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May 8

A Probabilistic Inference Approach to Inference-Time Scaling of LLMs using Particle-Based Monte Carlo Methods

Large language models (LLMs) have achieved significant performance gains via scaling up model sizes and/or data. However, recent evidence suggests diminishing returns from such approaches, motivating scaling the computation spent at inference time. Existing inference-time scaling methods, usually with reward models, cast the task as a search problem, which tends to be vulnerable to reward hacking as a consequence of approximation errors in reward models. In this paper, we instead cast inference-time scaling as a probabilistic inference task and leverage sampling-based techniques to explore the typical set of the state distribution of a state-space model with an approximate likelihood, rather than optimize for its mode directly. We propose a novel inference-time scaling approach by adapting particle-based Monte Carlo methods to this task. Our empirical evaluation demonstrates that our methods have a 4-16x better scaling rate over our deterministic search counterparts on various challenging mathematical reasoning tasks. Using our approach, we show that Qwen2.5-Math-1.5B-Instruct can surpass GPT-4o accuracy in only 4 rollouts, while Qwen2.5-Math-7B-Instruct scales to o1 level accuracy in only 32 rollouts. Our work not only presents an effective method to inference-time scaling, but also connects the rich literature in probabilistic inference with inference-time scaling of LLMs to develop more robust algorithms in future work. Code and further information is available at https://probabilistic-inference-scaling.github.io.

Accelerating Auto-regressive Text-to-Image Generation with Training-free Speculative Jacobi Decoding

The current large auto-regressive models can generate high-quality, high-resolution images, but these models require hundreds or even thousands of steps of next-token prediction during inference, resulting in substantial time consumption. In existing studies, Jacobi decoding, an iterative parallel decoding algorithm, has been used to accelerate the auto-regressive generation and can be executed without training. However, the Jacobi decoding relies on a deterministic criterion to determine the convergence of iterations. Thus, it works for greedy decoding but is incompatible with sampling-based decoding which is crucial for visual quality and diversity in the current auto-regressive text-to-image generation. In this paper, we propose a training-free probabilistic parallel decoding algorithm, Speculative Jacobi Decoding (SJD), to accelerate auto-regressive text-to-image generation. By introducing a probabilistic convergence criterion, our SJD accelerates the inference of auto-regressive text-to-image generation while maintaining the randomness in sampling-based token decoding and allowing the model to generate diverse images. Specifically, SJD facilitates the model to predict multiple tokens at each step and accepts tokens based on the probabilistic criterion, enabling the model to generate images with fewer steps than the conventional next-token-prediction paradigm. We also investigate the token initialization strategies that leverage the spatial locality of visual data to further improve the acceleration ratio under specific scenarios. We conduct experiments for our proposed SJD on multiple auto-regressive text-to-image generation models, showing the effectiveness of model acceleration without sacrificing the visual quality.

Locally Typical Sampling

Today's probabilistic language generators fall short when it comes to producing coherent and fluent text despite the fact that the underlying models perform well under standard metrics, e.g., perplexity. This discrepancy has puzzled the language generation community for the last few years. In this work, we posit that the abstraction of natural language generation as a discrete stochastic process--which allows for an information-theoretic analysis--can provide new insights into the behavior of probabilistic language generators, e.g., why high-probability texts can be dull or repetitive. Humans use language as a means of communicating information, aiming to do so in a simultaneously efficient and error-minimizing manner; in fact, psycholinguistics research suggests humans choose each word in a string with this subconscious goal in mind. We formally define the set of strings that meet this criterion: those for which each word has an information content close to the expected information content, i.e., the conditional entropy of our model. We then propose a simple and efficient procedure for enforcing this criterion when generating from probabilistic models, which we call locally typical sampling. Automatic and human evaluations show that, in comparison to nucleus and top-k sampling, locally typical sampling offers competitive performance (in both abstractive summarization and story generation) in terms of quality while consistently reducing degenerate repetitions.

Denotational validation of higher-order Bayesian inference

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.

On Sequential Bayesian Inference for Continual Learning

Sequential Bayesian inference can be used for continual learning to prevent catastrophic forgetting of past tasks and provide an informative prior when learning new tasks. We revisit sequential Bayesian inference and test whether having access to the true posterior is guaranteed to prevent catastrophic forgetting in Bayesian neural networks. To do this we perform sequential Bayesian inference using Hamiltonian Monte Carlo. We propagate the posterior as a prior for new tasks by fitting a density estimator on Hamiltonian Monte Carlo samples. We find that this approach fails to prevent catastrophic forgetting demonstrating the difficulty in performing sequential Bayesian inference in neural networks. From there we study simple analytical examples of sequential Bayesian inference and CL and highlight the issue of model misspecification which can lead to sub-optimal continual learning performance despite exact inference. Furthermore, we discuss how task data imbalances can cause forgetting. From these limitations, we argue that we need probabilistic models of the continual learning generative process rather than relying on sequential Bayesian inference over Bayesian neural network weights. In this vein, we also propose a simple baseline called Prototypical Bayesian Continual Learning, which is competitive with state-of-the-art Bayesian continual learning methods on class incremental continual learning vision benchmarks.

