assets/tilted_losses_variational.ipynb

{
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "view-in-github",
        "colab_type": "text"
      },
      "source": [
        "<a href=\"https://colab.research.google.com/github/alexshtf/alexshtf.github.io/blob/master/assets/tilted_losses_variational.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "cHCIbV_H14vw"
      },
      "source": [
        "# Syntetic dataset"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 35,
      "metadata": {
        "id": "dqeMILNjC4XU"
      },
      "outputs": [],
      "source": [
        "import numpy as np"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 36,
      "metadata": {
        "id": "D3iJpVHfDFVc"
      },
      "outputs": [],
      "source": [
        "true_w = np.array([0.8, -1])\n",
        "noise_strength = 0.3\n",
        "n = 100\n",
        "\n",
        "np.random.seed(42)\n",
        "x = np.stack([np.random.randn(n), np.ones(n)], axis=-1)\n",
        "y = x @ true_w + noise_strength * np.random.standard_t(df=3, size=n)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 37,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 434
        },
        "id": "U0TrdF1YDRDP",
        "outputId": "b4bab5b7-3e1b-425b-cc68-3adf1698a1d2"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import matplotlib.pyplot as plt\n",
        "\n",
        "plt.scatter(x[:, 0], y)\n",
        "plt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "he_bA4Q5LzvT"
      },
      "source": [
        "## Try stripped reformulation"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 38,
      "metadata": {
        "id": "H-bx4YczMcye"
      },
      "outputs": [],
      "source": [
        "class StrippedTiltedLoss:\n",
        "  def __init__(self, underlying_loss, t):\n",
        "    super().__init__()\n",
        "    self.underlying_loss = underlying_loss\n",
        "    self.t = t\n",
        "\n",
        "  def __call__(self, pred, target):\n",
        "    return torch.exp(self.t * self.underlying_loss(pred, target))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 39,
      "metadata": {
        "id": "mO03-7tcLy8Y"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "\n",
        "def pytorch_fit(x, y, criterion, make_optim_fn, n_epochs=100):\n",
        "  # convert numpy arrays to torch tensors\n",
        "  x = torch.as_tensor(x)\n",
        "  y = torch.as_tensor(y)\n",
        "\n",
        "  # define initial w to be the zero vector\n",
        "  w_fit = torch.nn.Parameter(torch.zeros_like(x[0]))\n",
        "\n",
        "  # create optimizer\n",
        "  optim = make_optim_fn([w_fit])\n",
        "\n",
        "  # regular PyTorch training loop.\n",
        "  for epoch in range(n_epochs):\n",
        "    for xi, yi in zip(x, y):\n",
        "      pred = torch.dot(xi, w_fit)\n",
        "      loss = criterion(pred, yi)\n",
        "\n",
        "      optim.zero_grad()\n",
        "      loss.backward()\n",
        "      optim.step()\n",
        "\n",
        "  return w_fit.detach()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 40,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iji6s_IsMUxj",
        "outputId": "d36b3b6c-6c5d-4454-e98e-b5935747f165"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor([nan, nan], dtype=torch.float64)"
            ]
          },
          "metadata": {},
          "execution_count": 40
        }
      ],
      "source": [
        "from torch.nn import MSELoss\n",
        "\n",
        "pytorch_fit(x, y,\n",
        "            criterion=StrippedTiltedLoss(MSELoss(), t=1),\n",
        "            make_optim_fn=lambda params: torch.optim.SGD(params, lr=1e-4))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 41,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "-XmD6U19RoQU",
        "outputId": "45171410-45f0-494a-aad8-e59ba58b9215"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "tensor([ 0.4708, -0.3032], dtype=torch.float64)"
            ]
          },
          "metadata": {},
          "execution_count": 41
        }
      ],
      "source": [
        "pytorch_fit(x, y,\n",
        "            criterion=StrippedTiltedLoss(MSELoss(), t=1),\n",
        "            make_optim_fn=lambda params: torch.optim.SGD(params, lr=1e-6))"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 42,
      "metadata": {
