LTFN 12: Bias and Variance

Part of the series Learn TensorFlow Now

In the last few posts we noticed a strange phenomenon: our test accuracy was about 10% worse than what we were getting on our training set. Let’s review the results from our last network:

Cost: 131.964
Accuracy: 11.9999997318 %
...
Cost: 0.47334
Accuracy: 83.9999973774 %
Test Cost: 1.04789093912
Test accuracy: 72.5600001812 %

Our neural network is getting ~84% accuracy on the training set but only ~73% on the test set. What’s going on and how do we fix it?

Bias and Variance

Two primary sources of error in any machine learning algorithm come from either underfitting or overfitting your training data. Underfitting occurs when an algorithm is unable to model the underlying trend of the data. Overfitting occurs when the algorithm essentially memorizes the training set but is unable to generalize and performs poorly on the test set.

Bias is error introduced by underfitting a dataset. It is characterized by poor performance on both the training set and the test set.

Variance is error introduced by overfitting a dataset. It is characterized by a good performance on the training set, but a poor performance on test set.

We can look at bias and variance visually by comparing the performance of our network on the training set and test set. Recall our training accuracy of 84% and test accuracy of 73%:

Visualization of bias and variance from our previous network’s results

The above image roughly demonstrates which portions of our error can be attributed to bias and variance. This visualization assumes that we could theoretically achieve 100% accuracy. In practice this may not always be the case as other sources of error (eg. noise or mislabelled examples) may creep into our dataset. As an aside, the lowest theoretical error rate on a given problem is called the Bayes Error Rate.

Reducing Error

Ideally we would have a high performance on both the test set and training set which would represent low bias and low variance. So what steps can we take to reduce each of these sources of error?

Reducing Bias

  • Create a larger neural network. Recall that high bias is a sign that our neural network is unable to properly capture the underlying trend in our dataset. In general the deeper a network, the more complex the functions it can represent.
  • Train it for a very long time. One sanity check for any neural network is to see whether or not it can memorize the dataset. A sufficiently deep neural network should be able to memorize your dataset given enough training time. Although this won’t fix any problems with variance it can be an assurance that your network isn’t completely broken in some way.
  • Use a different architecture.  Sometimes your chosen architecture may simply be unable to perform well on a given task. It may be worth considering other architectures to see if they perform better. A good place to start with Image Recognition tasks is to try different architectures submitted to previous ImageNet competitions.

Reducing Variance

  • Get more data. One nice property of neural networks is that they typically generalize better and better as you feed them more data. If your model is having problems handling out-of-sample data one obvious solution is to feed it more data.
  • Augment your existing data. While “Get more data” is a simple solution, it’s often not easy in practice. It can take months to curate, clean and verify a large dataset. One workaround is to artifically generate “new” data by augmenting your existing data. For image recognition tasks this might include flipping or rotating existing images, tweaking color settings or taking random crops of images. This is a topic we’ll explore in greater depth in future posts.
  • Regularization. High variance with low bias suggests our network has memorized the training set. Regularization describes a class of modifications we can make to our neural network that either penalizes memorization (eg. L2 regularization) or promotes redundant paths of learning in our network (ie. Dropout). We will dive deeper into various regularization approaches in future posts.
  • Use a different architecture. Like reducing bias, sometimes you get the most bang-for-your-buck when you switch architectures altogether. As the deep learning field grows, people are frequently discovering better architectures for certain tasks. Some recent papers have even suggested that the structure of a neural network is more important than any learned weights for that structure.

There’s a lot to unpack here and we’ve glossed over many of the solutions to the problems of bias and variance. In the next few posts we’re going to revisit some of these ideas and explore different areas of the TensorFlow API that allow us to tackle these problems.

LTFN 11: Image Pre-processing

Part of the series Learn TensorFlow Now

In previous posts, we simply passed raw images to our neural network. Other forms of machine learning pre-process input in various ways, so it seems reasonable to look at these approaches and see if they would work when applied to a neural network for image recognition.

Zero Centered Mean

One characteristic we desire from any learning algorithm is for it to generalize across different input distributions. For example, let’s imagine we design an algorithm for predicting whether or not the price of a house is “High” or “Low“. As input it takes:

  • Number of Rooms
  • Price of House

Below is some made-up data for the city of Boston. I’ve marked “High” in red, “Low” in blue and a reasonable decision boundary that our algorithm might learn in black. Our decision boundary correctly classifies all examples of “High” and “Low“.

Classification of house prices in Boston

What happens when we take this model and apply it to houses in New York where houses are much more expensive? Below we can see that the model does not generalize and incorrectly classifies many “Low” house prices as “High“.

