Theoretical guarantees in KL for Diffusion Flow Matching (2409.08311v1)

Abstract: Flow Matching (FM) (also referred to as stochastic interpolants or rectified flows) stands out as a class of generative models that aims to bridge in finite time the target distribution $\nu^\star$ with an auxiliary distribution $\mu$, leveraging a fixed coupling $\pi$ and a bridge which can either be deterministic or stochastic. These two ingredients define a path measure which can then be approximated by learning the drift of its Markovian projection. The main contribution of this paper is to provide relatively mild assumptions on $\nu^\star$, $\mu$ and $\pi$ to obtain non-asymptotics guarantees for Diffusion Flow Matching (DFM) models using as bridge the conditional distribution associated with the Brownian motion. More precisely, we establish bounds on the Kullback-Leibler divergence between the target distribution and the one generated by such DFM models under moment conditions on the score of $\nu^\star$, $\mu$ and $\pi$, and a standard $L^2$-drift-approximation error assumption.

• The paper derives explicit KL divergence bounds in non-early-stopping settings, ensuring error control under moment and integrability conditions.
• It introduces an early-stopping regime that achieves similar convergence guarantees with reduced computational complexity by limiting score assumptions.
• The findings offer practical insights for balancing drift-approximation error and sample efficiency in stochastic generative modeling.

Theoretical Guarantees in KL for Diffusion Flow Matching

The paper "Theoretical guarantees in KL for Diffusion Flow Matching (DFM)" by Marta Gentiloni Silveri, Giovanni Conforti, and Alain Durmus presents an in-depth investigation into the convergence guarantees for Diffusion Flow Matching models. This class of generative models focuses on bridging the target distribution $with an auxiliary distribution$\mu$$within a finite time by leveraging a fixed coupling$$\pi$ and a conditional distribution, specifically the Brownian bridge in this context. ### Summary of Contributions The main contributions of the paper are bifurcated into two specific scenarios: 1. **Non-Early-Stopping Regime:** The authors derive explicit bounds on the Kullback-Leibler (KL) divergence between the target distribution and the distribution generated by the DFM model. These bounds are obtained under moment conditions on the target and base distributions ($ and $\mu$), integrability conditions on the associated scores, and a standard $L^2$-drift-approximation error assumption.

1. Early-Stopping Regime: The paper also provides bounds in an early-stopping scenario where the coupling is independent. In this regime, the integrability conditions are required only on the score of the base distribution $\mu$. The result here bounds the KL divergence between a smoothed version of the target distribution and the early-stopped version of the DFM model.

Key Assumptions

The analysis rests on several key assumptions:

• Moment Conditions: Both $\mu$ and $$have bounded eighth moments. - **Integrable Scores:** The scores associated with$$\mu$,$, and the coupling $\pi$ are integrable to the eighth power.
• Drift Approximation: The drift of the Markovian projection must be approximable within an $\varepsilon^2$-precision.

Main Results and Theoretical Implications

In the non-early-stopping setting, the authors deliver the result that the KL divergence between the data distribution and the output of the DFM model is bounded by: $KL(|\nu^{\theta^{\star}_1) \lesssim \varepsilon^2 + h(h^{1/8}+1)\Big(d^4 + \text{moments and score norms}\Big),$ where $h$ is the step size of the Euler-Maruyama scheme. This result implies that by setting the step size appropriately in relation to $\varepsilon$ and the dimensions of the integrable norms, the approximation error can be controlled, ensuring the efficient generation of samples from $$.</p> <p>The early-stopping result indicates that similar performance can be achieved with the added benefit of potentially reduced computational complexity. The early-stopped model ensures convergence with bounds incorporating$$\delta$, which represents the stopping time, optimizing the balance between drift-estimation error and computational feasibility.

Practical and Theoretical Implications

The practical implications are substantial for the development of generative models. The findings offer a feasible and theoretically backed approach to balance the trade-offs between accuracy and computational efficiency in generative tasks. Additionally, they provide a foundation for further explorations in implementing DFMs in various scenarios, extending beyond Gaussian base distributions and exploring a richer space of couplings and interpolations.

Theoretically, the paper advances the understanding of the convergence properties of generative models, especially those based on stochastic transport dynamics. It aligns well with the ongoing research on SGMs and represents a significant step in comprehending and harnessing the potential of Diffusion-based generative models.

Future Directions and Open Questions

Several avenues for future research are posited by the findings of this paper:

1. Relaxation of Score Integrability Conditions: Investigating whether the integrability conditions on the scores can be further relaxed to include more distributions could enhance the applicability of DFM.
2. Statistical Analysis of Drift Estimation: Complementary statistical insights could solidify the theoretical framework, akin to the analyses seen in the deterministic FM models.
3. Dimensionality Reduction: Improved dimension dependence in the early-stopping procedure can pave the way for more practical implementations in high-dimensional data spaces.

In conclusion, this paper sets a rigorous foundation for the analysis of Diffusion Flow Matching models, offering both theoretical guarantees and practical methodologies for effective generative modeling. The insights gained here pave the way for more comprehensive and robust generative model designs in the future.