Anytime Distribution Matching (ADM)

• ADM is a class of probabilistic algorithms that iteratively aligns empirical distributions to a target with certified anytime optimality.
• It employs methods like interval refinement, kernel discrepancy minimization, and tailored loss functions to achieve provable convergence.
• ADM is applied in coding theory, sequential inference, self-supervised learning, and generative modeling, offering scalability and adaptivity.

Anytime Distribution Matching (ADM) designates a class of probabilistic algorithms and statistical procedures that iteratively align an empirical or synthetic distribution toward a target, maintaining validity and optimality guarantees throughout execution. The "anytime" property ensures that the quality of distributional matching provably improves as computation progresses, permitting intermediate termination with a certified level of divergence or approximation error. ADM appears across coding theory, sequential inference, self-supervised learning, adaptive optimization, distillation in generative modeling, and applied domains such as automated bidding systems. Implementations vary—from interval methods rooted in arithmetic coding, to kernel-based discrepancy minimization, likelihood-based alignment, and adversarial or non-adversarial losses—yet all share the goal of continually refining the empirical match between data (or function outputs) and a reference distribution, often with provable convergence rates.

1. Algorithmic Foundations

ADM algorithms rely on adaptive mechanisms that monitor and adjust empirical distributions:

• Arithmetic Distribution Matching (ADM) (Baur et al., 2014) utilizes interval refinements analogously to arithmetic coding to transform a discrete memoryless source (DMS) sequence into one that matches a target DMS distribution. The process maintains an online, invertible mapping, where the refinement of intervals according to the source and the target ensures one-to-one correspondence and decodability.
• Sequential Confidence Procedures (Waudby-Smith et al., 2023) establish anytime-valid inference and uniform convergence rates via probabilistic criteria (e.g., de la Vallée-Poussin uniform integrability), enabling variance and moment estimation that remains valid under arbitrary stopping rules.

In both cases, the procedural loop continually tightens the approximation to the target distribution. Arithmetic-based ADM guarantees that, as the input length increases, normalized divergence to the target DMS vanishes and the entropy rate of the output converges to that of the target.

2. Interval and Discrepancy-Based Methods

Interval representation is central in coding-theoretic ADM, while kernel discrepancies dominate distributional RL and self-supervised ADM:

• Interval Techniques: The (Baur et al., 2014) ADM maintains two overlapping interval lists: one for the source DMS (by sequential bit refinement), another for candidate codewords generated by refining the [0,1) interval using the target distribution. Output bits are produced whenever the refined source interval is fully contained within the current candidate, thus controlling the mapping.
• Kernel Discrepancy Minimization: In moment-matching RL (Nguyen et al., 2020), particles (deterministic samples) represent the empirical return distribution. The maximum mean discrepancy (MMD) between this and the Bellman target is minimized, ensuring implicit matching of all moments. The MMD objective, computed with Gaussian kernels, decomposes into attractive and repulsive terms—simultaneously guiding particles toward the target and preventing collapse.

Both enable flexible, incremental refinement of the matching process; interval methods handle variable-length codes and infinite input, kernels support particle-based learning with robust convergence guarantees $O(1/\sqrt{n})$ for $n$ samples.

3. Loss Functions and Theoretical Guarantees

ADM variants design loss functions that directly promote distributional alignment:

• Informational Divergence: In (Baur et al., 2014), $$D(P_Y \| P_Z) = \sum_{c} P_Y(c)\log_2(P_Y(c)/P_Z(c))$$ quantifies mismatch; bounds on the interval size ratios ensure that normalized divergence converges to zero with large input length.
• Neighborhood Likelihood Loss (NLL) (Li et al., 2021): In real-time bidding ADM, NLL aggregates three components that encourage probability mass concentration near observed (or censored) outcomes—directly refining the predicted win price distribution over discrete buckets. The objective is $$Loss_{nll} = \alpha Loss_1 + (1-\alpha)[\beta Loss_2 + (1-\beta)Loss_3]$$, where each loss targets different censoring cases in the auction dynamics.
• Variational Bounds: Non-adversarial ADM (Gong et al., 2023) employs the Variational Alignment Upper Bound (VAUB), optimizing over latent prior and decoder distributions to upper bound generalized Jensen-Shannon divergence between domain-conditioned representations.