        "id": "G7TzfFF-NvLe"
      },
      "outputs": [],
      "source": [
        "import math\n",
        "\n",
        "def compute_tilted_loss(w_fit, x, y, t):\n",
        "  x = torch.as_tensor(x)\n",
        "  y = torch.as_tensor(y)\n",
        "  n = x.shape[0]\n",
        "  return torch.logsumexp(t * torch.square(x @ w_fit - y) - math.log(n), dim=-1) / t"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 43,
      "metadata": {
        "id": "50i39vCAN4o7"
      },
      "outputs": [],
      "source": [
        "from tqdm.auto import tqdm\n",
        "from functools import partial\n",
        "\n",
        "def eval_sgd_exponential_loss(x, y, lrs, t=1):\n",
        "  losses = []\n",
        "  for lr in tqdm(lrs):\n",
        "      optim_factory = partial(torch.optim.SGD, lr=lr)\n",
        "      w_exp_fit = pytorch_fit(x, y, StrippedTiltedLoss(MSELoss(), t), optim_factory)\n",
        "      w_exp_loss = compute_tilted_loss(w_exp_fit, x, y, t).item()\n",
        "      losses.append(w_exp_loss)\n",
        "\n",
        "  return losses"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 44,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "23a14dd894fe4d54ad26a0b4e39f9cea",
            "d6c918bf55c9451a8b3b5241befe6fb2",
            "9393e44a95d747b9b7255cbf8363f475",
            "785e50beb38148d79febddd97b6d06c8",
            "7207f14369f5441c89a642315e6434dd",
            "b029b4a5a6c14cea8e8f0efb5d2d728d",
            "f7fd312e90424e0dbec3d83e8aecc829",
            "4b68b0269e994503912f1946cf481d8d",
            "265d99227e604142b74cc1eabf61972a",
            "288456f879194c06ae70a3f3c6f4ffc3",
            "5d3ca235c18949d29b8437b09bec9971"
          ]
        },
        "id": "8IiYZCWGOXNT",
        "outputId": "960fe045-6575-4942-80d7-4e3171031e4d"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/60 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "23a14dd894fe4d54ad26a0b4e39f9cea"
            }
          },
          "metadata": {}
        }
      ],
      "source": [
        "lrs = np.geomspace(1e-7, 1e-4, 60).tolist()\n",
        "losses = eval_sgd_exponential_loss(x, y, lrs)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 45,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 455
        },
        "id": "vUDC74VvO5eN",
        "outputId": "fc070b73-b0e7-4044-afa5-7e929ce0875a"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "(1e-07, 0.0001)"
            ]
          },
          "metadata": {},
          "execution_count": 45
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "plt.plot(lrs, losses)\n",
        "plt.xscale('log')\n",
        "plt.yscale('log')\n",
        "plt.xlim([np.min(lrs), np.max(lrs)])"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "L_e9acIDJH9O"
      },
      "source": [
        "## Fit with PyTorch"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 53,
      "metadata": {
        "id": "cGz6iD5Ot1rD"
      },
      "outputs": [],
      "source": [
        "import torch\n",
        "import math"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 54,
      "metadata": {
        "id": "7XbwDZaDcZe0"
      },
      "outputs": [],
      "source": [
        "from itertools import chain\n",
        "\n",
        "def pytorch_fit(x, y, criterion, make_optim_fn, n_epochs=100):\n",
        "  # convert numpy arrays to torch tensors\n",
        "  x = torch.as_tensor(x)\n",
        "  y = torch.as_tensor(y)\n",
        "  dim = x.shape[1]\n",
        "  dtype = x.dtype\n",
        "\n",
        "  # define initial w to be the zero vector\n",
        "  w_fit = torch.nn.Parameter(torch.zeros(dim, dtype=dtype))\n",
        "  criterion.initialize(x @ w_fit, y)\n",
        "\n",
        "  # create optimizer\n",
        "  parameters_to_learn = chain(criterion.parameters(), [w_fit])\n",
        "  optim = make_optim_fn(parameters_to_learn)\n",
        "\n",
        "  # regular PyTorch training loop.\n",
        "  for epoch in range(n_epochs):\n",
        "    for xi, yi in zip(x, y):\n",
        "      pred = torch.dot(xi, w_fit)\n",
        "      loss = criterion(pred, yi)\n",
        "\n",
        "      optim.zero_grad()\n",
        "      loss.backward()\n",
        "      optim.step()\n",
        "\n",
        "  return w_fit.detach()"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 55,
      "metadata": {