Classification of house prices in New York

In order to fix this, we want to take all of our data and zero-center it. To do this, we subtract the mean of each feature from from each data-point. For our examples this would look something like:

Zero centering the mean for Boston housing data
Zero centering the mean for New York housing data

Notice that we zero-center the mean for both the “Price” feature as well as the “Number of Rooms” feature. In general we don’t know which features might cause problems and which ones will not, so it’s easier just to zero-center them all.

Now that our data has a zero-centered mean, we can see how it would be easier to draw a single decision boundary that would accurately classify points from both Boston and New York. Zero centering our mean is one technique for handling data that comes from different distributions.

 

Changing Distributions in Images

It’s easy to see how the distribution of housing prices changes in different cities, but what would changes in distribution look like when we’re talking about images? Let’s imagine that we’re building an image classifier to distinguish between pictures of cats and pictures of dogs. Below is some sample data:

Training Data

Training images for our Cat vs. Dog classifier

Test Data

Test data for our Cat vs. Dog classifier

 

In the above classification task our cat images are coming from different distributions in our training and test sets. Our training set seems to contain exclusively black cats while our test set has a mix of colors. We would expect our classifier to fail on this task unless we take some time to fix our distribution problems. One way to fix this problem would be to fix our training set and ensure it contains many different colors of cats. Another approach we might take would be to zero-center the images, as we did with our housing prices.

 

Zero Centering Images

Now that we understand zero-centered means, how can we use this to improve our neural network? Recall that each pixel in an image is a feature, analogous to “Price” or “Number of Rooms” in our housing example. Therefore, we have to calculate the mean value for each pixel across the entire dataset. This gives us a 32x32x3 “mean image” which we can then subtract from every image we pass to our neural network.

You mean have noticed that the mean_image was automatically created for us when we called cifar_data_loader.load_data():

The mean image for the CIFAR-10 dataset looks something like:

Mean Image for CIFAR-10 dataset

Now we simply need to subtract the mean image from the input images in our neural network:

 

After running our network we’re greeted with the following output:

Cost: 131.964
Accuracy: 11.9999997318 %
Cost: 1.91737
Accuracy: 23.9999994636 %
Cost: 1.7101
Accuracy: 33.0000013113 %

...

Cost: 0.494887
Accuracy: 86.0000014305 %
Cost: 0.47334
Accuracy: 83.9999973774 %
Test Cost: 1.04789093912
Test accuracy: 72.5600001812 %

A test accuracy of 72.5% is a marginal increase over our previous result of 70.9% and it’s possible that our improvement is entirely due to chance. So why doesn’t zero centering the mean help much? Recall that zero-centering the mean leads to the biggest improvements when our data comes from different distributions. In the case of CIFAR-10, we have little reason to suspect that our portions of our images are obviously of different distributions.

Despite seeing only marginal improvements, we’ll continue to subtract the mean image from our input images. It imposes only a very small performance penalty and safeguards us against problems with distributions we might not anticipate in future datasets.

LTFN 10: CIFAR-10

Part of the series Learn TensorFlow Now

Over the last nine posts, we built a reasonably effective digit classifier. Now we’re ready to enter the big leagues and try out our VGGNet on a more challenging image recognition task. CIFAR-10 (Canadian Institute For Advanced Research) is a collection of 60,000 cropped images of planes, cars, birds, cats, deer, dogs, frogs, horses, ships, and trucks.

  • 50,000 images in the training set
  • 10,000 images in the test set
  • Size: 32×32 (1024 pixels)
  • 3 Channels (RGB)
  • 10 output classes
Sample images from CIFAR-10

CIFAR-10 is a natural next-step due to its similarities to the MNIST dataset. For starters, we have the same number of training images, testing images and output classes. CIFAR-10’s images are of size 32x32 which is convenient as we were paddding MNIST’s images to achieve the same size. These similarities make it easy to use our previous VGGNet architecture to classify these images.

Despite the similarities, there are some differences that make CIFAR-10 a more challenging image recognition problem. For starters, our images are RGB and therefore have 3 channels. Detecting lines might not be so easy when they can be drawn in any color. Another challenge is that our images are now 2-D depictions of 3-D objects. In the above image, the center two images represent the “truck” class, but are shown at different angles. This means our network has to learn enough about “trucks” to recognize them at angles it has never seen before.

