{"id":15647,"date":"2024-01-15T03:09:38","date_gmt":"2024-01-14T23:39:38","guid":{"rendered":"https:\/\/rasanegar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/"},"modified":"2024-01-15T03:09:38","modified_gmt":"2024-01-14T23:39:38","slug":"gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87","status":"publish","type":"post","link":"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/","title":{"rendered":"Gradient Descent \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646: \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0648 \u0646\u0638\u0631\u06cc\u0647"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_85 counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\"><p class=\"ez-toc-title\" style=\"cursor:inherit\">\u0633\u0631\u0641\u0635\u0644\u0647\u0627\u06cc \u0645\u0637\u0644\u0628<\/p>\n<\/div><nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d9%85%d8%b9%d8%b1%d9%81%db%8c\" >\u0645\u0639\u0631\u0641\u06cc<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#gradient_descent_%da%86%db%8c%d8%b3%d8%aa%d8%9f\" >Gradient Descent \u0686\u06cc\u0633\u062a\u061f<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d8%a7%d8%b6%d8%a7%d9%81%d9%87_%da%a9%d8%b1%d8%af%d9%86_%d9%85%d9%88%d9%85%d9%86%d8%aa%d9%88%d9%85\" >\u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u0646 \u0645\u0648\u0645\u0646\u062a\u0648\u0645<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d9%be%db%8c%d8%a7%d8%af%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_gradient_descent_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\" >\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc Gradient Descent \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d8%a8%d9%87%db%8c%d9%86%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_%d8%aa%d9%88%d8%a7%d8%a8%d8%b9_%d8%a8%d8%a7_gradient_descent\" >\u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u062a\u0648\u0627\u0628\u0639 \u0628\u0627 Gradient Descent<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d8%aa%d8%ac%d8%b3%d9%85_%d8%aa%d8%a7%d8%a8%d8%b9_%d9%87%d8%af%d9%81_fx\" >\u062a\u062c\u0633\u0645 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 f(x)<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d8%af%d8%b1_%d8%ad%d8%a7%d9%84_%d8%a7%d8%ac%d8%b1%d8%a7_gradient_descent_%d8%a8%d8%a7_%d9%87%d8%a7%db%8c%d9%be%d8%b1%d9%be%d8%a7%d8%b1%d8%a7%d9%85%d8%aa%d8%b1%d9%87%d8%a7%db%8c_%d9%85%d8%ae%d8%aa%d9%84%d9%81\" >\u062f\u0631 \u062d\u0627\u0644 \u0627\u062c\u0631\u0627 Gradient Descent \u0628\u0627 \u0647\u0627\u06cc\u067e\u0631\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#gradient_descent_%d8%a8%d8%b1%d8%a7%db%8c_%d8%a8%d9%87_%d8%ad%d8%af%d8%a7%d9%82%d9%84_%d8%b1%d8%b3%d8%a7%d9%86%d8%af%d9%86_%d8%ae%d8%b7%d8%a7%db%8c_%d9%85%db%8c%d8%a7%d9%86%da%af%db%8c%d9%86_%d9%85%d8%b1%d8%a8%d8%b9\" >Gradient Descent \u0628\u0631\u0627\u06cc \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0631\u0633\u0627\u0646\u062f\u0646 \u062e\u0637\u0627\u06cc \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d8%af%d8%b1_%d8%ad%d8%a7%d9%84_%d8%a7%d8%ac%d8%b1%d8%a7_gradient_descent_%d8%b1%d9%88%db%8c_ocr\" >\u062f\u0631 \u062d\u0627\u0644 \u0627\u062c\u0631\u0627 Gradient Descent \u0631\u0648\u06cc OCR<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d9%86%d8%b2%d9%88%d9%84_%da%af%d8%b1%d8%a7%d8%af%db%8c%d8%a7%d9%86_%d8%aa%d8%b5%d8%a7%d8%af%d9%81%db%8c_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\" >\u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0635\u0627\u062f\u0641\u06cc \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d9%85%d9%82%d8%a7%db%8c%d8%b3%d9%87_%d9%86%d8%b3%d8%ae%d9%87_%d9%87%d8%a7%db%8c_%d8%af%d8%b3%d8%aa%d9%87_%d8%a7%db%8c_%d9%88_%d8%aa%d8%b5%d8%a7%d8%af%d9%81%db%8c\" >\u0645\u0642\u0627\u06cc\u0633\u0647 \u0646\u0633\u062e\u0647 \u0647\u0627\u06cc \u062f\u0633\u062a\u0647 \u0627\u06cc \u0648 \u062a\u0635\u0627\u062f\u0641\u06cc<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/rasanegaar.com\/blog\/gradient-descent-%d8%af%d8%b1-%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86-%d9%be%db%8c%d8%a7%d8%af%d9%87-%d8%b3%d8%a7%d8%b2%db%8c-%d9%88-%d9%86%d8%b8%d8%b1%db%8c%d9%87\/#%d9%86%d8%aa%db%8c%d8%ac%d9%87_%da%af%db%8c%d8%b1%db%8c\" >\u0646\u062a\u06cc\u062c\u0647 \u06af\u06cc\u0631\u06cc<\/a><\/li><\/ul><\/nav><\/div>\n<span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u0632\u0645\u0627\u0646 \u0644\u0627\u0632\u0645 \u0628\u0631\u0627\u06cc \u0645\u0637\u0627\u0644\u0639\u0647: <\/span> <span class=\"rt-time\"> 9<\/span> <span class=\"rt-label rt-postfix\">\u062f\u0642\u06cc\u0642\u0647<\/span><\/span><p> <br \/>\n<\/p>\n<div><noscript><\/noscript><\/p>\n<h2 id=\"introduction\"><span class=\"ez-toc-section\" id=\"%d9%85%d8%b9%d8%b1%d9%81%db%8c\"><\/span>\u0645\u0639\u0631\u0641\u06cc<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0627\u06cc\u0646 \u0622\u0645\u0648\u0632\u0634 \u0645\u0642\u062f\u0645\u0647 \u0627\u06cc \u0627\u0633\u062a \u0628\u0631 \u06cc\u06a9 \u062a\u06a9\u0646\u06cc\u06a9 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0633\u0627\u062f\u0647 \u0628\u0647 \u0646\u0627\u0645 <strong>\u0634\u06cc\u0628 \u0646\u0632\u0648\u0644<\/strong>\u060c \u06a9\u0647 \u062f\u0631 \u067e\u06cc\u0634\u0631\u0641\u062a\u0647 \u062a\u0631\u06cc\u0646 \u0645\u062f\u0644 \u0647\u0627\u06cc \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0645\u0627\u0634\u06cc\u0646 \u06a9\u0627\u0631\u0628\u0631\u062f \u0639\u0645\u062f\u0647 \u0627\u06cc \u062f\u0627\u0634\u062a\u0647 \u0627\u0633\u062a.<\/p>\n<p>\u0645\u0627 \u06cc\u06a9 \u0631\u0648\u0627\u0644 \u0647\u062f\u0641 \u06a9\u0644\u06cc \u0628\u0631\u0627\u06cc \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u0648 \u0627\u0639\u0645\u0627\u0644 \u0622\u0646 \u0628\u0631\u0627\u06cc \u062d\u0644 \u0645\u0633\u0627\u0626\u0644 \u0645\u062e\u062a\u0644\u0641\u060c \u0627\u0632 \u062c\u0645\u0644\u0647 \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u0627\u0632 \u0637\u0631\u06cc\u0642 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0646\u0638\u0627\u0631\u062a \u0634\u062f\u0647\u060c \u0627\u06cc\u062c\u0627\u062f \u062e\u0648\u0627\u0647\u06cc\u0645 \u06a9\u0631\u062f.<\/p>\n<p>\u062f\u0631 \u0627\u06cc\u0646 process\u060c \u0645\u0627 \u0628\u06cc\u0646\u0634\u06cc \u062f\u0631 \u0645\u0648\u0631\u062f \u0639\u0645\u0644\u06a9\u0631\u062f \u0627\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0628\u0647 \u062f\u0633\u062a \u062e\u0648\u0627\u0647\u06cc\u0645 \u0622\u0648\u0631\u062f \u0648 \u062a\u0623\u062b\u06cc\u0631 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0641\u0631\u0627\u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0631\u0627 \u0645\u0637\u0627\u0644\u0639\u0647 \u062e\u0648\u0627\u0647\u06cc\u0645 \u06a9\u0631\u062f. \u0631\u0648\u06cc \u0639\u0645\u0644\u06a9\u0631\u062f \u0622\u0646  \u0645\u0627 \u0647\u0645 \u0645\u06cc\u0631\u06cc\u0645 <em>\u0646\u0632\u0648\u0644 \u0634\u06cc\u0628 \u062f\u0633\u062a\u0647 \u0627\u06cc \u0648 \u062a\u0635\u0627\u062f\u0641\u06cc<\/em> \u0627\u0646\u0648\u0627\u0639 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0646\u0645\u0648\u0646\u0647<\/p>\n<h2 id=\"whatisgradientdescent\"><span class=\"ez-toc-section\" id=\"gradient_descent_%da%86%db%8c%d8%b3%d8%aa%d8%9f\"><\/span>Gradient Descent \u0686\u06cc\u0633\u062a\u061f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u06cc\u06a9 \u062a\u06a9\u0646\u06cc\u06a9 \u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0645\u06cc \u062a\u0648\u0627\u0646\u062f \u0622\u0646 \u0631\u0627 \u067e\u06cc\u062f\u0627 \u06a9\u0646\u062f <em>\u06a9\u0645\u062a\u0631\u06cc\u0646<\/em> \u0627\u0632 \u06cc\u06a9 <strong>\u062a\u0627\u0628\u0639 \u0647\u062f\u0641<\/strong>.  \u0627\u06cc\u0646 \u06cc\u06a9 \u062a\u06a9\u0646\u06cc\u06a9 \u062d\u0631\u06cc\u0635\u0627\u0646\u0647 \u0627\u0633\u062a \u06a9\u0647 \u0628\u0627 \u06af\u0627\u0645 \u0628\u0631\u062f\u0627\u0634\u062a\u0646 \u062f\u0631 \u062c\u0647\u062a \u062d\u062f\u0627\u06a9\u062b\u0631 \u0646\u0631\u062e \u06a9\u0627\u0647\u0634 \u062a\u0627\u0628\u0639\u060c \u0631\u0627\u0647 \u062d\u0644 \u0628\u0647\u06cc\u0646\u0647 \u0631\u0627 \u067e\u06cc\u062f\u0627 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u062f\u0631 \u0645\u0642\u0627\u0628\u0644\u060c <em>\u0634\u06cc\u0628 \u0635\u0639\u0648\u062f<\/em> \u06cc\u06a9 \u0647\u0645\u062a\u0627\u06cc \u0646\u0632\u062f\u06cc\u06a9 \u0627\u0633\u062a \u06a9\u0647 \u0645\u06cc \u06cc\u0627\u0628\u062f <em>\u0628\u06cc\u0634\u062a\u0631\u06cc\u0646<\/em> \u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0628\u0627 \u067e\u06cc\u0631\u0648\u06cc \u0627\u0632 \u062c\u0647\u062a \u062d\u062f\u0627\u06a9\u062b\u0631 \u0646\u0631\u062e \u0627\u0641\u0632\u0627\u06cc\u0634 \u062a\u0627\u0628\u0639.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\u060c \u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0686\u0646\u062f \u0645\u062a\u063a\u06cc\u0631\u0647 \\(f(\\textbf{w})\\) \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f\u060c \u062c\u0627\u06cc\u06cc \u06a9\u0647 \\(\\textbf w = (w_1, w_2, \\ldots, w_n)^T \\).  \u0628\u0631\u0627\u06cc \u06cc\u0627\u0641\u062a\u0646 \\( \\textbf{w} \\) \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639 \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0645\u06cc \u0631\u0633\u062f\u060c \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u0627\u0632 \u0645\u0631\u0627\u062d\u0644 \u0632\u06cc\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u062f:<\/p>\n<ol>\n<li>\n<p>\u0645\u0642\u062f\u0627\u0631 \u062a\u0635\u0627\u062f\u0641\u06cc \u0627\u0648\u0644\u06cc\u0647 \\( \\textbf{w} \\) \u0631\u0627 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0646\u06cc\u062f<\/p>\n<\/li>\n<li>\n<p>\u062a\u0639\u062f\u0627\u062f \u062d\u062f\u0627\u06a9\u062b\u0631 \u062a\u06a9\u0631\u0627\u0631 \u0631\u0627 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0646\u06cc\u062f <code>T<\/code><\/p>\n<\/li>\n<li>\n<p>\u06cc\u06a9 \u0645\u0642\u062f\u0627\u0631 \u0628\u0631\u0627\u06cc <strong>\u0645\u06cc\u0632\u0627\u0646 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc<\/strong> \\( \\eta \\in (a,b) \\)<\/p>\n<\/li>\n<li>\n<p>\u062f\u0648 \u0645\u0631\u062d\u0644\u0647 \u0631\u0627 \u062a\u0627 \u0632\u0645\u0627\u0646\u06cc \u06a9\u0647 \\(f\\) \u062a\u063a\u06cc\u06cc\u0631 \u0646\u06a9\u0646\u062f \u06cc\u0627 \u062a\u06a9\u0631\u0627\u0631\u0647\u0627 \u0627\u0632 T \u0628\u06cc\u0634\u062a\u0631 \u0634\u0648\u062f \u062a\u06a9\u0631\u0627\u0631 \u06a9\u0646\u06cc\u062f<\/p>\n<p>a. \u0645\u062d\u0627\u0633\u0628\u0647: \\( \\Delta \\textbf{w} = &#8211; \\eta \\nabla_\\textbf{w} f(\\textbf{w}) \\)<\/p>\n<p>\u0628  \u0628\u0647 \u0631\u0648\u0632 \u0631\u0633\u0627\u0646\u06cc \\(\\textbf{w} \\) \u0628\u0647 \u0635\u0648\u0631\u062a: \\(\\textbf{w} \\lefttarrow \\textbf{w} + \\Delta \\textbf{w} \\)<\/p>\n<blockquote>\n<p>\u0646\u0645\u0627\u062f \u0641\u0644\u0634 \u0633\u0645\u062a \u0686\u067e \u0627\u0632 \u0646\u0638\u0631 \u0631\u06cc\u0627\u0636\u06cc \u0631\u0648\u0634\u06cc \u0635\u062d\u06cc\u062d \u0628\u0631\u0627\u06cc \u0646\u0648\u0634\u062a\u0646 \u0628\u06cc\u0627\u0646\u06cc\u0647 \u0627\u0646\u062a\u0633\u0627\u0628 \u0627\u0633\u062a.<\/p>\n<\/blockquote>\n<\/li>\n<\/ol>\n<p>\u062f\u0631 \u0627\u06cc\u0646\u062c\u0627 \\( \\nabla_\\textbf{w} f \\) \u0646\u0634\u0627\u0646 \u062f\u0647\u0646\u062f\u0647 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \\(f\\) \u0627\u0633\u062a \u06a9\u0647 \u062a\u0648\u0633\u0637:<br \/>$$<br \/>\\nabla_\\textbf{w} f(\\textbf{w}) =<br \/>\\begin{bmatrix}<br \/>\\frac{\\partial f(\\textbf{w})}{\\partial w_1} \\<br \/>\\frac{\\partial f(\\textbf{w})}{\\partial w_2} \\<br \/>\\vdots\\<br \/>\\frac{\\partial f(\\textbf{w})}{\\partial w_n}<br \/>\\end{bmatrix}<br \/>$$<\/p>\n<p>\u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0645\u062b\u0627\u0644 \u0627\u0632 \u062f\u0648 \u0645\u062a\u063a\u06cc\u0631 \\(f(w_1,w_2) = w_1^2+w_2^2 \\) \u0631\u0627 \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f\u060c \u0633\u067e\u0633 \u062f\u0631 \u0647\u0631 \u062a\u06a9\u0631\u0627\u0631 \\( ((w_1,w_2) \\) \u0628\u0647\u200c\u0635\u0648\u0631\u062a \u0628\u0647\u200c\u0631\u0648\u0632\u0631\u0633\u0627\u0646\u06cc \u0645\u06cc\u200c\u0634\u0648\u062f:<\/p>\n<p>$$<br \/>\\\u0634\u0631\u0648\u0639 {bmatrix}<br \/>w_1 \\ w_2<br \/>\\end {bmatrix} \\lefttarrow<br \/>\\\u0634\u0631\u0648\u0639 {bmatrix}<br \/>w_1 \\ w_2<br \/>\\end {bmatrix} &#8211; \\eta<br \/>\\\u0634\u0631\u0648\u0639 {bmatrix}<br \/>2w_1 \\ 2w_2<br \/>\\\u067e\u0627\u06cc\u0627\u0646 {bmatrix}<br \/>$$<\/p>\n<p>\u0634\u06a9\u0644 \u0632\u06cc\u0631 \u0631\u0648\u0634 \u0639\u0645\u0644\u06a9\u0631\u062f \u0634\u06cc\u0628 \u0646\u0632\u0648\u0644 \u0631\u0627 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f \u0631\u0648\u06cc \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639<\/p>\n<p>\u062f\u0627\u06cc\u0631\u0647 \u0647\u0627 \u062e\u0637\u0648\u0637 \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639 \u0647\u0633\u062a\u0646\u062f.  \u0627\u06af\u0631 \u062f\u0631 \u0627\u0645\u062a\u062f\u0627\u062f \u06cc\u06a9 \u06a9\u0627\u0646\u062a\u0648\u0631 \u062d\u0631\u06a9\u062a \u06a9\u0646\u06cc\u0645\u060c <em>\u0645\u0642\u062f\u0627\u0631 \u062a\u0627\u0628\u0639<\/em> \u062a\u063a\u06cc\u06cc\u0631 \u0646\u0645\u06cc \u06a9\u0646\u062f \u0648 \u0628\u0627\u0642\u06cc \u0645\u06cc \u0645\u0627\u0646\u062f a <a rel=\"nofollow noopener\" target=\"_blank\" href=\"https:\/\/www.khanacademy.org\/math\/geometry\/xff63fac4:hs-geo-conic-sections\/hs-geo-circle-expanded-equation\/a\/circle-equation-review#:~:text=We%20know%20that%20the%20general,and%20r%20is%20the%20radius.\">\u062b\u0627\u0628\u062a<\/a>.<\/p>\n<p>\u0627\u06cc\u0646 \u0628\u0627 \u062c\u0647\u062a \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0645\u062e\u0627\u0644\u0641 \u0627\u0633\u062a\u060c \u062c\u0627\u06cc\u06cc \u06a9\u0647 \u062a\u0627\u0628\u0639 \u0628\u0627 \u062d\u062f\u0627\u06a9\u062b\u0631 \u0646\u0631\u062e \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06cc \u06a9\u0646\u062f.  \u0628\u0646\u0627\u0628\u0631\u0627\u06cc\u0646 \u062c\u0647\u062a \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0627\u0628\u0639 \u062f\u0631 \u0647\u0631 \u0646\u0642\u0637\u0647 \u0646\u0633\u0628\u062a \u0628\u0647 \u0645\u0645\u0627\u0633 \u06a9\u0627\u0646\u062a\u0648\u0631 \u062f\u0631 \u0622\u0646 \u0646\u0642\u0637\u0647 \u0646\u0631\u0645\u0627\u0644 \u0627\u0633\u062a.<\/p>\n<p>\u0628\u0647 \u0639\u0628\u0627\u0631\u062a \u0633\u0627\u062f\u0647\u060c \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u06cc\u06a9 \u0641\u0644\u0634 \u062f\u0631 \u0646\u0638\u0631 \u06af\u0631\u0641\u062a \u06a9\u0647 \u062f\u0631 \u062c\u0647\u062a\u06cc \u0627\u0633\u062a \u06a9\u0647 \u062a\u0627\u0628\u0639 \u0628\u06cc\u0634\u062a\u0631\u06cc\u0646 \u062a\u063a\u06cc\u06cc\u0631 \u0631\u0627 \u062f\u0627\u0631\u062f.<\/p>\n<p>\u067e\u06cc\u0631\u0648\u06cc \u0627\u0632 \u062c\u0647\u062a \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0645\u0646\u0641\u06cc \u0645\u0646\u062c\u0631 \u0628\u0647 \u0646\u0642\u0627\u0637\u06cc \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u062a\u0627\u0628\u0639 \u0628\u0627 \u062d\u062f\u0627\u06a9\u062b\u0631 \u0646\u0631\u062e \u06a9\u0627\u0647\u0634 \u0645\u06cc \u06cc\u0627\u0628\u062f.  \u0627\u06cc\u0646 <strong>\u0645\u06cc\u0632\u0627\u0646 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc<\/strong>\u060c \u0647\u0645\u0686\u0646\u06cc\u0646 \u0628\u0647 \u0646\u0627\u0645 <strong>\u0627\u0646\u062f\u0627\u0632\u0647 \u06af\u0627\u0645<\/strong>\u060c \u062a\u0639\u06cc\u06cc\u0646 \u0645\u06cc \u06a9\u0646\u062f \u06a9\u0647 \u0686\u0642\u062f\u0631 \u0633\u0631\u06cc\u0639 \u06cc\u0627 \u0622\u0647\u0633\u062a\u0647 \u062f\u0631 \u062c\u0647\u062a \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062d\u0631\u06a9\u062a \u06a9\u0646\u06cc\u0645.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-1.png\" alt=\"\u0634\u06cc\u0628 \u0646\u0632\u0648\u0644\" title=\"\"><\/p>\n<h3 id=\"addingmomentum\"><span class=\"ez-toc-section\" id=\"%d8%a7%d8%b6%d8%a7%d9%81%d9%87_%da%a9%d8%b1%d8%af%d9%86_%d9%85%d9%88%d9%85%d9%86%d8%aa%d9%88%d9%85\"><\/span>\u0627\u0636\u0627\u0641\u0647 \u06a9\u0631\u062f\u0646 \u0645\u0648\u0645\u0646\u062a\u0648\u0645<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0647\u0646\u06af\u0627\u0645 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\u060c \u0628\u0627 \u0645\u0634\u06a9\u0644\u0627\u062a \u0632\u06cc\u0631 \u0645\u0648\u0627\u062c\u0647 \u0645\u06cc \u0634\u0648\u06cc\u0645:<\/p>\n<ol>\n<li>\n<p>\u06af\u0631\u0641\u062a\u0627\u0631 \u0634\u062f\u0646 \u062f\u0631 \u062d\u062f\u0627\u0642\u0644 \u0645\u062d\u0644\u06cc\u060c \u06a9\u0647 \u0646\u062a\u06cc\u062c\u0647 \u0645\u0633\u062a\u0642\u06cc\u0645 \u062d\u0631\u06cc\u0635 \u0628\u0648\u062f\u0646 \u0627\u06cc\u0646 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0627\u0633\u062a<\/p>\n<\/li>\n<li>\n<p>\u0628\u06cc\u0634 \u0627\u0632 \u062d\u062f \u0648 \u0627\u0632 \u062f\u0633\u062a \u062f\u0627\u062f\u0646 \u0628\u0647\u06cc\u0646\u0647 \u062c\u0647\u0627\u0646\u06cc\u060c \u0627\u06cc\u0646 \u0646\u062a\u06cc\u062c\u0647 \u0645\u0633\u062a\u0642\u06cc\u0645 \u062d\u0631\u06a9\u062a \u0628\u06cc\u0634 \u0627\u0632 \u062d\u062f \u0633\u0631\u06cc\u0639 \u062f\u0631 \u062c\u0647\u062a \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0627\u0633\u062a<\/p>\n<\/li>\n<li>\n<p>\u0646\u0648\u0633\u0627\u0646\u060c \u0627\u06cc\u0646 \u067e\u062f\u06cc\u062f\u0647\u200c\u0627\u06cc \u0627\u0633\u062a \u06a9\u0647 \u0632\u0645\u0627\u0646\u06cc \u0631\u062e \u0645\u06cc\u200c\u062f\u0647\u062f \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u062a\u0627\u0628\u0639 \u0628\u062f\u0648\u0646 \u062a\u0648\u062c\u0647 \u0628\u0647 \u062c\u0647\u062a\u06cc \u06a9\u0647 \u067e\u06cc\u0634\u0631\u0648\u06cc \u0645\u06cc\u200c\u06a9\u0646\u062f\u060c \u0628\u0647\u200c\u0637\u0648\u0631 \u0642\u0627\u0628\u0644 \u062a\u0648\u062c\u0647\u06cc \u062a\u063a\u06cc\u06cc\u0631 \u0646\u0645\u06cc\u200c\u06a9\u0646\u062f.  \u0634\u0645\u0627 \u0645\u06cc \u062a\u0648\u0627\u0646\u06cc\u062f \u0622\u0646 \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u062d\u0631\u06a9\u062a \u062f\u0631 \u06cc\u06a9 \u0641\u0644\u0627\u062a \u062f\u0631 \u0646\u0638\u0631 \u0628\u06af\u06cc\u0631\u06cc\u062f\u060c \u0645\u0647\u0645 \u0646\u06cc\u0633\u062a \u06a9\u0647 \u06a9\u062c\u0627 \u0628\u0631\u0648\u06cc\u062f \u062f\u0631 \u0647\u0645\u0627\u0646 \u0627\u0631\u062a\u0641\u0627\u0639 \u0647\u0633\u062a\u06cc\u062f<\/p>\n<\/li>\n<\/ol>\n<p>\u0628\u0631\u0627\u06cc \u0645\u0628\u0627\u0631\u0632\u0647 \u0628\u0627 \u0627\u06cc\u0646 \u0645\u0634\u06a9\u0644\u0627\u062a\u060c \u06cc\u06a9 \u0627\u0635\u0637\u0644\u0627\u062d \u062d\u0631\u06a9\u062a\u06cc \\( \\alpha \\) \u0628\u0647 \u0639\u0628\u0627\u0631\u062a \\(\\Delta \\textbf{w}\\) \u0627\u0636\u0627\u0641\u0647 \u0645\u06cc\u200c\u0634\u0648\u062f \u062a\u0627 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0631\u0627 \u0647\u0646\u06af\u0627\u0645 \u062d\u0631\u06a9\u062a \u0628\u0647 \u0633\u0645\u062a \u0645\u0642\u062f\u0627\u0631 \u0628\u0647\u06cc\u0646\u0647 \u062c\u0647\u0627\u0646\u06cc \u062a\u062b\u0628\u06cc\u062a \u06a9\u0646\u062f.