采样方法

Autoregressive (AR) models have recently achieved state-of-the-art performance in text and image generation. However, their primary limitation is slow generation speed due to the token-by-token process. We ask an ambitious question: can a pre-trained AR model be adapted to generate outputs in just one or two steps? If successful, this would significantly advance the development and deployment of AR models. We notice that existing works that attempt to speed up AR generation by generating multiple tokens at once fundamentally cannot capture the output distribution due to the conditional dependencies between tokens, limiting their effectiveness for few-step generation. To overcome this, we propose Distilled Decoding (DD), which leverages flow matching to create a deterministic mapping from Gaussian distribution to the output distribution of the pre-trained AR model. We then train a network to distill this mapping, enabling few-step generation. The entire training process of DD does not need the training data of the original AR model (as opposed to some other methods), thus making DD more practical. We evaluate DD on state-of-the-art image AR models and present promising results. For VAR, which requires 10-step generation (680 tokens), DD enables one-step generation (6.3$\times$ speed-up), with an acceptable increase in FID from 4.19 to 9.96. Similarly, for LlamaGen, DD reduces generation from 256 steps to 1, achieving an 217.8$\times$ speed-up with a comparable FID increase from 4.11 to 11.35. In both cases, baseline methods completely fail with FID scores $>$100. As the first work to demonstrate the possibility of one-step generation for image AR models, DD challenges the prevailing notion that AR models are inherently slow, and opens up new opportunities for efficient AR generation. The code and the pre-trained models will be released at https://github.com/imagination-research/distilled-decoding. The project website is at https://imagination-research.github.io/distilled-decoding.

Tokenization is associated with many poorly understood shortcomings in language models (LMs), yet remains an important component for long sequence scaling purposes. This work studies how tokenization impacts model performance by analyzing and comparing the stochastic behavior of tokenized models with their byte-level, or token-free, counterparts. We discover that, even when the two models are statistically equivalent, their predictive distributions over the next byte can be substantially different, a phenomenon we term as ``tokenization bias''. To fully characterize this phenomenon, we introduce the Byte-Token Representation Lemma, a framework that establishes a mapping between the learned token distribution and its equivalent byte-level distribution. From this result, we develop a next-byte sampling algorithm that eliminates tokenization bias without requiring further training or optimization. In other words, this enables zero-shot conversion of tokenized LMs into statistically equivalent token-free ones. We demonstrate its broad applicability with two use cases: fill-in-the-middle (FIM) tasks and model ensembles. In FIM tasks where input prompts may terminate mid-token, leading to out-of-distribution tokenization, our method mitigates performance degradation and achieves 18\% improvement in FIM coding benchmarks, while consistently outperforming the standard token healing fix. For model ensembles where each model employs a distinct vocabulary, our approach enables seamless integration, resulting in improved performance up to 3.7\% over individual models across various standard baselines in reasoning, knowledge, and coding. Code is available at:https: //github.com/facebookresearch/Exact-Byte-Level-Probabilities-from-Tokenized-LMs.

Given the limitations of backpropagation, perturbation-based gradient computation methods have recently gained focus for learning with only forward passes, also referred to as queries. Conventional forward learning consumes enormous queries on each data point for accurate gradient estimation through Monte Carlo sampling, which hinders the scalability of those algorithms. However, not all data points deserve equal queries for gradient estimation. In this paper, we study the problem of improving the forward learning efficiency from a novel perspective: how to reduce the gradient estimation variance with minimum cost? For this, we allocate the optimal number of queries within a set budget during training to balance estimation accuracy and computational efficiency. Specifically, with a simplified proxy objective and a reparameterization technique, we derive a novel plug-and-play query allocator with minimal parameters. Theoretical results are carried out to verify its optimality. We conduct extensive experiments for fine-tuning Vision Transformers on various datasets and further deploy the allocator to two black-box applications: prompt tuning and multimodal alignment for foundation models. All findings demonstrate that our proposed allocator significantly enhances the scalability of forward-learning algorithms, paving the way for real-world applications. The implementation is available at https://github.com/RTkenny/FLOPS-Forward-Learning-with-OPtimal-Sampling.

