236\u5230\u5bf9\u6bd4\u5b66\u4e60\u7684\u76f8\u4f3c\u5ea6\u5ea6\u91cf，\u6838\u7684\u5f71\u5b50\u65e0\u5904\u4e0d\u5728\u3002<\/p>\n<\/li>\n

          • \n

            \u4ece\u7edf\u8ba1\u5b66\u5230\u6df1\u5ea6\u5b66\u4e60\u7684\u6865\u6881：Vapnik \u7684\u7406\u8bba\u6846\u67b6\u2014\u2014\u7279\u522b\u662f VC \u7ef4\u548c\u7ed3\u6784\u98ce\u9669\u6700\u5c0f\u5316\u2014\u2014\u4e3a\u7406\u89e3\u6df1\u5ea6\u5b66\u4e60\u7684\u6cdb\u5316\u80fd\u529b\u63d0\u4f9b\u4e86\u91cd\u8981\u7684\u7406\u8bba\u5de5\u5177\u3002\u4e3a\u4ec0\u4e48\u8fc7\u53c2\u6570\u5316\u7684\u6df1\u5ea6\u7f51\u7edc\u4e0d\u4f1a\u8fc7\u62df\u5408？\u8fd9\u4e2a\u95ee\u9898\u81f3\u4eca\u4ecd\u662f\u7406\u8bba\u7814\u7a76\u7684\u524d\u6cbf，\u800c Vapnik \u7684\u7406\u8bba\u662f\u89e3\u7b54\u8fd9\u4e2a\u95ee\u9898\u7684\u91cd\u8981\u8d77\u70b9\u3002<\/p>\n<\/li>\n

            SVM \u65f6\u4ee3\u662f\u673a\u5668\u5b66\u4e60\u4ece"\u5de5\u7a0b"\u8d70\u5411"\u79d1\u5b66"\u7684\u5173\u952e\u65f6\u671f\u3002\u5b83\u544a\u8bc9\u6211\u4eec：\u6700\u597d\u7684\u7b97\u6cd5\u4e0d\u4ec5\u5e94\u8be5\u5728\u5b9e\u8df5\u4e2d\u6709\u6548，\u8fd8\u5e94\u8be5\u5728\u7406\u8bba\u4e0a\u53ef\u4ee5\u88ab\u7406\u89e3\u3002<\/p>\n

            \n

            \u672c\u6587\u662f"AI \u57fa\u7840\u7406\u8bba"\u7cfb\u5217\u7684\u7b2c\u4e5d\u7bc7\u3002\u4e0b\u4e00\u7bc7\u6211\u4eec\u5c06\u4ece\u66f4\u9ad8\u7684\u89c6\u89d2\u5ba1\u89c6 AI \u65b9\u6cd5\u8bba\u7684\u4e09\u6b21\u8303\u5f0f\u8f6c\u79fb\u2014\u2014\u4ece\u7b26\u53f7\u4e3b\u4e49\u5230\u8fde\u63a5\u4e3b\u4e49，\u518d\u5230\u7edf\u8ba1\u5b66\u4e60\u3002<\/p>\n

            \n

            \u672c\u7cfb\u5217\u8986\u76d6 AI \u5927\u6a21\u578b\u57fa\u7840\u3001Agent \u5f00\u53d1\u3001MCP \u534f\u8bae\u3001Skill \u5f00\u53d1\u3001RAG\u3001\u6a21\u578b\u5fae\u8c03\u3001\u90e8\u7f72\u63a8\u7406 \u4e03\u5927\u65b9\u5411，\u4ece\u5165\u95e8\u5230\u5b9e\u6218\u7684\u5168\u6808\u5185\u5bb9\u6301\u7eed\u66f4\u65b0\u4e2d\u3002<\/p>\n

            \u6240\u6709\u6587\u7ae0\u7684 Markdown \u6e90\u6587\u4ef6\u3001\u53ef\u8fd0\u884c\u4ee3\u7801\u3001\u9ad8\u6e05\u914d\u56fe\u5df2\u6574\u7406\u6210\u5b8c\u6574\u8d44\u6599\u5305\u3002<\/p>\n

            👍 \u70b9\u8d5e + \u2b50 \u5173\u6ce8，\u8bc4\u8bba\u533a\u6263\u300c1\u300d，\u6328\u4e2a\u53d1\u4f60\u9886\u53d6\u65b9\u5f0f 👇<\/p>\n","protected":false},"excerpt":{"rendered":"

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