End to End Learning for Self-Driving Cars and the Distribution Mismatch Problem

I recently came across this interesting paper by the NVIDIA autonomous driving team.

I wrote a summary and a few comments about it on my Twitter account. And I thought maybe I can repost it here, with some additional discussions, to rekindle this dormant blog. So here you are. As always, your comments are appreciated.


The NVIDIA group formulates the problem of learning how to drive as an imitation learning problem. It learns a mapping from the image input to the steering command by imitating how a human driver does that.

Their approach is essentially a modern (mid 2010s) version of ALVINN from late 1980s: more data, deeper neural networks, and more computation power.
The function approximator is a convolutional neural network (a normalization + 5 convolutional + 3 fully connected). They use a lot of collected data based on actual driver’s behaviour to train their network (about 70 hours of real driving, which I believe corresponds to about 2.5M data samples — not explicitly mentioned) and some data augmentation. You can see the video of the self-driving car here. Cool, isn’t it?!


It is exciting to see an end-to-end neural network learned how to perform relatively well. I congratulate them on this. But there are potential problems from machine learning perspective: Treating the imitation learning problem as a standard supervised learning problem may lead to lower performance than expected. This is due to the distribution mismatch caused by the dynamical nature of the agent-environment interaction: When an agent (e.g., self-driving car) makes a mistake at each time step, the distribution of the future states slightly changes compared to the distribution induced by the expert agent (e.g., human driver). This has a compounding effect and the difference in distributions can potentially grow as the agent makes more interactions with the environment. In the self-driving car example, it means that a series of small mistakes by a self-driving car moves the car to situations that are farther and farther away from the usual situation of a car driven by a human, e.g., the car gradually gets dangerously close to the shoulder.

As a result, as time passes, the agent is more likely to be in regions of the state space from which it doesn’t have much training data (generated by the expert agent). So the agent starts behaving in ways that are not predictable even though it might perform well on the training distribution. This difference between two distributions is called the distribution mismatch (or covariate shift) problem in the machine learning/statistics literature.

A solution to this problem is to use DAGGER-like algorithms:

The basic idea behind DAGGER is that instead of letting the agent only learn on a fixed training data coming from an expert agent (which is a human driver in this case), we should let it learn on the distribution that the agent itself actually encounters. So if it happens that the agent goes to regions of the state space which are not usually encountered by the expert (so not in the initial training data set), well, that’s OK, because we can ask the expert to tell us what to do then, and hopefully the expert also knows how to deal with those situations.  By keep training on the data from this distribution, the agent can learn a policy that is much better.

Of course, if the “expert” itself is not a real expert for certain situations, we cannot really hope to learn a useful agent even if we use DAGGER. For example, most drivers know how to drive a car that is a bit over the line to the centre of the lane, so they are expert in that situation and their expertise can be useful; but they may not be of any real use how to deal with a car that is in a ditch. Not being able to be better than the expert is a limitation of imitation learning. There are some solutions for that, but maybe that should be the topic of another post.

Aside the aforementioned work, which analyzes the phenomenon in the imitation learning setting, the analysis of how the distribution of the agent’s changes, in the context of reinforcement learning, has been done by several researchers, including myself. I only refer to three papers. See their references for further information.

Anyway, it is nice to see a self-driving car that is not based on a lot of manual design and engineering, but is heavily based on the principles of machine learning. Of course, there are a lot more to be done and I am sure that the NVIDIA team will improve their system.

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Advice for Graduate Students in Statistics

Advice for Graduate Students in Statistics by Michael Steel. I like this guy! (:

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Nonparametric Bayesian Methods

I collect a few references/tutorials to nonparametric Bayesian methods for inference.

I will try to collect a few papers that analyze the convergence rate of these nonparametric Bayesian methods in a future post. I remember I’d found a few! (;

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Reinforcement Learning blog

Satinder Singh started a new blog named Reinforcement Learning blog. Now, I can see that Michael Littman is one of the authors of the blog – though he hasn’t published any post yet. This is a good news for RL. Good luck to them!

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Bracketing Entropy Bounds for Distribution Function

Sh. Song and J. Wellner, “How many distribution functions are there? Bracketing entropy bounds for high-dimensional distribution functions,” 2008.

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Embedding, Metric Entropy, etc.

Dorothee D. Haroske, “Embeddings Of Some Weighted Function Spaces On R_n: Entropy and Approximation Numbers (A Survey of some recent results),” 1997.

Ridgway Scott, “Tutorial on Sobolev Spaces,

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Shannon Sampling and Learning Theory

These two papers by Steve Smale and Ding-Xuan Zhou:

Shannon Sampling and Function Reconstruction from Point Values
Shannon Sampling II. Connections to Learning Theory

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Statistical Performance of Support Vector Machines

Tonight, I found this paper by Blanchard, Bousquet, and Massart: “Statistical Performance of Support Vector Machines” (2008)

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Compression-related ideas in Machine Learning

Shuheng Zhou, John Lafferty, and Larry Wasserman, “Compressed Regression,” 2008.

Boris Ryabko, “Compression-based methods for nonparametric density estimation, online prediction, regression and classification for time series,”

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A few papers on estimation and control of robotic systems

These are papers that I want to read (or can be considered for our reading group).

Seems to be relevant to our IROS 2007 paper (A. M. Farahmand, A. Shademan, and M. Jagersand, “Global Visual-Motor Estimation for Uncalibrated Visual Servoing,” IROS 2007. Check later.

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