KL-minimax game

KL-minimax game.ipynb

{
  "cells": [
    {
      "metadata": {
        "scrolled": false,
        "trusted": true
      },
      "cell_type": "code",
      "source": "using Distributions\nusing Random\nusing QuadGK\nusing Memoize\nusing Optim\nusing Base.Threads\nusing Distributed\n\nusing Plots\ndefault(lw=1.2)\n\n⪆(x, y) = x > y || x ≈ y\n\nxlogy(x, y) = iszero(x) ? x : x*log(y)\nge(q, p) = -(xlogy(q, p) + xlogy(1-q, 1-p))\nkl(q, p) = ge(q, p) - ge(q, q)\nloss_kl(n, q, p) = kl(q, p)\n\nest_conjprior(n, k, a, b=a) = (k+a)/(n+a+b)\nest_jeffreys(n, k) = est_conjprior(n, k, 0.5)\nest_corrected_jeffreys(n, k) = est_conjprior(n, k, 0.51)\n\nfunction est_corrected_mle(n, k)\n    k == 0 ? 1/(2n) :\n    k == n ? 1 - 1/(2n) :\n    k/n\nend\n\nfunction Eloss(n, q, estfunc, lossfunc)\n    sum(lossfunc(n, q, estfunc(n, k))*pdf(Binomial(n, q), k) for k in 0:n)\nend\n\n@memoize function maximize_Eloss(n, estfunc, lossfunc)\n    o0 = optimize(q -> -Eloss(n, q, estfunc, lossfunc), eps(), 0.5)\n    o1 = optimize(q -> -Eloss(n, q, estfunc, lossfunc), 0.5, 1-eps())\n    mo0 = -o0.minimum\n    mo1 = -o1.minimum\n    m0 = Eloss(n, 0.0, estfunc, lossfunc)\n    m1 = Eloss(n, 1.0, estfunc, lossfunc)\n    maxEloss = max(mo0, mo1, m0, m1)\n    q_maximizer = mo0 ⪆ maxEloss ? o0.minimizer : \n        mo1 ⪆ maxEloss ? o1.minimizer :\n        m0 ⪆ maxEloss ? 0.0 : 1.0 \n    q_maximizer, maxEloss\nend\n\nfunction est_asymptbest(n, k, c)\n    k == 0   ? est_conjprior(n, k, 3/4-c, 3/4) :\n    k == 1   ? est_conjprior(n, k, 3/4+c, 3/4) :\n    k == n-1 ? est_conjprior(n, k, 3/4, 3/4+c) :\n    k == n   ? est_conjprior(n, k, 3/4, 3/4-c) :\n    est_conjprior(n, k, 3/4, 3/4)\nend\n\nest_asymptbest(n, k) = est_asymptbest(n, k, 1/4)\n\n@memoize function est_corrected_asymptbest(n, k)\n    c = minmax_param(n)[1]\n    est_asymptbest(n, k, c)\nend\n\nfunction est_corrected_asymptbest(n)\n    c = minmax_param(n)[1]\n    est(n, k) = est_asymptbest(n, k, c)\n    est\nend\n\nest_optimal(n, k) = (\n 0.04251155945436709,\n 0.17905246094661803,\n 0.22817364126399065,\n 0.32113227619309637,\n 0.42276759076040954,\n 0.5,\n 0.5772324092395904,\n 0.6788677238069036,\n 0.7718263587360094,\n 0.820947539053382,\n 0.9574884405456329,     \n)[1+k]\n\n@memoize function minmax_param(n)\n    maxEloss(c) = maximize_Eloss(n, (n, k)->est_asymptbest(n, k, c), loss_kl)[2]\n    o = optimize(maxEloss, 0.01, 0.74)\n    o.minimizer, o.minimum\nend\n\n@show c, _ = minmax_param(10)\n@show maximize_Eloss(10, est_optimal, loss_kl)\n@show maximize_Eloss(10, est_corrected_asymptbest, loss_kl)\n@show maximize_Eloss(10, est_asymptbest, loss_kl)\n@show maximize_Eloss(10, est_corrected_jeffreys, loss_kl)\n@show maximize_Eloss(10, est_jeffreys, loss_kl)\n@show maximize_Eloss(10, est_corrected_mle, loss_kl)\n;",
