singular_walks.ipynb

{
    "cells": [
        {
            "cell_type": "code",
            "execution_count": 51,
            "metadata": {},
            "outputs": [],
            "source": [
                "from typing import List, Callable\n",
                "\n",
                "import numpy as np\n",
                "import torch as t\n",
                "import pandas as pd\n",
                "from matplotlib import pyplot as plt\n",
                "from matplotlib import animation, cm, cbook\n",
                "from tqdm.notebook import tqdm \n",
                "\n",
                "from jax import grad, jit, vmap"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "# Why Singularities?\n",
                "\n",
                "I want to built up some intuition for [why singularities in the loss landscape can have such disproportionate effect over the global behavior of training](https://www.lesswrong.com/posts/fovfuFdpuEwQzJu2w/neural-networks-generalize-because-of-this-one-weird-trick).  \n",
                "\n",
                "## Simple Singularities\n",
                "\n",
                "As a simple proof of concept, let's study the effect of singularities on the behavior of a simple random walk. To start, let's look at a random walk on two or more intersecting 1D lines. \n",
                "We'll let these lines cross at one point (the singularity), and look at the stable distribution of the random walk as we increase the number of lines. To make this tractable, we'll put the lines on a torus."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 52,
            "metadata": {},
            "outputs": [],
            "source": [
                "from collections import Counter\n",
                "\n",
                "\n",
                "def walk_crossing_lattice(n: int, D=2, L=9) -> np.ndarray:\n",
                "    \"\"\"Generate a random walk on D x 1d random walks that intersect at the origin.\n",
                "\n",
                "    Parameters\n",
                "    ----------\n",
                "\n",
                "    n : (int) \n",
                "        Number of steps\n",
                "    D : (int, optional)\n",
                "        Number of dimensions. Defaults to 2.\n",
                "    L : (int, optional)\n",
                "        Length of the lattice. Defaults to 9.\n",
                "\n",
                "    Returns\n",
                "    -------\n",
                "    \n",
                "    np.ndarray [n, D]\n",
                "        The walk \n",
                "    \"\"\"\n",
                "    l = L // 2\n",
                "\n",
                "    walk = np.zeros((n, D))\n",
                "    origin = np.zeros(D)\n",
                "\n",
                "    for i in range(1, n):\n",
                "        direction = np.random.choice([-1, 1])\n",
                "\n",
                "        if np.all(walk[i - 1] == origin):\n",
                "            axis = np.random.randint(D)\n",
                "            walk[i, axis] = direction\n",
                "        else:\n",
                "            axis = np.abs(walk[i - 1]).argmax()\n",
                "            step = np.zeros(D)\n",
                "            step[axis] = direction\n",
                "            walk[i, :] = walk[i - 1] + step\n",
                "\n",
                "        # Toroidal boundary conditions (-l, l)\n",
                "        walk[i, :] = np.mod(walk[i, :] + l, L) - l\n",
                "\n",
                "    return walk\n",
                "\n",
                "walk_2d = walk_crossing_lattice(100_000, D=2)\n",
                "walk_3d = walk_crossing_lattice(100_000, D=3)\n",
                "walk_4d = walk_crossing_lattice(100_000, D=4)\n",
                "walk_10d = walk_crossing_lattice(100_000, D=10)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 77,
            "metadata": {},
            "outputs": [],
            "source": [
                "def plot_walk_2d(walk: np.ndarray, D=2, ax=None):\n",
                "    \"\"\"\n",
                "    Plot a 2D walk on a 2D lattice. The size of the circles is proportional to the number of visits.\n",
                "    \"\"\"\n",
                "    count = Counter([tuple(x) for x in walk])\n",
                "    \n",
                "    # Plot circles at each point with size proportional to the number of visits\n",
                "    \n",
                "    if ax is None:\n",
                "        fig, ax = plt.subplots(figsize=(5, 5))\n",
                "        \n",
                "    r = 1. / (max(count.values()) * 5)\n",
                "\n",
                "    for point, visits in count.items():\n",
                "        ax.add_artist(plt.Circle(point, radius=visits * r, color='blue', alpha=0.5))\n",
                "\n",
                "    ax.set_title(f'Random walk on {D}x1D intersecting lattices')\n",
                "    ax.set_xlim(-5, 5)\n",
                "    ax.set_ylim(-5, 5)\n",
                "\n",
                "    ax.axis(\"off\")\n",
                "    ax.set_aspect('equal')\n",
                "\n",
                "    print(\"Ratio of origin visits to other visits:\", count[(0, 0)] * len(count.values()) / (sum(count.values()) - count[(0, 0)]))\n"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 78,
            "metadata": {},
            "outputs": [],
            "source": [
                "def project_onto_diagonal(walk: np.ndarray, m=1):\n",
                "    \"\"\"Project the z axis onto the y = mx diagonal.\"\"\"\n",
                "    walk_2d = np.copy(walk)\n",
                "    walk_2d[:, 0] += m * walk[:, 2]\n",
                "    walk_2d[:, 1] += walk[:, 2]\n",
                "    return walk_2d[:, :2]\n",
                "\n",
                "def plot_walk_3d(walk: np.ndarray, ax=None):\n",
                "    \"\"\"\n",
                "    Project the z axis onto the x = y diagonal & plot the 2D walk.\n",
                "    \"\"\"\n",
                "    walk_2d = project_onto_diagonal(walk)\n",
                "    plot_walk_2d(walk_2d, D=3, ax=ax)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 79,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Ratio of origin visits to other visits: 2.1114308840723086\n",
                        "Ratio of origin visits to other visits: 3.125597668950465\n",
                        "Ratio of origin visits to other visits: 4.064492216456634\n",
                        "Ratio of origin visits to other visits: 9.848932805437476\n"
                    ]
                },
                {
                    "data": {
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                        "text/plain": [
                            "<Figure size 2000x1000 with 4 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "MS = (1, -1, 0.5, -0.5, 2, -2, 0.25, -0.25, 4, -4, 0.75, -0.75)\n",
                "\n",
                "def project_multiple_onto_diagonal(walk: np.ndarray, ms=MS):\n",
                "    \"\"\"Project the >2 axes onto the y = mx diagonal.\"\"\"\n",
                "    projected_walk = np.copy(walk)\n",
                "\n",
                "    if len(ms) <= walk.shape[1] - 2:\n",
                "        raise ValueError(\"Number of slopes must be greater than or equal to the number of dimensions - 2\")\n",
                "\n",
                "    for i, m in zip(range(2, walk.shape[1]), ms):\n",
                "        projected_walk[:, 0] += m * walk[:, i]\n",
                "        projected_walk[:, 1] += walk[:, i]\n",
                "        \n",
                "    return projected_walk[:, :2]\n",
                "\n",
                "\n",
                "def plot_walk(walk: np.ndarray,  ms=MS, ax=None):\n",
                "    \"\"\"\n",
                "    Project all higher-dim. axis onto x = m * y diagonals & plot the 2D walk.\n",
                "    \"\"\"\n",
                "    D=walk.shape[1]\n",
                "    walk = project_multiple_onto_diagonal(walk, ms=ms)\n",
                "    plot_walk_2d(walk, D=D, ax=ax)\n",
                "\n",
                "fig, axes = plt.subplots(1, 4, figsize=(20, 10))\n",
                "\n",
                "fig.tight_layout()\n",
                "\n",
                "plot_walk_2d(walk_2d, ax=axes[0])\n",
                "plot_walk_3d(walk_3d, ax=axes[1])\n",
                "plot_walk(walk_4d, ax=axes[2])\n",
                "plot_walk(walk_10d, ax=axes[3])"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "This is pretty decisive: the role of the singularity is that it acts as a kind of \"trap\". The random walk is as many times more likely to be at the singularity as there are lines.\n",
                "\n",
                "## Fancier singularities\n",
                "\n",
                "Is this true more broadly? What about cusps and other singularities?\n",
