\uc774\ubc88 \uae00\uc5d0\uc11c\ub294 \uba38\uc2e0\ub7ec\ub2dd\uc758 \uc911\uc694\ud55c \uae30\ubc95\uc778 \uc11c\ud3ec\ud2b8 \ubca1\ud130 \uba38\uc2e0(Support Vector Machine, SVM)\uc5d0 \ub300\ud574 \uc790\uc138\ud788 \uc54c\uc544\ubcf4\uace0, \uc774\ub97c \ud30c\uc774\ud1a0\uce58\ub97c \uc774\uc6a9\ud574\uc11c \uad6c\ud604\ud574\ubcf4\ub3c4\ub85d \ud558\uaca0\uc2b5\ub2c8\ub2e4. \uc11c\ud3ec\ud2b8 \ubca1\ud130 \uba38\uc2e0\uc740 \ud2b9\ud788 \ubd84\ub958 \ubb38\uc81c\uc5d0\uc11c \ub6f0\uc5b4\ub09c \uc131\ub2a5\uc744 \ubc1c\ud718\ud569\ub2c8\ub2e4. SVM\uc740 \ucd5c\ub300 \ub9c8\uc9c4 \uc6d0\uce59(maximum margin principle)\uc744 \uae30\ubc18\uc73c\ub85c \ud558\ub294 \ubd84\ub958 \uc54c\uace0\ub9ac\uc998\uc73c\ub85c, \uc8fc\ub85c \uc120\ud615 \ubd84\ub958\uae30(linear classifier)\ub85c \uc0ac\uc6a9\ub418\uc9c0\ub9cc, \ucee4\ub110 \uae30\ubc95(kernel trick)\uc744 \ud1b5\ud574 \ube44\uc120\ud615 \ub370\uc774\ud130\uc5d0 \ub300\ud574\uc11c\ub3c4 \ud6a8\uacfc\uc801\uc73c\ub85c \uc801\uc6a9\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n
\uc11c\ud3ec\ud2b8 \ubca1\ud130 \uba38\uc2e0\uc740 \ub450 \uac1c\uc758 \ud074\ub798\uc2a4 \uc0ac\uc774\ub97c \ubd84\ub9ac\ud558\ub294 \ucd5c\uc801\uc758 \ucd08\ud3c9\uba74(hyperplane)\uc744 \ucc3e\ub294 \uc54c\uace0\ub9ac\uc998\uc785\ub2c8\ub2e4. \uc5ec\uae30\uc11c ‘\ucd5c\uc801’\uc774\ub77c\ub294 \uac83\uc740 \ub9c8\uc9c4(margin)\uc744 \ucd5c\ub300\ud654\ud558\ub294 \uac83\uc744 \uc758\ubbf8\ud558\ub294\ub370, \ub9c8\uc9c4\uc740 \ucd08\ud3c9\uba74\uc5d0\uc11c \uac00\uc7a5 \uac00\uae4c\uc6b4 \ub370\uc774\ud130 \ud3ec\uc778\ud2b8(\uc11c\ud3ec\ud2b8 \ubca1\ud130)\uae4c\uc9c0\uc758 \uac70\ub9ac\ub97c \ub9d0\ud569\ub2c8\ub2e4. SVM\uc740 \uc774\ub7ec\ud55c \ub9c8\uc9c4\uc744 \ucd5c\ub300\ud654\ud558\uc5ec, \uc8fc\uc5b4\uc9c4 \ub370\uc774\ud130\uc5d0 \ub300\ud574 \uc77c\ubc18\ud654 \ub2a5\ub825\uc744 \ub192\uc774\ub3c4\ub85d \uc124\uacc4\ub418\uc5c8\uc2b5\ub2c8\ub2e4.<\/p>\n
SVM\uc758 \uae30\ubcf8 \ub3d9\uc791 \uc6d0\ub9ac\ub294 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4:<\/p>\n
SVM\uc758 \uae30\ubcf8 \ubaa9\ud45c\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ucd5c\uc801\ud654 \ubb38\uc81c\ub97c \ud574\uacb0\ud558\ub294 \uac83\uc785\ub2c8\ub2e4:<\/p>\n
