Multilevel Bregman Proximal Gradient Descent

TL;DR

The paper introduces ML-BPGD, a multilevel extension of Bregman Proximal Gradient Descent designed for constrained convex problems with relative smoothness. By embedding coarse models and transfer operators within a MGOPT-inspired framework, ML-BPGD achieves well-definedness and global linear convergence while handling convex constraints across all levels. The approach is validated on large-scale imaging tasks—deconvolution, tomography, and D-optimal design—where ML-BPGD significantly accelerates convergence over single-level BPGD, especially in early iterations and under KL/log-barrier geometries. The results demonstrate that exploiting multilevel structure yields substantial computational savings without sacrificing feasibility or robustness.

Abstract

We present the Multilevel Bregman Proximal Gradient Descent (ML BPGD) method, a novel multilevel optimization framework tailored to constrained convex problems with relative Lipschitz smoothness. Our approach extends the classical multilevel optimization framework (MGOPT) to handle Bregman-based geometries and constrained domains. We provide a rigorous analysis of ML BPGD for multiple coarse levels and establish a global linear convergence rate. We demonstrate the effectiveness of ML BPGD in the context of image reconstruction, providing theoretical guarantees for the well-posedness of the multilevel framework and validating its performance through numerical experiments.

Figures (6)

• Figure 3.1: Flowchart of \ref{['alg:Two-level-BPGD']}. Cooler and lighter colors refer to bigger function values.
• Figure 4.1: The reference images for the three experiments.Left: Crater Tycho on the Moon, taken by the Hubble Space Telescope, https://science.nasa.gov/image-detail/tycho-crater/, for Poisson-noisy deconvolution. Center: Walnut Phantom, tomographicxraydatawalnut, for tomographic reconstruction. Right: Jumping Mario, for D-optimal design in tomography.
• Figure 4.2: Normalized function value vs CPU time (in seconds). Deblurring performance across the various blur and noise conditions specified in \ref{['tbl:blur_vs_noise']}, using the Tycho Crater image. Yellow (SL): Single-level BPGD, with markers shown every 20 iterations. Blue (ML): ML-BPGD with two coarse levels; markers are shown every iteration. Violations of the coarse correction condition at the finest level are indicated by red 'x' markers in the plot. Our ML-BPGD far outperforms the single-level variant across all specified blur and noise variants.
• Figure 4.3: Deblurring results for the Crater Tycho image under varying noise and blur levels, as specified in \ref{['tbl:blur_vs_noise']}, after 60 iterations. The ML reconstructions far outperform their single-level counterparts, providing detailed images in few iterations, even in cases of severe degradation.
• Figure 4.4: Comparison of multilevel vs. single-level BPGD for tomographic reconstruction. Results are shown for the $1023 \times 1023$ Walnut Phantom from $20\%$ undersampled data. We use three discretization levels. Left: Normalized objective function values vs. cumulative CPU time (in seconds). The multilevel BPGD (ML-BPGD, blue, with markers at each multilevel iteration) rapidly catches up to the already fast single-level method (SL, yellow, with markers every 10 iterations), achieving comparable objective values in just seven iterations. While each ML iteration incurs extra overhead due to coarse model computations, it shows a more substantial reduction in function value per iteration: approximately four ML iterations match the effect of 10 single-level ones. The coarse correction condition consistently holds up to the final plotted iteration. Right: Selected iterations show consistently superior reconstructions from ML-BPGD. Despite higher per-iteration costs, ML achieves comparable visual quality about five times faster than the SL approach. This demonstrates ML-BPGD’s clear advantage in highly undersampled settings, where objective values alone may not fully reflect reconstruction quality.

Theorems & Definitions (29)

• Remark 2.1
• Lemma 2.2: Fixed point property of BPGD
• proof
• Lemma 2.3: Sufficient descent of BPGD, Teboulle_2018
• Remark 2.4
• Lemma 2.5
• proof
• Remark 3.1
• Lemma 3.2
• proof