Sample selection improves the efficiency and effectiveness of machine learning models by providing informative and representative samples. Typically, samples can be modeled as a sample graph, where nodes are samples and edges represent their similarities. Most existing methods are based on local information, such as the training difficulty of samples, thereby overlooking global information, such as connectivity patterns. This oversight can result in suboptimal selection because global information is crucial for ensuring that the selected samples well represent the structural properties of the graph. To address this issue, we employ structural entropy to quantify global information and losslessly decompose it from the whole graph to individual nodes using the Shapley value. Based on the decomposition, we present $\textbf{S}$tructural-$\textbf{E}$ntropy-based sample $\textbf{S}$election ($\textbf{SES}$), a method that integrates both global and local information to select informative and representative samples. SES begins by constructing a $k$NN-graph among samples based on their similarities. It then measures sample importance by combining structural entropy (global metric) with training difficulty (local metric). Finally, SES applies importance-biased blue noise sampling to select a set of diverse and representative samples. Comprehensive experiments on three learning scenarios --- supervised learning, active learning, and continual learning --- clearly demonstrate the effectiveness of our method.

Diffusion Probabilistic Models (DPMs) are generative models showing competitive performance in various domains, including image synthesis and 3D point cloud generation. Sampling from pre-trained DPMs involves multiple neural function evaluations (NFEs) to transform Gaussian noise samples into images, resulting in higher computational costs compared to single-step generative models such as GANs or VAEs. Therefore, reducing the number of NFEs while preserving generation quality is crucial. To address this, we propose LD3, a lightweight framework designed to learn the optimal time discretization for sampling. LD3 can be combined with various samplers and consistently improves generation quality without having to retrain resource-intensive neural networks. We demonstrate analytically and empirically that LD3 improves sampling efficiency with much less computational overhead. We evaluate our method with extensive experiments on 7 pre-trained models, covering unconditional and conditional sampling in both pixel-space and latent-space DPMs. We achieve FIDs of 2.38 (10 NFE), and 2.27 (10 NFE) on unconditional CIFAR10 and AFHQv2 in 5-10 minutes of training. LD3 offers an efficient approach to sampling from pre-trained diffusion models. Code is available at https://github.com/vinhsuhi/LD3.

Originally introduced in game theory, Shapley values have emerged as a central tool in explainable machine learning, where they are used to attribute model predictions to specific input features. However, computing Shapley values exactly is expensive: for a model with $n$ features, $O(2^n)$ model evaluations are necessary. To address this issue, approximation algorithms are widely used. One of the most popular is the Kernel SHAP algorithm, which is model agnostic and remarkably effective in practice. However, to the best of our knowledge, Kernel SHAP has no strong non-asymptotic complexity guarantees. We address this issue by introducing *Leverage SHAP*, a light-weight modification of Kernel SHAP that provides provably accurate Shapley value estimates with just $O(n\log n)$ model evaluations. Our approach takes advantage of a connection between Shapley value estimation and agnostic active learning by employing *leverage score sampling*, a powerful regression tool. Beyond theoretical guarantees, we find that Leverage SHAP achieves an approximately 50% reduction in error compared to the highly optimized implementation of Kernel SHAP in the widely used SHAP library [Lundberg & Lee, 2017].

