Near-optimality of the adaptive scaling function

Establish a formal optimality proof demonstrating that the adaptive layer-wise scaling function α_ℓ = α_base (1 + β [1 − |ξ_ℓ|]), where ξ_ℓ denotes the normalized position of layer ℓ within the effective layer set L_eff, is near-optimal for the weighted modification objective max_{ {α_ℓ} } ∑_{ℓ ∈ L_eff} w_ℓ α_ℓ^2 subject to ∑_{ℓ ∈ L_eff} α_ℓ = C, with w_ℓ reflecting layer importance proportional to the separability metric S_ℓ. Specify and justify the necessary assumptions on layer-wise contribution functions under which this near-optimality result holds.

Background

The paper introduces an adaptive scaling function for layer-wise weight modification in Gabliteration, designed to apply stronger scaling to middle layers and weaker scaling to boundary layers. The function αℓ = α_base (1 + β [1 − |ξℓ|]) uses ξ_ℓ to normalize the layer index within the set of effective layers, emphasizing layers where separability metrics are typically highest.

To motivate the design, the authors consider a simplified weighted objective that maximizes total modification strength subject to a budget constraint: max ∑{ℓ ∈ L_eff} wℓ αℓ^2 with ∑ αℓ = C, where w_ℓ represents layer importance proportional to S_ℓ. They argue heuristically that middle layers should receive higher α_ℓ and conjecture that their linear scaling function approximates the optimal structure, but they do not provide a formal proof and note that additional assumptions are required.