Why think step by step? Reasoning emerges from the locality of experience

Humans have a powerful and mysterious capacity to reason. By working through a series of purely mental steps, we can make inferences we would not be capable of making directly -- despite the fact that we get no additional data from the world. Similarly, when large language models generate a series of intermediate steps (a chain of thought) before answering a question, they often produce better answers than they otherwise would. We investigate why and how chain-of-thought reasoning is useful in language models, testing the hypothesis that reasoning is effective when training data consists of local clusters of variables that influence each other strongly. These training conditions enable the chaining of accurate local inferences in order to estimate relationships between variables that were not seen together in training. We prove that there will exist a "reasoning gap", where reasoning through intermediate variables improves inference, for the simple case of an autoregressive density estimator trained on local samples from a chain-structured probabilistic model. We then test our hypothesis empirically in more complex models, training an autoregressive language model on samples from Bayes nets but only including a subset of variables in each sample. We test language models' ability to match conditional probabilities with and without intermediate reasoning steps, finding that intermediate steps are only helpful when the training data is locally structured with respect to dependencies between variables and that the combination of locally-structured observations and reasoning is much more data-efficient than training on all variables. Our results illustrate how the effectiveness of reasoning step by step is rooted in the local statistical structure of the training data.

MIG: Automatic Data Selection for Instruction Tuning by Maximizing Information Gain in Semantic Space

Data quality and diversity are key to the construction of effective instruction-tuning datasets. % With the increasing availability of open-source instruction-tuning datasets, it is advantageous to automatically select high-quality and diverse subsets from a vast amount of data. % Existing methods typically prioritize instance quality and use heuristic rules to maintain diversity. % However, this absence of a comprehensive view of the entire collection often leads to suboptimal results. % Moreover, heuristic rules generally focus on distance or clustering within the embedding space, which fails to accurately capture the intent of complex instructions in the semantic space. % To bridge this gap, we propose a unified method for quantifying the information content of datasets. This method models the semantic space by constructing a label graph and quantifies diversity based on the distribution of information within the graph. % Based on such a measurement, we further introduce an efficient sampling method that selects data samples iteratively to Maximize the Information Gain (MIG) in semantic space. % Experiments on various datasets and base models demonstrate that MIG consistently outperforms state-of-the-art methods. % Notably, the model fine-tuned with 5\% Tulu3 data sampled by MIG achieves comparable performance to the official SFT model trained on the full dataset, with improvements of +5.73\% on AlpacaEval and +6.89\% on Wildbench.

ThinkSum: Probabilistic reasoning over sets using large language models

Large language models (LLMs) have a substantial capacity for high-level analogical reasoning: reproducing patterns in linear text that occur in their training data (zero-shot evaluation) or in the provided context (few-shot in-context learning). However, recent studies show that even the more advanced LLMs fail in scenarios that require reasoning over multiple objects or facts and making sequences of logical deductions. We propose a two-stage probabilistic inference paradigm, ThinkSum, which reasons over sets of objects or facts in a structured manner. In the first stage (Think - retrieval of associations), a LLM is queried in parallel over a set of phrases extracted from the prompt or an auxiliary model call. In the second stage (Sum - probabilistic inference or reasoning), the results of these queries are aggregated to make the final prediction. We demonstrate the possibilities and advantages of ThinkSum on the BIG-bench suite of LLM evaluation tasks, achieving improvements over the state of the art using GPT-family models on thirteen difficult tasks, often with far smaller model variants. We also compare and contrast ThinkSum with other proposed modifications to direct prompting of LLMs, such as variants of chain-of-thought prompting. Our results suggest that because the probabilistic inference in ThinkSum is performed outside of calls to the LLM, ThinkSum is less sensitive to prompt design, yields more interpretable predictions, and can be flexibly combined with latent variable models to extract structured knowledge from LLMs. Overall, our proposed paradigm represents a promising approach for enhancing the reasoning capabilities of LLMs.

Inference Scaling scriptsizeFLaws: The Limits of LLM Resampling with Imperfect Verifiers

Recent research has generated hope that inference scaling could allow weaker language models to match or exceed the accuracy of stronger models, such as by repeatedly sampling solutions to a coding problem until it passes unit tests. The central thesis of this paper is that there is no free lunch for inference scaling: indefinite accuracy improvement through resampling can only be realized if the "verifier" (in this case, a set of unit tests) is perfect. When the verifier is imperfect, as it almost always is in domains such as reasoning or coding (for example, unit tests have imperfect coverage), there is a nonzero probability of false positives: incorrect solutions that pass the verifier. Resampling cannot decrease this probability, so it imposes an upper bound to the accuracy of resampling-based inference scaling even with an infinite compute budget. We find that there is a very strong correlation between the model's single-sample accuracy (i.e. accuracy without unit tests) and its false positive rate on coding benchmarks HumanEval and MBPP, whose unit tests have limited coverage. Therefore, no amount of inference scaling of weaker models can enable them to match the single-sample accuracy of a sufficiently strong model (Fig. 1a). When we consider that false positives have a negative utility compared to abstaining from producing a solution, it bends the inference scaling curve further downward. Empirically, we find that the optimal number of samples can be less than 10 under realistic assumptions (Fig. 1b). Finally, we show that beyond accuracy, false positives may have other undesirable qualities, such as poor adherence to coding style conventions.

Distribution Transformers: Fast Approximate Bayesian Inference With On-The-Fly Prior Adaptation

While Bayesian inference provides a principled framework for reasoning under uncertainty, its widespread adoption is limited by the intractability of exact posterior computation, necessitating the use of approximate inference. However, existing methods are often computationally expensive, or demand costly retraining when priors change, limiting their utility, particularly in sequential inference problems such as real-time sensor fusion. To address these challenges, we introduce the Distribution Transformer -- a novel architecture that can learn arbitrary distribution-to-distribution mappings. Our method can be trained to map a prior to the corresponding posterior, conditioned on some dataset -- thus performing approximate Bayesian inference. Our novel architecture represents a prior distribution as a (universally-approximating) Gaussian Mixture Model (GMM), and transforms it into a GMM representation of the posterior. The components of the GMM attend to each other via self-attention, and to the datapoints via cross-attention. We demonstrate that Distribution Transformers both maintain flexibility to vary the prior, and significantly reduces computation times-from minutes to milliseconds-while achieving log-likelihood performance on par with or superior to existing approximate inference methods across tasks such as sequential inference, quantum system parameter inference, and Gaussian Process predictive posterior inference with hyperpriors.

BayesPrompt: Prompting Large-Scale Pre-Trained Language Models on Few-shot Inference via Debiased Domain Abstraction

As a novel and effective fine-tuning paradigm based on large-scale pre-trained language models (PLMs), prompt-tuning aims to reduce the gap between downstream tasks and pre-training objectives. While prompt-tuning has yielded continuous advancements in various tasks, such an approach still remains a persistent defect: prompt-tuning methods fail to generalize to specific few-shot patterns. From the perspective of distribution analyses, we disclose that the intrinsic issues behind the phenomenon are the over-multitudinous conceptual knowledge contained in PLMs and the abridged knowledge for target downstream domains, which jointly result in that PLMs mis-locate the knowledge distributions corresponding to the target domains in the universal knowledge embedding space. To this end, we intuitively explore to approximate the unabridged target domains of downstream tasks in a debiased manner, and then abstract such domains to generate discriminative prompts, thereby providing the de-ambiguous guidance for PLMs. Guided by such an intuition, we propose a simple yet effective approach, namely BayesPrompt, to learn prompts that contain the domain discriminative information against the interference from domain-irrelevant knowledge. BayesPrompt primitively leverages known distributions to approximate the debiased factual distributions of target domains and further uniformly samples certain representative features from the approximated distributions to generate the ultimate prompts for PLMs. We provide theoretical insights with the connection to domain adaptation. Empirically, our method achieves state-of-the-art performance on benchmarks.

Experts Don't Cheat: Learning What You Don't Know By Predicting Pairs

Identifying how much a model {p}_{theta}(Y|X) knows about the stochastic real-world process p(Y|X) it was trained on is important to ensure it avoids producing incorrect or "hallucinated" answers or taking unsafe actions. But this is difficult for generative models because probabilistic predictions do not distinguish between per-response noise (aleatoric uncertainty) and lack of knowledge about the process (epistemic uncertainty), and existing epistemic uncertainty quantification techniques tend to be overconfident when the model underfits. We propose a general strategy for teaching a model to both approximate p(Y|X) and also estimate the remaining gaps between {p}_{theta}(Y|X) and p(Y|X): train it to predict pairs of independent responses drawn from the true conditional distribution, allow it to "cheat" by observing one response while predicting the other, then measure how much it cheats. Remarkably, we prove that being good at cheating (i.e. cheating whenever it improves your prediction) is equivalent to being second-order calibrated, a principled extension of ordinary calibration that allows us to construct provably-correct frequentist confidence intervals for p(Y|X) and detect incorrect responses with high probability. We demonstrate empirically that our approach accurately estimates how much models don't know across ambiguous image classification, (synthetic) language modeling, and partially-observable navigation tasks, outperforming existing techniques.

Language Model Uncertainty Quantification with Attention Chain

Accurately quantifying a large language model's (LLM) predictive uncertainty is crucial for judging the reliability of its answers. While most existing research focuses on short, directly answerable questions with closed-form outputs (e.g., multiple-choice), involving intermediate reasoning steps in LLM responses is increasingly important. This added complexity complicates uncertainty quantification (UQ) because the probabilities assigned to answer tokens are conditioned on a vast space of preceding reasoning tokens. Direct marginalization is infeasible, and the dependency inflates probability estimates, causing overconfidence in UQ. To address this, we propose UQAC, an efficient method that narrows the reasoning space to a tractable size for marginalization. UQAC iteratively constructs an "attention chain" of tokens deemed "semantically crucial" to the final answer via a backtracking procedure. Starting from the answer tokens, it uses attention weights to identify the most influential predecessors, then iterates this process until reaching the input tokens. Similarity filtering and probability thresholding further refine the resulting chain, allowing us to approximate the marginal probabilities of the answer tokens, which serve as the LLM's confidence. We validate UQAC on multiple reasoning benchmarks with advanced open-source LLMs, demonstrating that it consistently delivers reliable UQ estimates with high computational efficiency.

Generating Pragmatic Examples to Train Neural Program Synthesizers

Programming-by-example is the task of synthesizing a program that is consistent with a set of user-provided input-output examples. As examples are often an under-specification of one's intent, a good synthesizer must choose the intended program from the many that are consistent with the given set of examples. Prior work frames program synthesis as a cooperative game between a listener (that synthesizes programs) and a speaker (a user choosing examples), and shows that models of computational pragmatic inference are effective in choosing the user intended programs. However, these models require counterfactual reasoning over a large set of programs and examples, which is infeasible in realistic program spaces. In this paper, we propose a novel way to amortize this search with neural networks. We sample pairs of programs and examples via self-play between listener and speaker models, and use pragmatic inference to choose informative training examples from this sample.We then use the informative dataset to train models to improve the synthesizer's ability to disambiguate user-provided examples without human supervision. We validate our method on the challenging task of synthesizing regular expressions from example strings, and find that our method (1) outperforms models trained without choosing pragmatic examples by 23% (a 51% relative increase) (2) matches the performance of supervised learning on a dataset of pragmatic examples provided by humans, despite using no human data in training.

Learning Math Reasoning from Self-Sampled Correct and Partially-Correct Solutions

Pretrained language models have shown superior performance on many natural language processing tasks, yet they still struggle at multi-step formal reasoning tasks like grade school math problems. One key challenge of finetuning them to solve such math reasoning problems is that many existing datasets only contain one reference solution for each problem, despite the fact that there are often alternative solutions resembling different reasoning paths to the final answer. This way, the finetuned models are biased towards the limited reference solutions, which limits their generalization to unseen examples. To mitigate this issue, we propose to let the model perform sampling during training and learn from both self-sampled fully-correct solutions, which yield the correct answer upon execution, and partially-correct solutions, whose intermediate state matches an intermediate state of a known correct solution. We show that our use of self-sampled correct and partially-correct solutions can benefit learning and help guide the sampling process, leading to more efficient exploration of the solution space. Additionally, we explore various training objectives to support learning from multiple solutions per example and find they greatly affect the performance. Experiments on two math reasoning datasets show the effectiveness of our method compared to learning from a single reference solution with MLE, where we improve PASS@100 from 35.5% to 44.5% for GSM8K, and 27.6% to 36.2% PASS@80 for MathQA. Such improvements are also consistent across different model sizes. Our code is available at https://github.com/microsoft/TraceCodegen.

Language Models (Mostly) Know What They Know

We study whether language models can evaluate the validity of their own claims and predict which questions they will be able to answer correctly. We first show that larger models are well-calibrated on diverse multiple choice and true/false questions when they are provided in the right format. Thus we can approach self-evaluation on open-ended sampling tasks by asking models to first propose answers, and then to evaluate the probability "P(True)" that their answers are correct. We find encouraging performance, calibration, and scaling for P(True) on a diverse array of tasks. Performance at self-evaluation further improves when we allow models to consider many of their own samples before predicting the validity of one specific possibility. Next, we investigate whether models can be trained to predict "P(IK)", the probability that "I know" the answer to a question, without reference to any particular proposed answer. Models perform well at predicting P(IK) and partially generalize across tasks, though they struggle with calibration of P(IK) on new tasks. The predicted P(IK) probabilities also increase appropriately in the presence of relevant source materials in the context, and in the presence of hints towards the solution of mathematical word problems. We hope these observations lay the groundwork for training more honest models, and for investigating how honesty generalizes to cases where models are trained on objectives other than the imitation of human writing.

Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts

While score-based generative models are the model of choice across diverse domains, there are limited tools available for controlling inference-time behavior in a principled manner, e.g. for composing multiple pretrained models. Existing classifier-free guidance methods use a simple heuristic to mix conditional and unconditional scores to approximately sample from conditional distributions. However, such methods do not approximate the intermediate distributions, necessitating additional 'corrector' steps. In this work, we provide an efficient and principled method for sampling from a sequence of annealed, geometric-averaged, or product distributions derived from pretrained score-based models. We derive a weighted simulation scheme which we call Feynman-Kac Correctors (FKCs) based on the celebrated Feynman-Kac formula by carefully accounting for terms in the appropriate partial differential equations (PDEs). To simulate these PDEs, we propose Sequential Monte Carlo (SMC) resampling algorithms that leverage inference-time scaling to improve sampling quality. We empirically demonstrate the utility of our methods by proposing amortized sampling via inference-time temperature annealing, improving multi-objective molecule generation using pretrained models, and improving classifier-free guidance for text-to-image generation. Our code is available at https://github.com/martaskrt/fkc-diffusion.

