on 23-Apr-2017 (Sun)

While most discussions of Ramsey conditionals and most “solutions” to Gardenfors’ challenge assume a purely syntactic, and purely propositional, perspective (following in this sense Gardenfors himself and the original AGM approach), in this paper we adopt a semantic, and more specifically modal, point of view. While Ramsey, Gardenfors and others only give conditions for “acceptance” (i.e. belief by an implicit agent) of a Ramsey conditional, we are primarily interested in its truth conditions. Indeed, from a semantic and modal p erspective, it makes no sense to ask for belief (or acceptance) of a statement whose extensional meaning was not yet defined. In this sense, Ramsey’s test cannot be taken in itself as a “definition” of a Ramsey con- ditional A > B, since it does not specify in what possible worlds might the conditional be true. So, in a semantic-modal setting, the important question is: are there any truth conditions for A > B that would be compatible with Ramsey’s test (given the usual modal semantics for belief, and given some reasonable semantics for belief revision)?

As usual in Kripke semantics, we do not identify a state of the world with its purely ontic content (represented by the valuation: the atomic sen- tences p satisfied by a state s represent the “ontic facts” of the world s). The world consists of more than just ontic facts: for instance, an (implicit) agent’s knowledge and beliefs at world s are also determined (via the doxastic accessibility relation) by the world s.

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Image Classification problem, which is the task of assigning an input image one label from a fixed set of categories.

Assigning a label to an image (aka image classification) is one of the core problems in Computer Vision that, despite its simplicity, has a large variety of practical applications.

Image classification model takes a single image and assigns probabilities to labels, {cat, dog, hat, mug, ...}. Our task is to turn this quarter of a million numbers into a single label, such as “cat”.

Challenges. Since this task of recognizing a visual concept (e.g. cat) is relatively trivial for a human to perform, it is worth considering the challenges involved from the perspective of a Computer Vision algorithm. As we present (an inexhaustive) list of challenges below, keep in mind the raw representation of images as a 3-D array of brightness values:

• Viewpoint variation. A single instance of an object can be oriented in many ways with respect to the camera.
• Scale variation. Visual classes often exhibit variation in their size (size in the real world, not only in terms of their extent in the image).
• Deformation. Many objects of interest are not rigid bodies and can be deformed in extreme ways.
• Occlusion. The objects of interest can be occluded. Sometimes only a small portion of an object (as little as few pixels) could be visible.
• Illumination conditions. The effects of illumination are drastic on the pixel level.
• Background clutter. The objects of interest may blend into their environment, making them hard to identify.
• Intra-class variation. The classes of interest can often be relatively broad, such as chair. There are many different types of these objects, each with their own appearance.

A good image classification model must be invariant to the cross product of all these variations, while simultaneously retaining sensitivity to the inter-class variations.

Data-Driven Approach: we’re going to provide the computer with many examples of each class and then develop learning algorithms that look at these examples and learn about the visual appearance of each class. This approach is referred to as a data-driven approach, since it relies on first accumulating a training dataset of labeled images.

The image classification pipeline. We’ve seen that the task in Image Classification is to take an array of pixels that represents a single image and assign a label to it. Our complete pipeline can be formalized as follows:

• Input: Our input consists of a set of N images, each labeled with one of K different classes. We refer to this data as the training set.
• Learning: Our task is to use the training set to learn what every one of the classes looks like. We refer to this step as training a classifier, or learning a model.
• Evaluation: In the end, we evaluate the quality of the classifier by asking it to predict labels for a new set of images that it has never seen before. We will then compare the true labels of these images to the ones predicted by the classifier. Intuitively, we’re hoping that a lot of the predictions match up with the true answers (which we call the ground truth).

image classification dataset: CIFAR-10. One popular toy image classification dataset is the CIFAR-10 dataset. This dataset consists of 60,000 tiny images that are 32 pixels high and wide. Each image is labeled with one of 10 classes (for example “airplane, automobile, bird, etc”). These 60,000 images are partitioned into a training set of 50,000 images and a test set of 10,000 images. In the image below you can see 10 random example images from each one of the 10 classes:

Left: Example images from the CIFAR-10 dataset. Right: first column shows a few test images and next to each we show the top 10 nearest neighbors in the training set according to pixel-wise difference.