\uc8fc\uc5b4\uc9c4 \ub370\uc774\ud130\uac00 \uc5ec\uae30\uc11c \ube44\uc120\ud615 \ub370\uc774\ud130\ub97c \ub2e4\ub8e8\uae30 \uc704\ud574 SVM\uc740 \ucee4\ub110 \ud568\uc218\ub97c \uc0ac\uc6a9\ud569\ub2c8\ub2e4. \ucee4\ub110 \ud568\uc218\ub294 \ub370\uc774\ud130\ub97c \uace0\ucc28\uc6d0 \uacf5\uac04\uc73c\ub85c \ub9e4\ud551\ud558\uc5ec \uc11c\ub85c \ubd84\ub9ac \uac00\ub2a5\ud55c \ud615\ud0dc\ub85c \ubcc0\ud658\ud558\ub294 \ud568\uc218\uc785\ub2c8\ub2e4. \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \ucee4\ub110 \ud568\uc218\ub294 \ub2e4\uc74c\uacfc \uac19\uc2b5\ub2c8\ub2e4:<\/p>\n \uc774\uc81c PyTorch\ub97c \uc0ac\uc6a9\ud558\uc5ec SVM\uc744 \uad6c\ud604\ud574\ubcf4\uaca0\uc2b5\ub2c8\ub2e4. PyTorch\ub294 \ub525\ub7ec\ub2dd \ud504\ub808\uc784\uc6cc\ud06c\uc774\uc9c0\ub9cc, \uc218\uce58 \uacc4\uc0b0\uc774 \uac00\ub2a5\ud558\uae30 \ub54c\ubb38\uc5d0 SVM\uacfc \uac19\uc740 \uc54c\uace0\ub9ac\uc998\ub3c4 \uc27d\uac8c \uad6c\ud604\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4. \ub2e4\uc74c \ub2e8\uacc4\ub85c \uc9c4\ud589\ud558\uaca0\uc2b5\ub2c8\ub2e4:<\/p>\n \uba3c\uc800 \ud544\uc694\ud55c \ud328\ud0a4\uc9c0\ub97c \uc124\uce58\ud558\uace0, \uc0ac\uc6a9\ud560 \ub370\uc774\ud130\ub97c \uc0dd\uc131\ud569\ub2c8\ub2e4.<\/p>\n \uc774\uc81c SVM \ubaa8\ub378\uc744 \uad6c\ucd95\ud558\uaca0\uc2b5\ub2c8\ub2e4. \ubaa8\ub378\uc740 \uc785\ub825 \ub370\uc774\ud130\uc640 \ub808\uc774\ube14\uc744 \uc774\uc6a9\ud558\uc5ec \uac00\uc911\uce58 \ubaa8\ub378\uc744 \ud559\uc2b5\ud558\uae30 \uc804\uc5d0 \uc635\ud2f0\ub9c8\uc774\uc800\uc640 \ud559\uc2b5\ub960\uc744 \uc124\uc815\ud569\ub2c8\ub2e4.<\/p>\n \ubaa8\ub378 \ud559\uc2b5\uc774 \uc644\ub8cc\ub418\uba74, \uacb0\uc815 \uacbd\uacc4\ub97c \uc2dc\uac01\ud654\ud558\uc5ec \ubaa8\ub378\uc758 \uc131\ub2a5\uc744 \ud3c9\uac00\ud560 \uc218 \uc788\uc2b5\ub2c8\ub2e4.<\/p>\n SVM\uc740 \ub180\ub77c\uc6b4 \uc131\ub2a5\uc744 \ubc1c\ud718\ud558\uc9c0\ub9cc, \ubaa8\ub4e0 \uc54c\uace0\ub9ac\uc998\uacfc \ub9c8\ucc2c\uac00\uc9c0\ub85c \uc7a5\ub2e8\uc810\uc774 \uc874\uc7ac\ud569\ub2c8\ub2e4.