Loading CIFAR-10

The CIFAR-10 dataset is hosted at: https://www.cs.toronto.edu/~kriz/cifar.html

In order to make it easier to work with, I’ve prepared a small script that downloads, shuffles and caches the dataset locally. You can find it on GitHub here.

After saving this file locally, we can use it to prepare our datasets:

Running this locally produces the following output:

Attempting to download: https://www.cs.toronto.edu/~kriz/cifar-10-python.tar.gz
0%....5%....10%....15%....20%....25%....30%....35%....40%....45%....50%....55%....60%....65%....70%....75%....80%....85%....90%....95%....100%
Download Complete!
(50000, 32, 32, 3)
(50000,)
(10000, 32, 32, 3)
(10000,)
(32, 32, 3)

The above output shows that we’ve downloaded the dataset and created a training set of size 50,000 and a test set of size 10,000. Note: Unlike MNIST, these labels are not 1-hot encoded (otherwise they’d be of size 50,000x10 and 10,000x10 respectively). We have to account for this difference in shape when we build VGGNet for this dataset.

Let’s start by adjusting input and labels to fit the CIFAR-10 dataset:

Next we have to adjust the first layer of our network. Recall from the post on convolutions that each convolutional filter must match the depth of the layer against which it is convolved. Previously we had defined our convolutional filter to be of shape [3, 3, 1, 64]. That is, a 64 3x3 convolutional filters, each with depth of 1, matching the depth of our grayscale input image. Now that we’re using RGB images, we must define it to be of shape [3, 3, 3, 64]:

Another change we must make is the calculation of cost. Previously we were using tf.nn.softmax_cross_entropy_with_logits() which is suitable only when our labels are 1-hot encoded. When we represent the labels as single integers, we can instead use tf.nn.sparse_softmax_cross_entropy_with_logits(). It is otherwise identical to our original softmax cross entropy function.

Finally, we must also modify our calculation of correction_prediction (used to calculate accuracy) to account for the change in label shape. We no longer have to take the tf.argmax of our labels because they’re already represented as a single number:

Note: We have to specify output_type=tf.int32 because tf.argmax() returns tf.int64 by default.

With that, we’ve got everything we need to test our VGGNet on CIFAR-10. The complete code is presented at the end of this post.

After running our network for 10,000 steps, we’re greeted with the following output:

Cost: 470.996
Accuracy: 9.00000035763 %
Cost: 2.00049
Accuracy: 25.0 %
...
Cost: 0.553867
Accuracy: 82.9999983311 %
Cost: 0.393799
Accuracy: 87.0000004768 %
Test Cost: 0.895597087741
Test accuracy: 70.9400003552 %

Our final test accuracy appears to be approximately 71%, which isn’t too great. On one hand this is disappointing as it means our VGGNet architecture (or the method in which we’re training it) doesn’t generalize very well. On the other hand, CIFAR-10 presents us with new opportunities to try out new neural network components and architectures. In the next few posts we’ll explore some of these approaches to build a neural network that can handle the more complex CIFAR-10 dataset.

If you look carefully at the previous results you may have noticed something interesting. For the first time, our test accuracy (71%) is much lower than our training accuracy (~82-87%). This is a problem we’ll discuss in future posts on bias and variance in deep learning.

Complete Code

LTFN 9: Saving and Restoring

Part of the series Learn TensorFlow Now

In the last post we looked at a modified version of VGGNet that achieved ~97.8% accuracy recognizing handwritten digits. Now that we’re relatively satisfied with our network, we’d like to save a trained version of the network that we can restore and use to classify digits whenever we’d like. We’ll do so by saving all of the tf.Variables() we’ve created to a checkpoint (.ckpt) file.

Saving a Checkpoint

When we save our computational graph, we serialize both the graph itself and the values of all of our parameters. When serializing nodes in our graph, TensorFlow keeps track of their names in order for us to interact with them later. Nodes that we don’t name will receive default names and be very hard to pick out. (While preparing this post I forgot to name input and labels which received the names Placeholder and Placeholder_1 instead). For this reason, we’ll take a minute to ensure that we give names to input, labels, cost, accuracy and predictions.

Saving a single checkpoint is straightforward. If we just want to save the state of our network after training then we simply add the following lines to the end of our previous network:

This snippet of code first creates a tf.train.Saver, an object that coordinates both saving and restoration of models. Next we call saver.save() passing in the current session. As a refresher, this session contains information about both the structure of the computational graph as well as the exact values of all parameters. By default the saver saves all tf.Variables() (weight/bias parameters) from our graph, but it also has the ability to save only portions of the graph.

