Softmax attention is the cornerstone of modern large language models, but its memory scales linearly and compute quadratically with sequence length. Linear recurrent models, such as linear attention and state space models, have become widely studied as alternatives to attention due to their linear compute and constant memory. While these sub-quadratic token mixing methods, or mixers, achieve promising efficiency gains and competitive results on a wide range of benchmarks, current linear recurrent models still lag behind on tasks that require long-context retrieval or in-context learning. A growing body of work studies hybrid architectures that attempt to mitigate these trade-offs by statically interleaving or merging attention and recurrent blocks. In this work, we explore a new axis of developing hybrid models: across the token sequence. We propose Oryx, a hybrid model that can, throughout a sequence, flexibly switch between different mixers, for example quadratic attention for rich context utilization and linear recurrences for efficient generation. Oryx ties at least 90

Group Relative Policy Optimization (GRPO) has emerged as a powerful algorithm for improving the reasoning capabilities of language models, but often fails to improve small models due to sparse rewards on difficult tasks. Existing works mitigate this issue by leveraging a larger model, either to provide hints for rollouts or to provide dense reward signals through knowledge distillation (KD). However, this assumes the existence of such an oracle, and training one can significantly increase total training time. In this work, we propose CoDistill-GRPO, a co-distillation algorithm that simultaneously trains a large and a small model by maximizing carefully designed GRPO objectives. The two models learn from each other: the small model uses an on-policy KD reward to learn from the large model's distribution, while the large model is updated using rollouts generated by the small model with importance reweighting, reducing the computational overhead of rollout generation. We show that CoDistill-GRPO substantially improves small model performance over standard GRPO on mathematical benchmarks across both Qwen and Llama models. Specifically, with Qwen2.5-Math-1.5B, we observe an accuracy increase of over 11.6 percentage points over the base model and an additional 6.0 percentage points over GRPO on the Minerva dataset. Interestingly, the larger model (Qwen2.5-Math-7B) trained with CoDistill-GRPO nearly matches standard GRPO performance despite training on small-model rollouts. This highlights CoDistill-GRPO as a cost-effective alternative to GRPO for larger models, yielding an approximate 18

We study a variant of the simple hypothesis testing problem where observed samples do not necessarily come from either of the specified distributions, but rather from a close variant of them. In this setting, we require a test that is robust to misspecification and identifies which distribution is closer in Hellinger distance. If the underlying distribution is nearly equidistant from both hypotheses, the problem becomes intractable. Our main result is a lower bound on the slack factor, which quantifies how much closer the underlying distribution must be to one hypothesis relative to the other for any test to remain robust. We also demonstrate the implications of this result for testing with respect to symmetric chi-squared distance. Finally, we study an alternative way to specify robustness, where each hypothesis is a Hellinger ball around a fixed distribution. We provide and analyze a test for this composite hypothesis testing problem.

A simple and effective method for the inference-time alignment and scaling test-time compute of generative models is best-of-n sampling, where n samples are drawn from a reference policy, ranked based on a reward function, and the highest ranking one is selected. A commonly used analytical expression in the literature claims that the KL divergence between the best-of-n policy and the reference policy is equal to log(n) - (n 1)/n. We disprove the validity of this claim, and show that it is an upper bound on the actual KL divergence. We also explore the tightness of this upper bound in different regimes, and propose a new estimator for the KL divergence and empirically show that it provides a tight approximation. We also show that the win rate of the best-of-n policy against the reference policy is upper bounded by n/(n + 1) and derive bounds on the tightness of this characterization. We conclude with analyzing the tradeoffs between win rate and KL divergence of the best-of-n alignment policy, which demonstrate that very good tradeoffs are achievable with n < 1000.