        "id": "R92uPgB_oVCj"
      },
      "outputs": [],
      "source": [
        "class StrippedTiltedLoss(torch.nn.Module):\n",
        "  def __init__(self, underlying_loss, t):\n",
        "    super().__init__()\n",
        "    self.underlying_loss = underlying_loss\n",
        "    self.t = t\n",
        "\n",
        "  def initialize(self, x, y):\n",
        "    pass\n",
        "\n",
        "  def forward(self, pred, target):\n",
        "    return torch.exp(self.t * self.underlying_loss(pred, target))\n",
        "\n",
        "\n",
        "class VariationalTiltedLoss(torch.nn.Module):\n",
        "  def __init__(self, underlying_loss, t):\n",
        "    super().__init__()\n",
        "    self.t = t\n",
        "    self.underlying_loss = underlying_loss\n",
        "    self.v = torch.nn.Parameter(torch.tensor(0.))\n",
        "\n",
        "  def initialize(self, preds, targets):\n",
        "    with torch.no_grad():\n",
        "      init_losses = self.underlying_loss(preds, targets)\n",
        "      n = init_losses.shape[0]\n",
        "\n",
        "      v_init = -torch.logsumexp(self.t * init_losses - math.log(n), dim=-1)\n",
        "      self.v.set_(v_init.to(self.v.dtype))\n",
        "\n",
        "  def forward(self, pred, target):\n",
        "    sample_loss = self.underlying_loss(pred, target)\n",
        "    return torch.exp(self.t * sample_loss + self.v) - self.v"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 56,
      "metadata": {
        "id": "X1CkbusnuBGy"
      },
      "outputs": [],
      "source": [
        "from functools import partial\n",
        "from itertools import product\n",
        "import pandas as pd\n",
        "\n",
        "def compare_tilted_formulations(x, y, ts, lrs, optim_ctor):\n",
        "  records = []\n",
        "  for t, lr in tqdm(list(product(ts, lrs))):\n",
        "      make_optim_fn = partial(optim_ctor, lr=lr)\n",
        "\n",
        "      mse_loss = MSELoss(reduction='none')\n",
        "      w_exp_fit = pytorch_fit(x, y, StrippedTiltedLoss(mse_loss, t), make_optim_fn)\n",
        "      w_tilted_fit = pytorch_fit(x, y, VariationalTiltedLoss(mse_loss, t), make_optim_fn)\n",
        "\n",
        "      w_exp_loss = compute_tilted_loss(w_exp_fit, x, y, t).item()\n",
        "      w_tilted_loss = compute_tilted_loss(w_tilted_fit, x, y, t).item()\n",
        "      records.append(dict(t=t, lr=lr, loss_type='stripped', value=w_exp_loss))\n",
        "      records.append(dict(t=t, lr=lr, loss_type='variational', value=w_tilted_loss))\n",
        "\n",
        "  return pd.DataFrame.from_records(records)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 57,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "4730bb605300408481693d26c4fe84a8",
            "f2c397f1d38d4e21b48178b5297f44f1",
            "4e5fba78abf245458f75b229f8afa9a4",
            "ea9d6bc884414a06969c05b4fcb5b6f4",
            "ed738d58308e460d8610b7729b5e12a3",
            "4c0c5c3f6f6b41199f1c82e651b00b36",
            "9404a89014bd4b5bbb73d3cddad8dfd1",
            "fa92ec9959d94657bf4462b107f311dc",
            "4d454580b90942789dc2c35ec082fb0e",
            "176c4b7d90de425c875c68473d674cca",
            "c388c8da409140f395b9d8b26902ef28"
          ]
        },
        "id": "vndlS5FlywpZ",
        "outputId": "fd1bf20c-50d0-4cbf-8cef-65b81abb820d"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/90 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "4730bb605300408481693d26c4fe84a8"
            }
          },
          "metadata": {}
        }
      ],
      "source": [
        "sgd_lrs = np.geomspace(1e-7, 1e-1, 30).tolist()\n",
        "ts = [0.5, 1, 2]\n",
        "\n",
        "sgd_eval_recs = compare_tilted_formulations(x, y, ts=ts, lrs=sgd_lrs, optim_ctor=torch.optim.SGD)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 58,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 521
        },
        "id": "1Org8MPF1u_x",
        "outputId": "150c27c1-7d44-476b-8187-06bb8124fa7e"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<seaborn.axisgrid.FacetGrid at 0x7c31e63cc610>"
            ]
          },
          "metadata": {},
          "execution_count": 58
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1636.25x500 with 3 Axes>"
            ],
