127.0.0.1:9200<br>URL: http://127.0.0.1:9200/likecs_art_db/_search<br>REQUEST:<pre>Array
(
    [query] => Array
        (
            [match] => Array
                (
                    [text] => Array
                        (
                            [query] => 逻辑回归损失函数推导
                        )

                )

        )

    [highlight] => Array
        (
            [fields] => Array
                (
                    [text] => stdClass Object
                        (
                        )

                )

            [pre_tags] => #em#
            [post_tags] => #/em#
        )

    [size] => 8
    [from] => 0
)
</pre>RESPONSE:<pre>string(7861) "{"took":22,"timed_out":false,"_shards":{"total":1,"successful":1,"skipped":0,"failed":0},"hits":{"total":{"value":10000,"relation":"gte"},"max_score":53.987785,"hits":[{"_index":"likecs_art_db","_type":"_doc","_id":"54860","_score":53.987785,"_source":{"id":"54860","text":"\u903b\u8f91\u56de\u5f52\u635f\u5931\u51fd\u6570\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"shayue","tagsname":"\u673a\u5668\u5b66\u4e60|\u903b\u8f91\u56de\u5f52","tagsid":"[\"171\",\"17849\"]","catesname":"\u673a\u5668\u5b66\u4e60\u57fa\u77f3","catesid":"[\"6757\"]","createtime":"1552404591"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em##em#损#/em##em#失#/em##em#函#/em##em#数#/em##em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"228931","_score":36.715263,"_source":{"id":"228931","text":"\u903b\u8f91\u56de\u5f52\u4ee3\u4ef7\u51fd\u6570\u7684\u8be6\u7ec6\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"loongofqiao","tagsname":"","tagsid":"","catesname":null,"catesid":"","createtime":"1629524609"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#代价#em#函#/em##em#数#/em#的详细#em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"961770","_score":35.710526,"_source":{"id":"961770","text":"\u903b\u8f91\u56de\u5f52\uff08Logistic Regression\uff09\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"lxs0731","tagsname":null,"tagsid":"","catesname":null,"catesid":"","createtime":"1639327343"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#（Logistic Regression）#em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"228930","_score":35.568886,"_source":{"id":"228930","text":"\u903b\u8f91\u56de\u5f52\u6a21\u578b\u53ca\u5176\u4ee3\u4ef7\u51fd\u6570\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"mtcnn","tagsname":"","tagsid":"","catesname":null,"catesid":"","createtime":"1629524599"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#模型及其代价#em#函#/em##em#数#/em##em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"203395422","_score":34.46506,"_source":{"id":"203395422","text":"\u56de\u5f52\u635f\u5931\u51fd\u6570: L1 Loss","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"","tagsname":null,"tagsid":"","catesname":null,"catesid":"","createtime":"1620240268"},"highlight":{"text":["#em#回#/em##em#归#/em##em#损#/em##em#失#/em##em#函#/em##em#数#/em#: L1 Loss"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"484269","_score":34.33963,"_source":{"id":"484269","text":"logstic\u56de\u5f52\u635f\u5931\u51fd\u6570\u53ca\u68af\u5ea6\u4e0b\u964d\u516c\u5f0f\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"peterwong666","tagsname":"","tagsid":"","catesname":null,"catesid":"","createtime":"1638879237"},"highlight":{"text":["logstic#em#回#/em##em#归#/em##em#损#/em##em#失#/em##em#函#/em##em#数#/em#及梯度下降公式#em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"228900","_score":33.338943,"_source":{"id":"228900","text":"\u903b\u8f91\u56de\u5f52\u2014\u2014\u4ee3\u4ef7\u51fd\u6570","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"qkloveslife","tagsname":"","tagsid":"","catesname":null,"catesid":"","createtime":"1629524491"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#——代价#em#函#/em##em#数#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"203428700","_score":33.338943,"_source":{"id":"203428700","text":"Sigmoid\u51fd\u6570\u4e0e\u903b\u8f91\u56de\u5f52","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"","tagsname":null,"tagsid":"","catesname":null,"catesid":"","createtime":"1617716193"},"highlight":{"text":["Sigmoid#em#函#/em##em#数#/em#与#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#"]}}]}}"
</pre>127.0.0.1:9200<br>URL: http://127.0.0.1:9200/likecs_art_db/_search<br>REQUEST:<pre>Array
(
    [query] => Array
        (
            [match] => Array
                (
                    [text] => Array
                        (
                            [query] => 逻辑回归损失函数推导
                        )

