From self-driving cars to model and dataset sizes

So I am done with teaching a vehicle to drive itself!

Errh, not quite there yet. I did it on a simulator, in an easy environment where there is only one lane, and no other traffic. This is very far from an actual self-driving vehicle.

Nevertheless, I had a lot of fun. It was actually way easier than I initially thought. It is simply a regression problem, where a CNN was trained to predict the steering angle. A vanila CNN with a significant amount of training data would do the job quite easily. Although it sounds simple, eventually this is how nVidia drives a car with their DAVE-2 system.

In practice, self-driving car is a bit more complicated. For example, nVidia’s paper didn’t show how they would handle traffic lights. I guess the Deep Learning way for that would be to collect a lot more data at crossroads, but I feel that would not be enough. At some point, you will need traditional engineering methods like sensor fusion to precisely locate the car on the road (more precise than what GPS provides), path finding for planning and all kinds of other signals.

However, every time I apply Deep Learning to a new domain, I learned something new. For this project, it is the following:

  • On the vehicle, there are 3 cameras: one in the middle, one on the left and one on the right. Normally you just need to train the CNN to map the image collected from the center camera to the steering angle, and be done with it. However, it turns out that you can use the side cameras to teach the vehicle to recover from mistakes. For example, if the car is taking a left turn, then you can use the image from the left camera to teach it to do a softer left turn, and the image from the right camera do a harder left turn. Using this approach, during inference, you only need to run inference on the center image. How much softer and harder should be empirically determined.
    You might think that you can read 3 images in the same time, and feed all three into the network, but that will require 3 images during inference, which might slow down the inference.
    In fact the above technique is used by nVidia in their paper, and it could help the vehicle to recover from mistake, for example when it is close to the edge of the road.
    Another data augmentation technique is to vertically flip the images, and reverse the steering angle. Using both techniques, you can augment the training set by a factor of 6.
  • Inference time is crucial. In the beginning, I struggled a lot making the model to work. Then at some point I realize that it took around 0.1 second to evaluate the model, which might be too slow to drive a car. I then reduce the size of the model, until the point where it takes 0.01 seconds to evaluate, then the vehicle starts driving smoothly.

So how small (or big) your model should be? This obviously depends on the training set,  but is there any rule of thumb? A related question that some people also asked me is how big the training set should be? We keep saying Deep Learning needs big datasets, but how big is big, or how big should it be to expect some sensible performance? I hope the rest of this post could answer those questions.

How big the model should be?

Let’s say you have a training set of N samples. Now if I use a simple array of bits to store those samples, then I would need N bits to store N samples (the first bit is ON given the first sample, and so on). More strictly, I could say I only need \log_2\left(N\right) bits to store N samples, because I could have N different configurations with that many bits.

In Deep Learning, we are well graduated from speaking in bits, but the same principle still holds. The easiest answer is you will need to construct your model so that it has N parameters to learn a training set of N samples.

That is still too lax though. Recall that a parameter in a neural net is a 32-bit floating point number, so a model of N parameters will have 32N bits in total. That’s why you would only need a model of \frac{N}{32} parameters?

Not that strict. Although the parameters in neural nets are floating points, their values are often small, typically in the range of -0.3 to 0.3 (depending on how you normalize the data). This is due to various tricks we apply to the nets like initialization and small learning rate, in order to make optimization easier.

Since their values are restricted, probably only a few bits in each parameters are carrying useful information. How many is that? Typically people think it is about 8 or 16 bits. The proof for that is when you quantize the nets to low-precision (of 8 or 16 bits), then the performance of the net doesn’t decrease much.

So, as a typical (wild) rule of thumb, you should be able to overfit a training set of size N with a model of \frac{N}{4} parameters. If you cannot overfit the training set, you are doing something really wrong with your initialization, learning rate and regularizer.

So you need to know how to count the number of parameters in a deep net. For fully connected layers, that simply is the size of the weight matrix and the biases. For convolutional layers, it is the size of the filter, multiplied by the number of filters. Most  modern Deep learning framework doesn’t use biases for convolutional layer, but in the past, people used to use a bias for each filter, so keep in mind that if you want to be very precise. The vanila RNN can be computed similarly.

LSTM is a bit more tricky, because there are a few variants of those: whether peephole is enabled, whether the forget bias is fixed, is it multi-dimensional LSTM, etc.. so the exact number might vary. However in general, the number of parameters of an LSTM layers of p units with inputs should be in the order of 5pq.

Some time ago I used to write a python script to compute the exact number of parameters in a MDLSTM cell, but looking at it now took me some time to understand it.

I hope this points out that the key advantage of Deep Learning, compared to traditional method, is we can engineer the model as big as we want, sometimes depending on the dataset. This is not easily doable with other models like SVM and the like.

How big is the training set?

Using a similar reasoning, you could also answer this pretty easily.

Assume that your input is a N-dimensional vector, then the maximum number of configuration in that space is 2^{32N}, which is enormous (sorry for using the word, you have Donald Trump to blame).

Of course that is the number of distinct configuration for all possible input. Your input domain is likely going to be a manifold in that high-dimensional space, meaning it will probably only take a tenth of that many degrees of freedom. So let’s say 2^{N/10}.

Now you don’t need every sample in your input domain to train a deep model. As long as your input domain is relatively smooth, and the training set covers the most important modes in the data distribution, the model should be able to figure out the missing regions. So again, probably you only need a fifth of those, meaning around 2^{N/50} samples.

For instance in MNIST, the input is of 28 * 28 = 784 dimensions, then you should have around 2^{784/50} \approx 32000 samples. In fact there are 50000 samples in the MNIST training set.

In general, I think the rule of thumb would be around tens of thousands samples for a typical problem so that you can expect some optimistic results.

Note that those calculations are very coarse, and should only be used to give some intuition. They shouldn’t be used as an exact calculation as-it-is.

The problem is worse with time series and sequential data in general. Using the same calculation, you would end up with pretty big numbers because you need to multiply the numbers by the length of the sequence. I don’t think the same calculation can be applied for sequential data, because in sequences, the correlation between consecutive elements also play a big role in learning, so that might lax or limit the degree of freedom of the data. However, I used to work with small sequence dataset of size around tens of thousands samples. For difficult datasets, we might need half a million of samples.

The more you work on modelling, the more you learn about it. As always, I would love to hear your experience!

 

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