In the very simple examples of reinforcement learning (gridworld, mountain car), we use real numbers or some elementary functions as reward functions.

When state spaces become larger and larger, and eventually continuous, the states become harder and computationally expensive to define. So here comes the idea of functional approximation, where we can use features to define.

I have always thought of 'features' (from the word itself) as qualities which I can measure. For example: how far is the agent from a certain obstacle, or how far the agent is from goal position, etc. But I have never seen this in examples/sample codes. In Sutton, there is talk of radial basis functions for features.

here are my questions:

  1. What is the role of radial basis functions in functional approximation?
  2. Can my idea of 'features' (per definition) work?
  3. Do you have some examples in github, or otherwise which shows this implementation?

2 Answers 2


I'm offering an answer this old post because it ranks in Google.

I like to think of kernels, or any feature transformation, as a way to increase the dimensionality of the input. You then pass this input to a model to do whatever it is you want to do.

For example, in a classification problem, you can inspect the decision boundary in the original domain of the data (imagine a 2D domain). The transformed data is in a much higher-dimensional space than the original. The linear model operates in this high-dimensional space. When you map that decision boundary back down to the original 2D space, it appears as if the boundary has significant complexity, like curves and circles and what-not.

The same is true for regression and is therefore also true in RL. They allow you to arbitrarily increase model complexity whilst retaining all the benefits of using a linear model (robustness, convergence guarantees, etc.).

Regarding examples, I find it easier to reason about this in terms of classification/regression problems. I recommend going through classification examples in sklearn's excellent documentation. For example the gaussian processes or support vector machines. Then you should be able to apply that knowledge to the world of RL.


Your idea of features is actually correct, I implemented a RL agent that learns to play Pac-Man: in a small environment it can use classic RL algorithms and actually store a Q table for each possible state/action tuple, but as you said when the space becomes larger it becomes intractable. In this case, I used features like the presence if ghosts or food nearby as an inout of a neural network, and I was able to train it to win every game. Radial Basis functions allow you to extend, in a sense, your input to a ML algorithm or a NN, if used on the input data as a kernel, to apply the "kernel trick". In this way you can exploit non linear aspects of the I put data to classify with non linear boundaries. A similar concept is thus applied in the field of RL, using the state input data.

  • $\begingroup$ Hi, thanks for your answer. I am not totally clear about how the radial basis functions can be used to work as features. I am not entirely sure how my idea of 'features' (which you said works) and how radial basis functions can work hand in hand. What is their connection? How are these two related? $\endgroup$
    – cgo
    Commented Mar 13, 2017 at 6:00

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