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## Pointwise adaptation via stagewise aggregation of local estimates for multiclass classification

We consider a problem of multiclass classification, where the training sample $$S_n = \- \{(X_i, Y_i)\}_{i=1}^n$$ is generated from the model $$P(Y = m | X = x) = \theta_m(x)$$, $$1 \leq m \leq M$$, and $$\theta_1(x), \dots, \theta_M(x)$$ are unknown Lipschitz functions. Given a test point $$X$$, our goal is to estimate $$\theta_1(X), \dots, \theta_M(X)$$. An approach based on nonparametric smoothing uses a localization technique, i.e. the weight of observation $$(X_i, Y_i)$$ depends on the distance between $$X_i$$ and $$X$$. However, local estimates strongly depend on localizing scheme. In our solution we fix several schemes $$W_1, \dots, W_K$$, compute corresponding local estimates $$\ttildei 1, \dots, \ttildei K$$ for each of them and apply an aggregation procedure. We propose an algorithm, which constructs a convex combination of the estimates $$\ttildei 1, \dots, \ttildei K$$ such that the aggregated estimate behaves approximately as well as the best one from the collection $$\ttildei 1, \dots, \ttildei K$$. We also study theoretical properties of the procedure, prove oracle results and establish rates of convergence under mild assumptions.