<\/p>\n<p>\u062f\u0631 \u0632\u06cc\u0631\u060c \u0645\u0627 \u0627\u0632 \u0628\u0627\u0644\u0627\u0646\u0648\u06cc\u0633 \\(i\\) \u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0634\u0645\u0627\u0631\u0647 \u062a\u06a9\u0631\u0627\u0631 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0645\u06cc \u06a9\u0646\u06cc\u0645:<br \/>$$<br \/>\\Delta \\textbf{w}^i = &#8211; \\eta \\nabla_\\textbf{w} f(\\textbf{w}^i) + \\alpha \\textbf{w}^{i-1}<br \/>$$<\/p>\n<h2 id=\"implementinggradientdescentinpython\"><span class=\"ez-toc-section\" id=\"%d9%be%db%8c%d8%a7%d8%af%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_gradient_descent_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\"><\/span>\u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc Gradient Descent \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0642\u0628\u0644 \u0627\u0632 \u0634\u0631\u0648\u0639 \u0646\u0648\u0634\u062a\u0646 \u06a9\u062f \u0648\u0627\u0642\u0639\u06cc \u0628\u0631\u0627\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\u060c \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f import \u0686\u0646\u062f \u06a9\u062a\u0627\u0628\u062e\u0627\u0646\u0647 \u06a9\u0647 \u0628\u0631\u0627\u06cc \u06a9\u0645\u06a9 \u0628\u0647 \u0645\u0627 \u0627\u0632 \u0622\u0646\u0647\u0627 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u062e\u0648\u0627\u0647\u06cc\u0645 \u06a9\u0631\u062f:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-keyword\">import<\/span> numpy <span class=\"hljs-keyword\">as<\/span> np\n<span class=\"hljs-keyword\">import<\/span> matplotlib\n<span class=\"hljs-keyword\">import<\/span> matplotlib.pyplot <span class=\"hljs-keyword\">as<\/span> plt\n<span class=\"hljs-keyword\">import<\/span> sklearn.datasets <span class=\"hljs-keyword\">as<\/span> dt\n<span class=\"hljs-keyword\">from<\/span> sklearn.model_selection <span class=\"hljs-keyword\">import<\/span> train_test_split\n<\/code><\/pre>\n<p>\u062d\u0627\u0644\u0627\u060c \u0628\u0627 \u0627\u06cc\u0646 \u06a9\u0647 \u0627\u0632 \u0631\u0627\u0647 \u0646\u06cc\u0633\u062a\u060c \u0628\u06cc\u0627\u06cc\u06cc\u062f \u062c\u0644\u0648 \u0628\u0631\u0648\u06cc\u0645 \u0648 a \u0631\u0627 \u062a\u0639\u0631\u06cc\u0641 \u06a9\u0646\u06cc\u0645 <code>gradient_descent()<\/code> \u062a\u0627\u0628\u0639.  \u062f\u0631 \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639\u060c \u062d\u0644\u0642\u0647 \u0632\u0645\u0627\u0646\u06cc \u0628\u0647 \u067e\u0627\u06cc\u0627\u0646 \u0645\u06cc \u0631\u0633\u062f \u06a9\u0647:<\/p>\n<ol>\n<li>\n<p>\u062a\u0639\u062f\u0627\u062f \u062a\u06a9\u0631\u0627\u0631\u0647\u0627 \u0627\u0632 \u062d\u062f\u0627\u06a9\u062b\u0631 \u0645\u0642\u062f\u0627\u0631 \u0628\u06cc\u0634\u062a\u0631 \u0627\u0633\u062a<\/p>\n<\/li>\n<li>\n<p>\u062a\u0641\u0627\u0648\u062a \u062f\u0631 \u0645\u0642\u0627\u062f\u06cc\u0631 \u062a\u0627\u0628\u0639 \u0628\u06cc\u0646 \u062f\u0648 \u062a\u06a9\u0631\u0627\u0631 \u0645\u062a\u0648\u0627\u0644\u06cc \u06a9\u0645\u062a\u0631 \u0627\u0632 \u06cc\u06a9 \u0622\u0633\u062a\u0627\u0646\u0647 \u0645\u0634\u062e\u0635 \u0627\u0633\u062a<\/p>\n<\/li>\n<\/ol>\n<p>\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627 \u062f\u0631 \u0647\u0631 \u062a\u06a9\u0631\u0627\u0631 \u0628\u0627 \u062a\u0648\u062c\u0647 \u0628\u0647 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u0628\u0647 \u0631\u0648\u0632 \u0645\u06cc \u0634\u0648\u0646\u062f.<\/p>\n<p>\u062a\u0627\u0628\u0639 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0632\u06cc\u0631 \u0631\u0627 \u0645\u06cc \u067e\u0630\u06cc\u0631\u062f:<\/p>\n<ul>\n<li>\n<p><code>max_iterations<\/code>: \u062d\u062f\u0627\u06a9\u062b\u0631 \u062a\u0639\u062f\u0627\u062f \u062a\u06a9\u0631\u0627\u0631 \u0628\u0631\u0627\u06cc \u0627\u062c\u0631\u0627<\/p>\n<\/li>\n<li>\n<p><code>threshold<\/code>: \u0627\u06af\u0631 \u062a\u0641\u0627\u0648\u062a \u062f\u0631 \u0645\u0642\u0627\u062f\u06cc\u0631 \u062a\u0627\u0628\u0639 \u0628\u06cc\u0646 \u062f\u0648 \u062a\u06a9\u0631\u0627\u0631 \u0645\u062a\u0648\u0627\u0644\u06cc \u06a9\u0645\u062a\u0631 \u0627\u0632 \u0627\u06cc\u0646 \u0622\u0633\u062a\u0627\u0646\u0647 \u0628\u0627\u0634\u062f\u060c \u0645\u062a\u0648\u0642\u0641 \u0634\u0648\u062f<\/p>\n<\/li>\n<li>\n<p><code>w_init<\/code>: \u0646\u0642\u0637\u0647 \u0627\u0648\u0644\u06cc\u0647 \u0627\u0632 \u062c\u0627\u06cc\u06cc \u06a9\u0647 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u0634\u0631\u0648\u0639 \u06a9\u0646\u06cc\u0645<\/p>\n<\/li>\n<li>\n<p><code>obj_func<\/code>: \u0627\u0634\u0627\u0631\u0647 \u0628\u0647 \u062a\u0627\u0628\u0639\u06cc \u06a9\u0647 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u0631\u0627 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f<\/p>\n<\/li>\n<li>\n<p><code>grad_func<\/code>: \u0627\u0634\u0627\u0631\u0647 \u0628\u0647 \u062a\u0627\u0628\u0639\u06cc \u06a9\u0647 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0627\u0628\u0639 \u0631\u0627 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f<\/p>\n<\/li>\n<li>\n<p><code>extra_param<\/code>: \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0627\u0636\u0627\u0641\u06cc (\u062f\u0631 \u0635\u0648\u0631\u062a \u0646\u06cc\u0627\u0632) \u0628\u0631\u0627\u06cc obj_func \u0648 grad_func<\/p>\n<\/li>\n<li>\n<p><code>learning_rate<\/code>: \u0627\u0646\u062f\u0627\u0632\u0647 \u067e\u0644\u0647 \u0628\u0631\u0627\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646.  \u0628\u0627\u06cc\u062f \u062f\u0631 (0\u060c1) \u0628\u0627\u0634\u062f<\/p>\n<\/li>\n<li>\n<p><code>momentum<\/code>: \u062d\u0631\u06a9\u062a \u0628\u0631\u0627\u06cc \u0627\u0633\u062a\u0641\u0627\u062f\u0647.  \u0628\u0627\u06cc\u062f \u062f\u0631 (0\u060c1) \u0628\u0627\u0634\u062f<\/p>\n<\/li>\n<\/ul>\n<p>\u0647\u0645\u0686\u0646\u06cc\u0646\u060c \u062a\u0627\u0628\u0639 \u0628\u0627\u0632 \u062e\u0648\u0627\u0647\u062f \u06af\u0634\u062a:<\/p>\n<ul>\n<li>\n<p><code>w_history<\/code>: \u062a\u0645\u0627\u0645 \u0646\u0642\u0627\u0637 \u062f\u0631 \u0641\u0636\u0627\u060c \u0628\u0627\u0632\u062f\u06cc\u062f \u0634\u062f\u0647 \u062a\u0648\u0633\u0637 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u0627\u0631\u0632\u06cc\u0627\u0628\u06cc \u0634\u062f<\/p>\n<\/li>\n<li>\n<p><code>f_history<\/code>: \u0645\u0642\u062f\u0627\u0631 \u0645\u062a\u0646\u0627\u0638\u0631 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u0645\u062d\u0627\u0633\u0628\u0647 \u0634\u062f\u0647 \u062f\u0631 \u0647\u0631 \u0646\u0642\u0637\u0647<\/p>\n<\/li>\n<\/ul>\n<pre><code class=\"hljs\">\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">gradient_descent<\/span>(<span class=\"hljs-params\">max_iterations,threshold,w_init,\n                     obj_func,grad_func,extra_param = (),\n                     learning_rate=<span class=\"hljs-number\">0.05<\/span>,momentum=<span class=\"hljs-number\">0.8<\/span><\/span>):<\/span>\n    \n    w = w_init\n    w_history = w\n    f_history = obj_func(w,extra_param)\n    delta_w = np.zeros(w.shape)\n    i = <span class=\"hljs-number\">0<\/span>\n    diff = <span class=\"hljs-number\">1.0e10<\/span>\n    \n    <span class=\"hljs-keyword\">while<\/span>  i&lt;max_iterations <span class=\"hljs-keyword\">and<\/span> diff&gt;threshold:\n        delta_w = -learning_rate*grad_func(w,extra_param) + momentum*delta_w\n        w = w+delta_w\n        \n        \n        w_history = np.vstack((w_history,w))\n        f_history = np.vstack((f_history,obj_func(w,extra_param)))\n        \n        \n        \n        i+=<span class=\"hljs-number\">1<\/span>\n        diff = np.absolute(f_history(-<span class=\"hljs-number\">1<\/span>)-f_history(-<span class=\"hljs-number\">2<\/span>))\n    \n    <span class=\"hljs-keyword\">return<\/span> w_history,f_history\n<\/code><\/pre>\n<h2 id=\"optimizingfunctionswithgradientdescent\"><span class=\"ez-toc-section\" id=\"%d8%a8%d9%87%db%8c%d9%86%d9%87_%d8%b3%d8%a7%d8%b2%db%8c_%d8%aa%d9%88%d8%a7%d8%a8%d8%b9_%d8%a8%d8%a7_gradient_descent\"><\/span>\u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u062a\u0648\u0627\u0628\u0639 \u0628\u0627 Gradient Descent<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u06a9\u0647 \u06cc\u06a9 \u067e\u06cc\u0627\u062f\u0647\u200c\u0633\u0627\u0632\u06cc \u0628\u0627 \u0647\u062f\u0641 \u06a9\u0644\u06cc \u0627\u0632 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u062f\u0627\u0631\u06cc\u0645\u060c \u0627\u062c\u0627\u0632\u0647 \u062f\u0647\u06cc\u062f \u0622\u0646 \u0631\u0627 \u0627\u062c\u0631\u0627 \u06a9\u0646\u06cc\u0645 \u0631\u0648\u06cc \u062a\u0627\u0628\u0639 \u062f\u0648 \u0628\u0639\u062f\u06cc \u0645\u062b\u0627\u0644 \u0645\u0627 \\(f(w_1,w_2) = w_1^2+w_2^2 \\) \u0628\u0627 \u062e\u0637\u0648\u0637 \u062f\u0627\u06cc\u0631\u0647 \u0627\u06cc.<\/p>\n<p>\u062a\u0627\u0628\u0639 \u062f\u0627\u0631\u0627\u06cc \u062d\u062f\u0627\u0642\u0644 \u0645\u0642\u062f\u0627\u0631 \u0635\u0641\u0631 \u062f\u0631 \u0645\u0628\u062f\u0627 \u0627\u0633\u062a.  \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0627\u0628\u062a\u062f\u0627 \u062a\u0627\u0628\u0639 \u0631\u0627 \u062a\u062c\u0633\u0645 \u06a9\u0646\u06cc\u0645 \u0648 \u0633\u067e\u0633 \u062d\u062f\u0627\u0642\u0644 \u0645\u0642\u062f\u0627\u0631 \u0622\u0646 \u0631\u0627 \u067e\u06cc\u062f\u0627 \u06a9\u0646\u06cc\u0645.<\/p>\n<h3 id=\"visualizingtheobjectivefunctionfx\"><span class=\"ez-toc-section\" id=\"%d8%aa%d8%ac%d8%b3%d9%85_%d8%aa%d8%a7%d8%a8%d8%b9_%d9%87%d8%af%d9%81_fx\"><\/span>\u062a\u062c\u0633\u0645 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 f(x)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0627\u06cc\u0646 <code>visualize_fw()<\/code> \u062a\u0627\u0628\u0639 \u0632\u06cc\u0631\u060c 2500 \u0646\u0642\u0637\u0647 \u0628\u0627 \u0641\u0627\u0635\u0644\u0647 \u0645\u0633\u0627\u0648\u06cc \u0627\u06cc\u062c\u0627\u062f \u0645\u06cc \u06a9\u0646\u062f \u0631\u0648\u06cc \u06cc\u06a9 \u0634\u0628\u06a9\u0647 \u0648 \u0645\u0642\u062f\u0627\u0631 \u062a\u0627\u0628\u0639 \u0631\u0627 \u062f\u0631 \u0647\u0631 \u0646\u0642\u0637\u0647 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0627\u06cc\u0646 <code>function_plot()<\/code> \u062a\u0627\u0628\u0639 \u0628\u0633\u062a\u0647 \u0628\u0647 \u0645\u0642\u062f\u0627\u0631 \\(f(\\textbf w)\\) \u062f\u0631 \u0622\u0646 \u0646\u0642\u0637\u0647\u060c \u062a\u0645\u0627\u0645 \u0646\u0642\u0627\u0637 \u0631\u0627 \u0628\u0627 \u0631\u0646\u06af \u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0646\u0645\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u062f.  \u0647\u0645\u0647 \u0646\u0642\u0627\u0637\u06cc \u06a9\u0647 \u0645\u0642\u062f\u0627\u0631 \u062a\u0627\u0628\u0639 \u062f\u0631 \u0622\u0646\u0647\u0627 \u06cc\u06a9\u0633\u0627\u0646 \u0627\u0633\u062a\u060c \u0631\u0646\u06af \u06cc\u06a9\u0633\u0627\u0646\u06cc \u062f\u0627\u0631\u0646\u062f:<\/p>\n<pre><code class=\"hljs\"><span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">visualize_fw<\/span>():<\/span>\n    xcoord = np.linspace(-<span class=\"hljs-number\">10.0<\/span>,<span class=\"hljs-number\">10.0<\/span>,<span class=\"hljs-number\">50<\/span>)\n    ycoord = np.linspace(-<span class=\"hljs-number\">10.0<\/span>,<span class=\"hljs-number\">10.0<\/span>,<span class=\"hljs-number\">50<\/span>)\n    w1,w2 = np.meshgrid(xcoord,ycoord)\n    pts = np.vstack((w1.flatten(),w2.flatten()))\n    \n    \n    pts = pts.transpose()\n    \n    \n    f_vals = np.<span class=\"hljs-built_in\">sum<\/span>(pts*pts,axis=<span class=\"hljs-number\">1<\/span>)\n    function_plot(pts,f_vals)\n    plt.title(<span class=\"hljs-string\">'Objective Function Shown in Color'<\/span>)\n    plt.show()\n    <span class=\"hljs-keyword\">return<\/span> pts,f_vals\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">annotate_pt<\/span>(<span class=\"hljs-params\">text,xy,xytext,color<\/span>):<\/span>\n    plt.plot(xy(<span class=\"hljs-number\">0<\/span>),xy(<span class=\"hljs-number\">1<\/span>),marker=<span class=\"hljs-string\">'P'<\/span>,markersize=<span class=\"hljs-number\">10<\/span>,c=color)\n    plt.annotate(text,xy=xy,xytext=xytext,\n                 \n                 arrowprops=<span class=\"hljs-built_in\">dict<\/span>(arrowstyle=<span class=\"hljs-string\">\"-&gt;\"<\/span>,\n                 color = color,\n                 connectionstyle=<span class=\"hljs-string\">'arc3'<\/span>))\n\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">function_plot<\/span>(<span class=\"hljs-params\">pts,f_val<\/span>):<\/span>\n    f_plot = plt.scatter(pts(:,<span class=\"hljs-number\">0<\/span>),pts(:,<span class=\"hljs-number\">1<\/span>),\n                         c=f_val,vmin=<span class=\"hljs-built_in\">min<\/span>(f_val),vmax=<span class=\"hljs-built_in\">max<\/span>(f_val),\n                         cmap=<span class=\"hljs-string\">'RdBu_r'<\/span>)\n    plt.colorbar(f_plot)\n    \n    annotate_pt(<span class=\"hljs-string\">'global minimum'<\/span>,(<span class=\"hljs-number\">0<\/span>,<span class=\"hljs-number\">0<\/span>),(-<span class=\"hljs-number\">5<\/span>,-<span class=\"hljs-number\">7<\/span>),<span class=\"hljs-string\">'yellow'<\/span>)    \n\npts,f_vals = visualize_fw()\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-2.png\" alt=\"\u062a\u062c\u0633\u0645 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\" title=\"\"><\/p>\n<h3 id=\"runninggradientdescentwithdifferenthyperparameters\"><span class=\"ez-toc-section\" id=\"%d8%af%d8%b1_%d8%ad%d8%a7%d9%84_%d8%a7%d8%ac%d8%b1%d8%a7_gradient_descent_%d8%a8%d8%a7_%d9%87%d8%a7%db%8c%d9%be%d8%b1%d9%be%d8%a7%d8%b1%d8%a7%d9%85%d8%aa%d8%b1%d9%87%d8%a7%db%8c_%d9%85%d8%ae%d8%aa%d9%84%d9%81\"><\/span>\u062f\u0631 \u062d\u0627\u0644 \u0627\u062c\u0631\u0627 Gradient Descent \u0628\u0627 \u0647\u0627\u06cc\u067e\u0631\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u0632\u0645\u0627\u0646 \u0627\u062c\u0631\u0627\u06cc \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u0627\u0633\u062a \u062a\u0627 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u062e\u0648\u062f \u0631\u0627 \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0628\u0631\u0633\u0627\u0646\u06cc\u0645.  \u062a\u0645\u0627\u0633 \u06af\u0631\u0641\u062a\u0646 <code>gradient_descent()<\/code>\u060c \u062f\u0648 \u062a\u0627\u0628\u0639 \u062a\u0639\u0631\u06cc\u0641 \u0645\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n<ul>\n<li><code>f()<\/code>: \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u0631\u0627 \u062f\u0631 \u0647\u0631 \u0646\u0642\u0637\u0647 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f <code>w<\/code><\/li>\n<li><code>grad()<\/code>: \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u062f\u0631 \u0647\u0631 \u0646\u0642\u0637\u0647 \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc \u06a9\u0646\u062f <code>w<\/code><\/li>\n<\/ul>\n<p>\u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u062a\u0623\u062b\u06cc\u0631 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0631\u0648\u06cc \u0634\u06cc\u0628 \u0646\u0632\u0648\u0644\u060c \u062a\u0627\u0628\u0639 <code>solve_fw()<\/code> \u062a\u0645\u0627\u0633 \u0645\u06cc \u06af\u06cc\u0631\u062f <code>gradient_descent()<\/code> \u0628\u0627 5 \u062a\u06a9\u0631\u0627\u0631 \u0628\u0631\u0627\u06cc \u0645\u0642\u0627\u062f\u06cc\u0631 \u0645\u062e\u062a\u0644\u0641 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0648 \u062d\u0631\u06a9\u062a.<\/p>\n<p>\u06a9\u0627\u0631\u06a9\u0631\u062f <code>visualize_learning()<\/code>\u060c \u0645\u0642\u0627\u062f\u06cc\u0631 \\((w_1,w_2) \\) \u0631\u0627 \u0631\u0633\u0645 \u0645\u06cc \u06a9\u0646\u062f \u0648 \u0645\u0642\u0627\u062f\u06cc\u0631 \u062a\u0627\u0628\u0639 \u0631\u0627 \u0628\u0627 \u0631\u0646\u06af \u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0646\u0634\u0627\u0646 \u0645\u06cc \u062f\u0647\u062f.  \u0641\u0644\u0634 \u0647\u0627\u06cc \u0645\u0648\u062c\u0648\u062f \u062f\u0631 \u0637\u0631\u062d \u0631\u062f\u06cc\u0627\u0628\u06cc \u0627\u06cc\u0646\u06a9\u0647 \u06a9\u062f\u0627\u0645 \u0646\u0642\u0637\u0647 \u0627\u0632 \u0622\u062e\u0631\u06cc\u0646 \u0646\u0642\u0637\u0647 \u0628\u0647 \u0631\u0648\u0632 \u0634\u062f\u0647 \u0627\u0633\u062a \u0631\u0627 \u0622\u0633\u0627\u0646 \u062a\u0631 \u0645\u06cc \u06a9\u0646\u062f:<\/p>\n<pre><code class=\"hljs\">\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">f<\/span>(<span class=\"hljs-params\">w,extra=()<\/span>):<\/span>\n    <span class=\"hljs-keyword\">return<\/span> np.<span class=\"hljs-built_in\">sum<\/span>(w*w)\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">grad<\/span>(<span class=\"hljs-params\">w,extra=()<\/span>):<\/span>\n    <span class=\"hljs-keyword\">return<\/span> <span class=\"hljs-number\">2<\/span>*w\n\n\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">visualize_learning<\/span>(<span class=\"hljs-params\">w_history<\/span>):<\/span>  \n    \n    \n    function_plot(pts,f_vals)\n    \n    \n    plt.plot(w_history(:,<span class=\"hljs-number\">0<\/span>),w_history(:,<span class=\"hljs-number\">1<\/span>),marker=<span class=\"hljs-string\">'o'<\/span>,c=<span class=\"hljs-string\">'magenta'<\/span>) \n    \n    \n    annotate_pt(<span class=\"hljs-string\">'minimum found'<\/span>,\n                (w_history(-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">0<\/span>),w_history(-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">1<\/span>)),\n                (-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">7<\/span>),<span class=\"hljs-string\">'green'<\/span>)\n    <span class=\"hljs-built_in\">iter<\/span> = w_history.shape(<span class=\"hljs-number\">0<\/span>)\n    <span class=\"hljs-keyword\">for<\/span> w,i <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">zip<\/span>(w_history,<span class=\"hljs-built_in\">range<\/span>(<span class=\"hljs-built_in\">iter<\/span>-<span class=\"hljs-number\">1<\/span>)):\n        \n        plt.annotate(<span class=\"hljs-string\">\"\"<\/span>,\n                    xy=w, xycoords=<span class=\"hljs-string\">'data'<\/span>,\n                    xytext=w_history(i+<span class=\"hljs-number\">1<\/span>,:), textcoords=<span class=\"hljs-string\">'data'<\/span>,\n                    arrowprops=<span class=\"hljs-built_in\">dict<\/span>(arrowstyle=<span class=\"hljs-string\">'&lt;-'<\/span>,\n                            connectionstyle=<span class=\"hljs-string\">'angle3'<\/span>))     \n    \n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">solve_fw<\/span>():<\/span>\n    \n    rand = np.random.RandomState(<span class=\"hljs-number\">19<\/span>)\n    w_init = rand.uniform(-<span class=\"hljs-number\">10<\/span>,<span class=\"hljs-number\">10<\/span>,<span class=\"hljs-number\">2<\/span>)\n    fig, ax = plt.subplots(nrows=<span class=\"hljs-number\">4<\/span>, ncols=<span class=\"hljs-number\">4<\/span>, figsize=(<span class=\"hljs-number\">18<\/span>, <span class=\"hljs-number\">12<\/span>))\n    learning_rates = (<span class=\"hljs-number\">0.05<\/span>,<span class=\"hljs-number\">0.2<\/span>,<span class=\"hljs-number\">0.5<\/span>,<span class=\"hljs-number\">0.8<\/span>)\n    momentum = (<span class=\"hljs-number\">0<\/span>,<span class=\"hljs-number\">0.5<\/span>,<span class=\"hljs-number\">0.9<\/span>)\n    ind = <span class=\"hljs-number\">1<\/span>\n    \n    \n    <span class=\"hljs-keyword\">for<\/span> alpha <span class=\"hljs-keyword\">in<\/span> momentum:\n        <span class=\"hljs-keyword\">for<\/span> eta,col <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">zip<\/span>(learning_rates,(<span class=\"hljs-number\">0<\/span>,<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">3<\/span>)):\n            plt.subplot(<span class=\"hljs-number\">3<\/span>,<span class=\"hljs-number\">4<\/span>,ind)        \n            w_history,f_history = gradient_descent(<span class=\"hljs-number\">5<\/span>,-<span class=\"hljs-number\">1<\/span>,w_init, f,grad,(),eta,alpha)\n            \n            visualize_learning(w_history)\n            ind = ind+<span class=\"hljs-number\">1<\/span>\n            plt.text(-<span class=\"hljs-number\">9<\/span>, <span class=\"hljs-number\">12<\/span>,<span class=\"hljs-string\">'Learning Rate = '<\/span>+<span class=\"hljs-built_in\">str<\/span>(eta),fontsize=<span class=\"hljs-number\">13<\/span>)\n            <span class=\"hljs-keyword\">if<\/span> col==<span class=\"hljs-number\">1<\/span>:\n                plt.text(<span class=\"hljs-number\">10<\/span>,<span class=\"hljs-number\">15<\/span>,<span class=\"hljs-string\">'momentum = '<\/span> + <span class=\"hljs-built_in\">str<\/span>(alpha),fontsize=<span class=\"hljs-number\">20<\/span>)\n\n    fig.subplots_adjust(hspace=<span class=\"hljs-number\">0.5<\/span>, wspace=<span class=\"hljs-number\">.3<\/span>)\n    plt.show()\n<\/code><\/pre>\n<p>\u0628\u0631\u06cc\u0645 \u0628\u062f\u0648\u06cc\u0645 <code>solve_fw()<\/code> \u0648 \u0628\u0628\u06cc\u0646\u06cc\u062f \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0648 \u062a\u06a9\u0627\u0646\u0647 \u0686\u06af\u0648\u0646\u0647 \u0628\u0631 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0623\u062b\u06cc\u0631 \u0645\u06cc \u06af\u0630\u0627\u0631\u062f:<\/p>\n<pre><code class=\"hljs\">solve_fw()\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/stackabuse.s3.amazonaws.com\/media\/gradient-descent-in-python-implementation-and-theory-3.png\" alt=\"\u062a\u062c\u0633\u0645 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0628\u0627 \u0641\u0631\u0627\u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641\" title=\"\"><\/p>\n<p>\u0627\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0646\u0642\u0634 \u062d\u0631\u06a9\u062a \u0648 \u0633\u0631\u0639\u062a \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0631\u0627 \u0631\u0648\u0634\u0646 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u062f\u0631 \u0646\u0645\u0648\u062f\u0627\u0631 \u0627\u0648\u0644\u060c \u0628\u0627 \u0627\u0646\u062f\u0627\u0632\u0647 \u062d\u0631\u06a9\u062a \u0635\u0641\u0631 \u0648 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc 0.05\u060c \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u06a9\u0646\u062f \u0627\u0633\u062a \u0648 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u062c\u0647\u0627\u0646\u06cc \u0646\u0645\u06cc \u0631\u0633\u062f.  \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0627\u0632 \u0646\u0645\u0648\u062f\u0627\u0631\u0647\u0627\u06cc \u0633\u062a\u0648\u0646 \u0627\u0648\u0644 \u0645\u06cc \u0628\u06cc\u0646\u06cc\u0645\u060c \u0627\u0641\u0632\u0627\u06cc\u0634 \u062a\u06a9\u0627\u0646\u0647 \u0633\u0631\u0639\u062a \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0631\u0627 \u0627\u0641\u0632\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u062f.  \u0627\u0641\u0631\u0627\u0637\u06cc \u062f\u06cc\u06af\u0631 \u0622\u062e\u0631\u06cc\u0646 \u0633\u062a\u0648\u0646 \u0627\u0633\u062a \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0628\u0627\u0644\u0627 \u0646\u06af\u0647 \u062f\u0627\u0634\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f.  \u0627\u06cc\u0646 \u0628\u0627\u0639\u062b \u0627\u06cc\u062c\u0627\u062f \u0646\u0648\u0633\u0627\u0646\u0627\u062a\u06cc \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u0628\u0627 \u0627\u0641\u0632\u0648\u062f\u0646 \u062a\u06a9\u0627\u0646\u0647 \u062a\u0627 \u062d\u062f\u06cc \u0645\u06cc \u062a\u0648\u0627\u0646 \u0622\u0646\u0647\u0627 \u0631\u0627 \u06a9\u0646\u062a\u0631\u0644 \u06a9\u0631\u062f.<\/p>\n<p>\u062f\u0633\u062a\u0648\u0631\u0627\u0644\u0639\u0645\u0644 \u06a9\u0644\u06cc \u0628\u0631\u0627\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u0627\u0632 \u0645\u0642\u0627\u062f\u06cc\u0631 \u06a9\u0648\u0686\u06a9 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0648 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0628\u0627\u0644\u0627\u062a\u0631 \u062a\u06a9\u0627\u0646\u0647 \u0627\u0633\u062a.<\/p>\n<h2 id=\"gradientdescentforminimizingmeansquareerror\"><span class=\"ez-toc-section\" id=\"gradient_descent_%d8%a8%d8%b1%d8%a7%db%8c_%d8%a8%d9%87_%d8%ad%d8%af%d8%a7%d9%82%d9%84_%d8%b1%d8%b3%d8%a7%d9%86%d8%af%d9%86_%d8%ae%d8%b7%d8%a7%db%8c_%d9%85%db%8c%d8%a7%d9%86%da%af%db%8c%d9%86_%d9%85%d8%b1%d8%a8%d8%b9\"><\/span>Gradient Descent \u0628\u0631\u0627\u06cc \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0631\u0633\u0627\u0646\u062f\u0646 \u062e\u0637\u0627\u06cc \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u06cc\u06a9 \u062a\u06a9\u0646\u06cc\u06a9 \u062e\u0648\u0628 \u0648 \u0633\u0627\u062f\u0647 \u0628\u0631\u0627\u06cc \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0631\u0633\u0627\u0646\u062f\u0646 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u062f\u0631 \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u0646\u0638\u0627\u0631\u062a \u0634\u062f\u0647 \u06cc\u0627 \u0645\u0633\u0626\u0644\u0647 \u0631\u06af\u0631\u0633\u06cc\u0648\u0646 \u0627\u0633\u062a.<\/p>\n<p>\u0641\u0631\u0636 \u06a9\u0646\u06cc\u062f \u0628\u0647 \u0645\u0627 \u0645\u062b\u0627\u0644 \u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634\u06cc \\(m\\) \\((x_{ij})\\) \u0628\u0627 \\(i=1\\ldots m\\) \u062f\u0627\u062f\u0647 \u0634\u062f\u0647 \u0627\u0633\u062a\u060c \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0647\u0631 \u0645\u062b\u0627\u0644 \u062f\u0627\u0631\u0627\u06cc \\(n\\) \u0648\u06cc\u0698\u06af\u06cc \u0627\u0633\u062a\u060c \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0645\u062b\u0627\u0644\u060c \\(j= 1\\ldots n \\).  \u0627\u06af\u0631 \u0645\u0642\u0627\u062f\u06cc\u0631 \u0647\u062f\u0641 \u0648 \u062e\u0631\u0648\u062c\u06cc \u0645\u0631\u0628\u0648\u0637\u0647 \u0628\u0631\u0627\u06cc \u0647\u0631 \u0645\u062b\u0627\u0644 \u0628\u0647 \u062a\u0631\u062a\u06cc\u0628 \\(t_i\\) \u0648 \\(o_i\\) \u0628\u0627\u0634\u062f\u060c \u0622\u0646\u06af\u0627\u0647 \u062a\u0627\u0628\u0639 \u0645\u0631\u0628\u0639 \u062e\u0637\u0627\u06cc \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \\(E\\) (\u062f\u0631 \u0627\u06cc\u0646 \u0645\u0648\u0631\u062f \u062a\u0627\u0628\u0639 \u0634\u06cc \u0645\u0627) \u0628\u0647 \u0635\u0648\u0631\u062a \u0632\u06cc\u0631 \u062a\u0639\u0631\u06cc\u0641 \u0645\u06cc \u0634\u0648\u062f:<\/p>\n<p>$$<br \/>E = \\frac{1}{m} \\Sigma_{i=1}^m (t_i &#8211; o_i)^2<br \/>$$<\/p>\n<p>\u062c\u0627\u06cc\u06cc \u06a9\u0647 \u062e\u0631\u0648\u062c\u06cc \\(o_i\\) \u062a\u0648\u0633\u0637 \u06cc\u06a9 \u062a\u0631\u06a9\u06cc\u0628 \u062e\u0637\u06cc \u0648\u0632\u0646\u06cc \u0627\u0632 \u0648\u0631\u0648\u062f\u06cc \u0647\u0627 \u062a\u0639\u06cc\u06cc\u0646 \u0645\u06cc \u0634\u0648\u062f \u06a9\u0647 \u062a\u0648\u0633\u0637:<\/p>\n<p>$$<br \/>o_i = w_0 + w_1 x_{i1} + w_2 x_{i2} + \\ldots + w_n x_{in}<br \/>$$<\/p>\n<p>\u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0645\u062c\u0647\u0648\u0644 \u062f\u0631 \u0645\u0639\u0627\u062f\u0644\u0647 \u0628\u0627\u0644\u0627 \u0628\u0631\u062f\u0627\u0631 \u0648\u0632\u0646 \\(\\textbf w = (w_0,w_1,\\ldots,w_n)^T\\) \u0627\u0633\u062a.<\/p>\n<p>\u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u062f\u0631 \u0627\u06cc\u0646 \u062d\u0627\u0644\u062a \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u0628\u0627 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0627\u0633\u062a \u06a9\u0647 \u062a\u0648\u0633\u0637:<\/p>\n<p>$$<br \/>\\nabla_{\\textbf w}E(\\textbf w) = -\\Sigma_{i=1}^{m} (t_i &#8211; o_i) \\textbf{x}_i<br \/>$$<\/p>\n<p>\u062c\u0627\u06cc\u06cc \u06a9\u0647 \\(x_{i}\\) i-\u0627\u0645\u06cc\u0646 \u0645\u062b\u0627\u0644 \u0627\u0633\u062a.  \u06cc\u0627 \u0645\u062c\u0645\u0648\u0639\u0647 \u0627\u06cc \u0627\u0632 \u0648\u06cc\u0698\u06af\u06cc \u0647\u0627\u06cc \u0627\u0646\u062f\u0627\u0632\u0647 <em>n<\/em>.<\/p>\n<p>\u062a\u0646\u0647\u0627 \u0686\u06cc\u0632\u06cc \u06a9\u0647 \u0627\u06a9\u0646\u0648\u0646 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u06cc\u0645 \u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0628\u0631\u0627\u06cc \u0645\u062d\u0627\u0633\u0628\u0647 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0648 \u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0628\u0631\u0627\u06cc \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u0627\u0633\u062a.<\/p>\n<p>\u0627\u06cc\u0646 <code>gradient_descent()<\/code> \u0633\u067e\u0633 \u0645\u06cc \u062a\u0648\u0627\u0646 \u0627\u0632 \u062a\u0627\u0628\u0639 \u0647\u0645\u0627\u0646\u0637\u0648\u0631 \u06a9\u0647 \u0647\u0633\u062a \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f.  \u062a\u0648\u062c\u0647 \u062f\u0627\u0634\u062a\u0647 \u0628\u0627\u0634\u06cc\u062f \u06a9\u0647 \u0647\u0646\u06af\u0627\u0645 \u0645\u062d\u0627\u0633\u0628\u0647 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\u060c \u062a\u0645\u0627\u0645 \u0645\u062b\u0627\u0644 \u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634\u06cc \u0628\u0627 \u0647\u0645 \u067e\u0631\u062f\u0627\u0632\u0634 \u0645\u06cc \u0634\u0648\u0646\u062f.  \u0627\u0632 \u0627\u06cc\u0646 \u0631\u0648\u060c \u0627\u06cc\u0646 \u0646\u0633\u062e\u0647 \u0627\u0632 \u0634\u06cc\u0628 \u0646\u0632\u0648\u0644 \u0628\u0631\u0627\u06cc \u0628\u0647 \u0631\u0648\u0632 \u0631\u0633\u0627\u0646\u06cc \u0648\u0632\u0646 \u0647\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u0646\u0627\u0645\u06cc\u062f\u0647 \u0645\u06cc \u0634\u0648\u062f <em>\u0628\u0647 \u0631\u0648\u0632 \u0631\u0633\u0627\u0646\u06cc \u062f\u0633\u062a\u0647 \u0627\u06cc<\/em> \u06cc\u0627 <em>\u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u062f\u0633\u062a\u0647 \u0627\u06cc<\/em>:<\/p>\n<pre><code class=\"hljs\">\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">grad_mse<\/span>(<span class=\"hljs-params\">w,xy<\/span>):<\/span>\n    (x,y) = xy\n    (rows,cols) = x.shape\n    \n    \n    o = np.<span class=\"hljs-built_in\">sum<\/span>(x*w,axis=<span class=\"hljs-number\">1<\/span>)\n    diff = y-o\n    diff = diff.reshape((rows,<span class=\"hljs-number\">1<\/span>))    \n    diff = np.tile(diff, (<span class=\"hljs-number\">1<\/span>, cols))\n    grad = diff*x\n    grad = -np.<span class=\"hljs-built_in\">sum<\/span>(grad,axis=<span class=\"hljs-number\">0<\/span>)\n    <span class=\"hljs-keyword\">return<\/span> grad\n\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">mse<\/span>(<span class=\"hljs-params\">w,xy<\/span>):<\/span>\n    (x,y) = xy\n    \n    \n    \n    \n    o = np.<span class=\"hljs-built_in\">sum<\/span>(x*w,axis=<span class=\"hljs-number\">1<\/span>)\n    mse = np.<span class=\"hljs-built_in\">sum<\/span>((y-o)*(y-o))\n    mse = mse\/<span class=\"hljs-number\">2<\/span>\n    <span class=\"hljs-keyword\">return<\/span> mse    \n<\/code><\/pre>\n<h3 id=\"runninggradientdescentonocr\"><span class=\"ez-toc-section\" id=\"%d8%af%d8%b1_%d8%ad%d8%a7%d9%84_%d8%a7%d8%ac%d8%b1%d8%a7_gradient_descent_%d8%b1%d9%88%db%8c_ocr\"><\/span>\u062f\u0631 \u062d\u0627\u0644 \u0627\u062c\u0631\u0627 Gradient Descent \u0631\u0648\u06cc OCR<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0628\u0631\u0627\u06cc \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u0646 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0648\u06cc \u06cc\u06a9 \u0645\u0634\u06a9\u0644 \u0637\u0628\u0642\u0647\u200c\u0628\u0646\u062f\u06cc\u060c \u0645\u0627 \u0645\u062c\u0645\u0648\u0639\u0647 \u062f\u0627\u062f\u0647\u200c\u0647\u0627\u06cc \u0631\u0642\u0645\u06cc \u0631\u0627 \u0627\u0646\u062a\u062e\u0627\u0628 \u06a9\u0631\u062f\u0647\u200c\u0627\u06cc\u0645 <code>sklearn.datasets<\/code>.