                )

        )

    [highlight] => Array
        (
            [fields] => Array
                (
                    [text] => stdClass Object
                        (
                        )

                )

            [pre_tags] => #em#
            [post_tags] => #/em#
        )

    [size] => 8
    [from] => 8
)
</pre>RESPONSE:<pre>string(8205) "{"took":18,"timed_out":false,"_shards":{"total":1,"successful":1,"skipped":0,"failed":0},"hits":{"total":{"value":10000,"relation":"gte"},"max_score":53.987785,"hits":[{"_index":"likecs_art_db","_type":"_doc","_id":"203283952","_score":33.25441,"_source":{"id":"203283952","text":"\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u903b\u8f91\u56de\u5f52(\u4ea4\u53c9\u71b5)\uff0csoftmax\u635f\u5931\u51fd\u6570\u7684\u63a8\u5bfc\uff0c\u4ee5\u53casigmoid, relu, tanh, softmax\u51fd\u6570\u7684\u7528\u5904","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"","tagsname":null,"tagsid":"","catesname":null,"catesid":"","createtime":"1636473242"},"highlight":{"text":["深度学习中，#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#(交叉熵)，softmax#em#损#/em##em#失#/em##em#函#/em##em#数#/em#的#em#推#/em##em#导#/em#，以及sigmoid, relu, tanh, softmax#em#函#/em##em#数#/em#的用处"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"203369750","_score":33.162243,"_source":{"id":"203369750","text":"\u903b\u8f91\u56de\u5f52\u68af\u5ea6\u4e0b\u964d\u63a8\u5bfc","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"","tagsname":null,"tagsid":"","catesname":null,"catesid":"","createtime":"1634437674"},"highlight":{"text":["#em#逻#/em##em#辑#/em##em#回#/em##em#归#/em#梯度下降#em#推#/em##em#导#/em#"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"54689","_score":32.005657,"_source":{"id":"54689","text":"\u7ebf\u6027\u56de\u5f52\u635f\u5931\u51fd\u6570\u6c42\u89e3","intro":"\u76ee\u5f55\n\nECharts\n\u5f02\u6b65\u52a0\u8f7d\n\n\n\nECharts\r\n\u6570\u636e\u53ef\u89c6\u5316\u5728\u8fc7\u53bb\u51e0\u5e74\u4e2d\u53d6\u5f97\u4e86\u5de8\u5927\u8fdb\u5c55\u3002\u5f00\u53d1\u4eba\u5458\u5bf9\u53ef\u89c6\u5316\u4ea7\u54c1\u7684\u671f\u671b\u4e0d\u518d\u662f\u7b80\u5355\u7684\u56fe\u8868\u521b\u5efa\u5de5\u5177\uff0c\u800c\u662f\u5728\u4ea4\u4e92\u3001\u6027\u80fd\u3001\u6570\u636e\u5904\u7406\u7b49\u65b9\u9762\u6709\u66f4\u9ad8\u7684\u8981\u6c42\u3002\r\nchart.setOption({\r\n    color: [\r\n        ","username":"shayue","tagsname":"\u673a\u5668\u5b66\u4e60|\u7ebf\u6027\u4ee3\u6570|\u6700\u5c0f\u4e8c\u4e58\u6cd5","tagsid":"[\"171\",\"2873\",\"18491\"]","catesname":"\u7ebf\u6027\u4ee3\u6570","catesid":"[\"3847\"]","createtime":"1552316788"},"highlight":{"text":["线性#em#回#/em##em#归#/em##em#损#/em##em#失#/em##em#函#/em##em#数#/em#求解"]}},{"_index":"likecs_art_db","_type":"_doc","_id":"755220","_score":32.000816,"_source":{"id":"755220","text":"\u3010353\u3011\u7ebf\u6027\u56de\u5f52\u635f\u5931\u51fd\u6570\u6c42\u5bfc\u4e3e\u4f8b - 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</pre>127.0.0.1:9200<br>URL: http://192.168.101.128/searchcore/index.php/cihere_cn_db/_search<br>REQUEST:<pre>Array
(
    [query] => Array
        (
            [match] => Array
                (
                    [title] => Array
                        (
                            [query] => 逻辑回归损失函数推导
                        )

                )

        )

    [highlight] => Array
        (
            [fields] => Array
                (
                    [title] => stdClass Object
                        (
                        )

