  Localized support vector machines solve SVMs on many spatially defined small
chunks and one of their main characteristics besides the computational benefit
compared to global SVMs is the freedom of choosing arbitrary kernel and
regularization parameter on each cell. We take advantage of this observation to
derive global learning rates for localized SVMs with Gaussian kernels and hinge
loss. Under certain assumptions our rates outperform known classification rates
for localized SVMs, for global SVMs, and other learning algorithms based on
e.g., plug-in rules, trees, or DNNs. These rates are achieved under a set of
margin conditions that describe the behavior of the data-generating
distribution, where no assumption on the existence of a density is made. We
observe that a margin condition that relates the distance to the decision
boundary to the amount of noise is crucial to obtain rates. The statistical
analysis relies on a careful analysis of the excess risk which includes a
separation of the input space into a subset that is close to the decision
boundary and into a subset that is sufficiently far away. Moreover, we show
that our rates are obtained adaptively, that is, without knowing the parameters
resulting from the margin conditions.
