Humboldt-Universität zu Berlin - High Dimensional Nonstationary Time Series

IRTG1792DP2018 029

Pointwise adaptation via stagewise aggregation of local estimates for multiclass classification

Nikita Puchkin
Vladimir Spokoiny

We consider a problem of multiclass classification, where the
training sample Sn = {(Xi, Yi)}n
i=1 is generated from the model P(Y =
m|X = x) = m(x), 1 6 m 6 M, and 1(x), . . . , M(x) are unknown Lip-
schitz functions. Given a test point X, our goal is to estimate 1(X), . . . ,
M(X). An approach based on nonparametric smoothing uses a localization
technique, i.e. the weight of observation (Xi, Yi) depends on the distance
between Xi and X. However, local estimates strongly depend on localiz-
ing scheme. In our solution we fix several schemes W1, . . . ,WK, compute
corresponding local estimates e(1), . . . , e(K) for each of them and apply an
aggregation procedure. We propose an algorithm, which constructs a con-
vex combination of the estimates e(1), . . . , e(K) such that the aggregated
estimate behaves approximately as well as the best one from the collection
e(1), . . . , e(K). We also study theoretical properties of the procedure, prove
oracle results and establish rates of convergence under mild assumptions.


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