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Bayesian optimization with Stheno
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{
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"cell_type": "markdown",
"metadata": {},
"source": [
"http://krasserm.github.io/2018/03/21/bayesian-optimization/\n",
"\n",
"Expected improvement:\n",
"\n",
"$$\n",
"\\operatorname{EI}(\\mathbf{x}) = \\mathbb{E}\\max(f(\\mathbf{x}) - f(\\mathbf{x}^+), 0)\\tag{1}\n",
"$$\n",
"\n",
"$$\n",
"\\operatorname{EI}(\\mathbf{x}) =\n",
"\\begin{cases}\n",
"(\\mu(\\mathbf{x}) - f(\\mathbf{x}^+) - \\xi)\\Phi(Z) + \\sigma(\\mathbf{x})\\phi(Z) &\\text{if}\\ \\sigma(\\mathbf{x}) > 0 \\\\\n",
"0 & \\text{if}\\ \\sigma(\\mathbf{x}) = 0\n",
"\\end{cases}\\tag{2}\n",
"$$\n",
"\n",
"$$\n",
"Z =\n",
"\\begin{cases}\n",
"\\frac{\\mu(\\mathbf{x}) - f(\\mathbf{x}^+) - \\xi}{\\sigma(\\mathbf{x})} &\\text{if}\\ \\sigma(\\mathbf{x}) > 0 \\\\\n",
"0 & \\text{if}\\ \\sigma(\\mathbf{x}) = 0\n",
"\\end{cases}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
" 21.054593 seconds (66.77 M allocations: 3.652 GiB, 5.30% gc time)\n"
]
}
],
"source": [
"@time begin\n",
" using Stheno\n",
" using Plots\n",
" using Optim\n",
" using Distributions\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"lower = [-1.]\n",
"upper = [2.]\n",
"noise = 0.2\n",
"\n",
"function f(X; noise=noise)\n",
" x = X[1]\n",
" -sin(3*x) - x^2 + 0.7*x + noise * randn()\n",
"end\n",
"\n",
"X_init = [-0.9, 1.1]\n",
"X_f_init = ColVecs(reshape(X_init, 1, length(X_init)))\n",
"Y_init = f.(X_init)\n",
"\n",
"# Dense grid of points within bounds\n",
"X = range(lower[1], upper[1], step=0.01)\n",
"X_f = ColVecs(reshape(X, 1, length(X)))\n",
"\n",
"# Noise-free objective function values at X \n",
"Y = f.(X_f, noise=0)\n",
";"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
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]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(X, Y, color=\"yellow\", linestyle=:dash, linewidth=2, label=\"Noise-free objective\")\n",
"scatter!(X, f.(X_f), markershape=:x, color=\"blue\", alpha=0.1, label=\"Noisy samples\")\n",
"scatter!(X_init[:], Y_init[:], markershape=:x, color=\"black\", markersize=5, label=\"Initial samples\")"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"expected_improvement (generic function with 1 method)"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function expected_improvement(X, max_sample_mu, gp; xi=0.01)\n",
" # X is a vector containing a test point\n",
" X_f = ColVecs(reshape(X, :, 1))\n",
" \n",
" m_gp = marginals(gp(X_f, 1e-9))\n",
" mu, sigma = mean.(m_gp)[1], std.(m_gp)[1]\n",
" \n",
" if sigma > 0\n",
" imp = mu - max_sample_mu - xi\n",
" Z = imp / sigma\n",
" return imp * cdf(Normal(), Z) + sigma * pdf(Normal(), Z)\n",
" end\n",
