In Part I, we saw a few examples of image classification. In particular counting objects seemed to be difficult for convolutional neural networks. After sharing my work on the fast.ai forums, I received a few suggestions and requests for further investigation.
The most common were:
Some transforms seemed uneccessary (eg. crop and zoom)
Some transforms might be more useful (eg. vertical flip)
Consider training the model from scratch (inputs come from a different distribution)
Try with more data
Try with different sizes
After regenerating our data we can look at it:
Now we can create a learner and train it on this new dataset.
Which gives us the following output:
Wow! Look at that, this time we’re getting 100% accuracy. It looks like if we throw enough data at it (and use proper transforms) this is a problem that can actually be trivially solved by convolutional neural networks. I honestly did not expect that at all going into this.
Different Sizes of Objects
One drawback of our previous dataset is that the objects we’re counting are all the same size. Is it possible this is making the task too easy? Let’s try creating a dataset with circles of various sizes.
Which allows us to create images that look something like:
Once again we can create a dataset this way and train a convolutional learner on it. Complete code on GitHub.
Still works! Once again I’m surprised. I had very little hope for this problem but these networks seem to have absolutely no issue with solving this.
This runs completely contrary to my expectations. I didn’t think we could count objects by classifying images. I should note that the network isn’t “counting” anything here, it’s simply putting each image into the class it thinks it would belong to. For example, if we showed it an example with 10 images, it would have to classify it as either “45”, “46”, “47”, “48” or “49”.
More generally, counting would probably make more sense as a regression problem than a classification problem. Still, this could be useful when trying to distinguish between object counts of a fixed and guaranteed range.
Over the last year I focused on what some call a “bottom-up” approach to studying deep learning. I reviewed linear algebra and calculus. I read Ian Goodfellow’s book “Deep Learning”. I built AlexNet, VGG and Inception architectures with TensorFlow.
While this approach helped me learn the bits and bytes of deep learning, I often felt too caught up in the details to create anything useful. For example, when reproducing a paper on superconvergence, I built my own ResNet from scratch. Instead of spending time running useful experiments, I found myself debugging my implementation and constantly unsure if I’d made some small mistake. It now looks like I did make some sort of implementation error as the paper was successfully reproduced by fast.ai and integrated into fast.ai’s framework for deep learning.
With all of this weighing on my mind I found it interesting that fast.ai advertised a “top-down” approach to deep learning. Instead of starting with the nuts and bolts of deep learning, they instead first seek to answer the question “How can you make the best/most accurate deep learning system?” and structure their course around this question.
The first lesson focuses on image classification via transfer learning. They provide a pre-trained ResNet-34 network that has learned weights using the ImageNet dataset. This has allowed it to learn various things about the natural world such as the existence of edges, corners, patterns and text.
After creating a competent pet classifier they recommend that students go out and try to use the same approach on a dataset of their own creation. For my part I’ve decided to try their approach on three different datasets, each chosen to be slightly more challenging than the last:
Our first step is simply to import everything that we’ll need from the fastai library:
Next we’ll take a look at the data itself. I’ve saved it in data/paintings. We’ll create an ImageDataBunch which automatically knows how to read labels for our data based off the folder structure. It also automatically creates a validation set for us.
Looking at the above images, it’s fairly easy to differentiate the solid lines of modernism from the soft edges and brush strokes of impressionist paintings. My hope is that this task will be just as easy for a pre-trained neural network that can already recognize edges and identify repeated patterns.
Now that we’ve prepped our dataset, we’ll prepare a learner and let it train for five epochs to get a sense of how well it does.
Looking good! With virtually no effort at all we have a classifier that reaches 95% accuracy. This task proved to be just as easy as expected. In the notebook we take things a further by choosing better learning rate and training for a little while longer before ultimately getting 100% accuracy.
The painting task ended up being as easy as we expected. For our second challenge we’re going to look at a dataset of about 180 cats and 180 kittens. Cats and kittens share many features (fur, whiskers, ears etc.) which seems like it would make this task harder. That said, a human can look at pictures of cats and kittens and easily differentiate between them.
