Virtual Logical Depth (VLD)

Virtual Logical Depth (VLD) is a scaling dimension for neural networks introduced by Zhu et al. in 2025. The three conventional scaling dimensions are depth (number of layers), width (hidden size), and parameter count. VLD adds a fourth: a network with L base layers is run for K additional virtual layers by reusing the same weights, yielding K*L effective algorithmic depth without changing the parameter count.

VLD is distinct from inference-time scaling methods that operate per token, like chain-of-thought or self-consistency. VLD alters the internal computation graph during both training and inference. It is related to looped transformers and weight-tied recurrent depth, but framed as a scaling law rather than a specific architecture.

The 2506.18233 paper's central finding is that VLD substantially improves reasoning capability (measured on GSM8K) at fixed parameter count, while leaving knowledge capacity nearly unchanged. Knowledge capacity, as measured by information bits recoverable from the network, still scales with parameters rather than with VLD. The two scaling dimensions are complementary: parameters scale capacity, VLD scales reasoning.

VLD is one of several test-time compute and architectural depth techniques explored in 2024-2026, including mixture-of-recursions (arXiv 2507.10524), recurrent depth, and SMELT. OpenAI's Astra model, reported in The Information in September 2026, was described as using a recurrent depth or looped transformer design, which is the architectural family that VLD formalizes as a scaling law.