            "image/png": 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C\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import seaborn as sns\n",
        "sns.set()\n",
        "\n",
        "g = sns.relplot(data=sgd_eval_recs,\n",
        "                hue='loss_type', col='t', x='lr', y='value', alpha=0.5)\n",
        "g.set(xscale='log')\n",
        "g.set(ylim=[0, 10])\n",
        "g.set(yscale='asinh')"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 59,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 49,
          "referenced_widgets": [
            "378fd4040af9498c9394aece86dac00e",
            "898d61f03e74491692efe98c7e22d6c7",
            "980a7837b7b147c8990eb96fb76d6e71",
            "c5b7d32d66434da28bfbbf1d0aced781",
            "0d44fc1ced164b24a9f2d575c70d7763",
            "47cf7b1759d0400eb53523f26b2be8bf",
            "3b4b45ea022249b0941da3e515effa48",
            "9596dd574a8e4663b8fe4ead0c4efa18",
            "99bdfb67294745ce8d12c5508e4dc3c4",
            "1c276efd171d4eb5a265d2807b0e32f4",
            "1a4a7b5ae0ca437a9880e97584ce304f"
          ]
        },
        "id": "p96k_UFfePXi",
        "outputId": "5d96eccc-75d4-4eb6-f5f4-2e6e94cdc9e8"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "  0%|          | 0/90 [00:00<?, ?it/s]"
            ],
            "application/vnd.jupyter.widget-view+json": {
              "version_major": 2,
              "version_minor": 0,
              "model_id": "378fd4040af9498c9394aece86dac00e"
            }
          },
          "metadata": {}
        }
      ],
      "source": [
        "adam_lrs = np.geomspace(1e-5, 1e2, 30)\n",
        "adam_eval_recs = compare_tilted_formulations(x, y, ts=ts, lrs=adam_lrs, optim_ctor=torch.optim.Adam)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 60,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 521
        },
        "id": "6uS9Nb5U6jaF",
        "outputId": "ce008ab7-a989-4969-aba7-c25af7399565"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<seaborn.axisgrid.FacetGrid at 0x7c31e4957d30>"
            ]
          },
          "metadata": {},
          "execution_count": 60
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 1636.25x500 with 3 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ],
      "source": [
        "g = sns.relplot(data=adam_eval_recs,\n",
        "                hue='loss_type', col='t', x='lr', y='value', alpha=0.5)\n",
        "g.set(xscale='log')\n",
        "g.set(ylim=[0, 10])\n",
        "g.set(yscale='asinh')"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 61,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "6ZJ-QsC_-FBu",
        "outputId": "c2406edf-5831-4a6c-a5e1-8c595610d06a"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                    value\n",
              "loss_type   t            \n",
              "stripped    0.5  0.179521\n",
              "            1.0  0.317836\n",
              "            2.0       NaN\n",
              "variational 0.5  0.179521\n",
              "            1.0  0.209373\n",
              "            2.0  0.291620"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-b64fc259-0b82-495b-a78e-f5a97d77cee5\" class=\"colab-df-container\">\n",
              "    <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th>value</th>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>loss_type</th>\n",
              "      <th>t</th>\n",
              "      <th></th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th rowspan=\"3\" valign=\"top\">stripped</th>\n",
              "      <th>0.5</th>\n",
              "      <td>0.179521</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1.0</th>\n",
              "      <td>0.317836</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2.0</th>\n",
              "      <td>NaN</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th rowspan=\"3\" valign=\"top\">variational</th>\n",
              "      <th>0.5</th>\n",
              "      <td>0.179521</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1.0</th>\n",
              "      <td>0.209373</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2.0</th>\n",
              "      <td>0.291620</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "    <div class=\"colab-df-buttons\">\n",
              "\n",
              "  <div class=\"colab-df-container\">\n",