<\/p>\n<p>\u0628\u0631\u0627\u06cc \u0633\u0627\u062f\u0647 \u0646\u06af\u0647 \u062f\u0627\u0634\u062a\u0646 \u0647\u0645\u0647 \u0686\u06cc\u0632\u060c \u0628\u06cc\u0627\u06cc\u06cc\u062f \u06cc\u06a9 \u0627\u062c\u0631\u0627\u06cc \u0622\u0632\u0645\u0627\u06cc\u0634\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u0627\u0646\u062c\u0627\u0645 \u062f\u0647\u06cc\u0645 \u0631\u0648\u06cc \u06cc\u06a9 \u0645\u0633\u0626\u0644\u0647 \u062f\u0648 \u06a9\u0644\u0627\u0633\u0647 (\u0631\u0642\u0645 0 \u062f\u0631 \u0645\u0642\u0627\u0628\u0644 \u0631\u0642\u0645 1).  \u06a9\u062f \u0632\u06cc\u0631 \u0627\u0631\u0642\u0627\u0645 \u0631\u0627 \u0628\u0627\u0631\u06af\u06cc\u0631\u06cc \u0645\u06cc \u06a9\u0646\u062f \u0648 10 \u0631\u0642\u0645 \u0627\u0648\u0644 \u0631\u0627 \u0646\u0645\u0627\u06cc\u0634 \u0645\u06cc \u062f\u0647\u062f.  \u0627\u06cc\u0646 \u0628\u0647 \u0645\u0627 \u0627\u06cc\u062f\u0647 \u0627\u06cc \u0627\u0632 \u0645\u0627\u0647\u06cc\u062a \u0646\u06a9\u0627\u062a \u0622\u0645\u0648\u0632\u0634\u06cc \u0645\u06cc \u062f\u0647\u062f:<\/p>\n<pre><code class=\"hljs\">\ndigits,target = dt.load_digits(n_class=<span class=\"hljs-number\">2<\/span>,return_X_y=<span class=\"hljs-literal\">True<\/span>)\nfig,ax = plt.subplots(nrows=<span class=\"hljs-number\">1<\/span>, ncols=<span class=\"hljs-number\">10<\/span>,figsize=(<span class=\"hljs-number\">12<\/span>,<span class=\"hljs-number\">4<\/span>),subplot_kw=<span class=\"hljs-built_in\">dict<\/span>(xticks=(), yticks=()))\n\n\n<span class=\"hljs-keyword\">for<\/span> i <span class=\"hljs-keyword\">in<\/span> np.arange(<span class=\"hljs-number\">10<\/span>):\n    ax(i).imshow(digits(i,:).reshape(<span class=\"hljs-number\">8<\/span>,<span class=\"hljs-number\">8<\/span>),cmap=plt.cm.gray)   \nplt.show()\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-4.png\" alt=\"\u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u0628\u0627 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\" title=\"\"><\/p>\n<p>\u0645\u0627 \u0647\u0645 \u0628\u0647 \u0631\u0648\u0634 \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u06cc\u0645 <code>train_test_split<\/code> \u0627\u0632 \u062c\u0627\u0646\u0628 <code>sklearn.model_selection<\/code> \u0628\u0631\u0627\u06cc \u062a\u0642\u0633\u06cc\u0645 \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634\u06cc \u0628\u0647 \u06cc\u06a9 \u0642\u0637\u0627\u0631 \u0648 \u06cc\u06a9 \u0645\u062c\u0645\u0648\u0639\u0647 \u0622\u0632\u0645\u0627\u06cc\u0634\u06cc.  \u06a9\u062f \u0632\u06cc\u0631 gradient descent \u0631\u0627 \u0627\u062c\u0631\u0627 \u0645\u06cc \u06a9\u0646\u062f \u0631\u0648\u06cc \u0645\u062c\u0645\u0648\u0639\u0647 \u0622\u0645\u0648\u0632\u0634\u06cc\u060c \u0648\u0632\u0646 \u0647\u0627 \u0631\u0627 \u06cc\u0627\u062f \u0645\u06cc \u06af\u06cc\u0631\u062f \u0648 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u0631\u0627 \u062f\u0631 \u062a\u06a9\u0631\u0627\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641 \u0631\u0633\u0645 \u0645\u06cc \u06a9\u0646\u062f.<\/p>\n<p>\u0647\u0646\u06af\u0627\u0645 \u0627\u062c\u0631\u0627\u06cc \u0634\u06cc\u0628 \u0646\u0632\u0648\u0644\u060c \u0633\u0631\u0639\u062a \u0648 \u062a\u06a9\u0627\u0646\u0647 \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0631\u0627 \u0628\u0633\u06cc\u0627\u0631 \u06a9\u0648\u0686\u06a9 \u0646\u06af\u0647 \u0645\u06cc\u200c\u062f\u0627\u0631\u06cc\u0645 \u0632\u06cc\u0631\u0627 \u0648\u0631\u0648\u062f\u06cc\u200c\u0647\u0627 \u0639\u0627\u062f\u06cc \u06cc\u0627 \u0627\u0633\u062a\u0627\u0646\u062f\u0627\u0631\u062f \u0646\u0634\u062f\u0647\u200c\u0627\u0646\u062f.  \u0647\u0645\u0686\u0646\u06cc\u0646\u060c \u0646\u0633\u062e\u0647 \u062f\u0633\u062a\u0647 \u0627\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0628\u0647 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u06a9\u0645\u062a\u0631\u06cc \u0646\u06cc\u0627\u0632 \u062f\u0627\u0631\u062f:<\/p>\n<pre><code class=\"hljs\">\nx_train, x_test, y_train, y_test = train_test_split(\n                        digits, target, test_size=<span class=\"hljs-number\">0.2<\/span>, random_state=<span class=\"hljs-number\">10<\/span>)\n\n\nx_train = np.hstack((np.ones((y_train.size,<span class=\"hljs-number\">1<\/span>)),x_train))\nx_test  = np.hstack((np.ones((y_test.size,<span class=\"hljs-number\">1<\/span>)),x_test))\n\n\nrand = np.random.RandomState(<span class=\"hljs-number\">19<\/span>)\nw_init = rand.uniform(-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">1<\/span>,x_train.shape(<span class=\"hljs-number\">1<\/span>))*<span class=\"hljs-number\">.000001<\/span>\nw_history,mse_history = gradient_descent(<span class=\"hljs-number\">100<\/span>,<span class=\"hljs-number\">0.1<\/span>,w_init,\n                              mse,grad_mse,(x_train,y_train),\n                             learning_rate=<span class=\"hljs-number\">1e-6<\/span>,momentum=<span class=\"hljs-number\">0.7<\/span>)\n\n\nplt.plot(np.arange(mse_history.size),mse_history)\nplt.xlabel(<span class=\"hljs-string\">'Iteration No.'<\/span>)\nplt.ylabel(<span class=\"hljs-string\">'Mean Square Error'<\/span>)\nplt.title(<span class=\"hljs-string\">'Gradient Descent \u0631\u0648\u06cc Digits Data (Batch Version)'<\/span>)\nplt.show()\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-7.png\" alt=\"\u0628\u0647\u06cc\u0646\u0647 \u0633\u0627\u0632\u06cc \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u0628\u0627 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646\" title=\"\"><\/p>\n<p>\u0627\u06cc\u0646 \u0639\u0627\u0644\u06cc \u0628\u0647 \u0646\u0638\u0631 \u0645\u06cc \u0631\u0633\u062f!  \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0645\u06cc\u0632\u0627\u0646 \u062e\u0637\u0627\u06cc OCR \u062e\u0648\u062f \u0631\u0627 \u0628\u0631\u0631\u0633\u06cc \u06a9\u0646\u06cc\u0645 \u0631\u0648\u06cc \u062f\u0627\u062f\u0647 \u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634 \u0648 \u0622\u0632\u0645\u0648\u0646  \u062f\u0631 \u0632\u06cc\u0631 \u062a\u0627\u0628\u0639 \u06a9\u0648\u0686\u06a9\u06cc \u0628\u0631\u0627\u06cc \u0645\u062d\u0627\u0633\u0628\u0647 \u0645\u06cc\u0632\u0627\u0646 \u062e\u0637\u0627\u06cc \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u0648\u062c\u0648\u062f \u062f\u0627\u0631\u062f \u06a9\u0647 \u0628\u0647 \u0622\u0646 \u06af\u0641\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f \u0631\u0648\u06cc \u0645\u062c\u0645\u0648\u0639\u0647 \u0622\u0645\u0648\u0632\u0634 \u0648 \u0622\u0632\u0645\u0648\u0646:<\/p>\n<pre><code class=\"hljs\">\n\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">error<\/span>(<span class=\"hljs-params\">w,xy<\/span>):<\/span>\n    (x,y) = xy\n    o = np.<span class=\"hljs-built_in\">sum<\/span>(x*w,axis=<span class=\"hljs-number\">1<\/span>)\n    \n    \n    ind_1 = np.where(o&gt;<span class=\"hljs-number\">0.5<\/span>)\n    ind_0 = np.where(o&lt;=<span class=\"hljs-number\">0.5<\/span>)\n    o(ind_1) = <span class=\"hljs-number\">1<\/span>\n    o(ind_0) = <span class=\"hljs-number\">0<\/span>\n    <span class=\"hljs-keyword\">return<\/span> np.<span class=\"hljs-built_in\">sum<\/span>((o-y)*(o-y))\/y.size*<span class=\"hljs-number\">100<\/span>\n    \ntrain_error = error(w_history(-<span class=\"hljs-number\">1<\/span>),(x_train,y_train))\ntest_error = error(w_history(-<span class=\"hljs-number\">1<\/span>),(x_test,y_test))\n\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Train Error Rate: \"<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(train_error))\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Test Error Rate: \"<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(test_error))\n<\/code><\/pre>\n<pre><code class=\"hljs\">Train Error Rate: 0.69\nTest Error Rate: 1.39\n<\/code><\/pre>\n<h3 id=\"stochasticgradientdescentinpython\"><span class=\"ez-toc-section\" id=\"%d9%86%d8%b2%d9%88%d9%84_%da%af%d8%b1%d8%a7%d8%af%db%8c%d8%a7%d9%86_%d8%aa%d8%b5%d8%a7%d8%af%d9%81%db%8c_%d8%af%d8%b1_%d9%be%d8%a7%db%8c%d8%aa%d9%88%d9%86\"><\/span>\u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u062a\u0635\u0627\u062f\u0641\u06cc \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u062f\u0631 \u0628\u062e\u0634 \u0642\u0628\u0644\u060c \u0627\u0632 \u0637\u0631\u062d \u0628\u0647\u200c\u0631\u0648\u0632\u0631\u0633\u0627\u0646\u06cc \u062f\u0633\u062a\u0647\u200c\u0627\u06cc \u0628\u0631\u0627\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0631\u062f\u06cc\u0645.<\/p>\n<p>\u0646\u0633\u062e\u0647 \u062f\u06cc\u06af\u0631\u06cc \u0627\u0632 \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0627\u0633\u062a <strong>\u0628\u0631\u062e\u0637<\/strong> \u06cc\u0627 <strong>\u062a\u0635\u0627\u062f\u0641\u06cc<\/strong> \u0637\u0631\u062d \u0628\u0647 \u0631\u0648\u0632 \u0631\u0633\u0627\u0646\u06cc\u060c \u06a9\u0647 \u062f\u0631 \u0622\u0646 \u0647\u0631 \u0646\u0645\u0648\u0646\u0647 \u062a\u0645\u0631\u06cc\u0646\u06cc \u06cc\u06a9 \u0628\u0647 \u06cc\u06a9 \u0628\u0631\u0627\u06cc \u0628\u0647 \u0631\u0648\u0632 \u0631\u0633\u0627\u0646\u06cc \u0648\u0632\u0646\u0647 \u0647\u0627 \u06af\u0631\u0641\u062a\u0647 \u0645\u06cc \u0634\u0648\u062f.<\/p>\n<p>\u0647\u0646\u06af\u0627\u0645\u06cc \u06a9\u0647 \u062a\u0645\u0627\u0645 \u0646\u0645\u0648\u0646\u0647 \u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634\u06cc \u0686\u0631\u062e\u0647 \u0627\u06cc \u0645\u06cc \u0634\u0648\u0646\u062f\u060c \u0645\u06cc \u06af\u0648\u06cc\u06cc\u0645 \u06a9\u0647 \u06cc\u06a9 \u062f\u0648\u0631\u0647 \u06a9\u0627\u0645\u0644 \u0634\u062f\u0647 \u0627\u0633\u062a.  \u0628\u0631\u0627\u06cc \u0646\u062a\u0627\u06cc\u062c \u0628\u0647\u062a\u0631\u060c \u0646\u0645\u0648\u0646\u0647\u200c\u0647\u0627\u06cc \u0622\u0645\u0648\u0632\u0634\u06cc \u0642\u0628\u0644 \u0627\u0632 \u0647\u0631 \u062f\u0648\u0631\u0647 \u0628\u0627 \u0647\u0645 \u0645\u062e\u0644\u0648\u0637 \u0645\u06cc\u200c\u0634\u0648\u0646\u062f.<\/p>\n<p>\u0642\u0637\u0639\u0647 \u06a9\u062f \u0632\u06cc\u0631 \u06cc\u06a9 \u062a\u063a\u06cc\u06cc\u0631 \u062c\u0632\u0626\u06cc \u062f\u0631 \u0622\u0646 \u0627\u0633\u062a <code>gradient_descent()<\/code> \u062a\u0627\u0628\u0639 \u0628\u0631\u0627\u06cc \u062a\u0631\u06a9\u06cc\u0628 \u0647\u0645\u062a\u0627\u06cc \u062a\u0635\u0627\u062f\u0641\u06cc \u062e\u0648\u062f.  \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639 \u0628\u0647 \u062c\u0627\u06cc \u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0627\u0636\u0627\u0641\u06cc\u060c (\u0645\u062c\u0645\u0648\u0639\u0647 \u0622\u0645\u0648\u0632\u0634\u06cc\u060c \u0647\u062f\u0641) \u0631\u0627 \u0628\u0647 \u0639\u0646\u0648\u0627\u0646 \u067e\u0627\u0631\u0627\u0645\u062a\u0631 \u0645\u06cc \u06af\u06cc\u0631\u062f.  \u0627\u0635\u0637\u0644\u0627\u062d &#8220;\u062a\u06a9\u0631\u0627\u0631&#8221; \u0628\u0647 &#8220;\u062f\u0648\u0631\u0627\u0646&#8221; \u062a\u063a\u06cc\u06cc\u0631 \u0646\u0627\u0645 \u062f\u0627\u062f\u0647 \u0627\u0633\u062a:<\/p>\n<pre><code class=\"hljs\">\n<span class=\"hljs-function\"><span class=\"hljs-keyword\">def<\/span> <span class=\"hljs-title\">stochastic_gradient_descent<\/span>(<span class=\"hljs-params\">max_epochs,threshold,w_init,\n                                obj_func,grad_func,xy,\n                                learning_rate=<span class=\"hljs-number\">0.05<\/span>,momentum=<span class=\"hljs-number\">0.8<\/span><\/span>):<\/span>\n    (x_train,y_train) = xy\n    w = w_init\n    w_history = w\n    f_history = obj_func(w,xy)\n    delta_w = np.zeros(w.shape)\n    i = <span class=\"hljs-number\">0<\/span>\n    diff = <span class=\"hljs-number\">1.0e10<\/span>\n    rows = x_train.shape(<span class=\"hljs-number\">0<\/span>)\n    \n    \n    <span class=\"hljs-keyword\">while<\/span>  i&lt;max_epochs <span class=\"hljs-keyword\">and<\/span> diff&gt;threshold:\n        \n        np.random.seed(i)\n        p = np.random.permutation(rows)\n        \n        \n        <span class=\"hljs-keyword\">for<\/span> x,y <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">zip<\/span>(x_train(p,:),y_train(p)):\n            delta_w = -learning_rate*grad_func(w,(np.array((x)),y)) + momentum*delta_w\n            w = w+delta_w\n            \n        i+=<span class=\"hljs-number\">1<\/span>\n        w_history = np.vstack((w_history,w))\n        f_history = np.vstack((f_history,obj_func(w,xy)))\n        diff = np.absolute(f_history(-<span class=\"hljs-number\">1<\/span>)-f_history(-<span class=\"hljs-number\">2<\/span>))\n        \n    <span class=\"hljs-keyword\">return<\/span> w_history,f_history\n<\/code><\/pre>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u06a9\u062f \u0631\u0627 \u0627\u062c\u0631\u0627 \u06a9\u0646\u06cc\u0645 \u062a\u0627 \u0628\u0628\u06cc\u0646\u06cc\u0645 \u0646\u062a\u0627\u06cc\u062c \u0628\u0631\u0627\u06cc \u0646\u0633\u062e\u0647 \u062a\u0635\u0627\u062f\u0641\u06cc \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u0686\u06af\u0648\u0646\u0647 \u0627\u0633\u062a:<\/p>\n<pre><code class=\"hljs\">rand = np.random.RandomState(<span class=\"hljs-number\">19<\/span>)\nw_init = rand.uniform(-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">1<\/span>,x_train.shape(<span class=\"hljs-number\">1<\/span>))*<span class=\"hljs-number\">.000001<\/span>\nw_history_stoch,mse_history_stoch = stochastic_gradient_descent(\n                                <span class=\"hljs-number\">100<\/span>,<span class=\"hljs-number\">0.1<\/span>,w_init,\n                              mse,grad_mse,(x_train,y_train),\n                             learning_rate=<span class=\"hljs-number\">1e-6<\/span>,momentum=<span class=\"hljs-number\">0.7<\/span>)\n\n\nplt.plot(np.arange(mse_history_stoch.size),mse_history_stoch)\nplt.xlabel(<span class=\"hljs-string\">'Iteration No.'<\/span>)\nplt.ylabel(<span class=\"hljs-string\">'Mean Square Error'<\/span>)\nplt.title(<span class=\"hljs-string\">'Gradient Descent \u0631\u0648\u06cc Digits Data (Stochastic Version)'<\/span>)\n\nplt.show()\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-5.png\" alt=\"\u0646\u0631\u062e \u062e\u0637\u0627 \u062a\u0635\u0627\u062f\u0641\u06cc \u062f\u0631 \u0645\u0642\u0627\u0628\u0644 \u0634\u06cc\u0628 \u062f\u0633\u062a\u0647 \u0627\u06cc \u0646\u0632\u0648\u0644\" title=\"\"><\/p>\n<p>\u0628\u06cc\u0627\u06cc\u06cc\u062f \u0645\u06cc\u0632\u0627\u0646 \u062e\u0637\u0627 \u0631\u0627 \u0646\u06cc\u0632 \u0628\u0631\u0631\u0633\u06cc \u06a9\u0646\u06cc\u0645:<\/p>\n<pre><code class=\"hljs\">train_error_stochastic = error(w_history_stoch(-<span class=\"hljs-number\">1<\/span>),(x_train,y_train))\ntest_error_stochastic = error(w_history_stoch(-<span class=\"hljs-number\">1<\/span>),(x_test,y_test))\n\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Train Error rate with Stochastic Gradient Descent: \"<\/span> + \n      <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(train_error_stochastic))\n<span class=\"hljs-built_in\">print<\/span>(<span class=\"hljs-string\">\"Test Error rate with Stochastic Gradient Descent: \"<\/span>  \n      + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(test_error_stochastic))\n<\/code><\/pre>\n<pre><code class=\"hljs\">Train Error rate with Stochastic Gradient Descent: 0.35\nTest Error rate with Stochastic Gradient Descent: 1.39\n<\/code><\/pre>\n<h3 id=\"comparingbatchandstochasticversions\"><span class=\"ez-toc-section\" id=\"%d9%85%d9%82%d8%a7%db%8c%d8%b3%d9%87_%d9%86%d8%b3%d8%ae%d9%87_%d9%87%d8%a7%db%8c_%d8%af%d8%b3%d8%aa%d9%87_%d8%a7%db%8c_%d9%88_%d8%aa%d8%b5%d8%a7%d8%af%d9%81%db%8c\"><\/span>\u0645\u0642\u0627\u06cc\u0633\u0647 \u0646\u0633\u062e\u0647 \u0647\u0627\u06cc \u062f\u0633\u062a\u0647 \u0627\u06cc \u0648 \u062a\u0635\u0627\u062f\u0641\u06cc<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>\u0627\u06a9\u0646\u0648\u0646 \u0628\u06cc\u0627\u06cc\u06cc\u062f \u0647\u0631 \u062f\u0648 \u0646\u0633\u062e\u0647 \u062f\u0633\u062a\u0647 \u0627\u06cc \u0648 \u062a\u0635\u0627\u062f\u0641\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u0628\u0627 \u0647\u0645 \u0645\u0642\u0627\u06cc\u0633\u0647 \u06a9\u0646\u06cc\u0645.<\/p>\n<p>\u0645\u0627 \u0646\u0631\u062e \u06cc\u0627\u062f\u06af\u06cc\u0631\u06cc \u0631\u0627 \u0628\u0631\u0627\u06cc \u0647\u0631 \u062f\u0648 \u0646\u0633\u062e\u0647 \u0628\u0647 \u06cc\u06a9 \u0645\u0642\u062f\u0627\u0631 \u062b\u0627\u0628\u062a \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u0648 \u062a\u06a9\u0627\u0646\u0647 \u0631\u0627 \u062a\u063a\u06cc\u06cc\u0631 \u0645\u06cc \u062f\u0647\u06cc\u0645 \u062a\u0627 \u0628\u0628\u06cc\u0646\u06cc\u0645 \u0647\u0631 \u062f\u0648 \u0628\u0627 \u0686\u0647 \u0633\u0631\u0639\u062a\u06cc \u0647\u0645\u06af\u0631\u0627 \u0645\u06cc \u0634\u0648\u0646\u062f.  \u0648\u0632\u0646 \u0647\u0627\u06cc \u0627\u0648\u0644\u06cc\u0647 \u0648 \u0645\u0639\u06cc\u0627\u0631\u0647\u0627\u06cc \u062a\u0648\u0642\u0641 \u0628\u0631\u0627\u06cc \u0647\u0631 \u062f\u0648 \u0627\u0644\u06af\u0648\u0631\u06cc\u062a\u0645 \u06cc\u06a9\u0633\u0627\u0646 \u0628\u0627\u0642\u06cc \u0645\u06cc \u0645\u0627\u0646\u0646\u062f:<\/p>\n<pre><code class=\"hljs\">fig, ax = plt.subplots(nrows=<span class=\"hljs-number\">3<\/span>, ncols=<span class=\"hljs-number\">1<\/span>, figsize=(<span class=\"hljs-number\">10<\/span>,<span class=\"hljs-number\">3<\/span>))\n\nrand = np.random.RandomState(<span class=\"hljs-number\">11<\/span>)\nw_init = rand.uniform(-<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">1<\/span>,x_train.shape(<span class=\"hljs-number\">1<\/span>))*<span class=\"hljs-number\">.000001<\/span>\neta = <span class=\"hljs-number\">1e-6<\/span>\n<span class=\"hljs-keyword\">for<\/span> alpha,ind <span class=\"hljs-keyword\">in<\/span> <span class=\"hljs-built_in\">zip<\/span>((<span class=\"hljs-number\">0<\/span>,<span class=\"hljs-number\">0.5<\/span>,<span class=\"hljs-number\">0.9<\/span>),(<span class=\"hljs-number\">1<\/span>,<span class=\"hljs-number\">2<\/span>,<span class=\"hljs-number\">3<\/span>)):\n    w_history,mse_history = gradient_descent(\n                                <span class=\"hljs-number\">100<\/span>,<span class=\"hljs-number\">0.01<\/span>,w_init,\n                              mse,grad_mse,(x_train,y_train),\n                             learning_rate=eta,momentum=alpha)\n\n    w_history_stoch,mse_history_stoch = stochastic_gradient_descent(\n                                <span class=\"hljs-number\">100<\/span>,<span class=\"hljs-number\">0.01<\/span>,w_init,\n                              mse,grad_mse,(x_train,y_train),\n                             learning_rate=eta,momentum=alpha)\n    \n    \n    plt.subplot(<span class=\"hljs-number\">130<\/span>+ind)\n    plt.plot(np.arange(mse_history.size),mse_history,color=<span class=\"hljs-string\">'green'<\/span>)\n    plt.plot(np.arange(mse_history_stoch.size),mse_history_stoch,color=<span class=\"hljs-string\">'blue'<\/span>)\n    plt.legend((<span class=\"hljs-string\">'batch'<\/span>,<span class=\"hljs-string\">'stochastic'<\/span>))\n    \n    \n    plt.text(<span class=\"hljs-number\">3<\/span>,-<span class=\"hljs-number\">30<\/span>,<span class=\"hljs-string\">'Batch: Iterations='<\/span>+\n             <span class=\"hljs-built_in\">str<\/span>(mse_history.size) )\n    plt.text(<span class=\"hljs-number\">3<\/span>,-<span class=\"hljs-number\">45<\/span>,<span class=\"hljs-string\">'Stochastic: Iterations='<\/span>+\n             <span class=\"hljs-built_in\">str<\/span>(mse_history_stoch.size))\n    plt.title(<span class=\"hljs-string\">'Momentum = '<\/span> + <span class=\"hljs-built_in\">str<\/span>(alpha))   \n    \n    \n    train_error = error(w_history(-<span class=\"hljs-number\">1<\/span>),(x_train,y_train))\n    test_error = error(w_history(-<span class=\"hljs-number\">1<\/span>),(x_test,y_test))\n    \n    train_error_stochastic = error(w_history_stoch(-<span class=\"hljs-number\">1<\/span>),(x_train,y_train))\n    test_error_stochastic = error(w_history_stoch(-<span class=\"hljs-number\">1<\/span>),(x_test,y_test))\n    \n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'Momentum = '<\/span>+<span class=\"hljs-built_in\">str<\/span>(alpha))\n    \n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\tBatch:'<\/span>)\n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\t\\tTrain error: '<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(train_error) )\n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\t\\tTest error: '<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(test_error) )\n    \n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\tStochastic:'<\/span>)\n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\t\\tTrain error: '<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(train_error_stochastic) )\n    <span class=\"hljs-built_in\">print<\/span> (<span class=\"hljs-string\">'\\t\\tTest error: '<\/span> + <span class=\"hljs-string\">\"{:.2f}\"<\/span>.