Variance reduction of diffusion model's gradients with Taylor approximation-based control variate (2408.12270v1)

Abstract: Score-based models, trained with denoising score matching, are remarkably effective in generating high dimensional data. However, the high variance of their training objective hinders optimisation. We attempt to reduce it with a control variate, derived via a $k$-th order Taylor expansion on the training objective and its gradient. We prove an equivalence between the two and demonstrate empirically the effectiveness of our approach on a low dimensional problem setting; and study its effect on larger problems.

• The paper introduces a Taylor series-based control variate to reduce gradient variance in score-based diffusion models, enhancing optimization efficiency.
• It derives an arbitrary order Taylor polynomial and leverages a regression coefficient to demonstrate effective variance reduction in low-dimensional settings.
• Empirical results show faster convergence and improved stability, highlighting both the benefits and limitations for complex high-dimensional models.

Variance Reduction in Diffusion Models Using Taylor Approximation-based Control Variate

The paper "Variance Reduction of Diffusion Model's Gradients with Taylor Approximation-based Control Variate" presents an innovative approach to addressing high variance in the training of score-based probabilistic generative models. Authored by Paul Jeha et al., this research explores the utilization of Taylor series-based control variate to mitigate variance in gradients, thereby enhancing optimization efficiency. This essay provides a comprehensive review of the paper's contributions, supported by theoretical foundations, empirical evidence, and a discussion on implications and future directions in AI research.

Introduction

Probabilistic generative models are pivotal in modeling unknown data distributions, with key methodologies including Variational Auto-Encoders (VAEs), Energy-Based Models (EBMs), and Normalizing Flows. Though effective, these methods encounter various constraints, notably non-tight ELBOs in VAEs, partition function estimation in EBMs, and specialized architectures in Normalizing Flows. Score-based models circumvent these constraints by modeling the gradient of the log-density (Stein score), making them appealing alternatives, especially in high-dimensional data contexts.

Problem Statement and Contribution

A key challenge in score-based models lies in the high variance of the training objective, primarily optimized via denoising score matching. This variance complicates optimization significantly. The paper proposes leveraging a control variate through a $k$-th order Taylor expansion of the training objective and its gradient to reduce this variance. The main contributions are:

1. Derivation of a control variate with arbitrary order Taylor polynomial.
2. Proof of equivalence between controlling the training objectives and its gradient.
3. Empirical demonstration of the necessity of a regression coefficient.
4. Efficacy demonstration of the control variate in low-dimensional settings.
5. Analysis of the impact in high-dimensional scenarios.
6. Identification and discussion of the limitations of Taylor-based control variate.

Related Work

Score Matching

Score matching, introduced primarily for training EBMs, focuses on aligning the model's gradients with those of the data. Various adaptations such as implicit score matching, sliced score matching, and denoising score matching have been explored. Denoising score matching, akin to diffusion models, corrupts data with noise and trains a denoiser, although it suffers from high variance, hence optimizing becomes arduous.

Control Variate

Control variate is a variance reduction technique extensively used in Monte Carlo integration problems. It leverages a correlated auxiliary problem with known expectation to reduce variance. Applications range from variational inference to reinforcement learning, with linear approximations previously utilized in score-based models.

Theoretical Framework

Denoising Score Matching

The paper starts with the weighted denoising score matching objective, where the loss function, $$L_{\boldsymbol{\theta}}(\mathbf{z}, \mathbf{x}, \sigma)$$, faces significant high variance. The control variate aims to stabilize this through a general Taylor series expansion.

Taylor Series and Control Variate

The core idea is to approximate the training objective using a Taylor series expansion around a point, usually the clean data $\mathbf{x}$. The paper derives the Taylor polynomial and utilizes it to construct a control variate, essentially a corrected form of the objective that has reduced variance.

Gradient Control Variate

Further, the paper establishes that controlling the variance in the training objective implicitly supports gradient variance reduction, a crucial aspect for stochastic optimization methods. It ensures that the corrected objective maintains unbiasedness, guaranteeing convergence properties.

Empirical Evidence

The empirical studies focus on several areas:

• Demonstrating the necessity of a regression coefficient ($\beta$) for effectively scaling the control variate.
• Showcasing variance reduction on a toy dataset, where proper control variate significantly reduces variance at different noise levels.
• Highlighting scenarios where control variate enables convergence under small batch sizes, which would otherwise be infeasible.
• Addressing the limitations through larger model evaluations on MNIST data.

Observations and Implications

Empirically, while the control variate technique shows promise in simple settings, its efficacy diminishes with increased model complexity. This hints towards the Taylor series unable to adequately approximate intricate models such as U-Nets. Improved understanding and possibly higher-order polynomial approximations or alternate variance reduction techniques might be necessary for larger models.

Theoretical and Practical Implications

Theoretical Implications: The equivalence between controlling the objective's variance and its gradient is a crucial insight, providing a basis for future optimization strategies in probabilistic models. This lays ground for deeper investigations into optimal control variates tailored for gradient variance reduction.

Practical Implications: Practically, models trained with reduced variance have faster and more stable convergence, crucial for deploying large-scale generative models. Variance control directly affects computational efficiency and robustness, impactful in real-world AI model training.

Future Directions

Future research may explore:

• Adaptive calculation of higher-order Taylor polynomials dynamically during training.
• Incorporating control variates across different types of neural networks and evaluating their broader efficacy.
• Developing hybrid methods combining Taylor-based approaches with other variance reduction techniques for enhanced performance.

Conclusion

The paper by Paul Jeha et al. presents a detailed and theoretically grounded approach to variance reduction in score-based models using Taylor approximation-based control variate. While demonstrating notable improvements in controlled settings, the technique's applicability to more complex models faces challenges. Future research addressing these limitations could further enhance training paradigms for probabilistic generative models. The theoretical underpinnings laid down in this work will serve as vital reference points for ongoing advancements in this domain.