Pushing the Limits of Rule Reasoning in Transformers through Natural Language Satisfiability

Investigating the reasoning abilities of transformer models, and discovering new challenging tasks for them, has been a topic of much interest. Recent studies have found these models to be surprisingly strong at performing deductive reasoning over formal logical theories expressed in natural language. A shortcoming of these studies, however, is that they do not take into account that logical theories, when sampled uniformly at random, do not necessarily lead to hard instances. We propose a new methodology for creating challenging algorithmic reasoning datasets that focus on natural language satisfiability (NLSat) problems. The key idea is to draw insights from empirical sampling of hard propositional SAT problems and from complexity-theoretic studies of language. This methodology allows us to distinguish easy from hard instances, and to systematically increase the complexity of existing reasoning benchmarks such as RuleTaker. We find that current transformers, given sufficient training data, are surprisingly robust at solving the resulting NLSat problems of substantially increased difficulty. They also exhibit some degree of scale-invariance - the ability to generalize to problems of larger size and scope. Our results, however, reveal important limitations too: a careful sampling of training data is crucial for building models that generalize to larger problems, and transformer models' limited scale-invariance suggests they are far from learning robust deductive reasoning algorithms.

Recovering Top-Two Answers and Confusion Probability in Multi-Choice Crowdsourcing

Crowdsourcing has emerged as an effective platform for labeling large amounts of data in a cost- and time-efficient manner. Most previous work has focused on designing an efficient algorithm to recover only the ground-truth labels of the data. In this paper, we consider multi-choice crowdsourcing tasks with the goal of recovering not only the ground truth, but also the most confusing answer and the confusion probability. The most confusing answer provides useful information about the task by revealing the most plausible answer other than the ground truth and how plausible it is. To theoretically analyze such scenarios, we propose a model in which there are the top two plausible answers for each task, distinguished from the rest of the choices. Task difficulty is quantified by the probability of confusion between the top two, and worker reliability is quantified by the probability of giving an answer among the top two. Under this model, we propose a two-stage inference algorithm to infer both the top two answers and the confusion probability. We show that our algorithm achieves the minimax optimal convergence rate. We conduct both synthetic and real data experiments and demonstrate that our algorithm outperforms other recent algorithms. We also show the applicability of our algorithms in inferring the difficulty of tasks and in training neural networks with top-two soft labels.

Probabilistic Tree-of-thought Reasoning for Answering Knowledge-intensive Complex Questions

Large language models (LLMs) are capable of answering knowledge-intensive complex questions with chain-of-thought (CoT) reasoning. However, they tend to generate factually incorrect reasoning steps when the required knowledge is not available or up-to-date in models' parameters. Recent works turn to retrieving external knowledge to augment CoT reasoning. Despite being promising, these chain-based methods suffer from: 1) Negative retrieval. Unnecessary or incorrect retrieval may mislead the reasoning; 2) Limited sight. Lacking the ability to look backward or forward, a local error in one step will propagate along the chain. In this paper, we propose a novel approach: Probabilistic Tree-of-thought Reasoning (ProbTree). First, LLMs translate a complex question into a query tree, in which each non-root node denotes a sub-question of its parent node. Then, probabilistic reasoning is conducted over the tree, by solving questions from leaf to root considering the confidence of both question decomposing and answering. During reasoning, for leaf nodes, LLMs choose a more confident answer from Closed-book QA that employs parametric knowledge and Open-book QA that employs retrieved external knowledge, thus eliminating the negative retrieval problem. For non-leaf nodes, with the hierarchical structure, LLMs have broader sights and are able to globally reason with the information from child nodes, thus recovering from local errors. The experiments on three Complex QA datasets under the open-domain setting show that our approach outperforms SOTA methods significantly, demonstrating the effect of probabilistic tree-of-thought reasoning.

Analysis on Riemann Hypothesis with Cross Entropy Optimization and Reasoning

In this paper, we present a novel framework for the analysis of Riemann Hypothesis [27], which is composed of three key components: a) probabilistic modeling with cross entropy optimization and reasoning; b) the application of the law of large numbers; c) the application of mathematical inductions. The analysis is mainly conducted by virtue of probabilistic modeling of cross entropy optimization and reasoning with rare event simulation techniques. The application of the law of large numbers [2, 3, 6] and the application of mathematical inductions make the analysis of Riemann Hypothesis self-contained and complete to make sure that the whole complex plane is covered as conjectured in Riemann Hypothesis. We also discuss the method of enhanced top-p sampling with large language models (LLMs) for reasoning, where next token prediction is not just based on the estimated probabilities of each possible token in the current round but also based on accumulated path probabilities among multiple top-k chain of thoughts (CoTs) paths. The probabilistic modeling of cross entropy optimization and reasoning may suit well with the analysis of Riemann Hypothesis as Riemann Zeta functions are inherently dealing with the sums of infinite components of a complex number series. We hope that our analysis in this paper could shed some light on some of the insights of Riemann Hypothesis. The framework and techniques presented in this paper, coupled with recent developments with chain of thought (CoT) or diagram of thought (DoT) reasoning in large language models (LLMs) with reinforcement learning (RL) [1, 7, 18, 21, 24, 34, 39-41], could pave the way for eventual proof of Riemann Hypothesis [27].

Vector representations of text data in deep learning

In this dissertation we report results of our research on dense distributed representations of text data. We propose two novel neural models for learning such representations. The first model learns representations at the document level, while the second model learns word-level representations. For document-level representations we propose Binary Paragraph Vector: a neural network models for learning binary representations of text documents, which can be used for fast document retrieval. We provide a thorough evaluation of these models and demonstrate that they outperform the seminal method in the field in the information retrieval task. We also report strong results in transfer learning settings, where our models are trained on a generic text corpus and then used to infer codes for documents from a domain-specific dataset. In contrast to previously proposed approaches, Binary Paragraph Vector models learn embeddings directly from raw text data. For word-level representations we propose Disambiguated Skip-gram: a neural network model for learning multi-sense word embeddings. Representations learned by this model can be used in downstream tasks, like part-of-speech tagging or identification of semantic relations. In the word sense induction task Disambiguated Skip-gram outperforms state-of-the-art models on three out of four benchmarks datasets. Our model has an elegant probabilistic interpretation. Furthermore, unlike previous models of this kind, it is differentiable with respect to all its parameters and can be trained with backpropagation. In addition to quantitative results, we present qualitative evaluation of Disambiguated Skip-gram, including two-dimensional visualisations of selected word-sense embeddings.

Quiet-STaR: Language Models Can Teach Themselves to Think Before Speaking

When writing and talking, people sometimes pause to think. Although reasoning-focused works have often framed reasoning as a method of answering questions or completing agentic tasks, reasoning is implicit in almost all written text. For example, this applies to the steps not stated between the lines of a proof or to the theory of mind underlying a conversation. In the Self-Taught Reasoner (STaR, Zelikman et al. 2022), useful thinking is learned by inferring rationales from few-shot examples in question-answering and learning from those that lead to a correct answer. This is a highly constrained setting -- ideally, a language model could instead learn to infer unstated rationales in arbitrary text. We present Quiet-STaR, a generalization of STaR in which LMs learn to generate rationales at each token to explain future text, improving their predictions. We address key challenges, including 1) the computational cost of generating continuations, 2) the fact that the LM does not initially know how to generate or use internal thoughts, and 3) the need to predict beyond individual next tokens. To resolve these, we propose a tokenwise parallel sampling algorithm, using learnable tokens indicating a thought's start and end, and an extended teacher-forcing technique. Encouragingly, generated rationales disproportionately help model difficult-to-predict tokens and improve the LM's ability to directly answer difficult questions. In particular, after continued pretraining of an LM on a corpus of internet text with Quiet-STaR, we find zero-shot improvements on GSM8K (5.9%rightarrow10.9%) and CommonsenseQA (36.3%rightarrow47.2%) and observe a perplexity improvement of difficult tokens in natural text. Crucially, these improvements require no fine-tuning on these tasks. Quiet-STaR marks a step towards LMs that can learn to reason in a more general and scalable way.

Internet-augmented language models through few-shot prompting for open-domain question answering

In this work, we aim to capitalize on the unique few-shot capabilities of large-scale language models (LSLMs) to overcome some of their challenges with respect to grounding to factual and up-to-date information. Motivated by semi-parametric language models (LMs), which ground their decisions in external retrieved evidence, we use few-shot prompting to learn to condition LMs on information returned from the web using Google Search, a broad and constantly updated knowledge source. Our approach does not involve fine-tuning or learning additional parameters, thus making it applicable to any LM, offering therefore a strong baseline. Indeed, we find that LMs conditioned on the web surpass performance of closed-book models of similar, or even larger, model sizes in open-domain question answering. Finally, we find that increasing the inference-time compute of models, achieved via using multiple retrieved evidences to generate multiple answers followed by a reranking stage that uses scores generated by the same LMs, leads to better performance and alleviates lower performance of smaller few-shot LMs. All in all, our findings suggest that it might be beneficial to slow down the race towards the biggest model and instead shift attention towards finding more effective ways to use models, including but not limited to, better prompting or increasing inference-time compute.

ProcBench: Benchmark for Multi-Step Reasoning and Following Procedure

Reasoning is central to a wide range of intellectual activities, and while the capabilities of large language models (LLMs) continue to advance, their performance in reasoning tasks remains limited. The processes and mechanisms underlying reasoning are not yet fully understood, but key elements include path exploration, selection of relevant knowledge, and multi-step inference. Problems are solved through the synthesis of these components. In this paper, we propose a benchmark that focuses on a specific aspect of reasoning ability: the direct evaluation of multi-step inference. To this end, we design a special reasoning task where multi-step inference is specifically focused by largely eliminating path exploration and implicit knowledge utilization. Our dataset comprises pairs of explicit instructions and corresponding questions, where the procedures necessary for solving the questions are entirely detailed within the instructions. This setup allows models to solve problems solely by following the provided directives. By constructing problems that require varying numbers of steps to solve and evaluating responses at each step, we enable a thorough assessment of state-of-the-art LLMs' ability to follow instructions. To ensure the robustness of our evaluation, we include multiple distinct tasks. Furthermore, by comparing accuracy across tasks, utilizing step-aware metrics, and applying separately defined measures of complexity, we conduct experiments that offer insights into the capabilities and limitations of LLMs in reasoning tasks. Our findings have significant implications for the development of LLMs and highlight areas for future research in advancing their reasoning abilities. Our dataset is available at https://huggingface.co/datasets/ifujisawa/procbench and code at https://github.com/ifujisawa/proc-bench.

