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added optimal transport example #43

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302 changes: 302 additions & 0 deletions examples/torch/optimal_transport.ipynb
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{
"cells": [
{
"cell_type": "code",
"execution_count": 186,
"metadata": {},
"outputs": [],
"source": [
"from cvxpylayers.torch import CvxpyLayer\n",
"import torch\n",
"import cvxpy as cp\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"torch.set_default_dtype(torch.double)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Differentiable optimal transport\n",
"\n",
"Suppose we have two one-dimensional probability distributions, $\\nu$ and $\\eta$, defined\n",
"by their supports $x\\in\\mathbf{R}^n$, $y\\in\\mathbf{R}^m$, and probability weight vectors $a\\in\\mathbf{R}_+^n$, $b\\in\\mathbf{R}_+^m$.\n",
"That is, $\\nu=x_i$ with probability $a_i$, and\n",
"$\\eta=y_j$ with probability $b_j$.\n",
"\n",
"In optimal transport, the goal is to find a matrix $P\\in\\mathbf{R}_+^{n\\times m}$ (the set of nonnegative $n$ by $m$ matrices) that *transports*\n",
"$\\nu$ into $\\eta$. The entry $P_{ij}$ tells us how much of the probability mass at $x_i$\n",
"that we transfer to $y_j$. We have two constraints:\n",
"* $\\sum_{j} P_{ij} = a_i$, $i=1,\\ldots,n$, which requires that the mass leaving $x_i$ is equal to the mass available, $a_i$.\n",
"* $\\sum_{i} P_{ij} = b_j$, $j=1,\\ldots,m$, which requires that the mass entering $y_i$ is equal to the mass there, $b_j$.\n",
"\n",
"For each mass transfer $P_{ij}$, we associate a cost $C_{ij} = h(x_i - y_j)$, where $h(d)$ measures\n",
"the cost of transferring a unit a distance $d$. An example is $h(d)=d^2$; in this case $C_{ij}$ measures\n",
"the squared difference between $x_i$ and $y_j$.\n",
"\n",
"We can find the (regularized) optimal transport between $\\nu$ and $\\eta$ by solving the optimization problem\n",
"\\begin{equation}\n",
"\\begin{array}{ll}\n",
"\\mbox{minimize} & \\mathbf{tr}(C^TP) - \\varepsilon H(P), \\\\\n",
"\\mbox{subject to} & P \\in \\mathbf{R}_+^{n \\times m}, \\\\\n",
"& P\\mathbf{1}=a, \\quad P^T\\mathbf{1}=b.\n",
"\\end{array}\n",
"\\end{equation}\n",
"where $H(P) = -\\sum_{ij} P_{ij} (\\log(P_{ij}) - 1)$ is an entropy regularization function and $\\varepsilon > 0$ is a hyper-parameter.\n",
"This is a convex optimization problem.\n",
"\n",
"See:\n",
"Leonid Kantorovich. On the transfer of masses (in russian). Doklady Akademii Nauk, 37(2): 227–229, 1942"
]
},
{
"cell_type": "code",
"execution_count": 187,
"metadata": {},
"outputs": [],
"source": [
"def get_regularized_ot(n, m):\n",
" C = cp.Parameter((n, m))\n",
" a = cp.Parameter(n)\n",
" b = cp.Parameter(m)\n",
" P = cp.Variable((n, m))\n",
" eps = cp.Parameter(1, nonneg=True)\n",
"\n",
" H = lambda P: cp.sum(cp.entr(P)) + cp.sum(P)\n",
" objective = cp.trace(P.T @ C) - eps * H(P)\n",
"\n",
" prob = cp.Problem(cp.Minimize(objective), [[email protected](m) == a, [email protected](n) == b, P >= 0])\n",
"\n",
" layer = CvxpyLayer(prob, [C, a, b, eps], [P])\n",
" def f(x, y, a, b, eps, h=lambda d: d.pow(2)):\n",
" \"\"\"Regularized optimal transport.\n",
"\n",
" x: n-vector\n",
" y: m-vector\n",
" a: n-vector of probabilities\n",
" b: m-vector of probabilities\n",
" eps: scalar, amount of regularization\n",
" \"\"\"\n",
" C = h(x[:,None] - y[None,:])\n",
" return layer(C, a, b, eps)[0]\n",
" return f"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example"
]
},
{
"cell_type": "code",
"execution_count": 188,
"metadata": {},
"outputs": [],
"source": [
"n, m = 3, 3\n",
"regularized_ot = get_regularized_ot(n, m)"
]
},
{
"cell_type": "code",
"execution_count": 189,
"metadata": {},
"outputs": [],
"source": [
"torch.manual_seed(6)\n",
"x = torch.randn(n, requires_grad=True)\n",
"y = torch.randn(m, requires_grad=True)\n",
"a = torch.full((n,), 1./n, requires_grad=True)\n",
"b = torch.full((m,), 1./m, requires_grad=True)\n",
"eps = torch.tensor([1.], requires_grad=True)"
]
},
{
"cell_type": "code",
"execution_count": 190,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<BarContainer object of 3 artists>"
]
},
"execution_count": 190,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.xlim(-2,2)\n",
"plt.bar(x.detach().numpy(), a.detach().numpy(), width=.05)\n",
"plt.bar(y.detach().numpy(), b.detach().numpy(), width=.05)"
]
},
{
"cell_type": "code",
"execution_count": 191,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"WARN: aa_init returned NULL, no acceleration applied.\n"
]
}
],
"source": [
"P = regularized_ot(x, y, a, b, eps)"
]
},
{
"cell_type": "code",
"execution_count": 192,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-1.8744, -0.9937, 0.7185], requires_grad=True)"
]
},
"execution_count": 192,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x"
]
},
{
"cell_type": "code",
"execution_count": 193,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.6985, -1.4716, 0.1777], requires_grad=True)"
]
},
"execution_count": 193,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"y"
]
},
{
"cell_type": "code",
"execution_count": 194,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tensor([[0.1007, 0.2269, 0.0058],\n",
" [0.1807, 0.1043, 0.0483],\n",
" [0.0520, 0.0021, 0.2792]], grad_fn=<_CvxpyLayerFnFnBackward>)\n"
]
}
],
"source": [
"print(P)"
]
},
{
"cell_type": "code",
"execution_count": 195,
"metadata": {},
"outputs": [],
"source": [
"# What would we have to do to increase the mass that we transfer from x = .7185 to y=.1777\n",
"P[2,2].backward()"
]
},
{
"cell_type": "code",
"execution_count": 196,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.0010, -0.0401, 0.0411])"
]
},
"execution_count": 196,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# increase x[2] and decrease x[0:2]\n",
"x.grad"
]
},
{
"cell_type": "code",
"execution_count": 197,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"tensor([-0.0826, 0.0007, 0.0819])"
]
},
"execution_count": 197,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# increase y[2] and decrease y[0], but increase y[1]?\n",
"y.grad"
]
}
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