<\/p>\n \uc11c\ud3ec\ud2b8 \ubca1\ud130 \uba38\uc2e0\uc740 \uac15\ub825\ud55c \uc131\ub2a5\uc744 \uac00\uc9c4 \ubd84\ub958 \uc54c\uace0\ub9ac\uc998\uc73c\ub85c, \ud2b9\ud788 \ud68c\uadc0\uac00 \uc544\ub2cc \ubd84\ub958 \ubb38\uc81c\uc5d0\uc11c \ub9e4\uc6b0 \uc720\uc6a9\ud558\uac8c \uc0ac\uc6a9\ub420 \uc218 \uc788\uc2b5\ub2c8\ub2e4. PyTorch\ub97c \ud65c\uc6a9\ud558\uc5ec SVM\uc744 \uad6c\ud604\ud574\ubcf4\uba70, \uae30\uacc4 \ud559\uc2b5\uc758 \uae30\ucd08\uc801\uc778 \uac1c\ub150\uc744 \ub2e4\uc2dc \ud55c\ubc88 \ub418\uc0c8\uae30\ub294 \uae30\ud68c\uac00 \ub418\uc5c8\uae30\ub97c \ubc14\ub78d\ub2c8\ub2e4. \ub354 \ub098\uc544\uac00 SVM\uc744 \ud65c\uc6a9\ud55c \uc2e4\uc804 \ud504\ub85c\uc81d\ud2b8\ub098 \uc5f0\uad6c\ub85c \ub098\uc544\uac08 \uc218 \uc788\ub294 \ubc1c\ud310\uc774 \ub418\uc5c8\uc73c\uba74 \uc88b\uaca0\uc2b5\ub2c8\ub2e4.<\/p>\n(x_i, y_i)<\/code> \ud615\ud0dc\ub85c \uc788\uc744 \ub54c, \uc5ec\uae30\uc11c x_i<\/code>\ub294 \uc785\ub825 \ub370\uc774\ud130\uc774\uace0 y_i<\/code>\ub294 \ud074\ub798\uc2a4 \ub808\uc774\ube14(1 \ub610\ub294 -1)\uc785\ub2c8\ub2e4. SVM\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \ucd5c\uc801\ud654 \ubb38\uc81c\ub97c \uc124\uc815\ud569\ub2c8\ub2e4:<\/p>\nminimize (1\/2) ||w||^2\nsubject to y_i (w * x_i + b) >= 1<\/code><\/pre>\nw<\/code>\ub294 \ucd08\ud3c9\uba74\uc758 \uae30\uc6b8\uae30 \ubca1\ud130, b<\/code>\ub294 \uc808\ud3b8(bias)\uc744 \uc758\ubbf8\ud569\ub2c8\ub2e4. \uc704\uc758 \uc2dd\uc740 \ucd5c\uc801\uc758 \uacbd\uacc4\ub97c \uc815\uc758\ud558\uace0 \ub9c8\uc9c4\uc744 \ucd5c\ub300\ud654\ud558\ub294 \uac83\uc785\ub2c8\ub2e4.<\/p>\n2.2. \ucee4\ub110 \uae30\ubc95<\/h3>\n
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K(x, x') = x * x'<\/code><\/li>\nK(x, x') = (alpha * (x * x') + c)^d<\/code><\/li>\nK(x, x') = exp(-gamma * ||x - x'||^2)<\/code><\/li>\n<\/ul>\n3. PyTorch\ub85c SVM \uad6c\ud604\ud558\uae30<\/h2>\n
3.1. \ud328\ud0a4\uc9c0 \uc124\uce58 \ubc0f \ub370\uc774\ud130 \uc900\ube44<\/h3>\n
import torch\nimport numpy as np\nimport matplotlib.pyplot as plt\nfrom sklearn.datasets import make_moons\nfrom sklearn.model_selection import train_test_split\n\n# \ub370\uc774\ud130 \uc0dd\uc131\nX, y = make_moons(n_samples=100, noise=0.1, random_state=42)\ny = np.where(y == 0, -1, 1) # \ub808\uc774\ube14\uc744 -1\uacfc 1\ub85c \ubcc0\ud658\n\n# \ub370\uc774\ud130 \ubd84\ud560\nX_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=42)\n\n# \ub370\uc774\ud130 \ud150\uc11c \ubcc0\ud658\nX_train_tensor = torch.FloatTensor(X_train)\ny_train_tensor = torch.FloatTensor(y_train)\nX_test_tensor = torch.FloatTensor(X_test)\ny_test_tensor = torch.FloatTensor(y_test)<\/code><\/pre>\n3.2. SVM \ubaa8\ub378 \uad6c\ucd95<\/h3>\n