After saving the checkpoint, the saver returns the save_path. Why return the save_path if we just provided it with a path? The saver also allows you to shard the saved checkpoint by device (eg. using multiple GPUs to train a model). In this situation, the returned save_path is appended with information on the number of shards created.

After running this code, we can navigate to the folder /tmp/vggnet/ and run ls -tralh to look at the contents:

-rw-rw-r--  1 jovarty jovarty 184M Mar 12 19:57 vgg_net.ckpt.data-00000-of-00001
-rw-rw-r--  1 jovarty jovarty 2.7K Mar 12 19:57 vgg_net.ckpt.index
-rw-rw-r--  1 jovarty jovarty  105 Mar 12 19:57 checkpoint
-rw-rw-r--  1 jovarty jovarty 188K Mar 12 19:57 vgg_net.ckpt.meta

The first file vgg_net.ckpt.data-00000-of-00001 is 184 MB in size and contains the values of all of our parameters. This is a reasonably large size and one of the reasons it’s nice to use networks with smaller numbers of parameters. This model is larger than most of the apps on my phone so it could be difficult to deploy to mobile devices.

The vgg_net.ckpt.meta file contains information on the structure of our computational graph and the names of all of our nodes. Later we’ll use this file to rebuild our computational graph from scratch.

 

Saving Multiple Checkpoints

Some neural networks are trained over the course of multiple weeks and we would like a way to periodically take checkpoints as our network learns. This allows us to go back in time and hand tune hyperparameters such as learning rate to try to squeeze the best performance out of our network. Fortunately, TensorFlow makes it easy to take checkpoints at any point during training. For example, we can modify our training loop to simply save a checkpoint whenever we print accuracy and cost.

The only real modification we’ve made here is to pass in global_step=step to track when each checkpoint was created. Be aware that this can eat up disk space relatively quickly depending on the size of your model. Each of our VGG checkpoints requires 184 MB of space.

 

Restoring a Model

Now that we know how to save our model’s parameters, how do we restore them? One way is to declare the original computational graph in Python and then restore the values to all the tf.Variables() (parameters) using tf.train.Saver.

For example, we could remove the training and testing code from our previous network and replace it with the following:

There are really only two additions to the code here:

  1. Create the tf.train.Saver()
  2. Restore the model to the current session. Note: This portion requires the graph to have been defined with identical names and parameters as when they were saved to a checkpoint.

Other than these changes, we test the network exactly as we would have before. If we wanted to test our network on new examples, we could load them into test_images and retrieve predictions from our graph instead of cost and accuracy.

This approach works well for networks we’ve built ourselves but it can be very cumbersome when we want to run networks designed by someone else. It takes hours to manually create each parameter and operation exactly as the original author had.

Restoring a Model from Scratch

One approach to using someone else’s neural network is to load up the computational graph defined in the .meta file before restoring the values to this graph from the .ckpt file. Below is a self-contained example of restoring a model from scratch:

There are a few subtle changes worth pointing out. First, we create our tf.train.Saver indirectly by importing the computational graph with tf.train.import_meta_graph(). Next, we restore the values to our computational graph with saver.restore() exactly as we had done previously.

Since we don’t have access to the input and labels nodes, we have to recover them from our graph with graph.get_tensor_by_name(). Notice that we are passing in the names that we had previously specified and appending :0 to these names. Some TensorFlow operations produce multiple outputs. When this happens, TensorFlow names them :0, :1 and so on until all the outputs have a unique name. All of the operations we’re using have only one output so we simply stick with :0.

Finally, the last change involves actually running the network. As in the previous step, we need to specify proper names for cost and accuracy because we don’t have direct access to the computational nodes. Fortunately, it’s simple to just pass in strings with the names 'cost:0' and 'accuracy:0' that specify which operations we want to run and return the values of. Alternatively, we could have recovered the nodes with graph.get_tensor_by_name() and passed them in directly.

Also note that if we had named our optimizer, we could have passed it into session.run() and continued to train our network. We could have even created a checkpoint of our saved network at this point if we decided it had improved in some way.

There are a variety of ways to save and restore models and we’ve really only scratched the surface. Below are a few self-contained examples of the various approaches we’ve looked at:

LTFN 8: Deeper ConvNets

Part of the series Learn TensorFlow Now

Now that we’ve got a handle on convolutions, max pooling and weight initialization the obvious question is: What’s next? How should we set up our network to achieve the maximum accuracy on image recognition tasks? For years this has been a focus of the ImageNet Large Scale Visual Recognition Challenge (ILSVRC) competitions. Since 2010 researchers have battled various architectures against one another in an attempt to categorize millions of images into 1,000 categories. When tackling any image recognition task it’s usually a good idea to pick one of the top performing architectures instead of trying to craft your own from scratch.