You may have noticed that we left unspecified the details of exactly how we compare two images, which in this case are just two blocks of 32 x 32 x 3. One of the simplest possibilities is to compare the images pixel by pixel and add up all the differences. In other words, given two images and representing them as vectors I1,I2I1,I2 , a reasonable choice for comparing them might be the L1 distance:\(d_1 (I_1, I_2) = \sum_{p} \left| I^p_1 - I^p_2 \right|\)

Where the sum is taken over all pixels. Here is the procedure visualized:

Let’s also look at how we might implement the classifier in code. First, let’s load the CIFAR-10 data into memory as 4 arrays: the training data/labels and the test data/labels. In the code below, Xtr (of size 50,000 x 32 x 32 x 3) holds all the images in the training set, and a corresponding 1-dimensional array Ytr (of length 50,000) holds the training labels (from 0 to 9):

Xtr, Ytr, Xte, Yte = load_CIFAR10('data/cifar10/') # a magic function we provide
# flatten out all images to be one-dimensional
Xtr_rows = Xtr.reshape(Xtr.shape[0], 32 * 32 * 3) # Xtr_rows becomes 50000 x 3072
Xte_rows = Xte.reshape(Xte.shape[0], 32 * 32 * 3) # Xte_rows becomes 10000 x 3072

Now that we have all images stretched out as rows, here is how we could train and evaluate a classifier:

nn = NearestNeighbor() # create a Nearest Neighbor classifier class
nn.train(Xtr_rows, Ytr) # train the classifier on the training images and labels
Yte_predict = nn.predict(Xte_rows) # predict labels on the test images
# and now print the classification accuracy, which is the average number
# of examples that are correctly predicted (i.e. label matches)
print 'accuracy: %f' % ( np.mean(Yte_predict == Yte) )

evaluation criterion, it is common to use the accuracy, which measures the fraction of predictions that were correct.

Notice that all classifiers we will build satisfy this one common API: they have a train(X,y) function that takes the data and the labels to learn from. Internally, the class should build some kind of model of the labels and how they can be predicted from the data. And then there is a predict(X) function, which takes new data and predicts the labels.

Here is an implementation of a simple Nearest Neighbor classifier with the L1 distance that satisfies this template:

import numpy as np class NearestNeighbor(object): def __init__(self): pass def train(self, X, y): """ X is N x D where each row is an example. Y is 1-dimension of size N """ # the nearest neighbor classifier simply remembers all the training data self.Xtr = X self.ytr = y def predict(self, X): """ X is N x D where each row is an example we wish to predict label for """ num_test = X.shape[0] # lets make sure that the output type matches the input type Ypred = np.zeros(num_test, dtype = self.ytr.dtype) # loop over all test rows for i in xrange(num_test): # find the nearest training image to the i'th test image # using the L1 distance (sum of absolute value differences) distances = np.sum(np.abs(self.Xtr - X[i,:]), axis = 1) min_index = np.argmin(distances) # get the index with smallest distance Ypred[i] = self.ytr[min_index] # predict the label of the nearest example return Ypred

If you ran this code, you would see that this classifier only achieves 38.6% on CIFAR-10. That’s more impressive than guessing at random (which would give 10% accuracy since there are 10 classes), but nowhere near human performance (which is estimated at about 94%) or near state-of-the-art Convolutional Neural Networks that achieve about 95%, matching human accuracy (see the leaderboard of a recent Kaggle competition on CIFAR-10).

The choice of distance. There are many other ways of computing distances between vectors. Another common choice could be to instead use the L2 distance, which has the geometric interpretation of computing the euclidean distance between two vectors. The distance takes the form:

\(d_2 (I_1, I_2) = \sqrt{\sum_{p} \left( I^p_1 - I^p_2 \right)^2}\)
In other words we would be computing the pixelwise difference as before, but this time we square all of them, add them up and finally take the square root. In numpy, using the code from above we would need to only replace a single line of code. The line that computes the distances:

distances = np.sqrt(np.sum(np.square(self.Xtr - X[i,:]), axis = 1))

Note that I included the np.sqrt call above, but in a practical nearest neighbor application we could leave out the square root operation because square root is a monotonic function. That is, it scales the absolute sizes of the distances but it preserves the ordering, so the nearest neighbors with or without it are identical. If you ran the Nearest Neighbor classifier on CIFAR-10 with this distance, you would obtain 35.4% accuracy (slightly lower than our L1 distance result).

L1 vs. L2. It is interesting to consider differences between the two metrics. In particular, the L2 distance is much more unforgiving than the L1 distance when it comes to differences between two vectors. That is, the L2 distance prefers many medium disagreements to one big one. L1 and L2 distances (or equivalently the L1/L2 norms of the differences between a pair of images) are the most commonly used special cases of a p-norm.

k - Nearest Neighbor Classifier

You may have noticed that it is strange to only use the label of the nearest image when we wish to make a prediction. Indeed, it is almost always the case that one can do better by using what’s called a k-Nearest Neighbor Classifier. The idea is very simple: instead of finding the single closest image in the training set, we will find the top k closest images, and have them vote on the label of the test image. In particular, when k = 1, we recover the Nearest Neighbor classifier. Intuitively, higher values of k have a smoothing effect that makes the classifier more resistant to outliers:

An example of the difference between Nearest Neighbor and a 5-Nearest Neighbor classifier, using 2-dimensional points and 3 classes (red, blue, green). The colored regions show the decision boundaries induced by the classifier with an L2 distance. The white regions show points that are ambiguously classified (i.e. class votes are tied for at least two classes). Notice that in the case of a NN classifier, outlier datapoints (e.g. green point in the middle of a cloud of blue points) create small islands of likely incorrect predictions, while the 5-NN classifier smooths over these irregularities, likely leading to better generalization on the test data (not shown). Also note that the gray regions in the 5-NN image are caused by ties in the votes among the nearest neighbors (e.g. 2 neighbors are red, next two neighbors are blue, last neighbor is green).

In practice, you will almost always want to use k-Nearest Neighbor.

Validation sets for Hyperparameter tuning

The k-nearest neighbor classifier requires a setting for k. But what number works best? Additionally, we saw that there are many different distance functions we could have used: L1 norm, L2 norm, there are many other choices we didn’t even consider (e.g. dot products). These choices are called hyperparameters and they come up very often in the design of many Machine Learning algorithms that learn from data. It’s often not obvious what values/settings one should choose.

You might be tempted to suggest that we should try out many different values and see what works best. That is a fine idea and that’s indeed what we will do, but this must be done very carefully. In particular, we cannot use the test set for the purpose of tweaking hyperparameters. Whenever you’re designing Machine Learning algorithms, you should think of the test set as a very precious resource that should ideally never be touched until one time at the very end. Otherwise, the very real danger is that you may tune your hyperparameters to work well on the test set, but if you were to deploy your model you could see a significantly reduced performance. In practice, we would say that you overfit to the test set. Another way of looking at it is that if you tune your hyperparameters on the test set, you are effectively using the test set as the training set, and therefore the performance you achieve on it will be too optimistic with respect to what you might actually observe when you deploy your model. But if you only use the test set once at end, it remains a good proxy for measuring the generalization of your classifier (we will see much more discussion surrounding generalization later in the class).

Evaluate on the test set only a single time, at the very end.

Luckily, there is a correct way of tuning the hyperparameters and it does not touch the test set at all. The idea is to split our training set in two: a slightly smaller training set, and what we call a validation set. Using CIFAR-10 as an example, we could for example use 49,000 of the training images for training, and leave 1,000 aside for validation. This validation set is essentially used as a fake test set to tune the hyper-parameters.