Sampling from diffusion models can be seen as solving the corresponding probability flow ordinary differential equation (ODE). The solving process requires a significant number of function evaluations (NFE), making it time-consuming. Recently, several solver search frameworks have attempted to find better-performing model-specific solvers. However, predicting the impact of intermediate solving strategies on final sample quality remains challenging, rendering the search process inefficient. In this paper, we propose a novel method for designing solving strategies. We first introduce a unified prediction formula for linear multistep solvers. Subsequently, we present a solver distillation framework, which enables a student solver to mimic the sampling trajectory generated by a teacher solver with more steps. We utilize the mean Euclidean distance between the student and teacher sampling trajectories as a metric, facilitating rapid adjustment and optimization of intermediate solving strategies. The design space of our framework encompasses multiple aspects, including prediction coefficients, time step schedules, and time scaling factors. Our framework has the ability to complete a solver search for Stable-Diffusion in under 12 total GPU hours. Compared to previous reinforcement learning-based search frameworks, our approach achieves over a 10$\times$ increase in search efficiency. With just 5 NFE, we achieve FID scores of 3.23 on CIFAR10, 7.16 on ImageNet-64, 5.44 on LSUN-Bedroom, and 12.52 on MS-COCO, resulting in a 2$\times$ sampling acceleration ratio compared to handcrafted solvers.

Diffusion probabilistic models (DPMs), while effective in generating high-quality samples, often suffer from high computational costs due to their iterative sampling process. To address this, we propose an enhanced ODE-based sampling method for DPMs inspired by Richardson extrapolation, which reduces numerical error and improves convergence rates. Our method, RX-DPM, leverages multiple ODE solutions at intermediate time steps to extrapolate the denoised prediction in DPMs. This significantly enhances the accuracy of estimations for the final sample while maintaining the number of function evaluations (NFEs). Unlike standard Richardson extrapolation, which assumes uniform discretization of the time grid, we develop a more general formulation tailored to arbitrary time step scheduling, guided by local truncation error derived from a baseline sampling method. The simplicity of our approach facilitates accurate estimation of numerical solutions without significant computational overhead, and allows for seamless and convenient integration into various DPMs and solvers. Additionally, RX-DPM provides explicit error estimates, effectively demonstrating the faster convergence as the leading error term's order increases. Through a series of experiments, we show that the proposed method improves the quality of generated samples without requiring additional sampling iterations.

We provide a framework for solving inverse problems with diffusion models learned from linearly corrupted data. Firstly, we extend the Ambient Diffusion framework to enable training directly from measurements corrupted in the Fourier domain. Subsequently, we train diffusion models for MRI with access only to Fourier subsampled multi-coil measurements at acceleration factors R$=2, 4, 6, 8$. Secondly, we propose $\textit{Ambient Diffusion Posterior Sampling}$ (A-DPS), a reconstruction algorithm that leverages generative models pre-trained on one type of corruption (e.g. image inpainting) to perform posterior sampling on measurements from a different forward process (e.g. image blurring). For MRI reconstruction in high acceleration regimes, we observe that A-DPS models trained on subsampled data are better suited to solving inverse problems than models trained on fully sampled data. We also test the efficacy of A-DPS on natural image datasets (CelebA, FFHQ, and AFHQ) and show that A-DPS can sometimes outperform models trained on clean data for several image restoration tasks in both speed and performance.

Are score function estimators a viable approach to learning with $k$-subset sampling? Sampling $k$-subsets is a fundamental operation that is not amenable to differentiable parametrization, impeding gradient-based optimization. Previous work has favored approximate pathwise gradients or relaxed sampling, dismissing score function estimators because of their high variance. Inspired by the success of score function estimators in variational inference and reinforcement learning, we revisit them for $k$-subset sampling. We demonstrate how to efficiently compute the distribution's score function using a discrete Fourier transform and reduce the estimator's variance with control variates. The resulting estimator provides both $k$-hot samples and unbiased gradient estimates while being applicable to non-differentiable downstream models, unlike existing methods. We validate our approach experimentally and find that it produces results comparable to those of recent state-of-the-art pathwise gradient estimators across a range of tasks.