This time our data is located in data/kittencat so we’ll go ahead and load it up.
Once again, let’s try a standard fastai CNN learner and run it for about 5 epochs to get a sense for how it’s doing.
So we’re looking at about 86% accuracy. Not quite the 95% we saw when classifying paintings but perhaps we can push it a little higher by choosing a good learning rate and running our model for longer.
Below we are going to use the “Learning Rate Finder” to (surprise, surprise) find a good learning rate. We’re looking for portions of the plot in which the graph steadily decreased.
It looks like there is a sweetspot between 1e-5 and 1e-3. We’ll shoot for the ‘middle’ and just use 1e-4. We’ll also run for 15 epochs this time to allow more time for learning.
Not bad! With a little bit of learning rate tuning, we were able to get a validation accuracy of about 92% which is much better than I expected considering we had less than 200 examples of each class. I imagine if we collected a larger dataset we could do even better.
For my last task I wanted to see whether or not we could train a ResNet to “count” identical objects. So far we have seen that these networks excel at distinguishing between different objects, but can these networks also identify multiple occurrences of something?
Note: I specifically chose this task because I don’t believe it should be possible for a vanilla ResNet to accomplish this task. A typical convolutional network is set up to differentiate between classes based on the features of those classes, but there is nothing in a convolutional network that suggests to me that it should be able to count objects with identical features.
For this challenge we are going to synthesize our own dataset using matplotlib. We’ll simply generate plots with the correct number of circles in them as shown below:
There are some things to note here:
When we create a dataset like this, we’re in uncharted territory as far as the pre-trained weights are concerned. Our network was trained on photographs of the natural world and expects its inputs to come from this distribution. We’re providing inputs from a completely different distribution (not necessarily a harder one!) so I wouldn’t expect transfer learning to work as flawlessly as it did in previous examples.
Our dataset might be trivially easy to learn. For example, if we wrote an algorithm that simply counted the number of “blue” pixels we could very accurately figure out how many circles were present as all circles are the same size.
We don’t need to hypothesize any further, though. We can just create our ImageDataBunch and pass it to a learner to see how well it does. For now we’ll just use a dataset with 1-5 elements.
Let’s create our learner and see how well it does with the defaults after 3 epochs.
So without any changes we’re sitting at over 85% accuracy. This surprised me as I thought this task would be harder for our neural network as each object it was counting has identical features. If we run this experiment again with a learning rate of 1e-4 and for 15 cycles things get even better:
Wow! We’ve pushed the accuracy up to 99%!
Ugh. This seems wrong to me…
I am not a deep learning pro but every fiber of my being screams out against convolutional networks being THIS GOOD at this task. I specifically chose this task to try to find a failure case! My understanding is that they should be able to identify composite features that occur in an image but there is nothing in there that says they should be able to count (or have any notion of what counting means!)
What I would guess is happening here is that there are certain visual patterns that can only occur for a given number of circles (for example, one circle can never create a line) and that our network uses these features to uniquely identify each class. I’m not sure how to prove this but I have an idea of how we might break it. Maybe we can put so many circles on the screen that the unique patterns will become very hard to find. For example, instead of trying 1-5 circles, let’s try counting images that have 45-50 circles.
After re-generating our data (see Notebook for details) we can visualize it below:
Now we can run our learner against this and see how it does:
Hah! That’s more like it. Now our network can only achieve ~25% accuracy which is slightly better than chance (1 in 5). Playing around with learning rate I was only able to achieve 27% on this task.
This makes more sense to me. There are no “features” in this image that would allow a network to look at it and instantly know how many circles are present. I suspect most humans can also not glance at one of these images and know whether or not there are 45 or 46 elements present. I suspect we would have to fall back to a different approach and manually count them out.
At the end of last year’s retrospective, I set a number of goals for myself. It feels (really) bad to look back and realize that I did complete a single one. I think it’s important to reflect on failures and shortcomings in order to understand them and hopefully overcome them going forward.