                "Let's extend this analysis to arbitrary 1D manifolds. We'll generate maps on an arbitrary analytic variety defined by the equation $f(x) = 0$. We'll then look at the stable distribution of the random walk on the manifold as we change the nature of the singularity."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 66,
            "metadata": {},
            "outputs": [],
            "source": [
                "def random_walk_on_curve(\n",
                "    fun: Callable[[np.ndarray], np.ndarray],\n",
                "    n_steps: int,\n",
                "    start: np.ndarray,\n",
                "    step_size: float = 1e-6,\n",
                ") -> np.ndarray:\n",
                "    \"\"\"Perform a random walk on a curve.\n",
                "\n",
                "    Parameters\n",
                "    ----------\n",
                "    f : Callable[[float, float], np.ndarray]\n",
                "        The curve to walk on.\n",
                "    n_steps : int\n",
                "        The number of steps to take.\n",
                "    start : np.ndarray\n",
                "        The starting point of the walk. Must be on the curve.\n",
                "    step_size : float, optional\n",
                "        The step size, by default 1.0\n",
                "\n",
                "    Returns\n",
                "    -------\n",
                "    np.ndarray\n",
                "        The trajectory of the walk.\n",
                "    \"\"\"\n",
                "    walk = [start]\n",
                "\n",
                "    # grad_fun = grad(fun, argnums=[0, 1])\n",
                "\n",
                "    for _ in tqdm(range(n_steps), desc=\"Random walk on curve\"):\n",
                "        dx = 0\n",
                "        step_size_ = step_size\n",
                "        curr_pt = walk[-1]\n",
                "\n",
                "        while dx < step_size:\n",
                "            # tangent = grad_fun(*curr_pt)\n",
                "            tangent = np.array(\n",
                "                [\n",
                "                    (fun(curr_pt[0] + step_size_, curr_pt[1]) - fun(curr_pt[0], curr_pt[1])) / step_size_,\n",
                "                    (fun(curr_pt[0], curr_pt[1] + step_size_) - fun(curr_pt[0], curr_pt[1])) / step_size_,\n",
                "                ]\n",
                "            )\n",
                "            tangent /= np.linalg.norm(tangent)\n",
                "\n",
                "            next_pt = curr_pt + step_size_ * tangent\n",
                "\n",
                "            # Check if the next point is on the curve\n",
                "            if np.allclose(fun(*next_pt), 0):\n",
                "                dx += step_size_\n",
                "                curr_pt = next_pt\n",
                "            else:\n",
                "                # Shrink step_size_ until next_pt is on the curve\n",
                "                step_size_ /= 2\n",
                "        \n",
                "        walk.append(curr_pt)\n",
                "\n",
                "    return np.array(walk)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 67,
            "metadata": {},
            "outputs": [
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "/var/folders/pw/6c492xhs4299155z1pvbwt8w0000gn/T/ipykernel_40307/2490885684.py:13: RuntimeWarning: invalid value encountered in sqrt\n",
                        "  ymins = np.sqrt(x ** 2 * (x + 1))\n"
                    ]
                },
                {
                    "data": {
                        "text/plain": [
                            "(<Figure size 500x500 with 1 Axes>,\n",
                            " <AxesSubplot: title={'center': 'Double Point singularity'}, xlabel='x', ylabel='y'>)"
                        ]
                    },
                    "execution_count": 67,
                    "metadata": {},
                    "output_type": "execute_result"
                },
                {
                    "data": {
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",
                        "text/plain": [
                            "<Figure size 500x500 with 1 Axes>"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "def plot_double_pt_singularity(xlim=(-5, 5), ylim=(-5, 5)):\n",
                "    # y2 − x2(x + 1) = 0\n",
                "\n",
                "    fig, ax = plt.subplots(figsize=(5, 5))\n",
                "    ax.set_xlim(xlim)\n",