Dual encoder (DE) models, where a pair of matching query and document are embedded into similar vector representations, are widely used in information retrieval due to their simplicity and scalability. However, the Euclidean geometry of the embedding space limits the expressive power of DEs, which may compromise their quality. This paper investigates such limitations in the context of hierarchical retrieval (HR), where the document set has a hierarchical structure and the matching documents for a query are all of its ancestors. We first prove that DEs are feasible for HR as long as the embedding dimension is linear in the depth of the hierarchy and logarithmic in the number of documents. Then we study the problem of learning such embeddings in a standard retrieval setup where DEs are trained on samples of matching query and document pairs. Our experiments reveal a lost-in-the-long-distance phenomenon, where retrieval accuracy degrades for documents further away in the hierarchy. To address this, we introduce a pretrain-finetune recipe that significantly improves long-distance retrieval without sacrificing performance on closer documents. We experiment on a realistic hierarchy from WordNet for retrieving documents at various levels of abstraction, and show that pretrain-finetune boosts the recall on long-distance pairs from 19% to 76%. Finally, we demonstrate that our method improves retrieval of relevant products on a shopping queries dataset.

In the private set union problem each user owns a bag of at most $k$ items (from some large universe of items), and we are interested in computing the union of the items in the bags of all of the users. This is trivial without privacy, but a differentially private algorithm must be careful about reporting items contained in only a small number of bags. We consider differentially private algorithms that always report a subset of the union, and define the utility of an algorithm to be the expected size of the subset that it reports. Because the achievable utility varies significantly with the dataset, we introduce the *utility ratio*, which normalizes utility by a dataset-specific upper bound and characterizes a mechanism by its lowest normalized utility across all datasets. We then develop algorithms with guaranteed utility ratios and complement them with bounds on the best possible utility ratio. Prior work has shown that a single algorithm can be simultaneously optimal for all datasets when $k=1$, but we show that instance-optimal algorithms do not exist when $k>1$, and characterize how performance degrades as $k$ grows. At the same time, we design a private algorithm that achieves the maximum possible utility, regardless of $k$, when the item histogram matches a prior prediction (for instance, from a previous data release) and degrades gracefully with the $L_\infty$ distance between the prediction and the actual histogram when the prediction is imperfect.

Post-training quantization is an effective method for reducing the serving cost of large language models, where the standard approach is to use a round-to-nearest quantization level scheme. However, this often introduces large errors due to outliers in the weights. Proposed mitigation mechanisms include applying adaptive rounding, random rotation transformations or committing to a post-training target using calibration data. Unfortunately, this reliance on calibration data can be severely limiting in some real-world scenarios as such data may be unavailable or subject to privacy regulations. In this paper, we propose algorithms to optimize transformations and adaptive rounding without access to any calibration data. The optimization is achieved by designing a suitable proxy function for the quantization loss without calibration data. To maintain inference efficiency, we perform structured matrix transformations for single matrices. For paired weights that interact directly in the computation graph, we use dual matrix transformations and adaptive rounding methods. We conduct experiments on Gemma 2 models, and observe consistent improvement over the baselines. For Gemma 2 9B quantization, our method improves the average benchmark score from 61.9 to 62.4 for 4-bit quantization and from 52.0 to 60.6 for 3-bit quantization, while adding less than 3

Language model alignment is a critical step in training modern generative language models. Alignment targets to improve win rate of a sample from the aligned model against the base model. Today, we are increasingly using inference-time algorithms (e.g., Best-of-N, controlled decoding, tree search) to decode from language models rather than standard sampling. We show that this train/test mismatch makes standard RL framework sub-optimal in view of such inference-time methods. To this end, we propose a framework for inference-aware alignment (InfAlign), which aims to optimize inference-time win rate of the aligned policy against the base model. We prove that for any inference-time decoding procedure, the optimal aligned policy is the solution to the standard RLHF problem with a transformation of the reward. This motivates us to provide the calibrate-and-transform RL (InfAlign-CTRL) algorithm to solve this problem, which involves a reward calibration step and a KL-regularized reward maximization step with a transformation of the calibrated reward. For best-of-N sampling and best-of-N jailbreaking, we propose specific transformations offering up to 3-8% improvement on inference-time win rates. Finally, we also show that our proposed reward calibration method is a strong baseline for optimizing standard win rate. [GRAPHICS] .

