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Rethinking Losses for Diffusion Bridge Samplers
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Argues that the log variance loss is not a valid objective for learnable-bridge diffusion samplers (violates the data processing inequality and differs from rKL gradients) and proposes training diffusion bridge samplers with reverse KL gradients via the log-derivative trick (rKL-LD), which outperforms LV and rKL-R on Bayesian and synthetic sampling benchmarks.
Problem
Training diffusion bridge samplers for unnormalized target distributions: the log variance (LV) loss is widely used and reported to beat reverse KL with reparameterization, yet it is unstable and may not respect the data processing inequality when forward processes or diffusion coefficients are lear
Approach
Show that on-policy LV gradients equal rKL log-derivative gradients only for non-learnable forward processes and fixed diffusion coefficients; propose optimizing the rKL objective with the log-derivative trick and control variate (rKL-LD), optionally learning diffusion coefficients, which is theoretically grounded via the DPI and empirically more stable and accurate.
Key results
- On Bayesian benchmarks, CMCD with rKL-LD (frozen diffusion coefficients) significantly outperforms its LV-trained counterpart on all tasks and the rKL-R version on 4 of 5 tasks.
- LV loss diverges in 3 of 5 cases for CMCD and 4 of 5 for DBS when learning diffusion coefficients, while rKL-LD does not diverge.
- On GMM-100D, DBS with rKL-LD reaches ELBO -198.53 versus -252.56 for DBS with LV.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use rKL-LD instead of LV | CMCD · Bayesian learning benchmarks | ELBO | Helped |
| Use rKL-LD instead of rKL-R | CMCD · Bayesian learning benchmarks | ELBO | Helped |
| Use rKL-LD instead of LV | DBS · Bayesian learning benchmarks | ELBO | Helped |
| Add learnable diffusion coefficients (ablation) | CMCD with rKL-LD · Bayesian learning benchmarks | ELBO: 4 out of 5 tasks | Helped |
| Add learnable diffusion coefficients (ablation) | DBS with rKL-LD · Bayesian learning benchmarks | ELBO: all tasks | Helped |
| Add learnable diffusion coefficients (ablation) | CMCD with LV · Bayesian learning benchmarks | ELBO: 4 out of 5 cases | Hurt |
| Add learnable diffusion coefficients (ablation) | DBS with LV · Bayesian learning benchmarks | ELBO: 4 out of 5 cases | Hurt |
| Add learnable diffusion coefficients (ablation) | CMCD with LV · Bayesian learning benchmarks | training stability: 3 out of 5 cases | Hurt |
| Add learnable diffusion coefficients (ablation) | DBS with LV · Bayesian learning benchmarks | training stability: 4 out of 5 cases | Hurt |
| Use rKL-LD instead of LV | CMCD · Synthetic targets | MMD: all cases | Helped |
| Use rKL-LD instead of LV | CMCD · Synthetic targets | ELBO: all 4 synthetic tasks | Helped |
| Use rKL-LD instead of LV | DBS · Synthetic targets | ELBO: all 4 cases | Helped |
Surprising findings
- The LV loss violates the data processing inequality, as shown by an explicit counterexample on the unit square.
- Learning diffusion coefficients worsens performance and increases divergence with the LV loss, while it consistently helps with rKL-LD.
Limitations
- rKL-LD remains susceptible to mode collapse when learning rates are not well tuned.
- Learnable diffusion coefficients are constant across time only; time-dependent coefficients left to future work.
Experimental setup
Datasets: Sonar German Credit Seeds Brownian LGCP GMM-40 MoS-10 Funnel Many Well
Benchmarks: Sonar Credit Seeds Brownian LGCP GMM-40 MoS-10 Funnel Many Well
Metrics: ELBO Sinkhorn distance entropic mode coverage maximum mean discrepancy log Z
Baselines: CMCD: rKL-R CMCD: LV DBS: rKL-R DBS: LV
Builds on: Controlled Monte Carlo Diffusion Denoising Bridge Samplers Log Variance loss reparameterization trick
Closest papers by meaning
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- Denoising Diffusion Bridge Models (ICLR 2024 · similarity 0.86)
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