                )

            [pre_tags] => #em#
            [post_tags] => #/em#
        )

    [from] => 0
)
</pre>RESPONSE:<pre>bool(false)
</pre>127.0.0.1:9200<br>URL: http://127.0.0.1:9200/likecs_down_db/_search<br>REQUEST:<pre>Array
(
    [query] => Array
        (
            [bool] => Array
                (
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                        (
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                                (
                                    [match] => Array
                                        (
                                            [title] => Array
                                                (
                                                    [query] => 逻辑回归损失函数推导
                                                )

                                        )

                                )

                        )

                    [must_not] => Array
                        (
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                                (
                                    [term] => Array
                                        (
                                            [cate1] => 电子书籍
                                        )

                                )

                        )

                )

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        (
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                (
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                        (
                        )

                )

            [pre_tags] => #em#
            [post_tags] => #/em#
        )

    [size] => 5
    [from] => 0
)
</pre>RESPONSE:<pre>string(2957) "{"took":10,"timed_out":false,"_shards":{"total":1,"successful":1,"skipped":0,"failed":0},"hits":{"total":{"value":2655,"relation":"eq"},"max_score":17.191544,"hits":[{"_index":"likecs_down_db","_type":"_doc","_id":"68876","_score":17.191544,"_source":{"id":"68876","title":"\u6613\u8bed\u8a00\u786c\u76d8\u903b\u8f91\u9501","spidertime":"1623080450","contenttime":"1624996047","pageimage":"https:\/\/img.jbzj.com\/file_images\/article\/201811\/20181106103934.jpg","tag":"\u6613\u8bed\u8a00|\u786c\u76d8|\u903b\u8f91\u9501","cate1":"\u6e90\u7801\u4e0b\u8f7d","cate2":"\u8f6f\u4ef6\u5f00\u53d1","cate3":"\u6613\u8bed\u8a00\u6e90\u7801","attr1":"49KB"},"highlight":{"title":["易语言硬盘#em#逻#/em##em#辑#/em#锁"]}},{"_index":"likecs_down_db","_type":"_doc","_id":"6222","_score":17.191544,"_source":{"id":"6222","title":"\u6613\u8bed\u8a00\u786c\u76d8\u903b\u8f91\u9501","spidertime":"1622872198","contenttime":"1671385795","pageimage":"https:\/\/img.jbzj.com\/file_images\/article\/201811\/20181106103934.jpg","tag":"\u6613\u8bed\u8a00|\u786c\u76d8|\u903b\u8f91\u9501","cate1":"\u6e90\u7801\u4e0b\u8f7d","cate2":"\u8f6f\u4ef6\u5f00\u53d1","cate3":"\u6613\u8bed\u8a00\u6e90\u7801","attr1":"49KB"},"highlight":{"title":["易语言硬盘#em#逻#/em##em#辑#/em#锁"]}},{"_index":"likecs_down_db","_type":"_doc","_id":"71965","_score":16.569603,"_source":{"id":"71965","title":"\u4fee\u590dWord\u627e\u56de\u4e22\u5931\u7684\u6570\u636e \u4e2d\u6587 PDF \u9ad8\u6e05\u7248","spidertime":"1623127820","contenttime":"1625124798","pageimage":"https:\/\/img.jbzj.com\/do\/uploads\/litimg\/120731\/1510331E260.gif","tag":"\u4fee\u590dWord|\u627e\u56de\u4e22\u5931\u7684\u6570\u636e","cate1":"\u7535\u5b50\u4e66\u7c4d","cate2":"\u529e\u516c\u6559\u80b2","attr1":"2.07MB"},"highlight":{"title":["修复Word找#em#回#/em#丢#em#失#/em#的#em#数#/em#据 中文 PDF 高清版"]}},{"_index":"likecs_down_db","_type":"_doc","_id":"58534","_score":14.904851,"_source":{"id":"58534","title":"Matlab\u4e2d\u7684\u903b\u8f91\u8fd0\u7b97 \u4e2d\u6587WORD\u7248","spidertime":"1623065564","contenttime":"1679248861","pageimage":"https:\/\/img.jbzj.com\/do\/uploads\/litimg\/161129\/155124593321.png","tag":"Matlab|\u903b\u8f91\u8fd0\u7b97","cate1":"\u7535\u5b50\u4e66\u7c4d","cate2":"\u7f16\u7a0b\u5f00\u53d1","cate3":"matlab","attr1":"8.51KB"},"highlight":{"title":["Matlab中的#em#逻#/em##em#辑#/em#运算 中文WORD版"]}},{"_index":"likecs_down_db","_type":"_doc","_id":"7264","_score":14.904851,"_source":{"id":"7264","title":"Matlab\u4e2d\u7684\u903b\u8f91\u8fd0\u7b97 \u4e2d\u6587WORD\u7248","spidertime":"1622873451","contenttime":"1679458801","pageimage":"https:\/\/img.jbzj.com\/do\/uploads\/litimg\/161129\/155124593321.png","tag":"Matlab|\u903b\u8f91\u8fd0\u7b97","cate1":"\u7535\u5b50\u4e66\u7c4d","cate2":"\u7f16\u7a0b\u5f00\u53d1","cate3":"matlab","attr1":"8.51KB"},"highlight":{"title":["Matlab中的#em#逻#/em##em#辑#/em#运算 中文WORD版"]}}]}}"
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                        <h1 class="page-title"><a href="/show-54860.html" title="逻辑回归损失函数推导">逻辑回归损失函数推导</a></h1>
            
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            <footer class="tc"> shayue 
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<h2 id="逻辑回归">逻辑回归</h2>
<h3 id="引言">引言</h3>
<p>假设今天希望将机器学习应用到医院中去，比如对于某一个患了心脏病的病人，求他3个月之后病危的概率。那么我们该选择哪一个模型，或者可以尝试已经学过的线性回归？</p>
<p>但是很遗憾的是，如果我们要利用线性回归，我们收集到的资料中应当包含病人3个月后病危的概率。这在实际中是很难得到的，因为对于一个患病的病人，你只能知道他3个月后到底是病危或者存活。所以线性回归并不适用这种场景。</p>
<h3 id="logistic函数">logistic函数</h3>
<p>上面提到我们最终的目标是一个概率值<span class="math inline">\(P(y|x)\)</span>，这里<span class="math inline">\(y=+1\)</span>指代病人3个月后病危这个事件;<span class="math inline">\(y=-1\)</span>指代病人3个月后存活这个事件。显然<span class="math inline">\(P(-1|x) = 1 - P(1|x)\)</span>.</p>
<p>我们先前学过线性回归，知道可以通过加权的方式求出各项特征的'分数'，那这个分数怎么转换为一个概率值？这里就需要引入一个logistic函数。它的表达式为：<span class="math display">\[
\theta(s)=\frac{1}{1+e^{-s}}
\]</span><br>
它的图像如下所示：</p>
<p><img src="/default/index/img?u=L2RlZmF1bHQvaW5kZXgvaW1nP3U9YUhSMGNITTZMeTlwYldjeU1ERTRMbU51WW14dlozTXVZMjl0TDJKc2IyY3ZNVFV6TVRBMk55OHlNREU1TURNdk1UVXpNVEEyTnkweU1ERTVNRE14TWpJek1qWTFPVEE0TkMweE16UXlOVEF4TlRRM0xuQnVadz09" height="50%" width="50%"></p>
<p>可以看到这个函数有十分不错的性质：</p>
<ol>
<li><span class="math inline">\(s(-∞)=0, \ s(+∞)=1\)</span></li>
<li><span class="math inline">\(1-\theta(s)=\theta(-s)\)</span></li>
</ol>
<p>也就是说我们可以把加权得到的'分数'通过logistic函数转化为一个概率值，并且加权得到的'分数'越大，这个概率值也越大。这真的还蛮有道理的。</p>
<p>好了，我们的模型已经定义完毕了，称它为逻辑回归模型：<span class="math display">\[
\begin{equation}
h(x) = \frac{1}{1+e^{-w^Tx}} \ \ \ \ \ w,x都是向量
\end{equation}
\]</span><br>
也就是说，我们获取到一个病人的特征<span class="math inline">\(x\)</span>，将它输入模型，就能知晓这个病人3个月后病危的概率。但是，还有最重要的一步，这个模型的参数<span class="math inline">\(w\)</span>如何确定？不同的参数<span class="math inline">\(w\)</span>会带来不同的模型<span class="math inline">\(h(x)\)</span>.经验告诉我们可以从已获得的资料中找到一些端倪获取最合适的<span class="math inline">\(w\)</span>。</p>
<h3 id="损失函数">损失函数</h3>
<p>线性回归中，我们定义了一个平方损失函数，通过对损失函数求导数得到最后的参数。那依样画葫芦，我们也为逻辑回归定义一个损失函数，然后试着对损失函数求梯度，是不是能解出最后的参数了。那么想一下，逻辑回归的损失函数如何定义？还用最小二乘法么？这显然不符合场景，毕竟已有的资料只告诉我们每一组数据对应的结果是哪一类的。</p>
<p>我们还是从数据的产生来分析，现在已有的数据是这些：<span class="math display">\[
D = {(x_1, 1), (x_2, 1), (x_3, 1), ... , (x_n, -1)}
\]</span><br>
当然，这些数据的产生是相互独立的，所以获得<span class="math inline">\(D\)</span>这笔资料的概率就是<span class="math display">\[
\begin{equation}
P(x_1, 1) * P(x_2, 1) * P(x_3, 1) * ... * P(x_n, -1)
\end{equation}\]</span><br>
再将(2)式写为条件概率分布<span class="math display">\[
\begin{equation}
P(x_1)P(1|x_1) * P(x_2)P(1|x_2) * P(x_3)P(1|x_3) * ... * P(x_n)P(-1|x_n)
\end{equation}\]</span><br>
再者，假设每一笔数据的产生服从0-1分布。<span class="math display">\[\begin{equation}
P(y|x_i) = 
\left \{
\begin{array}{lr}
f(x_i) \ \ \ \ \ \ \ \ \ \ y=+1 \\
1 - f(x_i) \ \ \ \ \ y=-1
\end{array}
\right.
\end{equation}\]</span><br>
所以最后写成的形式：<span class="math display">\[\begin{equation}
P(x_1)f(x_1) * P(x_2)f(x_2) * P(x_3)f(x_3) * ... * P(x_n)(1-f(x_n))
\end{equation}\]</span></p>
<p>也就说这笔资料<span class="math inline">\(D\)</span>由真正的模型<span class="math inline">\(f(x)\)</span>产生的话，概率是(5)这么大。但是我们不知道真正的模型f(x)长什么样子，我们现在只知道我们自己定义了一个模型<span class="math inline">\(h(x)\)</span>，它长成(1)这个样子。所以现在的任务就是从很多的<span class="math inline">\(h(x)_1, h(x)_2, h(x)_3, ..., h(x)_m\)</span>中找到其中一个最接近真正的模型<span class="math inline">\(f(x)\)</span>并将它作为我们最后的<span class="math inline">\(h(x)\)</span>。</p>
<p>所以如何衡量<span class="math inline">\(h(x)\)</span>与<span class="math inline">\(f(x)\)</span>的接近程度？如果我们现在用<span class="math inline">\(h(x)\)</span>代替<span class="math inline">\(f(x)\)</span>去产生这组数据集<span class="math inline">\(D\)</span>也能得到一个概率(6).<span class="math display">\[\begin{equation}
P(x_1)h(x_1) * P(x_2)h(x_2) * P(x_3)h(x_3) * ... * P(x_n)(1-h(x_n))
\end{equation}\]</span></p>
<p><font color="red"><strong>使得(6)式的概率最大的那个<span class="math inline">\(h(x)\)</span>我们会认为它与<span class="math inline">\(f(x)\)</span>最相似，这就是最大似然的思想。</strong></font>又因为对于所有的<span class="math inline">\(h(x)_i\)</span>产生的概率：<span class="math display">\[\begin{equation}
P(x_1) * P(x_2) * P(x_3) * ... * P(x_n)
\end{equation}\]</span><br>
这部分都是相同的，所以我们认为最接近<span class="math inline">\(f(x)\)</span>的<span class="math inline">\(h(x)\)</span>能使(8)最大即可<br><span class="math display">\[\begin{equation}
h(x_1) * h(x_2) * h(x_3) * ... * (1-h(x_n))
\end{equation}\]</span><br>
再由于logistic函数的第2个性质，可以将(8)变形：<br><span class="math display">\[\begin{equation}
h(x_1) * h(x_2) * h(x_3) * ... * h(-x_n)
\end{equation}\]</span><br>
最终的目标是解出下面这个优化问题：<span class="math display">\[\begin{equation}
 \mathop{max}\limits_{w} \ \ \prod_{i=1}^{n}h(y_ix_i)
\end{equation}\]</span><br>
再次变形，求一个式子的最大值，相当于求它相反数的最小：<span class="math display">\[\begin{equation}
 \mathop{min}\limits_{w} \ \ -\prod_{i=1}^{n}h(y_ix_i)
\end{equation}\]</span><br>
接下来我们要对(11)式取对数，一方面原因是因为对数函数的单调特性，另一方面是能将原来的连乘简化到连加，所以取对数后：<span class="math display">\[\begin{equation}
 \mathop{min}\limits_{w} \ \ -\sum_{i=1}^{n}\ln{h(y_ix_i)}
\end{equation}\]</span><br>
将<span class="math inline">\(h(x)\)</span>展开，能得到<span class="math display">\[\begin{equation}
 \mathop{min}\limits_{w} \ \ -\sum_{i=1}^{n}\ln{\frac{1}{1+e^{-y_iw^Tx_i}}} \ \ \ \ \ \ \ \ \ \ w与x_i都是向量,x_i表示第i笔数据
\end{equation}\]</span><br>
再一次<span class="math display">\[\begin{equation}
 \mathop{min}\limits_{w} \ \ \sum_{i=1}^{n}\ln{(1+e^{-y_iw^Tx_i})} \ \ \ \ \ \ \ \ \ \ w与x_i都是向量,x_i表示第i笔数据
\end{equation}\]</span><br>
大功告成，我们得到了逻辑回归的损失函数,它长成(15)式这个样子<span class="math display">\[\begin{equation}
 J(w)= \sum_{i=1}^{n}\ln{(1+e^{-y_iw^Tx_i})} \ \ \ \ \ \ \ \ \ \ w与x_i都是向量,x_i表示第i笔数据
\end{equation}\]</span><br>
我们的目标就是找到最小化<span class="math inline">\(J(w)\)</span>的那个<span class="math inline">\(w\)</span>.就像在线性回归中做的那样，接下来我们要利用链式法则对它求导：<span class="math display">\[\begin{equation}
\frac{\partial J(w)}{\partial w_j} = \sum_{i=1}^{n}\frac{\partial \ln{(1+e^{-y_iw^Tx_i})}}{\partial (-y_iw^Tx_i)} * \frac{\partial (-y_iw^Tx_i)}{\partial w_j}
\end{equation}\]</span><br>
化解得到<span class="math display">\[\begin{equation}
\frac{\partial J(w)}{\partial w_j} = \sum_{i=1}^{n}\frac{e^{-y_iw^Tx_i}}{1+e^{-y_iw^Tx_i}} * (-y_ix_{i,j}) \ \ \ \ x_{i,j}是个标量，是第i笔数据中第j个分量
\end{equation}\]</span><br>
所以对于整个向量<span class="math inline">\(w\)</span>的梯度为<span class="math display">\[
\begin{equation}
\frac{\partial J(w)}{\partial w} = \sum_{i=1}^{n}\frac{e^{-y_iw^Tx_i}}{1+e^{-y_iw^Tx_i}} * (-y_ix_i) \ \ \ \ 想象将对单个w_i的结果笔直堆成一个向量
\end{equation}\]</span><br>
而<span class="math inline">\(\frac{e^{-y_iww^Tx_i}}{1+e^{-y_iww^Tx_i}}\)</span>正好是逻辑回归函数，所以最终对<span class="math inline">\(w\)</span>的梯度写成下面这个样子<span class="math display">\[
\begin{equation}
\frac{\partial J(w)}{\partial w} = \sum_{i=1}^{n}h(-y_iw^Tx_i)(-y_ix_i) 
\end{equation}\]</span><br>
很遗憾，我们令(19)等于0的话，很难求解出<span class="math inline">\(w\)</span>。为此，我们需要用额外的方法求解这个问题。</p>
<h3 id="梯度下降">梯度下降</h3>
<p>这个可学习的资料太多了，思想就是假设函数上有一个点，它沿着各个方向都有它的方向导数，那么总是沿着方向导数最大的反方向走，也就是梯度的反方向走，这个点总是能走到最低点。每一次移动的距离用一个系数lr来表示，每次更新<span class="math inline">\(w\)</span>，数次迭代之后，<span class="math inline">\(w\)</span>趋近于最优解：<span class="math display">\[
\begin{equation}
w_{i+1} :=  w_{i} - lr * \sum_{i=1}^{n}\frac{e^{-y_iw^Tx_i}}{1+e^{-y_iw^Tx_i}} * (-y_ix_i) \ \ \ \ \ lr是大于0的系数
\end{equation}\]</span></p>
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