VGGNet

VGGNet is a nice starting point as it’s simply a deeper version of the network we’ve been building. Its debut in the 2013 ILSVRC competition was novel due to its exclusive use of 3x3 convolutional filters. Previous architectures had attempted to use a variety of filter sizes including 11x11, 7x7 and 5x5. Each of these filter sizes was a hyper-parameter that had to be tuned so it was a relief to see high performance with both a consistent and small filter size.

As with our previous network, VGG operates by staggering max-pooling layers between groups of convolutional layers. Below is a table listing the 16 layers of VGG alongside the intermediate shapes at each layer of the network and the number of trainable parameters (ie. weights, excluding biases) in the network.

Original VGGNet

Layers Parameters
Layer Shape Intermediate Shape
Input: 224x224x3
64 3×3 Conv Filters 224 x 224 x 64 64 * 3 * 3 * 3 = 1,728
64 3×3 Conv Filters 224 x 224 x 64 64 * 3 * 3 * 64 = 36,864
maxpool 2×2 112 x 112 x 64
128 3×3 Conv Filters 112 x 112 x 128 128 * 3 * 3 * 64 = 73,728
128 3×3 Conv Filters 112 x 112 x 128 128 * 3 * 3 * 128 = 147,456
maxpool 2×2 56 x 56 x 256
256 3×3 Conv Filters 56 x 56 x 256 256 * 3 * 3 * 128 = 294,912
256 3×3 Conv Filters 56 x 56 x 256 256 * 3 * 3 * 256 = 589,824
256 3×3 Conv Filters 56 x 56 x 256 256 * 3 * 3 * 256 = 589,824
maxpool 2×2 28 x 28 x 256
512 3×3 Conv Filters 28 x 28 x 512 512 * 3 * 3 * 256 = 1,179,648
512 3×3 Conv Filters 28 x 28 x 512 512 * 3 * 3 * 512 = 2,359,296
512 3×3 Conv Filters 28 x 28 x 512 512 * 3 * 3 * 512 = 2,359,296
maxpool 14 x 14 x 512
512 3×3 Conv Filters 14 x 14 x 512 512 * 3 * 3 * 512 = 2,359,296
512 3×3 Conv Filters 14 x 14 x 512 512 * 3 * 3 * 512 = 2,359,296
512 3×3 Conv Filters 14 x 14 x 512 512 * 3 * 3 * 512 = 2,359,296
maxpool 7 x 7 x 512
FC 4096 1 x 1 x 4096 7 * 7 * 512 * 4096 = 102,760,448
FC 4096 1 x 1 x 4096 4096 * 4096 = 16,777,216
FC 1000 1 x 1 x 1000 4096 * 1000 = 4,096,000

 

A few things to note about the VGG architecture:

  • It was originally built for images of size 224x224x3 and 1,000 output classes.
  • The number of parameters increases exponentially as we move through the network.
  • There are so many trainable parameters that we can only reasonably run such a network on a computer with a GPU.

There are a couple of modifications we’ll make to the VGG network in order to use it on our MNIST digits of shape 28x28x1. Notice that after each max_pooling layer we halve the width and height dimensions. Unfortunately, our images just aren’t big enough to go through so many max_pooling layers. For this reason, we’ll omit the final max_pooling layer and the final three 512 3x3 convolutional layers. We’ll also pad our 28x28 images to be of size 32x32 so the widths and heights divide by two cleanly.

Modified VGGNet

Layers Parameters
Layer Shape Intermediate Shape
Input: 28 x 28 x 1
Pad Image 32 x 32 x 1
64 3×3 Conv Filters 32 x 32 x 64 64 * 3 * 3 * 3 = 1,728
64 3×3 Conv Filters 32 x 32 x 64 64 * 3 * 3 * 64 = 36,864
maxpool 2×2 16 x 16 x 64
128 3×3 Conv Filters 16 x 16 x 128 128 * 3 * 3 * 64 = 73,728
128 3×3 Conv Filters 16 x 16 x 128 128 * 3 * 3 * 128 = 147,456
maxpool 2×2 8 x 8 x 256
256 3×3 Conv Filters 8 x 8 x 256 256 * 3 * 3 * 128 = 294,912
256 3×3 Conv Filters 8 x 8 x 256 256 * 3 * 3 * 256 = 589,824
256 3×3 Conv Filters 8 x 8 x 256 256 * 3 * 3 * 256 = 589,824
maxpool 2×2 4 x 4 x 256
512 3×3 Conv Filters 4 x 4 x 512 512 * 3 * 3 * 256 = 1,179,648
512 3×3 Conv Filters 4 x 4 x 512 512 * 3 * 3 * 512 = 2,359,296
512 3×3 Conv Filters 4 x 4 x 512 512 * 3 * 3 * 512 = 2,359,296
maxpool 2 x 2 x 512
FC 4096 1 x 1 x 4096 2 * 2 * 512 * 4096 = 8,388,608
FC 10 1 x 1 x 10 4096 * 10 = 40,960

 

In previous posts we’ve encountered fully connected layers, convolutional layers and max pooling operations. The only portion of this network we’ve not seen before is the initial padding step. TensorFlow makes this easy to accomplish via tf.image.resize_image_with_crop_or_pad.

We’ll also make use of the tf.train.AdamOptimizer discussed in the previous post:

With these two changes, we can create our modified version of VGGNet, presented in full at the end of this post.

Running our network gives us the following output:

Cost: 3.19188
Accuracy: 10.9999999404 %
Cost: 0.140771
Accuracy: 94.9999988079 %
Cost: 0.120058
Accuracy: 95.9999978542 %
Cost: 0.128447
Accuracy: 97.000002861 %
Cost: 0.0849798
Accuracy: 95.9999978542 %
Cost: 0.0180758
Accuracy: 99.0000009537 %
Cost: 0.0622907
Accuracy: 99.0000009537 %
Cost: 0.147945
Accuracy: 95.9999978542 %
Cost: 0.0502743
Accuracy: 99.0000009537 %
Cost: 0.149534
Accuracy: 99.0000009537 %
Test Cost: 0.0713789960416
Test accuracy: 97.8600007892 %

 

Running this network gives us a test accuracy of ~97.9% compared to our previous best of 97.3%. This is an improvement, but we’re starting to see fairly marginal improvements. In fact, I wouldn’t necessarily be convinced that our VGG network truly outperforms our previous best without running each network multiple times and comparing the average accuracies achieved. There’s a very real possibility that our small improvement may have just been due to chance. We won’t run this comparison here, but it’s something to consider when you’re starting to see very marginal improvements in your own networks.

Next week we’ll take a look at saving and restoring our model and we’ll take a look at some of the images on which our network is making mistakes in order to build a better intuition for what might be going on.

 

Complete Code

LTFN 7: A Quick Look at TensorFlow Optimizers

Part of the series Learn TensorFlow Now

So far we’ve managed to avoid the mathematics of optimization and treated our optimizer as a “black box” that does its best to find good weights for our network. In our last post we saw that it doesn’t always succeed: We had three networks with identical structures but different initial weights and our optimizer failed to find good weights for two of them (when the initial weights were too large in magnitude and when they were too small in magnitude).

I’ve avoided the mathematics primarily because I believe one can become a machine learning practitioner (but probably not researcher) without a deep understanding of the mathematics underlying deep learning. We’ll continue that tradition and avoid the bulk of the mathematics behind the optimization algorithms. That said, I’ll provide links to resources where you can dive into these topics if you’re interested.

There are three optimization algorithms you should be aware of:

  1. Stochastic Gradient Descent – The default optimizer we’ve been using so far
  2. Momentum Update – An improved version of stochastic gradient descent
  3. Adam Optimizer – Typically the best performing optimizer

 

Stochastic Gradient Descent

To keep things simple (and allow us to visualize what’s going on) let’s think about a network with just one weight. After we run our network on a batch of inputs we are given a cost. Our goal is to adjust the weight so as to minimize that cost. For example, the function could look something like the following (with our weight/cost highlighted):

A (completely made up) cost function, with a (completely made up) weight and corresponding cost

 

We can obviously look at this function and be confident that we want to increase weight_1. Ideally we’d just increase weight_1 to give us the cost at the bottom of the curve and be done after one step.

In reality, neither we nor the network have any idea of what the underlying function really looks like. We know three things:

  1. The value of weight_1
  2. The cost associated with our (one-weight) network
  3. A rough estimate of how much we should increase or decrease weight_1 to get a smaller cost

(That third piece of information is where I’ve hidden most of the math and complexities of neural networks away. It’s the gradient of the network and it is computed for all weights of the network via back-propagation)

With these three things in mind, a better visualization might be:

It’s a lot harder to tell how far we should increase weight_1 now, isn’t it?

 

So now we still know that we want to increase weight_1, but how much should we increase it? This is partially decided by learning_rate. Increasing learning_rate means that we adjust our weights by larger amounts.

The update step of stochastic gradient descent consists of:

  1. Find out which direction we should adjust the weights
  2. Adjust the weights by multiplying learning_rate by the gradient

We have been using this approach whenever we have been using tf.train.GradientDescentOptimizer.

 

Momentum Update

One problem with stochastic gradient descent is that it’s slow and can take a long time for the optimizer to converge on a good set of weights. One solution to this problem is to use momentum. Momentum simply means: “If we’ve been moving in the same direction for a long time, we should probably move faster and faster in that direction”.

We can accomplish this by adding a momentum factor (typically ~0.9) to our previous one-weight example:

We use velocity to keep track of the speed and direction in which weight_1 is increasing or decreasing. In general, momentum update works much better that stochastic gradient descent. For a math-focused look at why see: Why Momentum Works.

The TensorFlow momentum update optimizer is available at tf.train.MomentumOptimizer.

 

Adam Optimizer

The Adam Optimizer is my personal favorite optimizer simply because it seems to work the best. It combines the approaches of multiple optimizers we haven’t looked at so we’ll leave out the math and instead show a comparison of Adam, Momentum and SGD below:

 

Instead of using just one weight, this example uses two weights: x and y. Cost is represented on the z axis with blue colors representing smaller values and the star represented the global minimum.

Things to note:

  1. SGD is very slow. It doesn’t make it to the minima in the 120 training steps
  2. Momentum sometimes overshoots its target
  3. Adam seems to offer a somewhat reasonable balance between the two

The Adam Optimizer is available at tf.train.AdamOptimizer.

 

Additional Resources:

LTFN 6: Weight Initialization

Part of the series Learn TensorFlow Now

At the conclusion of the previous post, we realized that our first convolutional net wasn’t performing very well. It had a comparatively high cost (something we hadn’t seen before) and was performing slightly worse than a fully-connected network with the same number of layers.

Test results from 4-layer fully connected network:

Test Cost:  107.98408660641905
Test accuracy:  85.74999994039536 %

Test results from 4-layer Conv Net:

Test Cost: 15083.0833307
Test accuracy: 81.8799999356 %

 

As a refresher, here’s a visualization of the 4-layer ConvNet we built in the last post:

Visualization of layers from input through pool2 (Click to enlarge).

So how do we figure out what’s broken?

When writing any typical program we might fire up a debugger or even just use something like printf() to figure out what’s going on. Unfortunately neural networks makes this very difficult for us. We can’t really step through thousands of multiplication, addition and ReLU operations and expect to glean much insight. One common debugging technique is to visualize all of the intermediate outputs and try to see if there are any obvious problems.

Let’s take a look at a histogram of the outputs of each layer before they’re passed through the ReLU non-linearity. (Remember, the ReLU operation simply chops off all negative values).

If you look closely at the above plots you’ll notice that the variance increases substantially at each layer (TensorBoard doesn’t let me adjust the scales of each plot so it’s not immediately obvious). The majority of outputs at layer1_conv are within the range [-1,1], but by the time we get to layer4_conv the outputs vary between [-20,000, 20,000]. If we continue adding layers to our network this trend will continue and eventually our network will run into problems with overflow. In general we’d prefer our intermediate outputs to remain within some fixed range.

How does this relate to our high cost? Let’s take a look at the values of our logits and  predictions. Recall that these values are calculated via:

 

The first thing to notice is that like the previous layers, the values of logits have a large variance with some values in the hundreds of thousands. The second thing to notice is that once we take the softmax of logits to create predictions all of our values are reduced to either 1 or 0. Recall that tf.nn.softmax takes logits and ensures that the ten values add up to 1 and that each value represents the probability a given image is represented by each digit. When some of our logits are tens of thousands of times bigger than the others, these values end up dominating the probabilities.

The visualization of predictions tells us that our network is super confident about the predictions it’s making. Essentially our network is claiming that it is 99% sure of its predictions. Whenever our network makes a mistake it is making a huge mistake and receives a large cost penalty for it.

The problem with increasing (magnitude) intermediate outputs translates directly into an increased cost. So how do we fix this? We want restrict the magnitude of the intermediate outputs of our network so they don’t increase so drastically at each layer.

