Intelligent Robot Lab
Brown University, Providence RIHome  People  Research  Publications  Joining  Software 

The Fourier BasisThe Fourier basis is a simple, principled basis function scheme for linear value function approximation in reinforcement learning. It has performed well over a wide range of problems, despite its simplicity, and I now use it almost exclusively. My advice to fellow researchers who require function approximation is that the Fourier basis should be the first thing you try. If a problem is small enough, the Fourier basis avoids the need for extensive problemspecific feature engineering. For example, we have achieved good results on the Acrobot problem using a Fourier basis of order between 5 and 9; contrast this with the two paragraphs describing a problemspecific tilecoding originally used to solve the Acrobot. For larger problems, the Fourier basis provides a generic but complete basis function scheme suitable for feature selection. For a single continuous state variable, $s$, (scaled to $[0,\; 1]$), the $n$th order Fourier basis is the set of basis functions defined as: for $j\; \in \; [0,\; 1,\; ...,\; n]$. Thus, an order $n$ Fourier basis produces $n\; +\; 1$ basis functions, the first having value 1 everywhere. The following pictures show a few example basis functions; note that, as $j$ increases, the basis functions become higher frequency.
For multivariate state spaces, where $$s ∈ ℜ^{d}$(where\; each\; element$ s$$_{i} of $$s is scaled to $[0,\; 1]$) the $n$th order Fourier basis is defined by the set of basis functions: where each c_{j} is a length $d$coefficient vector, each element of which is an integer between $0$ and $n$. The basis is obtained by enumerating all such coefficient vectors. A few example basis functions for a 2D domain are shown below. Each element of the $$c_{j} vector specifies the basis function's frequency along the corresponding state variable; for example, in a 2D domain, $$c_{j} = [1, 2] results in a basis function with frequency $1$ along $s$_{1} and frequency $2$ along $s$_{2}. Setting a state variable's coefficient to zero results in a basis function that is invariant to changes along that state dimension.
One caveat: when performing gradientdescent style value function approximation (such as TD($\lambda $)), we must deal with the fact that some Fourier basis functions have much higher frequencies than others, and scale their gradient descent step sizes appropriately. In practice, we've found that setting $\alpha $_{i} = α_{0}/c_{i}_{2}, where $\alpha $_{0} is the base learning rate assigned to the basis function with $$c_{0}, performs reliably well. An $n$th order Fourier basis in a $d$dimensional space has $(n\; +\; 1)d$ basis functions, and thus suffers the combinatorial explosion in $d$ exhibited by all complete fixed basis methods. In a domain where $d$ is sufficiently small  perhaps less than 6 or 7  we may simply pick an order $n$ and enumerate all basis functions. This amounts to an expectation that components of the value function with frequency $n$ or less are signal, and components with higher frequency are noise. For higher dimensional domains, we may reduce the size of the basis by placing restrictions on the $$c_{j} vectors. For example, we can include only coefficient vectors with two or fewer nonzero elements; this is analogous to tile coding pairs of variables. The paper describing the Fourier Basis is:
Source Code:
Here is a list of papers using the Fourier basis, courtesy of Google Scholar.  