" return 0\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"propose_location (generic function with 1 method)"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function propose_location(acquisition, X_sample::ColVecs, gp, lower, upper; n_restarts=25)\n",
" # acquisition take expected_improvement usually\n",
" \n",
" min_val = +Inf # surrogate value\n",
" min_x = nothing\n",
" \n",
" mg = marginals(gp(X_sample, 1e-9))\n",
" max_sample_mu = maximum(mean.(mg))\n",
" \n",
" opt_f(X) = -acquisition(X, max_sample_mu, gp)\n",
" \n",
" for i in 1:n_restarts\n",
" x0 = rand(length(upper)) .* (upper - lower) + lower\n",
" res = optimize(opt_f, lower, upper, x0, Fminbox(LBFGS()))\n",
" if minimum(res) < min_val\n",
" min_val = minimum(res)\n",
" min_x = Optim.minimizer(res)\n",
" end\n",
" end\n",
" \n",
" return min_x\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"plot_acquisition (generic function with 1 method)"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function plot_approximation(gp, X, Y, X_sample, Y_sample, X_next; show_legend)\n",
" # plot surrogate function, X, Y are true function input and output\n",
" X_f = ColVecs(reshape(X, 1, :))\n",
" \n",
" mg = marginals(gp(X_f, 1e-9))\n",
" mu, sigma = mean.(mg), std.(mg)\n",
" \n",
" plt = plot(legend=show_legend, legendfontsize=4)\n",
" plot!(X, Y, color=\"yellow\", linestyle=:dash, linewidth=2, label=\"Noise-free objective\")\n",
" plot!(X, mu, ribbon=sigma*3, color=\"blue\", label=\"Surrogate function\")\n",
" scatter!(X_sample.X[:], Y_sample, markershape=:x, color=\"black\", markersize=5, label=\"Noisy sample\")\n",
" vline!(X_next, label=\"\")\n",
" \n",
" return plt\n",
"end\n",
"\n",
"function plot_acquisition(gp, X, ei, X_next; show_legend)\n",
" mg = marginals(gp(X_sample, 1e-9))\n",
" max_sample_mu = maximum(mean.(mg))\n",
" \n",
" ei_X = [ei([x], max_sample_mu, gp) for x in X]\n",
" \n",
" plt = plot(legend=show_legend, legendfontsize=4)\n",
" plot!(X, ei_X, label=\"Acquisition function\")\n",
" vline!(X_next, label=\"Next sampling location\")\n",
" \n",
" return plt\n",
"end"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
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" 1447.69,414.226 1450.94,413.823 1454.18,413.39 1457.42,412.925 1460.67,412.427 1463.91,411.896 1467.16,411.33 1470.4,410.729 1473.64,410.093 1476.89,409.42 \n",
" 1480.13,408.711 1483.38,407.965 1486.62,407.183 1489.86,406.364 1493.11,405.508 1496.35,404.616 1499.59,403.688 1502.84,402.723 1506.08,401.724 1509.33,400.689 \n",
" 1512.57,399.62 1515.81,398.518 1519.06,397.383 1522.3,396.216 1525.55,395.018 1528.79,393.789 1532.03,392.532 1535.28,391.246 1538.52,389.933 1541.77,388.594 \n",
" 1545.01,387.23 1548.25,385.842 1551.5,384.432 1554.74,383 1557.99,381.548 1561.23,380.077 1564.47,378.587 1567.72,377.081 1570.96,375.56 1574.21,374.024 \n",
