Inductive Moment Matching (2503.07565v7)

Abstract: Diffusion models and Flow Matching generate high-quality samples but are slow at inference, and distilling them into few-step models often leads to instability and extensive tuning. To resolve these trade-offs, we propose Inductive Moment Matching (IMM), a new class of generative models for one- or few-step sampling with a single-stage training procedure. Unlike distillation, IMM does not require pre-training initialization and optimization of two networks; and unlike Consistency Models, IMM guarantees distribution-level convergence and remains stable under various hyperparameters and standard model architectures. IMM surpasses diffusion models on ImageNet-256x256 with 1.99 FID using only 8 inference steps and achieves state-of-the-art 2-step FID of 1.98 on CIFAR-10 for a model trained from scratch.

• The paper presents IMM's main contribution by bridging high sample quality and fast inference through a single-stage training framework that ensures full distribution convergence.
• It introduces an inductive mapping operator based on time-dependent stochastic interpolants, enabling both one-step and multi-step sampling for efficient generation.
• Empirical results on ImageNet and CIFAR-10 with state-of-the-art FID scores validate IMM’s robustness and superior performance compared to traditional generative models.

Overview and Motivation

Inductive Moment Matching (IMM) addresses a key challenge in generative modeling: obtaining high-quality samples with minimal inference steps while maintaining robust training stability. Traditional approaches such as diffusion models and flow matching require many inference steps and often necessitate complex two-stage training procedures (e.g., distillation), which inherently bring various forms of instability and heavy tuning requirements. IMM circumvents these limitations by proposing a single-stage training framework that directly maps between time-dependent marginal distributions.

The core motivation is to reconcile the trade-offs between sample quality, inference speed, and training stability without relying on a pre-trained initialization. In contrast to step-reduction techniques like distillation and Consistency Models (CMs)—which may converge only on first-order statistics—IMM guarantees distribution-level convergence through moment matching, thereby integrating benefits across different generative paradigms.

Methodological Details

Stochastic Interpolants and Temporal Marginals

IMM leverages time-dependent stochastic interpolants that progressively transform the data distribution at $t=0$ into a prior (often Gaussian) at $t=1$. The process defines a continuum of marginal distributions over time. The key technical contribution is the formulation of a mapping operator that enables transitioning from any marginal at time $t$ to another marginal at some earlier time $s < t$. This mapping inherently supports both one-step and multi-step sampling:

• One-step generation: Direct mapping from the terminal marginal ($t=1$) to the data distribution ($s=0$).
• Multi-step generation: Recursive application of the mapping in multiple stages, where each step involves mapping from a distribution at time $t_i$ to a distribution at a lower time $t_{i-1}$ until reaching $t_0=0$.

Inductive Training Procedure

The training framework is built upon an inductive matching objective. For any triple of time indices $s < r < t$, IMM simultaneously:

• Generates one distribution by directly mapping from time $t$ to $s$.
• Generates an alternative distribution by composing mappings from $t$ to $r$ and then from $r$ to $s$.

The divergence between these two distributions is minimized using estimators such as the Maximum Mean Discrepancy (MMD). This procedure rigorously enforces that the mapping remains invariant irrespective of the intermediate time steps, effectively matching all moments between the generated and true distributions. The inductive structure of the training objective ensures convergence properties that are both robust and stable across hyperparameter configurations and model architectures.

Relationship with Consistency Models

While Consistency Models implicitly align first-order statistics through score-based methods, IMM generalizes this idea by enforcing convergence at the distribution level. The paper demonstrates that Consistency Models are a special case of IMM, particularly when moment matching is constrained to the first moment. However, by matching all moments, IMM provides a more comprehensive guarantee of distribution fidelity, which mathematically underpins its robustness during training. This comprehensive moment matching is a crucial factor in achieving state-of-the-art performance with fewer inference steps.

Key Contributions and Numerical Results

Single-Stage Training Without Distillation

Unlike traditional distillation methods that require a separate pre-training phase and involve the optimization of dual networks, IMM’s single-stage procedure simplifies the training pipeline. This reduction in procedural complexity not only lowers computational overhead but also eliminates the error accumulation often seen in multistage training processes.

Stability and Hyperparameter Robustness

A significant contribution of IMM is its robustness under varying hyperparameters and architecture choices. The inductive moment matching mechanism intrinsically stabilizes the training procedure, which traditionally suffers from instability in reduced inference step models such as Consistency Models. The convergence guarantee on the full distribution ensures that minor deviations or perturbations in hyperparameters do not dramatically affect the quality or fidelity of the generated samples.

State-of-the-Art Performance Metrics

The empirical results reported in the paper are particularly compelling:

• ImageNet-256×256: IMM achieves a FID of 1.99 using only 8 inference steps. This performance not only surpasses typical diffusion models but also marks a significant reduction in the computational cost during inference.
• CIFAR-10: The approach attains a state-of-the-art 2-step FID of 1.98, demonstrating that even with extremely few steps, a model can be trained from scratch and yield superior sample quality.

These results underscore the practical potential of IMM, especially in applications where high-quality generation and rapid inference are simultaneously required.

Implementation Considerations

Computational Requirements and Scalability

From an implementation standpoint, IMM offers a more efficient training process due to its single-stage framework. However, ensuring accurate moment matching requires careful numerical estimations, particularly when employing MMD as the divergence measure. Researchers should consider:

• Batch Size: Larger minibatches may be necessary to reduce variance in moment estimations.
• Kernel Choice in MMD: The selection of appropriate kernels can significantly impact the stability and convergence of the divergence estimator.
• Inverse Problem Complexity: The mapping function must accurately capture the continuous transformation between time marginals, which might require tailored neural network architectures depending on the application domain.

Practical Deployment Strategies

When deploying IMM in a production environment:

• Utilize mixed precision training and hardware accelerators (GPUs/TPUs) to efficiently train the model, particularly due to the potentially high dimensionality of the data.
• Leverage checkpointing strategies during training to monitor and validate the convergence of distribution moments.
• In inference, the direct mapping from $t=1$ to $s=0$ enables rapid sample generation, which makes IMM well-suited for time-sensitive or resource-constrained applications in real-time generative tasks.

Pseudocode Outline

The following pseudocode outlines the primary training loop for an IMM-based model:

for epoch in range(num_epochs): for batch in dataloader: # Sample a batch from the data distribution at t=0 x0 = batch # Generate stochastic samples at multiple time points xt = stochastic_interpolant(x0, t=1.0) xr = stochastic_interpolant(x0, t=r) # Compute mappings: direct and multi-step xs_direct = f_theta(xt, target_time=0.0) xs_indirect = f_theta(f_theta(xt, target_time=r), target_time=0.0) # Compute loss using MMD between the two outputs loss = MMD(xs_direct, xs_indirect) # Backpropagate and update parameters optimizer.zero_grad() loss.backward() optimizer.step()

This pseudocode encapsulates the inductive moment matching principle: enforcing consistency between direct and compositional mappings via the divergence estimation.

Conclusion

Inductive Moment Matching provides a robust alternative to conventional generative methods by ensuring full distribution-level convergence through an inductive training framework. Its single-stage training paradigm, equipped with rigorous moment matching, enhances stability and significantly reduces inference steps (as evidenced by the ImageNet (FID of 1.99 on 8 steps) and CIFAR-10 (FID of 1.98 on 2 steps) results). For practitioners aiming to deploy high-performance generative models in real-world applications, IMM offers both conceptual and practical advantages, simplifying the training process while guaranteeing high-fidelity sampling with minimal computational overhead.