Diffusion models have seen notable success in continuous domains, leading to the development of discrete diffusion models (DDMs) for discrete variables. Despite recent advances, DDMs face the challenge of slow sampling speeds. While parallel sampling methods like $\tau$-leaping accelerate this process, they introduce _Compounding Decoding Error_ (CDE), where discrepancies arise between the true distribution and the approximation from parallel token generation, leading to degraded sample quality. In this work, we present _Jump Your Steps_ (JYS), a novel approach that optimizes the allocation of discrete sampling timesteps by minimizing CDE without extra computational cost. More precisely, we derive a practical upper bound on CDE and propose an efficient algorithm for searching for the optimal sampling schedule. Extensive experiments across image, music, and text generation show that JYS significantly improves sampling quality, establishing it as a versatile framework for enhancing DDM performance for fast sampling.

Direct Preference Optimization (DPO) has been widely adopted for preference alignment of Large Language Models (LLMs) due to its simplicity and effectiveness. However, DPO is derived as a bandit problem in which the whole response is treated as a single arm, ignoring the importance differences between tokens, which may affect optimization efficiency and make it difficult to achieve optimal results. In this work, we propose that the optimal data for DPO has equal expected rewards for each token in winning and losing responses, as there is no difference in token importance. However, since the optimal dataset is unavailable in practice, we propose using the original dataset for importance sampling to achieve unbiased optimization. Accordingly, we propose a token-level importance sampling DPO objective named TIS-DPO that assigns importance weights to each token based on its reward. Inspired by previous works, we estimate the token importance weights using the difference in prediction probabilities from a pair of contrastive LLMs. We explore three methods to construct these contrastive LLMs: (1) guiding the original LLM with contrastive prompts, (2) training two separate LLMs using winning and losing responses, and (3) performing forward and reverse DPO training with winning and losing responses. Experiments show that TIS-DPO significantly outperforms various baseline methods on harmlessness and helpfulness alignment and summarization tasks. We also visualize the estimated weights, demonstrating their ability to identify key token positions.

The performance of large language models (LLMs) is closely linked to their underlying size, leading to ever-growing networks and hence slower inference. Speculative decoding has been proposed as a technique to accelerate autoregressive generation, leveraging a fast draft model to propose candidate tokens, which are then verified in parallel based on their likelihood under the target model. While this approach guarantees to reproduce the target output, it incurs a substantial penalty: many high-quality draft tokens are rejected, even when they represent objectively valid continuations. Indeed, we show that even powerful draft models such as GPT-4o, as well as human text cannot achieve high acceptance rates under the standard verification scheme. This severely limits the speedup potential of current speculative decoding methods, as an early rejection becomes overwhelmingly likely when solely relying on alignment of draft and target. We thus ask the following question: Can we adapt verification to recognize correct, but non-aligned replies? To this end, we draw inspiration from the LLM-as-a-judge framework, which demonstrated that LLMs are able to rate answers in a versatile way. We carefully design a dataset coined TokenCourt to elicit the same capability in the target model by training a compact module on top of the embeddings to produce ``judgements" of the current continuation. We showcase our strategy on the Llama-3.1 family, where our 8B/405B-Judge achieves a speedup of $9\times$ over Llama-405B, while maintaining its quality on a large range of benchmarks. These benefits remain present even in optimized inference frameworks, where our method reaches up to $141$ tokens/s for 8B/70B-Judge and $129$ tokens/s for 8B/405B on $2$ and $8$ H100s respectively.

As trajectories sampled by policies used by reinforcement learning (RL) and generative flow networks (GFlowNets) grow longer, credit assignment and exploration become more challenging, and the long planning horizon hinders mode discovery and generalization. The challenge is particularly pronounced in entropy-seeking RL methods, such as generative flow networks, where the agent must learn to sample from a structured distribution and discover multiple high-reward states, each of which take many steps to reach. To tackle this challenge, we propose an approach to incorporate the discovery of action abstractions, or high-level actions, into the policy optimization process. Our approach involves iteratively extracting action subsequences commonly used across many high-reward trajectories and `chunking' them into a single action that is added to the action space. In empirical evaluation on synthetic and real-world environments, our approach demonstrates improved sample efficiency performance in discovering diverse high-reward objects, especially on harder exploration problems. We also observe that the abstracted high-order actions are potentially interpretable, capturing the latent structure of the reward landscape of the action space. This work provides a cognitively motivated approach to action abstraction in RL and is the first demonstration of hierarchical planning in amortized sequential sampling.

