Gradient Checkng

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      "$$\\frac{d}{d\\theta}J(\\theta) = lim_{\\epsilon\\rightarrow 0} \\frac{J(\\theta+\\epsilon) - J(\\theta - \\epsilon)}{2\\epsilon}$$\n",
      "\n",
      "Thus we can numerically approximate the derivatives $g(\\theta) = \\frac{d}{d\\theta}J(\\theta)$.\n",
      "\n",
      "When $\\mathbf\\theta$ is a vector, we compute $g(\\theta_i) = \\frac{d}{d\\theta}J(\\theta_i)$, which is the derivative with respetivve to the $i$-th element of $\\theta$.\n",
      "\n",
      "Then we combine $g(\\theta_i)$ vertically to get $g(\\theta) = [g(\\theta_1), \\cdots, g(\\theta_m)]^T$.\n",
      "\n",
      "To compute $g(\\theta_i)$, we use two intermediate vectors $\\theta_i^{(+)}$ and $\\theta_i^{(-)}$. \n",
      "\n",
      "$\\theta_i^{(+)} = \\theta + e_i$\n",
      "\n",
      "$\\theta_i^{(-)} = \\theta - e_i$\n",
      "\n",
      "$e_i = [0, 0, \\cdots, \\epsilon, \\cdots, 0]^T$, where  $e_i$ is a vector with a \"1\" in the $i$-th position and \"0\"s everywhere else."
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      "```matlab\n",
      "function numgrad = computeNumericalGradient(J, theta)\n",
      "% numgrad = computeNumericalGradient(J, theta)\n",
      "% theta: a vector of parameters\n",
      "% J: a function that outputs a real-number. Calling y = J(theta) will return the\n",
      "% function value at theta. \n",
      "  \n",
      "% Initialize numgrad with zeros\n",
      "numgrad = zeros(size(theta));\n",
      "\n",
      "%% ---------- YOUR CODE HERE --------------------------------------\n",
      "% Instructions: \n",
      "% Implement numerical gradient checking, and return the result in numgrad.  \n",
      "% (See Section 2.3 of the lecture notes.)\n",
      "% You should write code so that numgrad(i) is (the numerical approximation to) the \n",
      "% partial derivative of J with respect to the i-th input argument, evaluated at theta.  \n",
      "% I.e., numgrad(i) should be the (approximately) the partial derivative of J with \n",
      "% respect to theta(i).\n",
      "%                \n",
      "% Hint: You will probably want to compute the elements of numgrad one at a time. \n",
      "\n",
      "e = zeros(size(theta));\n",
      "epsilon = 1e-4;\n",
      "\n",
      "for i = 1:length(numgrad)\n",
      "    ei = e;\n",
      "    ei(i) = 1;\n",
      "    theta_pos = J(theta + epsilon .* ei);\n",
      "    theta_neg = J(theta - epsilon .* ei);\n",
      "\n",
      "    numgrad(i) = (theta_pos - theta_neg)./(2*epsilon);\n",
      "end\n",
      "\n",
      "\n",
      "disp(' numerical comp done!')\n",
      "\n",
      "%% ---------------------------------------------------------------\n",
      "end\n",
      "```"
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     "input": [
      "matlab checkNumericalGradient()"
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       "text": [
        "     2     1\n",
        "\n",
        " numerical comp done!\n",
        "   38.0000   38.0000\n",
        "   12.0000   12.0000\n",
        "\n",
        "The above two columns you get should be very similar.\n",
        "(Left-Your Numerical Gradient, Right-Analytical Gradient)\n",
        "\n",
        "   2.1452e-12\n",
        "\n",
        "Norm of the difference between numerical and analytical gradient (should be < 1e-9)\n",
        "\n"
       ]
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