Breaking Dimensions: Adaptive Scoring with Sparse Grids - Realtime Data Mining - page 137

music data set in 166 dimensions. However, the prediction quality of formulation
( 7.33 ) is somewhat lower compared to our more complex one ( 7.32 ).
Moreover, sparse grids are not restricted to regularization network problems
( 7.1 ) at all! In fact they can be used to a wide family of high-dimensional
differential and integral equations. Of course, the function f needs to be smooth.
This usually applies to most scoring problems. But even the non-smooth case can be
handled, depending on the problem, using adaptive sparse grids. Also the concept of
adaptive sparse grids can be generalized (see [Heg03]), arriving at the following
core ideas: grids that are sparse, hierarchically organized, and adaptively refined.
This results in a very powerful approximation approach which, however, requires
much further research. At this, one of the most complex problems is the develop-
ment of error estimation techniques required for grid refinement; see [Gar12b].
7.2.6.5 Sparse Grids for Regression in Reinforcement Learning
As stated in Sect. 6.1.1 and used in Chap. 10 , for the representation of the state-
value function v and action-value function q in reinforcement learning, often
regression models are used which can handle large state spaces S and can also
serve for regularization. Here, sparse grids are a good candidate, especially because
they can solve complex operator equations on large data volumes and are well
suited for adaptivity. For the continuous counterpart of the Bellman equation, the
Hamilton-Bellman-Jacobi equation, promising results have been obtained recently
[BGGK12].
7.3 Experimental Results
We now apply our approach to different data sets. Both synthetic and real data from
practical data mining applications are used. All the data sets are rescaled to [0,1] d .
To evaluate our method, we give the correctness rates on testing data sets, if
available, or the tenfold cross-validation results otherwise. For a critical discussion
on the evaluation of the quality of classification algorithms, see [Diet98, Sal97].
The results are mostly based on the offline Algorithm 7.1 since the adaptive
Algorithm 7.3 yields the same results. The equivalence of the results of both offline
and online algorithms will be demonstrated in the last example.
7.3.1 Two-Dimensional Problems
Example 7.2 The first example is the spiral data set proposed by Alexis Wieland of
MITRE Corp [Wie88]. Here, 194 data points describe two intertwined spirals; see
Fig. 7.3 . This is surely an artificial problem which does not appear in practical