Using Imperfect Surrogates for Downstream Inference: Design-based Supervised Learning for Social Science Applications of Large Language Models

In computational social science (CSS), researchers analyze documents to explain social and political phenomena. In most scenarios, CSS researchers first obtain labels for documents and then explain labels using interpretable regression analyses in the second step. One increasingly common way to annotate documents cheaply at scale is through large language models (LLMs). However, like other scalable ways of producing annotations, such surrogate labels are often imperfect and biased. We present a new algorithm for using imperfect annotation surrogates for downstream statistical analyses while guaranteeing statistical properties -- like asymptotic unbiasedness and proper uncertainty quantification -- which are fundamental to CSS research. We show that direct use of surrogate labels in downstream statistical analyses leads to substantial bias and invalid confidence intervals, even with high surrogate accuracy of 80-90%. To address this, we build on debiased machine learning to propose the design-based supervised learning (DSL) estimator. DSL employs a doubly-robust procedure to combine surrogate labels with a smaller number of high-quality, gold-standard labels. Our approach guarantees valid inference for downstream statistical analyses, even when surrogates are arbitrarily biased and without requiring stringent assumptions, by controlling the probability of sampling documents for gold-standard labeling. Both our theoretical analysis and experimental results show that DSL provides valid statistical inference while achieving root mean squared errors comparable to existing alternatives that focus only on prediction without inferential guarantees.

Fast Controlled Generation from Language Models with Adaptive Weighted Rejection Sampling

The dominant approach to generating from language models subject to some constraint is locally constrained decoding (LCD), incrementally sampling tokens at each time step such that the constraint is never violated. Typically, this is achieved through token masking: looping over the vocabulary and excluding non-conforming tokens. There are two important problems with this approach. (i) Evaluating the constraint on every token can be prohibitively expensive -- LM vocabularies often exceed 100,000 tokens. (ii) LCD can distort the global distribution over strings, sampling tokens based only on local information, even if they lead down dead-end paths. This work introduces a new algorithm that addresses both these problems. First, to avoid evaluating a constraint on the full vocabulary at each step of generation, we propose an adaptive rejection sampling algorithm that typically requires orders of magnitude fewer constraint evaluations. Second, we show how this algorithm can be extended to produce low-variance, unbiased estimates of importance weights at a very small additional cost -- estimates that can be soundly used within previously proposed sequential Monte Carlo algorithms to correct for the myopic behavior of local constraint enforcement. Through extensive empirical evaluation in text-to-SQL, molecular synthesis, goal inference, pattern matching, and JSON domains, we show that our approach is superior to state-of-the-art baselines, supporting a broader class of constraints and improving both runtime and performance. Additional theoretical and empirical analyses show that our method's runtime efficiency is driven by its dynamic use of computation, scaling with the divergence between the unconstrained and constrained LM, and as a consequence, runtime improvements are greater for better models.

Optimizing Chain-of-Thought Reasoners via Gradient Variance Minimization in Rejection Sampling and RL

Chain-of-thought (CoT) reasoning in large language models (LLMs) can be formalized as a latent variable problem, where the model needs to generate intermediate reasoning steps. While prior approaches such as iterative reward-ranked fine-tuning (RAFT) have relied on such formulations, they typically apply uniform inference budgets across prompts, which fails to account for variability in difficulty and convergence behavior. This work identifies the main bottleneck in CoT training as inefficient stochastic gradient estimation due to static sampling strategies. We propose GVM-RAFT, a prompt-specific Dynamic Sample Allocation Strategy designed to minimize stochastic gradient variance under a computational budget constraint. The method dynamically allocates computational resources by monitoring prompt acceptance rates and stochastic gradient norms, ensuring that the resulting gradient variance is minimized. Our theoretical analysis shows that the proposed dynamic sampling strategy leads to accelerated convergence guarantees under suitable conditions. Experiments on mathematical reasoning show that GVM-RAFT achieves a 2-4x speedup and considerable accuracy improvements over vanilla RAFT. The proposed dynamic sampling strategy is general and can be incorporated into other reinforcement learning algorithms, such as GRPO, leading to similar improvements in convergence and test accuracy. Our code is available at https://github.com/RLHFlow/GVM.

