Chapter 4 corrections and fix (fastai#463)

Co-authored-by: Jeremy Howard <j@fast.ai>

04_mnist_basics.ipynb

@@ -3281,7 +3281,7 @@
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-    "> a: Understanding this bit depends on remembering recent history. To calculate the gradients we call `backward` on the `loss`. But this `loss` was itself calculated by `mse`, which in turn took `preds` as an input, which was calculated using `f` taking as an input `params`, which was the object on which we originally called `required_grads_`—which is the original call that now allows us to call `backward` on `loss`. This chain of function calls represents the mathematical composition of functions, which enables PyTorch to use calculus's chain rule under the hood to calculate these gradients."
+    "> a: Understanding this bit depends on remembering recent history. To calculate the gradients we call `backward` on the `loss`. But this `loss` was itself calculated by `mse`, which in turn took `preds` as an input, which was calculated using `f` taking as an input `params`, which was the object on which we originally called `requires_grad_`—which is the original call that now allows us to call `backward` on `loss`. This chain of function calls represents the mathematical composition of functions, which enables PyTorch to use calculus's chain rule under the hood to calculate these gradients."
    ]
   },
   {
@@ -3789,7 +3789,7 @@
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    "source": [
-    "This image shows two matrices, `A` and `B`, being multiplied together. Each item of the result, which we'll call `AB`, contains each item of its corresponding row of `A` multiplied by each item of its corresponding column of `B`, added together. For instance, row 1, column 2 (the orange dot with a red border) is calculated as $a_{1,1} * b_{1,2} + a_{1,2} * b_{2,2}$. If you need a refresher on matrix multiplication, we suggest you take a look at the [Intro to Matrix Multiplication](https://youtu.be/kT4Mp9EdVqs) on *Khan Academy*, since this is the most important mathematical operation in deep learning.\n",
+    "This image shows two matrices, `A` and `B`, being multiplied together. Each item of the result, which we'll call `AB`, contains each item of its corresponding row of `A` multiplied by each item of its corresponding column of `B`, added together. For instance, row 1, column 2 (the yellow dot with a red border) is calculated as $a_{1,1} * b_{1,2} + a_{1,2} * b_{2,2}$. If you need a refresher on matrix multiplication, we suggest you take a look at the [Intro to Matrix Multiplication](https://youtu.be/kT4Mp9EdVqs) on *Khan Academy*, since this is the most important mathematical operation in deep learning.\n",
     "\n",
     "In Python, matrix multiplication is represented with the `@` operator. Let's try it:"
    ]
@@ -3896,7 +3896,6 @@
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    "outputs": [],
    "source": [
-    "with torch.no_grad(): weights[0] *= 1.0001"
    ]
   },
   {
@@ -4154,7 +4153,7 @@
     "\n",
     "Having defined a loss function, now is a good moment to recapitulate why we did this. After all, we already had a metric, which was overall accuracy. So why did we define a loss?\n",
     "\n",
-    "The key difference is that the metric is to drive human understanding and the loss is to drive automated learning. To drive automated learning, the loss must be a function that has a meaningful derivative. It can't have big flat sections and large jumps, but instead must be reasonably smooth. This is why we designed a loss function that would respond to small changes in confidence level. This requirement means that sometimes it does not really reflect exactly what we are trying to achieve, but is rather a compromise between our real goal, and a function that can be optimized using its gradient. The loss function is calculated for each item in our dataset, and then at the end of an epoch the loss values are all averaged and the overall mean is reported for the epoch.\n",
+    "The key difference is that the metric is to drive human understanding and the loss is to drive automated learning. To drive automated learning, the loss must be a function that has a meaningful derivative. It can't have big flat sections and large jumps, but instead must be reasonably smooth. This is why we designed a loss function that would respond to small changes in confidence level. This requirement means that sometimes it does not really reflect exactly what we are trying to achieve, but is rather a compromise between our real goal and a function that can be optimized using its gradient. The loss function is calculated for each item in our dataset, and then at the end of an epoch the loss values are all averaged and the overall mean is reported for the epoch.\n",
     "\n",
     "Metrics, on the other hand, are the numbers that we really care about. These are the values that are printed at the end of each epoch that tell us how our model is really doing. It is important that we learn to focus on these metrics, rather than the loss, when judging the performance of a model."
    ]
@@ -5664,7 +5663,7 @@
     "\n",
     "That means that we can train the model more quickly, and it will take up less memory. In the 1990s researchers were so focused on the universal approximation theorem that very few were experimenting with more than one nonlinearity. This theoretical but not practical foundation held back the field for years. Some researchers, however, did experiment with deep models, and eventually were able to show that these models could perform much better in practice. Eventually, theoretical results were developed which showed why this happens. Today, it is extremely unusual to find anybody using a neural network with just one nonlinearity.\n",
     "\n",
-    "Here what happens when we train an 18-layer model using the same approach we saw in <<chapter_intro>>:"
+    "Here is what happens when we train an 18-layer model using the same approach we saw in <<chapter_intro>>:"
    ]
   },
   {