{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# softmax \n", "- $softmax(x_i) = \\frac{e^{x_i}}{\\sum_j{e^{x_j}}}$\n", "\n", "- \\begin{align} (softmax(x + c))_{i}= \\frac{e^{x_{i} + c}}{\\sum_{j} e^{x_{j} + c}} = \\frac{e^{x_{i}} \\times e^{c}}{e^{c} \\times \\sum_{j} e^{x_{j}}} \\\\ = \\frac{e^{x_{i}} \\times {e^{c}}}{{e^{c}} \\times \\sum_{j} e^{x_{j}}} = (softmax(x))_{i} \\end{align}\n", "\n", "so:\n", "\n", "- $softmax(x) = softmax(x + c)$" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "def softmax(x):\n", " orig_shape = x.shape\n", "\n", " if len(x.shape) > 1:\n", " # Matrix\n", " ### YOUR CODE HERE\n", " x_max = np.max(x, axis=1).reshape(x.shape[0], 1)\n", " x -= x_max\n", " exp_sum = np.sum(np.exp(x), axis=1).reshape(x.shape[0], 1)\n", " x = np.exp(x) / exp_sum \n", " ### END YOUR CODE\n", " else:\n", " # Vector\n", " ### YOUR CODE HERE\n", " x_max = np.max(x)\n", " x -= x_max\n", " exp_sum = np.sum(np.exp(x))\n", " x = np.exp(x) / exp_sum\n", " ### END YOUR CODE\n", " #or: x = (np.exp(x)/sum(np.exp(x))) \n", "\n", " assert x.shape == orig_shape\n", " return x" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Running basic tests...\n", "[0.26894142 0.73105858]\n", "[[0.26894142 0.73105858]\n", " [0.26894142 0.73105858]]\n", "[[0.73105858 0.26894142]]\n", "You should be able to verify these results by hand!\n", "\n" ] } ], "source": [ "def test_softmax_basic():\n", " \"\"\"\n", " Some simple tests to get you started.\n", " Warning: these are not exhaustive.\n", " \"\"\"\n", " print(\"Running basic tests...\")\n", " test1 = softmax(np.array([1,2]))\n", " print(test1)\n", " ans1 = np.array([0.26894142, 0.73105858])\n", " assert np.allclose(test1, ans1, rtol=1e-05, atol=1e-06)\n", "\n", " test2 = softmax(np.array([[1001,1002],[3,4]]))\n", " print(test2)\n", " ans2 = np.array([\n", " [0.26894142, 0.73105858],\n", " [0.26894142, 0.73105858]])\n", " assert np.allclose(test2, ans2, rtol=1e-05, atol=1e-06)\n", "\n", " test3 = softmax(np.array([[-1001,-1002]]))\n", " print(test3)\n", " ans3 = np.array([0.73105858, 0.26894142])\n", " assert np.allclose(test3, ans3, rtol=1e-05, atol=1e-06)\n", "\n", " print(\"You should be able to verify these results by hand!\\n\")\n", "\n", "if __name__ == \"__main__\":\n", " test_softmax_basic()" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "x = np.array([[1,2],[4,3]])" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([2, 4])" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "np.max(x, axis=1)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 2 }