" 1577.45,372.475 1580.69,370.914 1583.94,369.343 1587.18,367.762 1590.43,366.172 1593.67,364.576 1596.91,362.973 1600.16,361.365 1603.4,359.753 1606.65,358.139 \n",
" 1609.89,356.523 1613.13,354.907 1616.38,353.29 1619.62,351.676 1622.87,350.064 1626.11,348.455 1629.35,346.851 1632.6,345.253 1635.84,343.661 1639.09,342.076 \n",
" 1642.33,340.5 1645.57,338.933 1648.82,337.376 1652.06,335.83 1655.3,334.295 1658.55,332.774 1661.79,331.265 1665.04,329.771 1668.28,328.292 1671.52,326.829 \n",
" 1674.77,325.382 1678.01,323.953 1681.26,322.541 1684.5,321.148 1687.74,319.774 1690.99,318.42 1694.23,317.087 1697.48,315.775 1700.72,314.485 1703.96,313.217 \n",
" 1707.21,311.973 1710.45,310.752 1713.7,309.555 1716.94,308.383 1720.18,307.236 1723.43,306.115 1726.67,305.02 1729.92,303.951 1733.16,302.91 1736.4,301.897 \n",
" 1739.65,300.912 1742.89,299.955 1746.14,299.027 1749.38,298.129 1752.62,297.26 1755.87,296.421 1759.11,295.613 1762.36,294.835 1765.6,294.089 1768.84,293.374 \n",
" 1772.09,292.69 1775.33,292.039 1778.58,291.419 1781.82,290.833 1785.06,290.278 1788.31,289.757 1791.55,289.269 1794.8,288.814 1798.04,288.392 1801.28,288.004 \n",
" 1804.53,287.649 1807.77,287.328 1811.01,287.041 1814.26,286.787 1817.5,286.568 1820.75,286.382 1823.99,286.229 1827.23,286.111 1830.48,286.026 1833.72,285.974 \n",
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" 1901.85,291.989 1905.09,292.56 1908.33,293.148 1911.58,293.753 1914.82,294.373 1918.07,295.006 1921.31,295.651 1924.55,296.306 1927.8,296.968 1931.04,297.637 \n",
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" 1350.37,2786.47 1353.62,2787.18 1356.86,2787.79 1360.1,2788.31 1363.35,2788.74 1366.59,2789.08 1369.84,2789.33 1373.08,2789.48 1376.32,2789.54 1379.57,2789.51 \n",
" 1382.81,2789.39 1386.06,2789.17 1389.3,2788.86 1392.54,2788.45 1395.79,2787.94 1399.03,2787.32 1402.28,2786.61 1405.52,2785.79 1408.76,2784.87 1412.01,2783.83 \n",
" 1415.25,2782.68 1418.5,2781.42 1421.74,2780.05 1424.98,2778.55 1428.23,2776.94 1431.47,2775.2 1434.72,2773.34 1437.96,2771.36 1441.2,2769.25 1444.45,2767.01 \n",
" 1447.69,2764.65 1450.94,2762.16 1454.18,2759.55 1457.42,2756.81 1460.67,2753.95 1463.91,2750.97 1467.16,2747.87 1470.4,2744.65 1473.64,2741.32 1476.89,2737.88 \n",
" 1480.13,2734.34 1483.38,2730.69 1486.62,2726.95 1489.86,2723.11 1493.11,2719.2 1496.35,2715.2 1499.59,2711.12 1502.84,2706.98 1506.08,2702.78 1509.33,2698.52 \n",
" 1512.57,2694.22 1515.81,2689.87 1519.06,2685.49 1522.3,2681.08 1525.55,2676.65 1528.79,2672.22 1532.03,2667.77 1535.28,2663.33 1538.52,2658.9 1541.77,2654.49 \n",
" 1545.01,2650.11 1548.25,2645.75 1551.5,2641.44 1554.74,2637.17 1557.99,2632.96 1561.23,2628.81 1564.47,2624.73 1567.72,2620.73 1570.96,2616.8 1574.21,2612.97 \n",
" 1577.45,2609.23 1580.69,2605.59 1583.94,2602.07 1587.18,2598.65 1590.43,2595.36 1593.67,2592.19 1596.91,2589.15 1600.16,2586.25 1603.4,2583.48 1606.65,2580.87 \n",
