Merge pull request #151 from davidcotton/master

Fix typo in 'Gradient Descent, revisited' section

math_differential_calculus.ipynb

@@ -1191,7 +1191,7 @@
     "\n",
     "In Deep Learning, the letter $\\mathbf{x}$ is generally used to represent the input data. When you _use_ a neural network to make predictions, you feed the neural network the inputs $\\mathbf{x}$, and you get back a prediction $\\hat{y} = f(\\mathbf{x})$. The function $f$ treats the model parameters as constants. We can use more explicit notation by writing $\\hat{y} = f_\\mathbf{w}(\\mathbf{x})$, where $\\mathbf{w}$ represents the model parameters and indicates that the function relies on them, but treats them as constants.\n",
     "\n",
-    "However, when _training_ a neural network, we do quite the opposite: all the training examples are grouped in a matrix $\\mathbf{X}$, all the labels are grouped in a vector $\\mathbf{y}$, and both $\\mathbf{X}$ and $\\mathbf{y}$ are treated as constants, while $\\mathbf{w}$ is treated as variable: specifically, we try to minimize the cost function $\\mathcal L_{\\mathbf{X}, \\mathbf{y}}(\\mathbf{w}) = g(f_{\\mathbf{X}}(\\mathbf{w}), \\mathbf{y})$, where $g$ is a function that measures the \"discrepancy\" between the predictions $f_{\\mathbf{X}}(\\mathbf{w})$ and the labels $\\mathbf{w}$, where $f_{\\mathbf{X}}(\\mathbf{w})$ represents the vector containing the predictions for each training example. Minimizing the loss function is usually performed using Gradient Descent (or a variant of GD): we start with random model parameters $\\mathbf{w}_0$, then we compute $\\nabla \\mathcal L(\\mathbf{w}_0)$ and we use this gradient vector to perform a Gradient Descent step, then we repeat the process until convergence. It is crucial to understand that the gradient of the loss function is with regards to the model parameters $\\mathbf{w}$ (_not_ the inputs $\\mathbf{x}$)."
+    "However, when _training_ a neural network, we do quite the opposite: all the training examples are grouped in a matrix $\\mathbf{X}$, all the labels are grouped in a vector $\\mathbf{y}$, and both $\\mathbf{X}$ and $\\mathbf{y}$ are treated as constants, while $\\mathbf{w}$ is treated as variable: specifically, we try to minimize the cost function $\\mathcal L_{\\mathbf{X}, \\mathbf{y}}(\\mathbf{w}) = g(f_{\\mathbf{X}}(\\mathbf{w}), \\mathbf{y})$, where $g$ is a function that measures the \"discrepancy\" between the predictions $f_{\\mathbf{X}}(\\mathbf{w})$ and the labels $\\mathbf{y}$, where $f_{\\mathbf{X}}(\\mathbf{w})$ represents the vector containing the predictions for each training example. Minimizing the loss function is usually performed using Gradient Descent (or a variant of GD): we start with random model parameters $\\mathbf{w}_0$, then we compute $\\nabla \\mathcal L(\\mathbf{w}_0)$ and we use this gradient vector to perform a Gradient Descent step, then we repeat the process until convergence. It is crucial to understand that the gradient of the loss function is with regards to the model parameters $\\mathbf{w}$ (_not_ the inputs $\\mathbf{x}$)."
    ]
   },
   {
@@ -1781,6 +1781,15 @@
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