Minimax-Optimal Multiclass Classification with Cyclic Oblique Forests

Oblique splits can represent classification boundaries that are poorly aligned with the input coordinates. A key challenge is to combine this geometric flexibility with sharp statistical guarantees that exploit smooth class probabilities. We introduce the Cyclic Oblique Forest, a purely randomized approach to multiclass classification that combines cyclic oblique splitting with regularized local polynomial estimation. Under uniform design, we establish minimax rates for expected multiclass excess risk over H\"older classes of conditional class probabilities for every fixed finite smoothness order, without additional logarithmic factors. These rates are attained with midpoint cuts along a basis of splitting directions, with depth and polynomial degree chosen for the specified smoothness. We further derive a sharper excess-risk bound under an additional multiclass margin condition with a positive exponent. Real data analysis on classic UCI datasets shows that the proposed method achieves lower mean test error than existing random forest baselines.