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1231/2016 CS231n Convolutional Neural Networks for Visual Recognition Interpreting a linear classifier Notice that a linear classifier computes the score of a class as a weighted sum of all of its pixel values across all 3 of its color channels. Depending on precisely what values we set for these weights, the function has the capacity to like or dislike(depending on the sign of each weight certain colors at certain positions in the image. For instance, you can imagine that the"ship"class might be more likely if there is a lot of blue on the sides of an image(which could likely correspond to water). You might expect that the"ship"classifier would then have a lot of positive weights across its blue channel weights(presence of blue increases score of ship), and negative weights in the red/green channels(presence of red/green decreases the score of ship stretch pixels into single column 56 96.8 cat score 15132100231+32 437.9 dog score 00.25020.3 24 -12 input image 61.95 ship score W f(i; W, b i An example of mapping an image to class scores For the sake of visualization, we assume the image only has 4 pixels(4 monochrome pixels, we are not considering color channels in this example for brevity ), and that we have 3 classes(red(cat), green(dog), blue(ship) class).(Clarification: in particular, the colors here simply indicate 3 classes and are not related to the RGB channels. )We stretch the image pixels into a column and perform matrix multiplication to get the scores for each class. Note that this particular set of weights W is not good at all: the weights assign our cat image a very low cat score. In particular, this set of weights seems convinced that it's looking at a dog Analogy of images as high-dimensional points. Since the images are stretched into high- dimensional column vectors, we can interpret each image as a single point in this space (e.g each image in CIFAR-10 is a point in 3072-dimensional space of 32X 3 pixels ). Analogously, the entire dataset is a(labeled set of points Since we defined the score of each class as a weighted sum of all image pixels, each class score is a linear function over this space. We cannot visualize 3072-dimensional spaces, but if we imagine squashing all those dimensions into only two dimensions, then we can try to visualize what the classifier might be doing tt p: c$231n.github. o/linear-classifyl 3/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition car classifier 0 airplane classifier/ deer classifier Cartoon representation of the image space, where each image is a single point, and three classifiers are visualized. Using the example of the car classifier(in red), the red line shows all points in the space that get a score of zero for the car class. The red arrow shows the direction of increase, so all points to the right of the red line have positive(and linearly increasing) scores, and all points to the left have a negative(and linearly decreasing) scores As We saw above, every row of w is a classifier for one of the classes. The geometric interpretation of these numbers is that as we change one of the rows of w, the corresponding line in the pixel space will rotate in different directions. the biases b, on the other hand, allow our classifiers to translate the lines. In particular, note that without the bias terms, plugging in i=0 would always give score of zero regardless of the weights, so all lines would be forced to cross the origin Interpretation of linear classifiers as template matching. Another interpretation for the weights w is that each row of w corresponds to a template or sometimes also called a prototype)for one of the classes. The score of each class for an image is then obtained by comparing each template with the image using an inner product (or dot product) one by one to find the one that fits"best. With this terminology, the linear classifier is doing template matching where the templates are learned. Another way to think of it is that we are still effectively doing Nearest tt p: c$231n.github. o/linear-classifyl 4/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition Neighbor, but instead of having thousands of training images we are only using a single image per class (although we will learn it, and it does not necessarily have to be one of the images in the training set), and we use the(negative) inner product as the distance instead of the l1 or L2 distance gda cat deet horse sh甲p truck Skipping ahead a bit: Example learned weights at the end of learning for CIFAR- 10. Note that, for example, the ship template contains a lot of blue pixels as expected. this template will therefore give a high score once it is matched against images of ships on the ocean with an inner product Additionally, note that the horse template seems to contain a two-headed horse, which is due to both left and right facing horses in the dataset. The linear classifier merges these two modes of horses in the data into a single template. Similarly, the car classifier seems to have merged several modes into a single template which has to identify cars from all sides and of all colors. In particular, this template ended up being red, which hints that there are more red cars in the CIFAR-10 dataset than of any other color. The linear classifier is too weak to properly account for different-colored cars, but as we will see later neural networks will allow us to perform this task ooking ahead a bit, a neural network will be able to develop intermediate neurons in its hidden layers that could detect specific car types(e.g. green car facing left, blue car facing front, etc ) and neurons on the next layer could combine these into a more accurate car score through a weighted sum of the individual car detectors Bias trick. Before moving on we want to mention a common simplifying trick to representing the two parameters w, b as one. Recall that we defined the score function as f(xi, w,b=Wxit b As we proceed through the material it is a little cumbersome to keep track of two sets of parameters(the biases b and weights W) separately. a commonly used trick is to combine the two sets of parameters into a single matrix that holds both of them by extending the vector x with one additional dimension that always holds the constant 1-a default