ML_Maschinelles_Lernen_Grundlagen/Exercise_2
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Pascal Schindler 2021-11-25 12:10:55 +01:00
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exercise.ipynb
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@ -86,7 +86,7 @@
},
"outputs": [],
"source": [
"data = np.load(\"iris_data.npz\")\n",
"data = np.load(\"/Users/pascalschindler/Desktop/ML_Übungsblätter/Exercise_2/iris_data.npz\")\n",
"train_samples = data[\"train_features\"]\n",
"train_labels = data[\"train_labels\"]\n",
"test_samples = data[\"test_features\"]\n",
@ -192,13 +192,16 @@
"\n",
"\\begin{align} \n",
"\\mathcal L_{\\mathrm{cat-NLL}} \n",
"=& - \\sum_{i=1}^N \\log p(c_i | \\boldsymbol x_i) \\\\ \n",
"=& - \\sum_{i=1}^N \\sum_{k=1}^K h_{i, k} \\log(p_{i,k}) \\\\ \n",
"=& - \\sum_{i=1}^N \\sum_{k=1}^K h_{i, k} \\log(\\textrm{softmax}(\\boldsymbol{w}_k^T \\boldsymbol \\phi(\\boldsymbol{x}_i))_k)\\\\ \n",
"=& - \\sum_{i=1}^N \\log p(c_i | \\boldsymbol x_i) \\\\\n",
"=& - \\sum_{i=1}^N \\sum_{k=1}^K h_{i, k} \\log(p_{i,k}) \\\\\n",
"=& - \\sum_{i=1}^N \\sum_{k=1}^K h_{i, k} \\log(\\textrm{softmax}(\\boldsymbol{w}_k^T \\boldsymbol \\phi(\\boldsymbol{x}_i))_k)\\\\\n",
"=& - \\sum_{i=1}^N \\left(\\sum_{k=1}^K h_{i,k}\\boldsymbol{w}^T_k \\boldsymbol \\phi(\\boldsymbol{x}_i) - \\log \\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))\\right).\n",
"\\end{align}\n",
"\n",
"In order to use gradient based optimization for this, we of course also need to derive the gradient.\n",
"This gives us the loss for a single sample. To get the loss for all samples we will need to sum over loss for a single sample\n",
"\n",
"\n",
"\n",
"### 1.1) Derivation (4 Points)\n",
"Derive the gradient $\\dfrac{\\partial \\mathcal L_{\\mathrm{cat-NLL}}}{\\partial \\boldsymbol{w}}$ of the loss function w.r.t. the full weight vector $\\boldsymbol w \\equiv \\begin{pmatrix} \\boldsymbol w_1^T & \\dots & \\boldsymbol w_K^T \\end{pmatrix}^T$, which is obtained by stacking the class-specific weight vectors $\\boldsymbol w_k$.\n",
@ -206,9 +209,73 @@
"**Hint 1:** Follow the steps in the derivation of the gradient of the loss for the binary classification in the lecture.\n",
"\n",
"**Hint 2:** Derive the gradient not for the whole vector $\\boldsymbol w$ but only for $\\boldsymbol w_k$ i.e., $\\dfrac{\\partial \\mathcal L_{\\mathrm{cat-NLL}}}{\\partial \\boldsymbol{w}_k}$. The gradients for the individual\n",
"$\\boldsymbol w_k$ can then be stacked to obtain the full gradient.\n"
"$\\boldsymbol w_k$ can then be stacked to obtain the full gradient."
]
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"cell_type": "markdown",
"source": [
"\\begin{align}\n",
"\\mathcal L_{\\mathrm{cat-NLL}}\n",
"=& - \\sum_{i=1}^N \\left(\\sum_{k=1}^K h_{i,k}\\boldsymbol{w}^T_k \\boldsymbol \\phi(\\boldsymbol{x}_i) - \\log \\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))\\right) \\\\\n",
"=& - \\sum_{i=1}^N \\left(\\sum_{k=1}^K h_{i,k}\\boldsymbol{w}^T_k \\boldsymbol \\phi(\\boldsymbol{x}_i) \\right) + \\sum_{i=1}^N \\log \\left(\\sum_{j=1}^K h_{i,k} \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i)\\right) \\\\\n",
"\\end{align}\n"
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"Take $\\nabla_{w_j}\\mathcal L_{\\mathrm{cat-NLL}}$ with respect to a particular weight vector $\\boldsymbol{w}_j$, which leads to the fact that the sum $\\sum_{k=1}^K$ is no longer necessary and $\\sum_{i=1}^N\\log \\left(\\sum_{j=1}^K h_{i,k} \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i)\\right) = \\sum_{i=1}^N \\log \\left( \\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i)\\right)$, since $\\sum_{k=1}^K h_{i,k} = 1$\n"
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"cell_type": "markdown",
"source": [
"\\begin{align}\n",
"\\dfrac{\\partial \\mathcal L_{\\mathrm{cat-NLL}}}{\\partial \\boldsymbol{w}_k}\n",
"=& \\dfrac{\\partial}{\\partial \\boldsymbol{w}_k} \\left(- \\sum_{i=1}^N \\left( h_{i,k}\\boldsymbol{w}^T_k \\boldsymbol \\phi(\\boldsymbol{x}_i) \\right) + \\sum_{i=1}^N \\log \\left(\\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i)\\right)\\right) \\\\\n",
"=& - \\sum_{i=1}^N \\left( h_{i,k}\\boldsymbol \\phi(\\boldsymbol{x}_i) \\right) + \\sum_{i=1}^N \\left( \\frac{1}{\\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))} \\right) \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i)) * \\phi(\\boldsymbol{x}_i)\\\\\n",
"=& \\sum_{i=1}^N \\left[ \\left( \\frac{\\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))}{\\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))} - h_{i,k} \\right ) \\right] \\phi(\\boldsymbol{x}_i))\n",
"\\end{align}"
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"cell_type": "markdown",
"source": [
"This is the derivation for one specific weight vector/class. This has to be done for all weight vectors\n"
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"cell_type": "markdown",
"source": [
"\\begin{align}\n",
"\\dfrac{\\partial \\mathcal L_{\\mathrm{cat-NLL}}}{\\partial \\boldsymbol{w}}\n",
"=& \\begin{pmatrix}\n",
"\\sum_{i=1}^N \\left[ \\left( \\frac{\\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))}{\\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))} - h_{i,1} \\right ) \\right] \\phi(\\boldsymbol{x}_i)) \\\\\n",
"\\vdots \\\\\n",
"\\sum_{i=1}^N \\left[ \\left( \\frac{\\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))}{\\sum_{j=1}^K \\exp(\\boldsymbol{w}_j^T \\boldsymbol \\phi(\\boldsymbol{x}_i))} - h_{i,K} \\right ) \\right] \\phi(\\boldsymbol{x}_i))\n",
"\\end{pmatrix}\n",
"\\end{align}"
],
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@ -1030,4 +1097,4 @@
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