Speculative decoding is an effective method for lossless acceleration of large language models during inference. It uses a fast model to draft a block of tokens which are then verified in parallel by the target model, and provides a guarantee that the output is distributed identically to a sample from the target model. In prior works, draft verification is performed independently token-by-token. Surprisingly, we show that this approach is not optimal. We propose *Block Verification*, a simple draft verification algorithm that verifies the entire block jointly and provides additional wall-clock speedup. We prove that the proposed mechanism is optimal in the expected number of tokens produced each iteration and specifically is never worse than the standard token-level verification.Empirically, block verification provides modest but consistent wall-clock speedups over the standard token verification algorithm of 5\%-8\% in a range of tasks and datasets.Given that block verification does not increase code complexity, maintains the strong lossless guarantee of the standard speculative decoding verification algorithm, cannot deteriorate performance, and, in fact, consistently improves it, it can be used as a good default in speculative decoding implementations.

Suppose Alice has a distribution P and Bob has a distribution Q. Alice wants to draw a sample a∼ P and Bob a sample b ∼ Q such that a = b with as high of probability as possible. It is well-known that, by sampling from an optimal coupling between the distributions, Alice and Bob can achieve [a = b] = 1 - D_TV(P,Q), where D_TV(P,Q) is the total variation distance between P and Q. What if Alice and Bob must solve this same problem without communicating at all? Perhaps surprisingly, with access to public randomness, they can still achieve [a = b] ≥1 - D_TV(P,Q)/1 + D_TV(P,Q)≥ 1-2D_TV(P,Q) using a simple protocol based on the Weighted MinHash algorithm. This bound was shown to be optimal in the worst-case by [Bavarian et al., 2020]. In this work, we revisit the communication-free coupling problem. We provide a simpler proof of the optimality result from [Bavarian et al., 2020]. We show that, while the worst-case success probability of Weighted MinHash cannot be improved, an equally simple protocol based on Gumbel sampling offers a Pareto improvement: for every pair of distributions P, Q, Gumbel sampling achieves an equal or higher value of [a = b] than Weighted MinHash. Importantly, this improvement translates to practice. We demonstrate an application of communication-free coupling to speculative decoding, a recent method for accelerating autoregressive large language models [Leviathan, Kalman, Matias, ICML 2023]. We show that communication-free protocols can be used to contruct schemes, which have the desirable property that their output is fixed given a fixed random seed, regardless of what drafter is used for speculation. In experiments on a language generation task, Gumbel sampling outperforms Weighted MinHash. Code is available at https://github.com/majid-daliri/DISD.

In real-world applications of large language models, outputs are often required to be confined: selecting items from predefined product or document sets, generating phrases that comply with safety standards, or conforming to specialized formatting styles. To control the generation, constrained decoding has been widely adopted. However, existing prefix-tree-based constrained decoding is inefficient under GPU-based model inference paradigms, and it introduces unintended biases into the output distribution. This paper introduces Dynamic Importance Sampling for Constrained Decoding (DISC) with GPU-based Parallel Prefix-Verification (PPV), a novel algorithm that leverages dynamic importance sampling to achieve theoretically guaranteed asymptotic unbiasedness and overcomes the inefficiency of prefix-tree. Extensive experiments demonstrate the superiority of our method over existing methods in both efficiency and output quality. These results highlight the potential of our methods to improve constrained generation in applications where adherence to specific constraints is essential.

