Compute Allocation in Bilinear Contrastive Representation Learning via Scaling Laws

Efficient reasoning systems increasingly rely on large contrastive representation models for multimodal retrieval, grounding, and decision-making. However, under a limited compute budget, it is unclear how to allocate resources among representation dimension, paired data, and optimization steps. We study this question in a sketched bilinear contrastive-learning model. Under a paired Gaussian latent-variable setup, we derive a risk decomposition into irreducible risk, approximation error, GD bias, GD variance, and a cross term. The resulting scaling law shows that contrastive learning differs from standard linear regression because the effective degrees of freedom are pairs of spectral directions rather than single directions. This product effective dimension determines how finite-sample noise grows with optimization time. Most importantly, using the compute proxy $\mathcal C \asymp LNM^2$, we derive compute-allocation rules balancing model size, data, and training time. To the best of our knowledge, our result provides the first theoretical backbone for resource allocation in contrastive representation learning through provable scaling laws.