The diffusion sampling process faces a persistent challenge stemming from its incoherence, attributable to varying noise directions across different timesteps. Our Representative Guidance (RepG) offers a new perspective to address this issue by reformulating the sampling process with a coherent direction toward a representative target. From this perspective, classic classifier guidance reveals its drawback in lacking meaningful representative information, as the features it relies on are optimized for discrimination and tend to highlight only a narrow set of class-specific cues. This focus often sacrifices diversity and increases the risk of adversarial generation. In contrast, we leverage self-supervised representations as the coherent target and treat sampling as a downstream task—one that focuses on refining image details and correcting generation errors, rather than settling for oversimplified outputs. Our Representative Guidance achieves superior performance and demonstrates the potential of pre-trained self-supervised models in guiding diffusion sampling. Our findings show that RepG not only significantly improves vanilla diffusion sampling, but also surpasses state-of-the-art benchmarks when combined with classifier-free guidance.

We revisit the well-studied problem of approximating a matrix product, $\bv{A}^T\bv{B}$, based on small space sketches $\mathcal{S}(\bv{A})$ and $\mathcal{S}(\bv{B})$ of $\bv{A} \in \R^{n \times d}$ and $\bv{B}\in \R^{n \times m}$. We are interested in the setting where the sketches must be computed independently of each other, except for the use of a shared random seed. We prove that, when $\bv{A}$ and $\bv{B}$ are sparse, methods based on \emph{coordinated random sampling} can outperform classical linear sketching approaches, like Johnson-Lindenstrauss Projection or CountSketch. For example, to obtain Frobenius norm error $\epsilon\|\bv{A}\|_F\|\bv{B}\|_F$, coordinated sampling requires sketches of size $O(s/\epsilon^2)$ when $\bv{A}$ and $\bv{B}$ have at most $s \leq d,m$ non-zeros per row. In contrast, linear sketching leads to sketches of size $O(d/\epsilon^2)$ and $O(m/\epsilon^2)$ for $\bv{A}$ and $\bv{B}$. We empirically evaluate our approach on two applications: 1) distributed linear regression in databases, a problem motivated by tasks like dataset discovery and augmentation, and 2) approximating attention matrices in transformer-based language models. In both cases, our sampling algorithms yield an order of magnitude improvement over linear sketching.

The majority of online continual learning (CL) advocates single-epoch training and imposes restrictions on the size of replay memory. However, single-epoch training would incur a different amount of computations per CL algorithm, and the additional storage cost to store logit or model in addition to replay memory is largely ignored in calculating the storage budget. Arguing different computational and storage budgets hinder fair comparison among CL algorithms in practice, we propose to use floating point operations (FLOPs) and total memory size in Byte as a metric for computational and memory budgets, respectively, to compare and develop CL algorithms in the same ‘total resource budget.’ To improve a CL method in a limited total budget, we propose adaptive layer freezing that does not update the layers for less informative batches to reduce computational costs with a negligible loss of accuracy. In addition, we propose a memory retrieval method that allows the model to learn the same amount of knowledge as using random retrieval in fewer iterations. Empirical validations on the CIFAR-10/100, CLEAR-10/100, and ImageNet-1K datasets demonstrate that the proposed approach outperforms the state-of-the-art methods within the same total budget. Furthermore, we validate its effectiveness in the Multi-modal Concept incremental Learning setup with multimodal large language models, such as LLaVA-1.5-7B. Code is available at https://github.com/snumprlab/budgeted-cl.