Theoretical analyses in these models demonstrate that normalized divergence or Wasserstein distance to the target drops with more data or computation; population and sample theorems (Jiao et al., 20 Feb 2025) connect small loss values with separability and transfer learning guarantees.

4. Expressive Priors and Geometry Regularization

Recent ADM approaches employ expressive priors and geometry preservation:

• Score-Based Priors (Gong et al., 17 Jun 2025): ADM algorithms may forgo explicit prior forms for the latent distribution, instead training flexible score networks $\nabla_z\log Q(z)$ via denoising score matching. This enables more powerful and bias-free representation of invariances required in fairness and domain adaptation.
• Geometry-Preserving Regularization: Gromov-Wasserstein objectives align intrinsic pairwise distances in the original and latent spaces, either under Euclidean or semantic (e.g., CLIP-embedding) metrics. This regularization stabilizes learning and improves disentanglement, crucial for high-dimensional practical deployment.

The “Score Function Substitution” (SFS) trick allows gradients to be computed efficiently using the score network without explicit density modeling, further improving scalability and computational efficiency.

5. Adversarial and Non-Adversarial Matching

ADM spans both adversarial and non-adversarial regimes:

• Non-Adversarial Approaches (Gong et al., 2023, Gong et al., 17 Jun 2025): VAE-based, likelihood-centered ADM is optimized via cooperative (min-min) objectives, supporting plug-and-play drop-in alignment in fairness, robustness, and domain adaptation without requiring extra discriminator networks.
• Adversarial Distribution Matching (Lu et al., 24 Jul 2025): In generative model distillation, diffusion-based discriminators judge the alignment of score predictions from student and teacher models, employing adversarial hinge losses on both latent and pixel spaces. The DMDX pipeline combines adversarial distillation pretraining with ADM fine-tuning for efficient, high-fidelity synthesis.

Notably, adversarial ADM avoids mode collapse tied to reverse KL-divergence minimization, instead permitting learnable, data-driven discrepancy measures for richer alignment and improved generative diversity.

6. Applications and Empirical Performance

ADM is deployed in diverse contexts:

• Probabilistic Shaping and Rate Adaptation (Baur et al., 2014): ADM is used to shape transmission statistics in communication systems, providing online adaptation to channel constraints.
• Distributional Reinforcement Learning (Nguyen et al., 2020): Moment-matching ADM surpasses C51, QR-DQN, and other baselines on Atari benchmarks via flexible empirical particle-based return modeling.
• RTB Inventory Pricing (Li et al., 2021): ADM’s NLL yields competitive business metrics (auction wins, average price) and algorithmic performance (MAE, ANLP, C-Index) under strict time constraints.
• Self-Supervised Transfer Learning (Jiao et al., 20 Feb 2025): Wasserstein-based ADM with structured reference distributions supports interpretable representation learning with robust theoretical and empirical accuracy.
• Efficient Image and Video Synthesis (Lu et al., 24 Jul 2025): Adversarial ADM distillation sets new GPU efficiency and fidelity benchmarks in generative modeling through one-step and multi-step diffusion compression.

7. Comparison with Related Distribution Matching Paradigms

ADM improves on prior distribution matching by offering:

• Online, Anytime Guarantees: Contrasted with offline codebook-based approaches, ADM refines its match dynamically and is not restricted to fixed-length input.
• Adaptivity: Lower confidence bounds and empirical loss tailoring ensure sampling and computational effort is concentrated where it most benefits overall approximation.
• Scalability and Plug-and-Play Applicability: Through score-based priors, gradient-efficient objectives, and semantic regularization, ADM scales to high-dimensional, complex domains and model pipelines.
• Diversity Preservation: KL-divergence-based ADM (Yao et al., 30 Jan 2025) and adversarial constraints ensure that optimization avoids mode collapse and maintains distributional breadth.

In summary, ADM represents a flexible and theoretically sound approach to distributional alignment, applicable across streaming, sequential, and batch-processing paradigms in probabilistic modeling, reinforcement learning, generative synthesis, and commercial systems. Its defining characteristics—online refinement, theoretical convergence, computational adaptivity, and empirical diversity preservation—distinguish it from classic offline, parametric, or strictly adversarial distribution matching algorithms.