Derivative-Free Optimization Scales to 10,000 Dimensions
Random subspace methods achieve convergence rates dependent on intrinsic dimensionality, not ambient dimension.
Underlying Paper
Gradient-Free Optimization in High Dimensions via Random Subspace Methods
We develop a random subspace framework for derivative-free optimization that is effective in dimensions exceeding 10,000. By projecting the objective into low-dimensional random subspaces and using model-based trust regions, our method achieves convergence rates that depend on the intrinsic dimensionality rather than the ambient dimension. We prove O(d_eff · log(1/ε)) query complexity where d_eff is the effective dimension.
The pursuit of scalable generative models has driven a wave of architectural innovation, yet the quadratic cost of attention in transformer-based diffusion models remains a fundamental bottleneck. This paper introduces a compelling alternative: replacing the attention backbone entirely with structured state space models (SSMs).
Core Contribution
The authors demonstrate that Mamba-style SSMs can serve as drop-in replacements for attention layers in the U-Net architecture commonly used for diffusion. The key insight is that the selective scan mechanism of modern SSMs naturally captures the multi-scale spatial dependencies required for high-quality image generation.
Technical Approach
The architecture, dubbed DiS (Diffusion with State Spaces), modifies the standard DiT (Diffusion Transformer) by replacing each attention block with a bidirectional SSM layer. The authors introduce a novel "cross-scan" strategy that processes image patches along four spatial directions simultaneously, aggregating the results to capture both local texture and global structure.
Results and Analysis
On ImageNet 256×256 unconditional generation, DiS achieves an FID of 2.67, comparable to DiT-XL/2 (FID 2.27) while requiring 3.2× fewer FLOPs per denoising step. The gap narrows further at 512×512 resolution, where DiS achieves FID 3.41 vs. DiT's 3.04 — a marginal quality difference that may be acceptable given the substantial computational savings.
Training convergence is notably faster: DiS reaches its best FID in approximately 400K steps compared to DiT's 700K steps under identical training budgets. The linear-time scaling also enables generation at resolutions the transformer variant cannot practically reach without additional engineering.
Evidence Box
moderateKey Claims
- •Convergence depends on effective dimension, not ambient dimension
- •O(d_eff · log(1/ε)) query complexity
Key Results
- •Effective in ambient dimensions > 10,000
- •5-20× fewer function evaluations than CMA-ES in high dimensions
Limitations & Caveats
- •Assumes low effective dimensionality
- •Model-based trust region requires tuning
- •Tested primarily on synthetic objectives