      "execution_count": 1,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(c, _) = minmax_param(10) = (0.26448095780701575, 0.044174348921259446)\nmaximize_Eloss(10, est_optimal, loss_kl) = (0.15822440677808142, 0.04344217318800149)\nmaximize_Eloss(10, est_corrected_asymptbest, loss_kl) = (0.1631009148142917, 0.044174348921259446)\nmaximize_Eloss(10, est_asymptbest, loss_kl) = (0.0, 0.04546237407675729)\nmaximize_Eloss(10, est_corrected_jeffreys, loss_kl) = (0.41513956692681325, 0.04789857741650475)\nmaximize_Eloss(10, est_jeffreys, loss_kl) = (0.4013569799241886, 0.048128342454167854)\nmaximize_Eloss(10, est_corrected_mle, loss_kl) = (0.43013229806950265, 0.060054858078406825)\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "plot(; legend=:bottom, ytick=0:0.002:1)\nplot!(q -> Eloss(10, q, est_optimal, loss_kl), 0, 1; label=\"optimal\")\nplot!(q -> Eloss(10, q, est_corrected_asymptbest, loss_kl), 0, 1; label=\"corrected asymptbest\", ls=:auto)\nplot!(q -> Eloss(10, q, est_asymptbest, loss_kl), 0, 1; label=\"asymptbest\", ls=:auto)\nplot!(q -> Eloss(10, q, est_corrected_jeffreys, loss_kl), 0, 1; label=\"corrected_jeffreys\", ls=:auto)\nplot!(q -> Eloss(10, q, est_jeffreys, loss_kl), 0, 1; label=\"jeffreys\", ls=:auto)\nplot!(q -> Eloss(10, q, est_corrected_mle, loss_kl), 0, 1; label=\"corrected_mle\", ls=:auto)",
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 2,
          "data": {
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        {
          "output_type": "execute_result",
          "execution_count": 8,
          "data": {
            "text/plain": "11-element Vector{Float64}:\n 0.05\n 0.1\n 0.2\n 0.3\n 0.4\n 0.5\n 0.6\n 0.7\n 0.8\n 0.9\n 0.95"
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      "source": "function random_loss!(loss = zeros(1000);\n        estfunc = est_jeffreys, \n        maxlossfunc = maximize_Eloss, \n        lossfunc = loss_kl, \n        n = 10, \n        rng = Random.default_rng()\n    )\n    q = maxlossfunc(n, estfunc, lossfunc)[1]\n    bin = Binomial(n, q)\n    for l in eachindex(loss)\n        k = rand(rng, bin)\n        p = estfunc(n, k)\n        loss[l] = lossfunc(n, q, p)\n    end\n    loss\nend\n\nfunction simulate_loss(;\n        estfunc = est_jeffreys, \n        maxlossfunc = maximize_Eloss, \n        lossfunc = loss_kl, \n        n = 10, \n        L = 1000, \n        niters = 10^5,\n        kwargs...\n    )\n    loss = zeros(niters)\n    ranges = Distributed.splitrange(1, niters, nthreads())\n    @threads for ran in ranges\n        rng = Random.default_rng()\n        losstmp = zeros(L)\n        for i in ran\n            random_loss!(losstmp; estfunc, maxlossfunc, lossfunc, n, rng)\n            loss[i] = sum(losstmp)\n        end\n    end\n    m, s = mean(loss), std(loss)\n    (μ = m, σ = s) |> println\n    histogram(loss; norm=true, alpha=0.3, label=\"loss\")\n    plot!(; xtick=-200:200, ytick=0:0.01:1, tickfontsize=6)\n    title!(\"Player($estfunc)\"; titlefontsize=11)\n    vline!([m-2s, m, m+2s]; label=\"μ ± 2σ\")\n    plot!(; kwargs...)\nend\n\nfunction simulate_games(;\n        estfunc0 = est_jeffreys, \n        estfunc1 = est_asymptbest, \n        maxlossfunc = maximize_Eloss, \n        lossfunc = loss_kl, \n        n = 10, \n        L = 1000, \n        niters = 10^5,\n        kwargs...\n    )\n    score = zeros(niters)\n    ranges = Distributed.splitrange(1, niters, nthreads())\n    @threads for ran in ranges\n        rng = Random.default_rng()\n        losstmp0 = zeros(L)\n        losstmp1 = zeros(L)\n        for i in ran\n            random_loss!(losstmp0; estfunc=estfunc0, maxlossfunc, lossfunc, n, rng)\n            random_loss!(losstmp1; estfunc=estfunc1, maxlossfunc, lossfunc, n, rng)\n            score[i] = sum(losstmp0) - sum(losstmp1)\n        end\n    end\n    m, s, pow = mean(score), std(score), mean(>(0), score)\n    (μ = m, σ = s, prob_of_win = pow) |> println\n    histogram(score; norm=true, alpha=0.3, label=\"score\")\n    plot!(; xtick=-200:200, ytick=0:0.01:1, tickfontsize=6)\n    title!(\"Player($estfunc0) vs. Player($estfunc1)\"; titlefontsize=11)\n    vline!([m-2s, m, m+2s]; label=\"μ ± 2σ\")\n    vline!([0]; label=\"\", color=:black, ls=:dash, lw=1.5)\n    plot!(; kwargs...)\nend",