<span class=\"hljs-built_in\">format<\/span>(test_error_stochastic) )\n    \n        \nplt.show()\n<\/code><\/pre>\n<pre><code class=\"hljs\">Momentum = 0\n    Batch:\n        Train error: 0.35\n        Test error: 1.39\n    Stochastic:\n        Train error: 0.35\n        Test error: 1.39\nMomentum = 0.5\n    Batch:\n        Train error: 0.00\n        Test error: 1.39\n    Stochastic:\n        Train error: 0.35\n        Test error: 1.39\nMomentum = 0.9\n    Batch:\n        Train error: 0.00\n        Test error: 1.39\n    Stochastic:\n        Train error: 0.00\n        Test error: 1.39\n<\/code><\/pre>\n<p><img decoding=\"async\" src=\"https:\/\/rasanegar.com\/blog\/wp-content\/uploads\/2024\/01\/gradient-descent-in-python-implementation-and-theory-6.png\" alt=\"\u062e\u0631\u0648\u062c\u06cc_28_1\" title=\"\"><\/p>\n<p>\u062f\u0631 \u062d\u0627\u0644\u06cc \u06a9\u0647 \u062a\u0641\u0627\u0648\u062a \u0642\u0627\u0628\u0644\u200c\u062a\u0648\u062c\u0647\u06cc \u062f\u0631 \u062f\u0642\u062a \u0628\u06cc\u0646 \u062f\u0648 \u0646\u0633\u062e\u0647 \u0637\u0628\u0642\u0647\u200c\u0628\u0646\u062f\u06cc\u200c\u06a9\u0646\u0646\u062f\u0647 \u0648\u062c\u0648\u062f \u0646\u062f\u0627\u0631\u062f\u060c \u0646\u0633\u062e\u0647 \u062a\u0635\u0627\u062f\u0641\u06cc \u0628\u0631\u0646\u062f\u0647 \u0622\u0634\u06a9\u0627\u0631\u06cc \u062f\u0631 \u0645\u0648\u0631\u062f \u0633\u0631\u0639\u062a \u0647\u0645\u200c\u06af\u0631\u0627\u06cc\u06cc \u0627\u0633\u062a.  \u062a\u06a9\u0631\u0627\u0631\u0647\u0627\u06cc \u06a9\u0645\u062a\u0631\u06cc \u0637\u0648\u0644 \u0645\u06cc\u200c\u06a9\u0634\u062f \u062a\u0627 \u0628\u0647 \u0646\u062a\u06cc\u062c\u0647\u200c\u0627\u06cc \u0645\u0634\u0627\u0628\u0647 \u0628\u0627 \u0647\u0645\u062a\u0627\u06cc \u062f\u0633\u062a\u0647\u200c\u0627\u06cc \u062e\u0648\u062f \u0628\u0631\u0633\u06cc\u062f.<\/p>\n<h2 id=\"conclusions\"><span class=\"ez-toc-section\" id=\"%d9%86%d8%aa%db%8c%d8%ac%d9%87_%da%af%db%8c%d8%b1%db%8c\"><\/span>\u0646\u062a\u06cc\u062c\u0647 \u06af\u06cc\u0631\u06cc<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u0634\u06cc\u0628 \u0646\u0632\u0648\u0644 \u06cc\u06a9 \u062a\u06a9\u0646\u06cc\u06a9 \u0633\u0627\u062f\u0647 \u0648 \u0622\u0633\u0627\u0646 \u0628\u0631\u0627\u06cc \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0627\u0633\u062a.<\/p>\n<p>\u062f\u0631 \u0627\u06cc\u0646 \u0622\u0645\u0648\u0632\u0634\u060c \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0631\u0627 \u0646\u0634\u0627\u0646 \u062f\u0627\u062f\u06cc\u0645 \u0631\u0648\u06cc \u062a\u0627\u0628\u0639\u06cc \u0627\u0632 \u062f\u0648 \u0645\u062a\u063a\u06cc\u0631 \u0628\u0627 \u062e\u0637\u0648\u0637 \u062f\u0627\u06cc\u0631\u0647 \u0627\u06cc.  \u0633\u067e\u0633 \u0645\u062b\u0627\u0644 \u062e\u0648\u062f \u0631\u0627 \u0628\u0631\u0627\u06cc \u0628\u0647 \u062d\u062f\u0627\u0642\u0644 \u0631\u0633\u0627\u0646\u062f\u0646 \u0645\u06cc\u0627\u0646\u06af\u06cc\u0646 \u0645\u0631\u0628\u0639\u0627\u062a \u062e\u0637\u0627 \u062f\u0631 \u06cc\u06a9 \u0645\u0633\u0626\u0644\u0647 \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u06af\u0633\u062a\u0631\u0634 \u062f\u0627\u062f\u06cc\u0645 \u0648 \u06cc\u06a9 \u0633\u06cc\u0633\u062a\u0645 OCR \u0633\u0627\u062f\u0647 \u0633\u0627\u062e\u062a\u06cc\u0645.  \u0645\u0627 \u0647\u0645\u0686\u0646\u06cc\u0646 \u062f\u0631 \u0645\u0648\u0631\u062f \u0646\u0633\u062e\u0647 \u062a\u0635\u0627\u062f\u0641\u06cc \u0646\u0632\u0648\u0644 \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0628\u062d\u062b \u06a9\u0631\u062f\u06cc\u0645.<\/p>\n<p>\u06cc\u06a9 \u062a\u0627\u0628\u0639 \u0647\u062f\u0641 \u06a9\u0644\u06cc \u0628\u0631\u0627\u06cc \u0627\u062c\u0631\u0627\u06cc \u06af\u0631\u0627\u062f\u06cc\u0627\u0646 \u0646\u0632\u0648\u0644 \u062f\u0631 \u0627\u06cc\u0646 \u0622\u0645\u0648\u0632\u0634 \u062a\u0648\u0633\u0639\u0647 \u062f\u0627\u062f\u0647 \u0634\u062f.  \u0645\u0627 \u062e\u0648\u0627\u0646\u0646\u062f\u06af\u0627\u0646 \u0631\u0627 \u062a\u0634\u0648\u06cc\u0642 \u0645\u06cc \u06a9\u0646\u06cc\u0645 \u06a9\u0647 \u0627\u0632 \u0627\u06cc\u0646 \u062a\u0627\u0628\u0639 \u0627\u0633\u062a\u0641\u0627\u062f\u0647 \u06a9\u0646\u0646\u062f \u0631\u0648\u06cc \u0645\u0634\u06a9\u0644\u0627\u062a \u0631\u06af\u0631\u0633\u06cc\u0648\u0646 \u0648 \u0637\u0628\u0642\u0647 \u0628\u0646\u062f\u06cc \u0645\u062e\u062a\u0644\u0641\u060c \u0628\u0627 \u067e\u0627\u0631\u0627\u0645\u062a\u0631\u0647\u0627\u06cc \u0645\u062e\u062a\u0644\u0641\u060c \u0628\u0631\u0627\u06cc \u062f\u0631\u06a9 \u0628\u0647\u062a\u0631 \u0639\u0645\u0644\u06a9\u0631\u062f \u0622\u0646.<\/p>\n<\/div>\n<p><script>\n                        !function(f,b,e,v,n,t,s)\n                        {if(f.fbq)return;n=f.fbq=function(){n.callMethod?\n                        n.callMethod.apply(n,arguments):n.queue.push(arguments)};\n                        if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0';\n                        n.queue=();t=b.createElement(e);t.async=!0;\n                        t.src=v;s=b.getElementsByTagName(e)(0);\n                        s.parentNode.insertBefore(t,s)}(window, document,'script',\n                        'https:\/\/connect.facebook.net\/en_US\/fbevents.js');\n                        fbq('init', '525232124909042');\n                        fbq('track', 'PageView');\n                    <\/script>    (\u0628\u0631\u0686\u0633\u0628\u200c\u0647\u0627 \u0628\u0647 \u062a\u0631\u062c\u0645\u0647)# python<br \/>\n<br \/><br \/>\n<br \/>\u0645\u0646\u062a\u0634\u0631 \u0634\u062f\u0647 \u062f\u0631 1403-01-15 03:09:04<br \/>\n<\/p>\n\n\n<div class=\"kk-star-ratings kksr-auto kksr-align-center kksr-valign-bottom\"\n    data-payload='{&quot;align&quot;:&quot;center&quot;,&quot;id&quot;:&quot;15647&quot;,&quot;slug&quot;:&quot;default&quot;,&quot;valign&quot;:&quot;bottom&quot;,&quot;ignore&quot;:&quot;&quot;,&quot;reference&quot;:&quot;auto&quot;,&quot;class&quot;:&quot;&quot;,&quot;count&quot;:&quot;0&quot;,&quot;legendonly&quot;:&quot;&quot;,&quot;readonly&quot;:&quot;&quot;,&quot;score&quot;:&quot;0&quot;,&quot;starsonly&quot;:&quot;&quot;,&quot;best&quot;:&quot;5&quot;,&quot;gap&quot;:&quot;5&quot;,&quot;greet&quot;:&quot;\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628&quot;,&quot;legend&quot;:&quot;0\\\/5 (0 \u0631\u0627\u06cc)&quot;,&quot;size&quot;:&quot;30&quot;,&quot;title&quot;:&quot;Gradient Descent \u062f\u0631 \u067e\u0627\u06cc\u062a\u0648\u0646: \u067e\u06cc\u0627\u062f\u0647 \u0633\u0627\u0632\u06cc \u0648 \u0646\u0638\u0631\u06cc\u0647&quot;,&quot;width&quot;:&quot;0&quot;,&quot;_legend&quot;:&quot;{score}\\\/{best} ({count} \u0631\u0627\u06cc)&quot;,&quot;font_factor&quot;:&quot;1.25&quot;}'>\n            \n<div class=\"kksr-stars\">\n    \n<div class=\"kksr-stars-inactive\">\n            <div class=\"kksr-star\" data-star=\"1\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"2\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"3\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"4\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" data-star=\"5\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n    <\/div>\n    \n<div class=\"kksr-stars-active\" style=\"width: 0px;\">\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n            <div class=\"kksr-star\" style=\"padding-left: 5px\">\n            \n\n<div class=\"kksr-icon\" style=\"width: 30px; height: 30px;\"><\/div>\n        <\/div>\n    <\/div>\n<\/div>\n                \n\n<div class=\"kksr-legend\" style=\"font-size: 24px;\">\n            <span class=\"kksr-muted\">\u0627\u0645\u062a\u06cc\u0627\u0632 \u0634\u0645\u0627 \u0628\u0647 \u0627\u06cc\u0646 \u0645\u0637\u0644\u0628<\/span>\n    <\/div>\n    <\/div>\n","protected":false},"excerpt":{"rendered":"<p><span class=\"span-reading-time rt-reading-time\" style=\"display: block;\"><span class=\"rt-label rt-prefix\">\u0632\u0645\u0627\u0646 \u0644\u0627\u0632\u0645 \u0628\u0631\u0627\u06cc \u0645\u0637\u0627\u0644\u0639\u0647: 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[&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":15648,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1743,620],"tags":[],"class_list":["post-15647","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-python","category-programming"],"acf":[],"_links":{"self":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/15647","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/comments?post=15647"}],"version-history":[{"count":0,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/posts\/15647\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media\/15648"}],"wp:attachment":[{"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/media?parent=15647"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/categories?post=15647"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rasanegaar.com\/blog\/wp-json\/wp\/v2\/tags?post=15647"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}