w<\/code>\uc640 \uc808\ud3b8 b<\/code>\ub97c \ud559\uc2b5\ud569\ub2c8\ub2e4.<\/p>\nclass SVM(torch.nn.Module):\n def __init__(self):\n super(SVM, self).__init__()\n self.w = torch.nn.Parameter(torch.randn(2, requires_grad=True))\n self.b = torch.nn.Parameter(torch.randn(1, requires_grad=True))\n \n def forward(self, x):\n return torch.matmul(x, self.w) + self.b\n \n def hinge_loss(self, y, output):\n return torch.mean(torch.clamp(1 - y * output, min=0))<\/code><\/pre>\n3.3. \ud2b8\ub808\uc774\ub2dd \ubc0f \ud14c\uc2a4\ud2b8<\/h3>\n
# \ud558\uc774\ud37c\ud30c\ub77c\ubbf8\ud130 \uc124\uc815\nlearning_rate = 0.01\nnum_epochs = 1000\n\nmodel = SVM()\noptimizer = torch.optim.SGD(model.parameters(), lr=learning_rate)\n\n# \ud559\uc2b5 \uacfc\uc815\nfor epoch in range(num_epochs):\n optimizer.zero_grad()\n \n # \ubaa8\ub378 \uc608\uce21\n output = model(X_train_tensor)\n \n # \uc190\uc2e4 \uacc4\uc0b0 (Hinge Loss)\n loss = model.hinge_loss(y_train_tensor, output)\n \n # \uc5ed\uc804\ud30c\n loss.backward()\n optimizer.step()\n\n if (epoch+1) % 100 == 0:\n print(f'Epoch [{epoch+1}\/{num_epochs}], Loss: {loss.item():.4f}')<\/code><\/pre>\n3.4. \uacb0\uacfc \uc2dc\uac01\ud654<\/h3>\n
# \uacb0\uc815 \uacbd\uacc4 \uc2dc\uac01\ud654\ndef plot_decision_boundary(model, X, y):\n x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1\n y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1\n xx, yy = np.meshgrid(np.linspace(x_min, x_max, 100), np.linspace(y_min, y_max, 100))\n grid = torch.FloatTensor(np.c_[xx.ravel(), yy.ravel()])\n \n with torch.no_grad():\n model.eval()\n Z = model(grid)\n Z = Z.view(xx.shape)\n plt.contourf(xx, yy, Z.data.numpy(), levels=50, alpha=0.5)\n \n plt.scatter(X[:, 0], X[:, 1], c=y, s=20, edgecolor='k')\n plt.title(\"SVM Decision Boundary\")\n plt.xlabel(\"Feature 1\")\n plt.ylabel(\"Feature 2\")\n plt.show()\n\nplot_decision_boundary(model, X, y)<\/code><\/pre>\n4. SVM\uc758 \uc7a5\ub2e8\uc810<\/h2>\n
4.1. \uc7a5\uc810<\/h3>\n
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4.2. \ub2e8\uc810<\/h3>\n
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5. \uacb0\ub860<\/h2>\n
6. References<\/h2>\n