 

Smaller Initial Weights

Recall that each convolution operation takes the dot product of our weights with a portion of the input. Basically, we’re multiplying and adding up a bunch of numbers similar to the following:

w0*i0 + w1*i1 + w2*i2 + … wn*in = output

Where:

  • wx – Represents a single weight
  • ix – Represents a single input (eg. pixel)
  • n – The number of weights

One way to reduce the magnitude of this expression is to reduce the magnitude of all of our weights by some factor:

0.01*w0*i0 + 0.01*w1*i1 + 0.01*w2*i2 + … 0.01*wn*in = 0.01*output

Let’s try it and see if it works! We’ll modify the creation of our weights by multiplying them all by 0.01. Therefore layer1_weights would now be defined as:

After changing all five sets of weights (don’t forget about the fully-connected layer at the end), we can run our network and see the following test cost and accuracies:

Test Cost:  2.3025865221
Test accuracy:  5.01999998465 %

Yikes! The cost has decreased quite a bit, but that accuracy is abysmal… What’s going on this time? Let’s take a look at the intermediate outputs of the network:

If you look closely at the scales, you’ll see that this time the intermediate outputs are decreasing! The first layer’s outputs lie largely within the interval [-0.02, 0.02] while the fourth layer generates outputs that lie within [-0.0002, 0.0002]. This is essentially the opposite of the problem we saw before.

Let’s also examine the logits and predictions as we did before:

 

This time the logits vary over a very small interval [-0.003, 0.003] and predictions are completely uniform. The predictions appear to be centered around 0.10 which seems to indicate that our network is simply predicting each of the ten digits with 10% probability. In other words, our network is learning nothing at all and we’re in an even worse state than before!

 

Choosing the Perfect Initial Weights

What we’ve learned so far:

  1. Large initial weights lead to very large output in intermediate layers and an over-confident network.
  2. Small initial weights lead to very small output in intermediate layers and a network that doesn’t learn anything.

So how do we choose initial weights that are not too small and not too large? In 2013, Xavier Glorot and Yoshua Bengio published Understanding the difficulty of training deep forward neural networks in which they proposed initializing a set of weights based on how many input and output nerons are present for a given weight. For more on this initialization scheme see An Explanation of Xavier Initialization. This initialization scheme is called Xavier Initialization.

It turns out that Xavier Initialization does not work for layers using the asymmetric ReLU activation function. So while we can use it on our fully connected layer we can’t use it for our intermediate layers. However in 2015 Microsoft Research (Kaiming He et al.) published Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification. In this paper they introduced a modified version of Xavier Initialization called Variance Scaling Initialization.

The math behind these initialization schemes is out of scope for this post, but TensorFlow makes them easy to use. I recommend simply remembering:

  1. Use Xavier Initialization in the fully-connected layers of your network. (Or layers that use softmax/tanh activation functions)
  2. Use Variance Scaling Initialization in the intermediate layer of your network that use ReLU activation functions.

We can modify the initialization of layer1_weights from tf.random.normal to use tf.contrib.layers.variance_scaling_initializer() as follows:

We can also modify the fully connected layer’s weights to use tf.contrib.xavier_initializer as follows:

There are a few small changes to note here. First, we use tf.get_variable instead of calling tf.Variable directly. This allows us to pass in a custom initializer for our weights. Second, we have to provide a unique name for our variable. Typically I just use the same name as my variable name.

If we continue changing all the weights in our network and run it, we can see the following output:

Cost: 2.49579
Accuracy: 9.00000035763 %
Cost: 1.05762
Accuracy: 77.999997139 %
...
Cost: 0.110656
Accuracy: 94.9999988079 %
Test Cost: 0.0945288215741
Test accuracy: 97.2900004387 %

 

Much better! This is a big improvement over our previous results and we can see that both cost and accuracy have improved substantially. For the sake of curiosity, let’s look at the intermediate outputs of our network:

 

 

This looks much better. The variance of the intermediate values appears to increase only slightly as we move through the layers and all values are within about an order of magnitude of one another. While we can’t make any claims about the intermediate outputs being “perfect” or even “good”, we can at least rest assured that there is no glaringly obvious problems with them. (Sidenote: This seems to be a common theme in deep learning: We usually can’t prove we’ve done things correctly, we can only look for signs that we’ve done them incorrectly).

 

Thoughts on Weights

Hopefully I’ve managed to convince you of the importance of choosing good initial weights for a neural network. Fortunately when it comes to image recognition, there are well-known initialization schemes that pretty much solve this problem for us.

The problems with weight initialization should highlight the fragility of deep neural networks. After all, we would hope that even if we choose poor initial weights, after enough time our gradient descent optimizer would manage to correct them and settle on good values for our weights. Unfortunately that doesn’t seem to be the case, and our optimizer instead settles into a relatively poor local minima.

 

Complete Code