" 1609.89,2578.4 1613.13,2576.09 1616.38,2573.93 1619.62,2571.94 1622.87,2570.11 1626.11,2568.45 1629.35,2566.96 1632.6,2565.64 1635.84,2564.5 1639.09,2563.53 \n",
" 1642.33,2562.74 1645.57,2562.14 1648.82,2561.71 1652.06,2561.46 1655.3,2561.39 1658.55,2561.51 1661.79,2561.8 1665.04,2562.28 1668.28,2562.93 1671.52,2563.75 \n",
" 1674.77,2564.75 1678.01,2565.92 1681.26,2567.26 1684.5,2568.76 1687.74,2570.43 1690.99,2572.25 1694.23,2574.22 1697.48,2576.34 1700.72,2578.61 1703.96,2581.01 \n",
" 1707.21,2583.55 1710.45,2586.21 1713.7,2588.99 1716.94,2591.89 1720.18,2594.9 1723.43,2598.01 1726.67,2601.21 1729.92,2604.5 1733.16,2607.88 1736.4,2611.33 \n",
" 1739.65,2614.84 1742.89,2618.42 1746.14,2622.05 1749.38,2625.73 1752.62,2629.44 1755.87,2633.19 1759.11,2636.96 1762.36,2640.74 1765.6,2644.54 1768.84,2648.34 \n",
" 1772.09,2652.14 1775.33,2655.93 1778.58,2659.7 1781.82,2663.45 1785.06,2667.18 1788.31,2670.86 1791.55,2674.51 1794.8,2678.12 1798.04,2681.67 1801.28,2685.17 \n",
" 1804.53,2688.61 1807.77,2691.98 1811.01,2695.3 1814.26,2698.54 1817.5,2701.71 1820.75,2704.8 1823.99,2707.82 1827.23,2710.76 1830.48,2713.62 1833.72,2716.39 \n",
" 1836.97,2719.09 1840.21,2721.71 1843.45,2724.24 1846.7,2726.7 1849.94,2729.07 1853.19,2731.37 1856.43,2733.6 1859.67,2735.75 1862.92,2737.83 1866.16,2739.83 \n",
" 1869.41,2741.78 1872.65,2743.65 1875.89,2745.47 1879.14,2747.22 1882.38,2748.91 1885.63,2750.55 1888.87,2752.14 1892.11,2753.67 1895.36,2755.16 1898.6,2756.59 \n",
" 1901.85,2757.98 1905.09,2759.33 1908.33,2760.64 1911.58,2761.9 1914.82,2763.12 1918.07,2764.31 1921.31,2765.46 1924.55,2766.57 1927.8,2767.65 1931.04,2768.69 \n",
" 1934.29,2769.7 1937.53,2770.68 1940.77,2771.63 1944.02,2772.55 1947.26,2773.44 1950.51,2774.3 1953.75,2775.14 1956.99,2775.95 1960.24,2776.74 1963.48,2777.51 \n",
" 1966.72,2778.26 1969.97,2778.98 1973.21,2779.69 1976.46,2780.39 1979.7,2781.07 1982.94,2781.74 1986.19,2782.4 1989.43,2783.05 1992.68,2783.7 1995.92,2784.34 \n",
" 1999.16,2784.98 2002.41,2785.61 2005.65,2786.25 2008.9,2786.89 2012.14,2787.53 2015.38,2788.17 2018.63,2788.82 2021.87,2789.47 2025.12,2790.13 2028.36,2790.8 \n",
" 2031.6,2791.48 2034.85,2792.16 2038.09,2792.85 2041.34,2793.55 2044.58,2794.25 2047.82,2794.96 2051.07,2795.68 2054.31,2796.41 2057.56,2797.14 2060.8,2797.87 \n",
" 2064.04,2798.61 2067.29,2799.34 2070.53,2800.08 2073.78,2800.82 2077.02,2801.55 2080.26,2802.28 2083.51,2803.01 2086.75,2803.72 2090,2804.43 2093.24,2805.13 \n",
" 2096.48,2805.81 2099.73,2806.49 2102.97,2807.14 2106.22,2807.79 2109.46,2808.41 2112.7,2809.01 2115.95,2809.6 2119.19,2810.16 2122.43,2810.7 2125.68,2811.22 \n",
" 2128.92,2811.72 2132.17,2812.19 2135.41,2812.64 2138.65,2813.06 2141.9,2813.46 2145.14,2813.84 2148.39,2814.19 2151.63,2814.52 2154.87,2814.82 2158.12,2815.1 \n",
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