bias dimension With the extra dimension, the new score function will simplify to a single matrix multiply f(xi, w)=wxi With our CIFAR-10 example, x i is now [3073X 1] instead of [3072 x 1]-(with the extra dimension holding the constant 1), and w is now [10 x 3073] instead of [10 x 3072]. The extra column that w now corresponds to the bias b. An illustration might help clarify tt p: c$231n.github. o/linear-classifyl 5/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition 02-0.50.120 6 1.1 0.2-‖1.1 56 15132100231+321513210032231 00250.2-0.3 24 1.2 00.250.2-0.3-12 24 2 W 2 new, single W Ilustration of the bias trick. Doing a matrix multiplication and then adding a bias vector(left) is equivalent to adding a bias dimension with a constant of 1 to all input vectors and extending the weight matrix by 1 column-a bias column(right). Thus, if we preprocess our data by appending ones to all vectors we only have to learn a single matrix of weights instead of two matrices that hold the weights and the biases Image data preprocessing. As a quick note, in the examples above we used the raw pixel values (Which range from [0..255) In Machine Learning, it is a very common practice to always perform normalization of your input features (in the case of images, every pixel is thought of as a feature). In particular, it is important to center your data by subtracting the mean from every feature. In the case of images, this corresponds to computing a mean image across the training images and subtracting it from every image to get images where the pixels range from approximately [-127... 127]. Further common preprocessing is to scale each input feature so that its values range from[ 1, 1]. of these, zero mean centering is arguably more important but we will have to wait for its justification until we understand the dynamics of gradient descent Loss function In the previous section we defined a function from the pixel values to class scores, which was parameterized by a set of weights W. Moreover, we saw that we don't have control over the data (xi, yi)(it is fixed and given), but we do have control over these weights and we want to set them so that the predicted class scores are consistent with the ground truth labels in the training data For example, going back to the example image of a cat and its scores for the classes cat dog and"", we saw that the particular set of weights in that example was not very good at all: We fed in the pixels that depict a cat but the cat score came out very low(96. 8)compared to the other classes(dog score 437.9 and ship score 61.95). We are going to measure our unhappiness with outcomes such as this one with a loss function (or sometimes also referred to as the cost function or the objective). Intuitively, the loss will be high if we're doing a poor job of classifying the training data, and it will be low if we're doing well tt p: c$231n.github. o/linear-classifyl 6/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition Multiclass Support Vector Machine loss There are several ways to define the details of the loss function. as a first example we will first develop a commonly used loss called the Multiclass Support Vector Machine(SVM)loss. The SVM loss is set up so that the svm wants" the correct class for each image to a have a score higher than the incorrect classes by some fixed margin A Notice that it's sometimes helpful to anthropomorphise the loss functions as we did above: The Svm"wants"a certain outcome in the sense that the outcome would yield a lower loss(which is good) et's now get more precise. Recall that for the i-th example we are given the pixels of image x and the label yi that specifies the index of the correct class. The score function takes the pixels and computes the vector f(xi, w)of class scores, Which we will abbreviate to S(short for scores). For example, the score for the j-th class is the jth element: S=f(xi, W)j.The Multiclass SVM loss for the i-th example is then formalized as follows ∑ max(U. S +△) ≠v2 Example. Lets unpack this with an example to see how it works. Suppose that we have three classes that receive the scores s=[13, -7,11, and that the first class is the true class (i.e Yi=o). Also assume that a(a hyperparameter we will go into more detail about soon) is 10 The expression above sums over all incorrect classes (i* yi, so we get two terms Lz=max(0,-7-13+10)+max(0,1l-13+10) You can see that the first term gives zero since [7-13+10 gives a negative number, which is then thresholded to zero with the max(o, - function We get zero loss for this pair because the correct class score(13) was greater than the incorrect class score(7 )by at least the margin 10 In fact the difference was 20, which is much greater than 10 but the Svm only cares that the difference is at least 10: Any additional difference above the margin is clamped at zero with the max operation. The second term computes [11-13+10 which gives 8. That is, even though the correct class had a higher score than the incorrect class(13>11), it was not greater by the desired margin of 10. the difference was only 2, which is why the loss comes out to 8(i.e. how much higher the difference would have to be to meet the margin). In summary, the SvM loss function wants the score of the correct class y i to be larger than the incorrect class scores by at least by A(delta If this is not the case, we will accumulate loss Note that in this particular module we are working with linear score functions(f(xi,w)=Wxi so we can also rewrite the loss function in this equivalent form ∑ max(0,w2x;-w1x;+△) j≠y tt p: c$231n.github. o/linear-classifyl 7/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition where wi is the j-th row of w reshaped as a column. However, this will not necessarily be the case once we start to consider more complex forms of the score function f A last piece of terminology we'll mention before we finish with this section is that the threshold at zero max(o, - function is often called the hinge loss. You'll sometimes hear about people instead using the squared hinge loss SVM(or L2-SVM), which uses the form max(, -) that penalizes violated margins more strongly(quadratically instead of linearly The unsquared ersion is more standard, but in some datasets the squared hinge loss can work better. this can be