      "execution_count": 9,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 9,
          "data": {
            "text/plain": "simulate_games (generic function with 1 method)"
          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time loss = simulate_loss(; estfunc=est_corrected_mle)",
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 60.046249243119455, σ = 2.9842570142229152)\n  5.670381 seconds (4.44 M allocations: 265.184 MiB, 1.50% gc time, 10.82% compilation time)\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 10,
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          "metadata": {}
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    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time loss = simulate_loss(; estfunc=est_jeffreys)",
      "execution_count": 11,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 48.13141589455698, σ = 2.4346588097571544)\n  3.395510 seconds (195.02 k allocations: 10.301 MiB, 2.18% compilation time)\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 11,
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    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time simulate_loss(; estfunc=est_corrected_jeffreys)",
      "execution_count": 12,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 47.89219416971975, σ = 2.429885336073135)\n  3.417961 seconds (195.07 k allocations: 10.304 MiB, 2.15% compilation time)\n",
          "name": "stdout"
        },
        {
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          },
          "metadata": {}
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time simulate_loss(; estfunc=est_asymptbest);",
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 45.46237407675723, σ = 1.421092577000849e-14)\n  1.631957 seconds (193.42 k allocations: 10.223 MiB, 4.55% compilation time)\n",
          "name": "stdout"
        }
      ]
    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time simulate_loss(; estfunc=est_corrected_asymptbest(10))",
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 44.17270008612336, σ = 1.6913369336917667)\n  3.252354 seconds (958.71 k allocations: 58.388 MiB, 0.00% compilation time)\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
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    },
    {
      "metadata": {
        "trusted": true
      },
      "cell_type": "code",
      "source": "@time simulate_loss(; estfunc=est_optimal)",
      "execution_count": 15,
      "outputs": [
        {
          "output_type": "stream",
          "text": "(μ = 43.45063130106135, σ = 1.6880304292528594)\n  2.928236 seconds (195.31 k allocations: 10.304 MiB, 1.10% gc time, 3.60% compilation time)\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 15,
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