                "    ax.set_ylim(ylim)\n",
                "    ax.set_aspect(\"equal\")\n",
                "    ax.set_xlabel(\"x\")\n",
                "    ax.set_ylabel(\"y\")\n",
                "    ax.set_title(\"Double Point singularity\")\n",
                "\n",
                "    x = np.linspace(*xlim, 10000)\n",
                "    ymins = np.sqrt(x ** 2 * (x + 1))\n",
                "    ymaxs = -ymins\n",
                "\n",
                "    ax.plot(x, ymins, color=\"black\")\n",
                "    ax.plot(x, ymaxs, color=\"black\")\n",
                "\n",
                "    return fig, ax\n",
                "\n",
                "plot_double_pt_singularity()"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 68,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "application/vnd.jupyter.widget-view+json": {
                            "model_id": "013d4e04a89747f2bb59ce79ebe4af43",
                            "version_major": 2,
                            "version_minor": 0
                        },
                        "text/plain": [
                            "Random walk on curve:   0%|          | 0/100 [00:00<?, ?it/s]"
                        ]
                    },
                    "metadata": {},
                    "output_type": "display_data"
                },
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "/var/folders/pw/6c492xhs4299155z1pvbwt8w0000gn/T/ipykernel_40307/416782329.py:42: RuntimeWarning: invalid value encountered in divide\n",
                        "  tangent /= np.linalg.norm(tangent)\n",
                        "/var/folders/pw/6c492xhs4299155z1pvbwt8w0000gn/T/ipykernel_40307/416782329.py:38: RuntimeWarning: invalid value encountered in scalar divide\n",
                        "  (fun(curr_pt[0] + step_size_, curr_pt[1]) - fun(curr_pt[0], curr_pt[1])) / step_size_,\n",
                        "/var/folders/pw/6c492xhs4299155z1pvbwt8w0000gn/T/ipykernel_40307/416782329.py:39: RuntimeWarning: invalid value encountered in scalar divide\n",
                        "  (fun(curr_pt[0], curr_pt[1] + step_size_) - fun(curr_pt[0], curr_pt[1])) / step_size_,\n"
                    ]
                },
                {
                    "ename": "KeyboardInterrupt",
                    "evalue": "",
                    "output_type": "error",
                    "traceback": [
                        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
                        "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
                        "Cell \u001b[0;32mIn[68], line 17\u001b[0m\n\u001b[1;32m     13\u001b[0m     ax\u001b[39m.\u001b[39mplot(trajectory[:, \u001b[39m0\u001b[39m], trajectory[:, \u001b[39m1\u001b[39m], color\u001b[39m=\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mblue\u001b[39m\u001b[39m\"\u001b[39m)\n\u001b[1;32m     15\u001b[0m     \u001b[39mreturn\u001b[39;00m fig, ax\n\u001b[0;32m---> 17\u001b[0m plot_random_walk_on_curve(\n\u001b[1;32m     18\u001b[0m     \u001b[39mlambda\u001b[39;49;00m x, y: y \u001b[39m-\u001b[39;49m np\u001b[39m.\u001b[39;49msqrt(x \u001b[39m*\u001b[39;49m\u001b[39m*\u001b[39;49m \u001b[39m2\u001b[39;49m \u001b[39m*\u001b[39;49m (x \u001b[39m+\u001b[39;49m \u001b[39m1\u001b[39;49m)),\n\u001b[1;32m     19\u001b[0m     \u001b[39m100\u001b[39;49m,\n\u001b[1;32m     20\u001b[0m     np\u001b[39m.\u001b[39;49marray([\u001b[39m0.\u001b[39;49m, \u001b[39m0.\u001b[39;49m]),\n\u001b[1;32m     21\u001b[0m     step_size\u001b[39m=\u001b[39;49m\u001b[39m0.1\u001b[39;49m,\n\u001b[1;32m     22\u001b[0m )\n",
                        "Cell \u001b[0;32mIn[68], line 9\u001b[0m, in \u001b[0;36mplot_random_walk_on_curve\u001b[0;34m(fun, n_steps, start, step_size, xlim, ylim)\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[39mdef\u001b[39;00m \u001b[39mplot_random_walk_on_curve\u001b[39m(\n\u001b[1;32m      2\u001b[0m     fun: Callable[[np\u001b[39m.\u001b[39mndarray], np\u001b[39m.\u001b[39mndarray],\n\u001b[1;32m      3\u001b[0m     n_steps: \u001b[39mint\u001b[39m,\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m      7\u001b[0m     ylim\u001b[39m=\u001b[39m(\u001b[39m-\u001b[39m\u001b[39m5\u001b[39m, \u001b[39m5\u001b[39m),\n\u001b[1;32m      8\u001b[0m ):\n\u001b[0;32m----> 9\u001b[0m     trajectory \u001b[39m=\u001b[39m random_walk_on_curve(fun, n_steps, start, step_size)\n\u001b[1;32m     11\u001b[0m     fig, ax \u001b[39m=\u001b[39m plot_double_pt_singularity(xlim, ylim)\n\u001b[1;32m     13\u001b[0m     ax\u001b[39m.\u001b[39mplot(trajectory[:, \u001b[39m0\u001b[39m], trajectory[:, \u001b[39m1\u001b[39m], color\u001b[39m=\u001b[39m\u001b[39m\"\u001b[39m\u001b[39mblue\u001b[39m\u001b[39m\"\u001b[39m)\n",