determined during cross-validation The loss function quantifies our unhappiness with predictions on the training set ++‖ delta score scores for other classes score for correct class The Multiclass Support Vector Machine wants the score of the correct class to be higher than all other scores by at least a margin of delta. If any class has a score inside the red region(or higher), then there will be accumulated loss. otherwise the loss will be zero. Our objective will be to find the weights that will simultaneously satisfy this constraint for all examples in the training data and give a total loss that is as low as possible Regularization. There is one bug with the loss function we presented above. Suppose that we have a dataset and a set of parameters w that correctly classify every example (i.e. all scores are so that all the margins are met, and Li =o for all i. the issue is that this set of w is not necessarily unique: there might be many similar w that correctly classify the examples. One easy way to see this is that if some parameters w correctly classify all examples(so loss is zero for each example), then any multiple of these parameters nw where n> I will also give zero loss because this transformation uniformly stretches all score magnitudes and hence also their absolute differences. For example, if the difference in scores between a correct class and a nearest incorrect class was 15, then multiplying all elements of w by 2 would make the new difference 30 In other words, we wish to encode some preference for a certain set of weights w over others to remove this ambiguity. We can do so by extending the loss function with a regularization penalty R(W The most common regularization penalty is the l2 norm that discourages large weights through an elementwise quadratic penalty over all parameters R(W)=∑∑W2 tt p: c$231n.github. o/linear-classifyl 8/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition In the expression above, we are summing up all the squared elements of w. notice that the regularization function is not a function of the data, it is only based on the weights. Including the regularization penalty completes the full Multiclass Support Vector Machine loss, which is made up of two components: the data loss(which is the average loss Li over all examples) and the regularization loss. That is, the full Multiclass SvM loss becomes N∑+xR(W regularization loss data lo Or expanding this out in its full form L=N∑∑ma(O/;w)-/(x;Wy+4)+2∑W ij≠=y k l Where N is the number of training examples. As you can see, we append the regularization penalty to the loss objective, weighted by a hyperparameter n. there is no simple way of setting this hyperparameter and it is usually determined by cross-validation In addition to the motivation we provided above there are many desirable properties to include the regularization penalty, many of which we will come back to in later sections for example it turns out that including the L2 penalty leads to the appealing max margin property in SVMs(See CS229 lecture notes for full details if you are interested The most appealing property is that penalizing large weights tends to improve generalization, because it means that no input dimension can have a very large influence on the scores all b itself. For example, suppose that we have some input vector x=l, 1, l, 1 and two weight vectors w1=[1,0O,0,O],w2=[0.25,0.25,0.25,0.25. Then wix=w2x=1 so both weight vectors lead to the same dot product, but the l2 penalty of wi is 1.0 while the l2 penalty of w2 is only 0.25. Therefore, according to the l2 penalty the weight vector w2 would be preferred since it achieves a lower regularization loss. Intuitively, this is because the weights in w2 are smaller and more diffuse. Since the L2 penalty prefers smaller and more diffuse weight vectors, the final classifier is encouraged to take into account all input dimensions to small amounts rather than a few input dimensions and very strongly. As we will see later in the class, this effect can improve the generalization performance of the classifiers on test images and lead to less overfitting Note that biases do not have the same effect since, unlike the weights, they do not control the strength of influence of an input dimension. Therefore, it is common to only regularize the weights w but not the biases b. However, in practice this often turns out to have a negligible tt p: c$231n.github. o/linear-classifyl 9/18 1231/2016 CS231n Convolutional Neural Networks for Visual Recognition effect. Lastly, note that due to the regularization penalty we can never achieve loss of exactly 0.0 on all examples, because this would only be possible in the pathological setting of W=o Code. Here is the loss function (without regularization) implemented in Python, in both unvectorized and half-vectorized form defL_ i(x, y, w) unvectorized version Compute the multiclass svm loss for a single example (x, y) x is a column vector representing an image(e.g. 3073 x I in CIFAR-10) with an appended bias dimension in the 3073-rd position (i.e. bias trick) y is an integer giving index of correct class(e. g. between 0 and 9 in CIFAR-10 W is the weight matrix(e.g. 10 x 3073 in CIFAR-10) delta = 1.0 #t see notes about delta later in this section scores= W.dot(x)# scores becomes of size 10x 1, the scores for each class correct_class_score= scoresby] D=W shape[0] number of classes, e. g. 10 loss i=0.0 for j in xrange(D): #f iterate over all wrong classes skip for the true class to only loop over incorrect classes continue f accumulate loss for the i-th example loss_i+= max(0, scoresll-correct class score t delta) return loss i defl_i_vectorized(x, y, w) A faster half-vectorized implementation. half-vectorized refers to the fact that for a single example the implementation contains no for loops, but there is still one loop over the examples(outside this function) delta = 1.0 scores W. dot(x) compute the margins for all classes in one vector operation margins np maximum(0, scores-scores[y]+ delta) f on y-th position scores/y/-scoresly/ canceled and gave delta. Wewant f to ignore the y-th position and only consider margin on max wrong class margins] loss_i=np. sum(margins) return loss i tt p: c$231n.github. o/linear-classifyl 10/18

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