Goal 1: Write one blog post every week
Result: 13 posts / 52 weeks
In January 2018 I began the blog series Learn TensorFlow Now which walked users through the very basics of TensorFlow. For three months I stuck to my goal of writing one blog post every week and I’m very proud of how my published posts turned out. Unfortunately during April I took on a consulting project and my posts completely halted. Once I missed a single week I basically gave up on blogging altogether. While I don’t regret taking on a consulting project, I do regret that I used it as an excuse to stop blogging.
This year I would like to start over and try once again to write one blog post per week (off to a rough start considering it’s already the end of January!). I don’t really have a new strategy other than I will resolve not to quit entirely if I miss a week.
When I first started reading this book I was very intimidated by the first few chapters covering the background mathematics of deep learning. While my linear algebra was solid, my calculus was very weak. I put the book away for three months and grinded through Khan Academy’s calculus modules. I say “grinded” because I didn’t enjoy this process at all. Every day felt like a slog and my progress felt painfully slow. Even knowing calculus would ultimately be applicable to deep learning, I struggled to stay focused and interested in the work.
When I came back to the book in the second half of 2018 I realized it was a mistake to stop reading. While the review chapters were mathematically challenging, the actual deep learning portions were much less difficult and most of the insights could be reached without worrying about the math at all. For example, I cannot prove to you that L1 regularization results in sparse weight matrices, but I am aware that such a proof exists (at least in the case of linear regression).
This year I would like to finish this book. I think it might be worth my time to try to implement some of the basic algorithms illustrated in the book without the use of PyTorch or TensorFlow, but that will remain a stretch goal.
Goal 3: Contribute to TensorFlow
Result: 1 Contribution?
In February one of my revised PRs ended up making it into TensorFlow. Since I opened it in December of the previous year I’ve only marked it as half a contribution. Other than this PR I didn’t actively seek out any other places where I could contribute to TensorFlow.
On the plus side, I recently submitted a pull request to PyTorch. It’s a small PR that helps bring the C++ API closer to the Python API. Since it’s not yet merged I guess I should only count this as half a contribution? At least that puts me at one full contribution to deep learning libraries for the year.
Goal 4: Compete in a more Challenging Kaggle competition
Result: 0 attempts
There’s not much to say here other than that I didn’t really seek out or attempt any Kaggle competitions. In the later half of 2018 I began to focus on reinforcement learning so I was interested in other competitive environments such as OpenAI Gym and Halite.io. Unfortunately my RL agents were not very competitive when it came to Halite, but I’m hoping this year I will improve my RL knowledge and be able to submit some results to other competitions.
Goal 5: Work on HackerRank problems to strengthen my interview skills
Result: 3 months / 12 months
While I started off strong and completed lots of problems, I tapered off around the same time I stopped blogging. While I don’t feel super bad about stopping these exercises (I had started working, after all) I am a little sad because it didn’t really feel like I improved at solving questions. This remains an area I want to improve in but I don’t think I’m going to make it an explicit goal in 2019.
Goal 6: Get a job related to ML/AI
Result: 0 jobs
I did not receive (or apply to) any jobs in ML/AI during 2018. After focusing on consulting for most of the year I didn’t feel like I could demonstrate that I was proficient enough to be hired into the field. My understanding is that an end-to-end personal project is probably the best way to demonstrate true proficiency and something I want to pursue during 2019.
Goals for 2019
While I’m obviously not thrilled with my progress in 2018 I try not to consider failure a terminal state. I’m going to regroup and try to be more disciplined and consistent when it comes to my work this year. One activity that I’ve found both fun and productive is streaming on Twitch. I spent about 100 hours streaming and had a pretty consistent schedule during November and December.
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:
Accuracy: 11.9999997318 %
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%:
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.
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?
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.
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.
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.
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“.
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“.
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:
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:
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:
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:
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.
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
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.
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 643x3 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]:
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:
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.
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:
Create the tf.train.Saver()
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: