Correction to “Nonparametric regression using deep neural networks with ReLU activation function”

Correction: Condition (ii) of Theorem 1 in [1] should be changed to (Equation presented). Moreover, the constants C, C^' in Theorem 1 also depend on the implicit constants that appear in conditions (ii)-(iv). There are large regimes where the new condition (ii') is weaker than (ii). Explanation: Rather than choosing m and N in the proof of Theorem 1 globally, one should instead apply Theorem 5 individually to each i with (Equation presented) and (Equation presented), where 0 < c ≤ 1/2 is a sufficiently small constant. As mentioned at the beginning of the proof of Theorem 1, it is sufficient to prove the result for sufficiently large n. Therefore, we can assume that m_i ≥ 1 for all i = 0,..., q and (Equation presented). The latter implies (Equation presented). If we now define L'_i := 8 + (m_i + 5)(1 + ⌈log₂(t_i ∨ β_i)⌉), then there exists a network (Equation presented)_ij ∈ F(L'_i, (t_i, 6(t_i + ⌈β_i⌉)N_i,..., 6(t_i + ⌈β_i⌉)N_i, 1), s_i) with s_i ≤ 141(t_i + β_i + 1)^3+tiN_i(m_i + 6), such that (Equation presented), where Q_i is any upper bound of the Hölder norms of h_ij, j = 1,..., d_i+1. We can now argue as in the original proof to show that the composite network f^∗ is in the class F(E, (d, 6r_i max_i N_i,..., 6r_i max_i N_i, 1), (Equation presented)), with (Equation presented). Using the definition of L'_i above, it can be shown as in the original proof that (Equation presented) (log₂(4) + log₂(t_i ∨ β_i))log₂(n) for all sufficiently large n. All remaining steps are the same as in the original proof of Theorem 1. The constant c in the definition of N_i will also depend on the implicit constant in the conditions L ≾ nφ_n, nφ_n ≾ min_{i=1,..., L} p_i and s nφ_n log n. Further comments: - Lemma 1 requires that the constant K is large enough such that Theorem 3 is applicable. - First display on page 1886: The value t₂ is N not Nd. - Equation (18) also requires that the inputs are nonnegative. - In Lemma 3, the L^∞-norms should be replaced by the supremum, that is, f _L∞_(A) should be changed to sup_x∈A |f (x)|. - In the proof of Theorem 1, r_i does not depend on i and should be named r. Three lines after equation (26), C should be replaced by C^'. - In the proof of Theorem 3, β^∗ in the first line on page 1893 should be β^∗∗. It is sufficient to check that the Hölder constant of φ_w is bounded by (β^∗ + 1)^t∗ (t^∗ + 1) as all later arguments of the proof carry over. Moreover, g_i(x) = (x₁,..., x_di ) should be g_i(x) = (x₁,..., x_di+₁ ) if d_i ≥ d_i+₁ and g_i(x) = (x₁,..., x_di, 0,..., 0) if d_i < d_i+1. Finally, ψ_u²₂ should be replaced by ψ_u^B2₂. - In the proof of Lemma 2, one can simply take the constant function h_{j, α} = K if μ₀/= 0. This immediately gives d_{j, k} = Kμ^d₀ 2^−jd/2 for all wavelet coefficients d_{j, k}. For the case μ₀ = 0, one should replace the binomial coefficient (Equation presented) by the multinomial coefficient (Equation presented) = (dr)!/(r!)^d. - To verify the last inequality in (B.8) of the Supplementary Material, one can replace ≤ 1/M by < 1/M. - In the proof of Lemma 4, some F are missing. In particular, it should be (Equation presented). Also, on page 12 of the Supplementary Material it should be P (T ≥ t) ≤ 1 ∧ 2N_n max_j (Equation presented). Since r_j ≥ F n⁻¹ log N_n, we can argue as before and obtain P (T ≥ t) ≤ 2N_n exp(−3t√log N_n/(16√n)) for all t ≥ 6√n log N_n. The conclusion of (I) is still valid. - In the proof of Lemma 5, we always work with the | · |_∞-norm for vectors. The grid size of an individual parameter should be taken as δ/((L + 1)V).