An effective approach for sampling from unnormalized densities is based on the idea of gradually transporting samples from an easy prior to the complicated target distribution. Two popular methods are (1) Sequential Monte Carlo (SMC), where the transport is performed through successive annealed densities via prescribed Markov chains and resampling steps, and (2) recently developed diffusion-based sampling methods, where a learned dynamical transport is used. Despite the common goal, both approaches have different, often complementary, advantages and drawbacks. The resampling steps in SMC allow focusing on promising regions of the space, often leading to robust performance. While the algorithm enjoys asymptotic guarantees, the lack of flexible, learnable transitions can lead to slow convergence. On the other hand, diffusion-based samplers are learned and can potentially better adapt themselves to the target at hand, yet often suffer from training instabilities. In this work, we present a principled framework for combining SMC with diffusion-based samplers by viewing both methods in continuous time and considering measures on path space. This culminates in the new Sequential Controlled Langevin Diffusion (SCLD) sampling method, which is able to utilize the benefits of both methods and reaches improved performance on multiple benchmark problems, in many cases using only 10% of the training budget of previous diffusion-based samplers.

Traditional parameter-space posterior inference for Bayesian neural networks faces several challenges, such as the difficulty in specifying meaningful prior, the potential pathologies in deep models and the intractability for multi-modal posterior. To address these issues, functional variational inference (fVI) and functional Markov Chain Monte Carlo (fMCMC) are two recently emerged Bayesian inference schemes that perform posterior inference directly in function space by incorporating more informative functional priors. Similar to their parameter-space counterparts, fVI and fMCMC have their own strengths and weaknesses. For instance, fVI is computationally efficient but imposes strong distributional assumptions, while fMCMC is asymptotically exact but suffers from slow mixing in high dimensions. To inherit the complementary benefits of both schemes, this work proposes a novel hybrid inference method for functional posterior inference. Specifically, it combines fVI and fMCMC successively by an elaborate linking mechanism to form an alternating approximation process. We also provide theoretical justification for the soundness of such a hybrid inference through the lens of Wasserstein gradient flows in the function space. We evaluate our method on several benchmark tasks and observe improvements in both predictive accuracy and uncertainty quantification compared to parameter/function-space VI and MCMC.

Investigating critical phenomena or phase transitions is of high interest in physics and chemistry, for which Monte Carlo (MC) simulations, a crucial tool for numerically analyzing macroscopic properties of given systems, are often hindered by an emerging divergence of correlation length---known as scale invariance at criticality (SIC) in the renormalization group theory. SIC causes the system to behave the same at any length scale, from which many existing sampling methods suffer: long-range correlations cause critical slowing down in Markov chain Monte Carlo (MCMC), and require intractably large receptive fields for generative samplers. In this paper, we propose a Renormalization-informed Generative Critical Sampler (RiGCS)---a novel sampler specialized for near-critical systems, where SIC is leveraged as an advantage rather than a nuisance. Specifically, RiGCS builds on MultiLevel Monte Carlo (MLMC) with Heat Bath (HB) algorithms, which perform ancestral sampling from low-resolution to high-resolution lattice configurations with site wise-independent conditional HB sampling. Although MLMC-HB is highly efficient under exact SIC, it suffers from a low acceptance rate under slight SIC violation. Notably, SIC violation always occurs in finite-size systems, and may induce long-range and higher-order interactions in the renormalized distributions, which are not considered by independent HB samplers. RiGCS enhances MLMC-HB by replacing a part of the conditional HB sampler with generative models that capture those residual interactions and improve the sampling efficiency. Our experiments show that the effective sample size of RiGCS is a few orders of magnitude higher than state-of-the-art generative model baselines in sampling configurations for $128 \times 128$ two-dimensional Ising systems. SIC also allows us to adopt a specialized sequential training protocol with model transfer, which significantly accelerates training.