              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-b64fc259-0b82-495b-a78e-f5a97d77cee5')\"\n",
              "            title=\"Convert this dataframe to an interactive table.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
              "    <path d=\"M120-120v-720h720v720H120Zm60-500h600v-160H180v160Zm220 220h160v-160H400v160Zm0 220h160v-160H400v160ZM180-400h160v-160H180v160Zm440 0h160v-160H620v160ZM180-180h160v-160H180v160Zm440 0h160v-160H620v160Z\"/>\n",
              "  </svg>\n",
              "    </button>\n",
              "\n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    .colab-df-buttons div {\n",
              "      margin-bottom: 4px;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-b64fc259-0b82-495b-a78e-f5a97d77cee5 button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-b64fc259-0b82-495b-a78e-f5a97d77cee5');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "\n",
              "<div id=\"df-2241b4e1-a955-4475-b3f7-dce8aa8723cd\">\n",
              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-2241b4e1-a955-4475-b3f7-dce8aa8723cd')\"\n",
              "            title=\"Suggest charts\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "     width=\"24px\">\n",
              "    <g>\n",
              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    80% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "    90% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "    }\n",
              "  }\n",
              "</style>\n",
              "\n",
              "  <script>\n",
              "    async function quickchart(key) {\n",
              "      const quickchartButtonEl =\n",
              "        document.querySelector('#' + key + ' button');\n",
              "      quickchartButtonEl.disabled = true;  // To prevent multiple clicks.\n",
              "      quickchartButtonEl.classList.add('colab-df-spinner');\n",
              "      try {\n",
              "        const charts = await google.colab.kernel.invokeFunction(\n",
              "            'suggestCharts', [key], {});\n",
              "      } catch (error) {\n",
              "        console.error('Error during call to suggestCharts:', error);\n",
              "      }\n",
              "      quickchartButtonEl.classList.remove('colab-df-spinner');\n",
              "      quickchartButtonEl.classList.add('colab-df-quickchart-complete');\n",
              "    }\n",
              "    (() => {\n",
              "      let quickchartButtonEl =\n",
              "        document.querySelector('#df-2241b4e1-a955-4475-b3f7-dce8aa8723cd button');\n",
              "      quickchartButtonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "    })();\n",
              "  </script>\n",
              "</div>\n",
              "    </div>\n",
              "  </div>\n"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "dataframe",
              "summary": "{\n  \"name\": \"sgd_eval_recs\",\n  \"rows\": 6,\n  \"fields\": [\n    {\n      \"column\": \"value\",\n      \"properties\": {\n        \"dtype\": \"number\",\n        \"std\": 0.06495855880922873,\n        \"min\": 0.1795205789159846,\n        \"max\": 0.3178356659277082,\n        \"num_unique_values\": 5,\n        \"samples\": [\n          0.3178356659277082,\n          0.2916196308659833,\n          0.17952134959645516\n        ],\n        \"semantic_type\": \"\",\n        \"description\": \"\"\n      }\n    }\n  ]\n}"
            }
          },
          "metadata": {},
          "execution_count": 61
        }
      ],
      "source": [
        "sgd_eval_recs.groupby(['loss_type', 't']).agg({'value': 'min'})"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 62,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 269
        },
        "id": "2GzEfs2o-Qsd",
        "outputId": "5524bc71-f6a8-4e2d-9962-f8b98c06c4aa"
      },
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "                    value\n",
              "loss_type   t            \n",
              "stripped    0.5  0.179530\n",
              "            1.0  0.209426\n",
              "            2.0  0.293253\n",
              "variational 0.5  0.179524\n",
              "            1.0  0.209382\n",
              "            2.0  0.291761"
            ],
            "text/html": [
              "\n",
