{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Exponential Machines\n",
"\n",
"[Exponential machines (Novikov et al., 2017)](https://arxiv.org/pdf/1605.03795.pdf) are predictors that are able to model all $2^N$ interactions between $N$ features. Those interactions are limited to being of degree 1, i.e. the regressor is of the form \n",
"\n",
"$$f(x_1, \\dots, x_N) = \\sum_{\\alpha \\subseteq \\{1, \\dots, N\\}} \\left( w_{\\alpha} \\cdot \\prod_{n \\in \\alpha} x_n \\right)$$\n",
"\n",
"and fitting the regressor means finding the best $w_{\\alpha}$ for all $2^N$ possible $\\alpha$'s. Besides the $L^2$ loss on the training data, the original paper also adds an $L^2$ regularization term (\"ridge\") on the set of weights $w$.\n",
"\n",
"We note that an exponential machine is equivalent to a TT-Tucker model with Tucker rank equal to 2: the first column of the Tucker factors should be a constant ($1$), and the second column should be linear ($x$). To this end, we can use for example a Legendre expansion of polynomials truncated to 2 basis elements. Note that this can be seen as a particular case of a [polynomial chaos expansion](pce.ipynb).\n",
"\n",
"We will try here a synthetic model with noise: $f^{\\mathrm{true}}(x_1, \\dots, x_5) = x_1 x_2 x_3 + x_1 x_2 x_3 x_4 x_5 + \\varepsilon$:"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import torch\n",
"torch.set_default_dtype(torch.float64)\n",
"import tntorch as tn\n",
"\n",
"P = 100\n",
"ntrain = int(P*0.75)\n",
"N = 5\n",
"ticks = 32 # We will use a 32^5 tensor\n",
"\n",
"X = torch.rand(P, N)*2 - 1 # Features between -1 and 1\n",
"y = torch.prod(X, dim=1) + torch.prod(X[:, :3], dim=1)\n",
"y += torch.randn(y.shape)*torch.std(y)/10 # Gaussian noise: 1/5th of the clean signal's sigma\n",
"X = (X+1)/2*(ticks-1) # Make feature between 0 and ticks-1, i.e. indexable by the tensor\n",
"\n",
"# Split into train/test\n",
"X_train = X[:ntrain]\n",
"y_train = y[:ntrain]\n",
"X_test = X[ntrain:]\n",
"y_test = y[ntrain:]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"A TT-rank of 2 should be enough to fit this data set, since it arises from the sum of exactly two interactions."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"t = tn.rand(shape=[ticks]*N, ranks_tt=2, ranks_tucker=2, requires_grad=True)\n",
"t.set_factors('legendre', requires_grad=False) # We set the factors to Legendre polynomials, and fix them (won't be changed during optimization) "
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"iter: 0 | loss: 7.956270 | total time: 0.0022\n",
"iter: 500 | loss: 0.475733 | total time: 0.9688\n",
"iter: 1000 | loss: 0.071103 | total time: 1.8799\n",
"iter: 1500 | loss: 0.024970 | total time: 2.8605\n",
"iter: 2000 | loss: 0.013503 | total time: 3.9351\n",
"iter: 2500 | loss: 0.009449 | total time: 5.0269\n",
"iter: 3000 | loss: 0.007776 | total time: 6.1072\n",
"iter: 3500 | loss: 0.007118 | total time: 7.1863\n",
"iter: 3532 | loss: 0.007094 | total time: 7.2544 <- converged (tol=0.0001)\n"
]
}
],
"source": [
"def loss(t):\n",
" return tn.relative_error(y_train, t[X_train])**2\n",
"tn.optimize(t, loss)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Test relative error: tensor(0.1051, grad_fn=<DivBackward1>)\n"
]
}
],
"source": [
"print('Test relative error:', tn.relative_error(y_test, t[X_test]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The tensor of weights ($\\mathcal{W}$ in the original paper) can be retrieved as the TT part of our TT-Tucker tensor:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"5D TT tensor:\n",
"\n",
" 2 2 2 2 2\n",
" | | | | |\n",
" (0) (1) (2) (3) (4)\n",
" / \\ / \\ / \\ / \\ / \\\n",
"1 2 2 2 2 1"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"core = tn.Tensor(t.cores)\n",
"core"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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gNVW14V64XGH2q1ncDijgNPCLT+5fbzRJfhT4R+CLwHe607/B4r71JDz/K82/jwn4HCR5MYsvmG5i8cL4SFXd3P07Pgw8B/g88Pqq+vb4Jl008XGXJD3VpG/LSJKWYdwlqUHGXZIaZNwlqUHGXZIaZNwlqUHGXZIaZNwlqUH/Byo1Kx61k9XSAAAAAElFTkSuQmCC\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"import numpy as np\n",
"\n",
"plt.figure()\n",
"plt.bar(np.arange(core.size), np.sort(core.numpy().flatten()))\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Despite the noise, the model correctly retrieved the two interacting components. Those indeed correspond to $x_1 x_2 x_3$ and $x_1 x_2 x_3 x_4 x_5$:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tensor(1.0280, grad_fn=<SqueezeBackward0>)\n",
"tensor(0.9253, grad_fn=<SqueezeBackward0>)\n"
]
}
],
"source": [
"print(core[1, 1, 1, 0, 0])\n",
"print(core[1, 1, 1, 1, 1])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The orders of participating interactions (namely, 3 and 5) can be revealed as well by the [dimension distribution](sobol.ipynb):"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.figure()\n",
"plt.bar(np.arange(1, N+1), tn.dimension_distribution(t).numpy())\n",
"plt.xlabel('Order')\n",
"plt.ylabel('Variance contribution')\n",
"plt.title('Dimension distribution')\n",
"plt.show()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