Given the ease of creating synthetic data from machine learning models, new models can be potentially trained on synthetic data generated by previous models. This recursive training process raises concerns about the long-term impact on model quality. As models are recursively trained on generated data from previous rounds, their ability to capture the nuances of the original human-generated data may degrade. This is often referred to as model collapse. In this work, we ask how fast model collapse occurs for some well-studied distribution families under maximum likelihood (ML or near ML) estimation during recursive training. Surprisingly, even for fundamental distributions such as discrete and Gaussian distributions, the exact rate of model collapse is unknown. In this work, we theoretically characterize the rate of collapse in these fundamental settings and complement it with experimental evaluations. Our results show that for discrete distributions, the time to forget a symbol is approximately linearly dependent on the number of times it occurred in the original corpus, and for Gaussian models, the standard deviation reduces to zero roughly at n iterations, where n is the number of samples at each iteration. Both of these findings imply that model forgetting, at least in these simple distributions under near ML estimation with many samples, takes a long time.

Despite the remarkable strides made by autoregressive language models, their potential is often hampered by the slow inference speeds inherent in sequential token generation. Blockwise parallel decoding (BPD) was proposed by Stern et al. (2018) as a way to improve inference speed of language models. In this paper, we make two contributions to understanding and improving BPD drafts. We first offer an analysis of the token distributions produced by the BPD prediction heads. Secondly, we use this analysis to inform algorithms to improve BPD inference speed by refining the BPD drafts using small n-gram or neural language models. We empirically show that these refined BPD drafts yield a higher average verified prefix length across tasks.

Autoregressive language models have achieved remarkable advancements, yet their potential is often limited by the slow inference speeds associated with sequential token generation. Blockwise parallel decoding (BPD) was proposed by Stern et al. [42] as a method to improve inference speed of language models by simultaneously predicting multiple future tokens, termed block drafts, which are subsequently verified by the autoregressive model. This paper advances the understanding and improvement of block drafts in two ways. First, we analyze token distributions generated across multiple prediction heads. Second, leveraging these insights, we propose algorithms to improve BPD inference speed by refining the block drafts using task-independent \ngram and neural language models as lightweight rescorers. Experiments demonstrate that by refining block drafts of open-sourced Vicuna and Medusa LLMs, the mean accepted token length are increased by 5-25% relative. This results in over a 3x speedup in wall clock time compared to standard autoregressive decoding in open-source 7B and 13B LLMs.

Training machine learning models with differential privacy (DP) has received increasing interest in recent years. One of the most popular algorithms for training differentially private models is differentially private stochastic gradient descent (DPSGD) and its variants, where at each step gradients are clipped and combined with some noise. Given the increasing usage of DPSGD, we ask the question: is DPSGD alone sufficient to find a good minimizer for every dataset under privacy constraints? As a first step towards answering this question, we show that even for the simple case of linear classification, unlike non-private optimization, (private) feature preprocessing is vital for differentially private optimization. In detail, we first show theoretically that there exists an example where without feature preprocessing, DPSGD incurs a privacy error proportional to the maximum norm of features over all samples. We then propose an algorithm called *DPSGD-F*, which combines DPSGD with feature preprocessing and prove that for classification tasks, it incurs a privacy error proportional to the diameter of the features $\max_{x, x' \in D} \|x - x'\|_2$. We then demonstrate the practicality of our algorithm on image classification benchmarks.

Communication efficient distributed mean estimation is an important primitive that arises in many distributed learning and optimization scenarios such as federated learning. Without any probabilistic assumptions on the underlying data, we study the problem of distributed mean estimation where the server has access to side information. We propose Wyner-Ziv estimators , which are communication and computationally efficient and near-optimal when an upper bound for the distance between the side information and the data is known. As a corollary, we also show that our algorithms provide efficient schemes for the classic Wyner-Ziv problem in information theory. In a different direction, when there is no knowledge assumed about the distance between side information and the data, we present an alternative Wyner-Ziv estimator that uses correlated sampling. This latter setting offers universal recovery guarantees , and perhaps will be of interest in practice when the number of users is large and keeping track of the distances between the data and the side information may not be possible. With this mean estimator at our disposal, we revisit basic problems in decentralized optimization and compression where our Wyner-Ziv estimator yields algorithms with almost optimal performance. First, we consider the problem of communication constrained distributed optimization and provide an algorithm which attains the optimal convergence rate by exploiting the fact that the gradient estimates are close to each other. Specifically, the gradient compression scheme in our algorithm first uses half of the parties to form side information and then uses our Wyner-Ziv estimator to compress the remaining half of the gradient estimates. Finally, we apply our Wynzer-Ziv estimators to the classic Wyner-Ziv compression problem in information theory to get compression schemes that are computationally efficient and are almost optimal under much more relaxed assumptions than the standard probabilistic setting.

