For multiclass PAC learning, the optimal excess risk at any fixed oracle error L* equals the square root of L* times the Natarajan dimension over n, plus the realizable DS-dimension rate, with matching upper and lower bounds.
Optimal PAC bounds without uniform convergence
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Optimistic Rates for Multiclass PAC Learning
For multiclass PAC learning, the optimal excess risk at any fixed oracle error L* equals the square root of L* times the Natarajan dimension over n, plus the realizable DS-dimension rate, with matching upper and lower bounds.