                        "Cell \u001b[0;32mIn[66], line 42\u001b[0m, in \u001b[0;36mrandom_walk_on_curve\u001b[0;34m(fun, n_steps, start, step_size)\u001b[0m\n\u001b[1;32m     34\u001b[0m \u001b[39mwhile\u001b[39;00m dx \u001b[39m<\u001b[39m step_size:\n\u001b[1;32m     35\u001b[0m     \u001b[39m# tangent = grad_fun(*curr_pt)\u001b[39;00m\n\u001b[1;32m     36\u001b[0m     tangent \u001b[39m=\u001b[39m np\u001b[39m.\u001b[39marray(\n\u001b[1;32m     37\u001b[0m         [\n\u001b[1;32m     38\u001b[0m             (fun(curr_pt[\u001b[39m0\u001b[39m] \u001b[39m+\u001b[39m step_size_, curr_pt[\u001b[39m1\u001b[39m]) \u001b[39m-\u001b[39m fun(curr_pt[\u001b[39m0\u001b[39m], curr_pt[\u001b[39m1\u001b[39m])) \u001b[39m/\u001b[39m step_size_,\n\u001b[1;32m     39\u001b[0m             (fun(curr_pt[\u001b[39m0\u001b[39m], curr_pt[\u001b[39m1\u001b[39m] \u001b[39m+\u001b[39m step_size_) \u001b[39m-\u001b[39m fun(curr_pt[\u001b[39m0\u001b[39m], curr_pt[\u001b[39m1\u001b[39m])) \u001b[39m/\u001b[39m step_size_,\n\u001b[1;32m     40\u001b[0m         ]\n\u001b[1;32m     41\u001b[0m     )\n\u001b[0;32m---> 42\u001b[0m     tangent \u001b[39m/\u001b[39m\u001b[39m=\u001b[39m np\u001b[39m.\u001b[39;49mlinalg\u001b[39m.\u001b[39;49mnorm(tangent)\n\u001b[1;32m     44\u001b[0m     next_pt \u001b[39m=\u001b[39m curr_pt \u001b[39m+\u001b[39m step_size_ \u001b[39m*\u001b[39m tangent\n\u001b[1;32m     46\u001b[0m     \u001b[39m# Check if the next point is on the curve\u001b[39;00m\n",
                        "File \u001b[0;32m<__array_function__ internals>:200\u001b[0m, in \u001b[0;36mnorm\u001b[0;34m(*args, **kwargs)\u001b[0m\n",
                        "File \u001b[0;32m~/.virtualenvs/notes/lib/python3.9/site-packages/numpy/linalg/linalg.py:2500\u001b[0m, in \u001b[0;36mnorm\u001b[0;34m(x, ord, axis, keepdims)\u001b[0m\n\u001b[1;32m   2498\u001b[0m \u001b[39m# Immediately handle some default, simple, fast, and common cases.\u001b[39;00m\n\u001b[1;32m   2499\u001b[0m \u001b[39mif\u001b[39;00m axis \u001b[39mis\u001b[39;00m \u001b[39mNone\u001b[39;00m:\n\u001b[0;32m-> 2500\u001b[0m     ndim \u001b[39m=\u001b[39m x\u001b[39m.\u001b[39;49mndim\n\u001b[1;32m   2501\u001b[0m     \u001b[39mif\u001b[39;00m ((\u001b[39mord\u001b[39m \u001b[39mis\u001b[39;00m \u001b[39mNone\u001b[39;00m) \u001b[39mor\u001b[39;00m\n\u001b[1;32m   2502\u001b[0m         (\u001b[39mord\u001b[39m \u001b[39min\u001b[39;00m (\u001b[39m'\u001b[39m\u001b[39mf\u001b[39m\u001b[39m'\u001b[39m, \u001b[39m'\u001b[39m\u001b[39mfro\u001b[39m\u001b[39m'\u001b[39m) \u001b[39mand\u001b[39;00m ndim \u001b[39m==\u001b[39m \u001b[39m2\u001b[39m) \u001b[39mor\u001b[39;00m\n\u001b[1;32m   2503\u001b[0m         (\u001b[39mord\u001b[39m \u001b[39m==\u001b[39m \u001b[39m2\u001b[39m \u001b[39mand\u001b[39;00m ndim \u001b[39m==\u001b[39m \u001b[39m1\u001b[39m)):\n\u001b[1;32m   2505\u001b[0m         x \u001b[39m=\u001b[39m x\u001b[39m.\u001b[39mravel(order\u001b[39m=\u001b[39m\u001b[39m'\u001b[39m\u001b[39mK\u001b[39m\u001b[39m'\u001b[39m)\n",
                        "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
                    ]
                }
            ],
            "source": [
                "def plot_random_walk_on_curve(\n",
                "    fun: Callable[[np.ndarray], np.ndarray],\n",
                "    n_steps: int,\n",
                "    start: np.ndarray,\n",
                "    step_size: float = 1.0,\n",
                "    xlim=(-5, 5),\n",
                "    ylim=(-5, 5),\n",
                "):\n",
                "    trajectory = random_walk_on_curve(fun, n_steps, start, step_size)\n",
                "\n",
                "    fig, ax = plot_double_pt_singularity(xlim, ylim)\n",
                "\n",
                "    ax.plot(trajectory[:, 0], trajectory[:, 1], color=\"blue\")\n",
                "\n",
                "    return fig, ax\n",
                "\n",
                "plot_random_walk_on_curve(\n",
                "    lambda x, y: y - np.sqrt(x ** 2 * (x + 1)),\n",
                "    100,\n",
                "    np.array([0., 0.]),\n",
                "    step_size=0.1,\n",
                ")"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": null,
            "metadata": {},
            "outputs": [],
            "source": []
        }
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