Machine learning (ML) models used in prediction and classification tasks may display performance disparities across population groups determined by sensitive attributes (e.g., race, sex, age). We consider the problem of evaluating the performance of a fixed ML model across population groups defined by multiple sensitive attributes (e.g., race and sex and age). Here, the sample complexity for estimating the worst-case performance gap across groups (e.g., the largest difference in error rates) increases exponentially with the number of group-denoting sensitive attributes. To address this issue, we propose an approach to test for performance disparities based on Conditional Value-at-Risk (CVaR). By allowing a small probabilistic slack on the groups over which a model has approximately equal performance, we show that the sample complexity required for discovering performance violations is reduced exponentially to be at most upper bounded by the square root of the number of groups. As a byproduct of our analysis, when the groups are weighted by a specific prior distribution, we show that Rényi entropy of order $2/3$ of the prior distribution captures the sample complexity of the proposed CVaR test algorithm. Finally, we also show that there exists a non-i.i.d. data collection strategy that results in a sample complexity independent of the number of groups.

Cross-device federated learning (FL) is a technique that trains a model on data distributed across typically millions of edge devices without data leaving the devices. SGD is the standard client optimizer for on device training in cross-device FL, favored for its memory and computational efficiency. However, in centralized training of neural language models, adaptive optimizers are preferred as they offer improved stability and performance. In light of this, we ask if language models can be modified such that they can be efficiently trained with SGD client optimizers and answer this affirmatively. We propose a scale-invariant Coupled Input Forget Gate (SI CIFG) recurrent network by modifying the sigmoid and tanh activations in the recurrent cell and show that this new model converges faster and achieves better utility than the standard CIFG recurrent model in cross-device FL in large scale experiments. We further show that the proposed scale invariant modification also helps in federated learning of larger transformer models. Finally, we demonstrate the scale invariant modification is also compatible with other non-adaptive algorithms. Particularly, our results suggest an improved privacy utility trade-off in federated learning with differential privacy.

Let $p$ denote a generative language model. Let $r$ denote a reward model that returns a scalar that captures the degree at which a draw from $p$ is preferred. The goal of language model alignment is to alter $p$ to a new distribution $\phi$ that results in a higher expected reward while keeping $\phi$ close to $p.$ A popular alignment method is the KL-constrained reinforcement learning (RL), which chooses a distribution $\phi_\Delta$ that maximizes $E_{\phi_{\Delta}} r(y)$ subject to a relative entropy constraint $KL(\phi_\Delta || p) \leq \Delta.$ Another simple alignment method is best-of-$N$, where $N$ samples are drawn from $p$ and one with highest reward is selected. In this paper, we offer a closed-form characterization of the optimal KL-constrained RL solution. We demonstrate that any alignment method that achieves a comparable trade-off between KL divergence and reward must approximate the optimal KL-constrained RL solution in terms of relative entropy. To further analyze the properties of alignment methods, we introduce two simplifying assumptions: we let the language model be memoryless, and the reward model be linear. Although these assumptions may not reflect complex real-world scenarios, they enable a precise characterization of the asymptotic behavior of both the best-of-$N$ alignment, and the KL-constrained RL method, in terms of information-theoretic quantities. We prove that the reward of the optimal KL-constrained RL solution satisfies a large deviation principle, and we fully characterize its rate function. We also show that the rate of growth of the scaled cumulants of the reward is characterized by a proper Renyi cross entropy. Finally, we show that best-of-$N$ is asymptotically equivalent to KL-constrained RL solution by proving that their expected rewards are asymptotically equal, and concluding that the two distributions must be close in KL divergence.