              "  <div id=\"df-5b3d72aa-7b9b-4f90-9c46-948ae1dd1664\" class=\"colab-df-container\">\n",
              "    <div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th></th>\n",
              "      <th>value</th>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>loss_type</th>\n",
              "      <th>t</th>\n",
              "      <th></th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th rowspan=\"3\" valign=\"top\">stripped</th>\n",
              "      <th>0.5</th>\n",
              "      <td>0.179530</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1.0</th>\n",
              "      <td>0.209426</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2.0</th>\n",
              "      <td>0.293253</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th rowspan=\"3\" valign=\"top\">variational</th>\n",
              "      <th>0.5</th>\n",
              "      <td>0.179524</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1.0</th>\n",
              "      <td>0.209382</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2.0</th>\n",
              "      <td>0.291761</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>\n",
              "    <div class=\"colab-df-buttons\">\n",
              "\n",
              "  <div class=\"colab-df-container\">\n",
              "    <button class=\"colab-df-convert\" onclick=\"convertToInteractive('df-5b3d72aa-7b9b-4f90-9c46-948ae1dd1664')\"\n",
              "            title=\"Convert this dataframe to an interactive table.\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "  <svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\" viewBox=\"0 -960 960 960\">\n",
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              "  </svg>\n",
              "    </button>\n",
              "\n",
              "  <style>\n",
              "    .colab-df-container {\n",
              "      display:flex;\n",
              "      gap: 12px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert {\n",
              "      background-color: #E8F0FE;\n",
              "      border: none;\n",
              "      border-radius: 50%;\n",
              "      cursor: pointer;\n",
              "      display: none;\n",
              "      fill: #1967D2;\n",
              "      height: 32px;\n",
              "      padding: 0 0 0 0;\n",
              "      width: 32px;\n",
              "    }\n",
              "\n",
              "    .colab-df-convert:hover {\n",
              "      background-color: #E2EBFA;\n",
              "      box-shadow: 0px 1px 2px rgba(60, 64, 67, 0.3), 0px 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "      fill: #174EA6;\n",
              "    }\n",
              "\n",
              "    .colab-df-buttons div {\n",
              "      margin-bottom: 4px;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert {\n",
              "      background-color: #3B4455;\n",
              "      fill: #D2E3FC;\n",
              "    }\n",
              "\n",
              "    [theme=dark] .colab-df-convert:hover {\n",
              "      background-color: #434B5C;\n",
              "      box-shadow: 0px 1px 3px 1px rgba(0, 0, 0, 0.15);\n",
              "      filter: drop-shadow(0px 1px 2px rgba(0, 0, 0, 0.3));\n",
              "      fill: #FFFFFF;\n",
              "    }\n",
              "  </style>\n",
              "\n",
              "    <script>\n",
              "      const buttonEl =\n",
              "        document.querySelector('#df-5b3d72aa-7b9b-4f90-9c46-948ae1dd1664 button.colab-df-convert');\n",
              "      buttonEl.style.display =\n",
              "        google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "      async function convertToInteractive(key) {\n",
              "        const element = document.querySelector('#df-5b3d72aa-7b9b-4f90-9c46-948ae1dd1664');\n",
              "        const dataTable =\n",
              "          await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                    [key], {});\n",
              "        if (!dataTable) return;\n",
              "\n",
              "        const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "          '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "          + ' to learn more about interactive tables.';\n",
              "        element.innerHTML = '';\n",
              "        dataTable['output_type'] = 'display_data';\n",
              "        await google.colab.output.renderOutput(dataTable, element);\n",
              "        const docLink = document.createElement('div');\n",
              "        docLink.innerHTML = docLinkHtml;\n",
              "        element.appendChild(docLink);\n",
              "      }\n",
              "    </script>\n",
              "  </div>\n",
              "\n",
              "\n",
              "<div id=\"df-f5609d07-d7e4-4f05-949e-5a16d592b736\">\n",
              "  <button class=\"colab-df-quickchart\" onclick=\"quickchart('df-f5609d07-d7e4-4f05-949e-5a16d592b736')\"\n",
              "            title=\"Suggest charts\"\n",
              "            style=\"display:none;\">\n",
              "\n",
              "<svg xmlns=\"http://www.w3.org/2000/svg\" height=\"24px\"viewBox=\"0 0 24 24\"\n",
              "     width=\"24px\">\n",
              "    <g>\n",
              "        <path d=\"M19 3H5c-1.1 0-2 .9-2 2v14c0 1.1.9 2 2 2h14c1.1 0 2-.9 2-2V5c0-1.1-.9-2-2-2zM9 17H7v-7h2v7zm4 0h-2V7h2v10zm4 0h-2v-4h2v4z\"/>\n",
              "    </g>\n",
              "</svg>\n",
              "  </button>\n",
              "\n",
              "<style>\n",
              "  .colab-df-quickchart {\n",
              "      --bg-color: #E8F0FE;\n",
              "      --fill-color: #1967D2;\n",
              "      --hover-bg-color: #E2EBFA;\n",
              "      --hover-fill-color: #174EA6;\n",
              "      --disabled-fill-color: #AAA;\n",
              "      --disabled-bg-color: #DDD;\n",
              "  }\n",
              "\n",
              "  [theme=dark] .colab-df-quickchart {\n",
              "      --bg-color: #3B4455;\n",
              "      --fill-color: #D2E3FC;\n",
              "      --hover-bg-color: #434B5C;\n",
              "      --hover-fill-color: #FFFFFF;\n",
              "      --disabled-bg-color: #3B4455;\n",
              "      --disabled-fill-color: #666;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart {\n",
              "    background-color: var(--bg-color);\n",
              "    border: none;\n",
              "    border-radius: 50%;\n",
              "    cursor: pointer;\n",
              "    display: none;\n",
              "    fill: var(--fill-color);\n",
              "    height: 32px;\n",
              "    padding: 0;\n",
              "    width: 32px;\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart:hover {\n",
              "    background-color: var(--hover-bg-color);\n",
              "    box-shadow: 0 1px 2px rgba(60, 64, 67, 0.3), 0 1px 3px 1px rgba(60, 64, 67, 0.15);\n",
              "    fill: var(--button-hover-fill-color);\n",
              "  }\n",
              "\n",
              "  .colab-df-quickchart-complete:disabled,\n",
              "  .colab-df-quickchart-complete:disabled:hover {\n",
              "    background-color: var(--disabled-bg-color);\n",
              "    fill: var(--disabled-fill-color);\n",
              "    box-shadow: none;\n",
              "  }\n",
              "\n",
              "  .colab-df-spinner {\n",
              "    border: 2px solid var(--fill-color);\n",
              "    border-color: transparent;\n",
              "    border-bottom-color: var(--fill-color);\n",
              "    animation:\n",
              "      spin 1s steps(1) infinite;\n",
              "  }\n",
              "\n",
              "  @keyframes spin {\n",
              "    0% {\n",
              "      border-color: transparent;\n",
              "      border-bottom-color: var(--fill-color);\n",
              "      border-left-color: var(--fill-color);\n",
              "    }\n",
              "    20% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    30% {\n",
              "      border-color: transparent;\n",
              "      border-left-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "      border-right-color: var(--fill-color);\n",
              "    }\n",
              "    40% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
              "      border-top-color: var(--fill-color);\n",
              "    }\n",
              "    60% {\n",
              "      border-color: transparent;\n",
              "      border-right-color: var(--fill-color);\n",
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