Here is what this might look like in the case of CIFAR-10:

# assume we have Xtr_rows, Ytr, Xte_rows, Yte as before
# recall Xtr_rows is 50,000 x 3072 matrix
Xval_rows = Xtr_rows[:1000, :] # take first 1000 for validation
Yval = Ytr[:1000]
Xtr_rows = Xtr_rows[1000:, :] # keep last 49,000 for train
Ytr = Ytr[1000:] # find hyperparameters that work best on the validation set
validation_accuracies = []
for k in [1, 3, 5, 10, 20, 50, 100]: # use a particular value of k and evaluation on validation data nn = NearestNeighbor() nn.train(Xtr_rows, Ytr) # here we assume a modified NearestNeighbor class that can take a k as input Yval_predict = nn.predict(Xval_rows, k = k) acc = np.mean(Yval_predict == Yval) print 'accuracy: %f' % (acc,) # keep track of what works on the validation set validation_accuracies.append((k, acc))

By the end of this procedure, we could plot a graph that shows which values of k work best. We would then stick with this value and evaluate once on the actual test set.

Split your training set into training set and a validation set. Use validation set to tune all hyperparameters. At the end run a single time on the test set and report performance.

Cross-validation. In cases where the size of your training data (and therefore also the validation data) might be small, people sometimes use a more sophisticated technique for hyperparameter tuning called cross-validation. Working with our previous example, the idea is that instead of arbitrarily picking the first 1000 datapoints to be the validation set and rest training set, you can get a better and less noisy estimate of how well a certain value of k works by iterating over different validation sets and averaging the performance across these. For example, in 5-fold cross-validation, we would split the training data into 5 equal folds, use 4 of them for training, and 1 for validation. We would then iterate over which fold is the validation fold, evaluate the performance, and finally average the performance across the different folds.

Example of a 5-fold cross-validation run for the parameter k. For each value of k we train on 4 folds and evaluate on the 5th. Hence, for each k we receive 5 accuracies on the validation fold (accuracy is the y-axis, each result is a point). The trend line is drawn through the average of the results for each k and the error bars indicate the standard deviation. Note that in this particular case, the cross-validation suggests that a value of about k = 7 works best on this particular dataset (corresponding to the peak in the plot). If we used more than 5 folds, we might expect to see a smoother (i.e. less noisy) curve.

In practice. In practice, people prefer to avoid cross-validation in favor of having a single validation split, since cross-validation can be computationally expensive. The splits people tend to use is between 50%-90% of the training data for training and rest for validation. However, this depends on multiple factors: For example if the number of hyperparameters is large you may prefer to use bigger validation splits. If the number of examples in the validation set is small (perhaps only a few hundred or so), it is safer to use cross-validation. Typical number of folds you can see in practice would be 3-fold, 5-fold or 10-fold cross-validation.

Common data splits. A training and test set is given. The training set is split into folds (for example 5 folds here). The folds 1-4 become the training set. One fold (e.g. fold 5 here in yellow) is denoted as the Validation fold and is used to tune the hyperparameters. Cross-validation goes a step further and iterates over the choice of which fold is the validation fold, separately from 1-5. This would be referred to as 5-fold cross-validation. In the very end once the model is trained and all the best hyperparameters were determined, the model is evaluated a single time on the test data (red).

Pros and Cons of a Nearest Neighbor classifier.

Advantages and drawbacks of the Nearest Neighbor classifier:

Advantages:

• very simple to implement and understand.
• the classifier takes no time to train, since all that is required is to store and possibly index the training data

Disadvantages:

• we pay a computational cost at test time, since classifying a test example requires a comparison to every single training example.
• This is backward since in practice we often care about the test time efficiency much more than the efficiency at training time.

Side note: deep neural networks shift the tradeoff:

• They are very expensive to train, but once the training is finished it is very cheap to classify a new test example.
• Expensive training fast classification is much more desirable in practice.

• Nearest Neighbor classifier is an active area of research,
• There are several Approximate Nearest Neighbor (ANN) algorithms and libraries exist that can accelerate the nearest neighbor lookup in a dataset (e.g. FLANN).
• These algorithms allow one to trade off the correctness of the nearest neighbor retrieval with its space/time complexity during retrieval and usually rely on a pre-processing/indexing stage that involves building a kD-tree, or running the k-means algorithm.
• The Nearest Neighbor Classifier may sometimes be a good choice in some settings (especially if the data is low-dimensional)
• But Nearest Neighbor Classifiers are rarely appropriate for use in practical image classification settings.
• One problem is that images are high-dimensional objects (i.e. they often contain many pixels), and distances over high-dimensional spaces can be very counter-intuitive.