Shifting Attention to Relevance: Towards the Uncertainty Estimation of Large Language Models

While Large Language Models (LLMs) have demonstrated remarkable potential in natural language generation and instruction following, a persistent challenge lies in their susceptibility to "hallucinations", which erodes trust in their outputs. Although Uncertainty Quantification (UQ) presents a promising solution, its accurate implementation within the context of LLMs remains a significant hurdle. To address this critical roadblock, our research originates from a fundamental heuristic insight: tokens within auto-regressive LLM-generated text do not equally reflect the underlying meaning. Some tokens carry greater relevance and representativeness than others, owing to the phenomenon of "linguistic redundancy", wherein a select few keywords suffice to convey the essence of lengthy sentences. Regrettably, existing methodologies treat all tokens with equal importance when estimating uncertainty, disregarding these inherent generative inequalities. Our analysis reveals a significant issue with state-of-the-art: numerous tokens (and sentences) of limited semantic significance receive equal or even excessive weighting during uncertainty estimation. To rectify this bias, we propose to jointly Shifting Attention to more Relevant (SAR) components, at both the token- and the sentence-levels for accurate uncertainty estimation. We conduct extensive experiments involving a range of popular "off-the-shelf" LLMs, including instruction-tuned LLMs such as Vicuna, WizardLM, and LLaMA-2-chat, as well as pretrained LLMs like OPT and LLaMA, with model sizes extending up to 33B parameters. We carry out evaluation across various free-form question-answering tasks, encompassing domains such as reading comprehension, science Q&A, and medical Q&A. Our experimental results demonstrate the superior performance of SAR in addressing the challenges of uncertainty estimation within the realm of LLMs.

Adaptive Inference-Time Compute: LLMs Can Predict if They Can Do Better, Even Mid-Generation

Inference-time computation is a powerful paradigm to enhance the performance of large language models (LLMs), with Best-of-N sampling being a widely used technique. However, this method is computationally expensive, requiring both (1) an external reward model and (2) the generation of multiple samples. In this work, we introduce a new generative self-evaluation scheme designed to adaptively reduce the number of generated samples while maintaining or even improving performance. We use a generative reward model formulation, allowing the LLM to predict mid-generation the probability that restarting the generation will yield a better response. These predictions are obtained without an external reward model and can be used to decide whether or not to generate more samples, prune unpromising samples early on, or to pick the best sample. This capability is very inexpensive as it involves generating a single predefined token. Trained using a dataset constructed with real unfiltered LMSYS user prompts, Llama 3.1 8B's win rate against GPT-4 on AlpacaEval increases from 21% to 34% with 16 samples and math performance on GSM8K improves from 84% to 91%. By sampling only when the LLM determines that it is beneficial to do so and adaptively adjusting temperature annealing, we demonstrate that 74% of the improvement from using 16 samples can be achieved with only 1.2 samples on average. We further demonstrate that 50-75% of samples can be pruned early in generation with minimal degradation in performance. Overall, our methods enable more efficient and scalable compute utilization during inference for LLMs.

AriEL: volume coding for sentence generation

Mapping sequences of discrete data to a point in a continuous space makes it difficult to retrieve those sequences via random sampling. Mapping the input to a volume would make it easier to retrieve at test time, and that's the strategy followed by the family of approaches based on Variational Autoencoder. However the fact that they are at the same time optimizing for prediction and for smoothness of representation, forces them to trade-off between the two. We improve on the performance of some of the standard methods in deep learning to generate sentences by uniformly sampling a continuous space. We do it by proposing AriEL, that constructs volumes in a continuous space, without the need of encouraging the creation of volumes through the loss function. We first benchmark on a toy grammar, that allows to automatically evaluate the language learned and generated by the models. Then, we benchmark on a real dataset of human dialogues. Our results indicate that the random access to the stored information is dramatically improved, and our method AriEL is able to generate a wider variety of correct language by randomly sampling the latent space. VAE follows in performance for the toy dataset while, AE and Transformer follow for the real dataset. This partially supports to the hypothesis that encoding information into volumes instead of into points, can lead to improved retrieval of learned information with random sampling. This can lead to better generators and we also discuss potential disadvantages.

Semantic Volume: Quantifying and Detecting both External and Internal Uncertainty in LLMs

Large language models (LLMs) have demonstrated remarkable performance across diverse tasks by encoding vast amounts of factual knowledge. However, they are still prone to hallucinations, generating incorrect or misleading information, often accompanied by high uncertainty. Existing methods for hallucination detection primarily focus on quantifying internal uncertainty, which arises from missing or conflicting knowledge within the model. However, hallucinations can also stem from external uncertainty, where ambiguous user queries lead to multiple possible interpretations. In this work, we introduce Semantic Volume, a novel mathematical measure for quantifying both external and internal uncertainty in LLMs. Our approach perturbs queries and responses, embeds them in a semantic space, and computes the determinant of the Gram matrix of the embedding vectors, capturing their dispersion as a measure of uncertainty. Our framework provides a generalizable and unsupervised uncertainty detection method without requiring white-box access to LLMs. We conduct extensive experiments on both external and internal uncertainty detection, demonstrating that our Semantic Volume method consistently outperforms existing baselines in both tasks. Additionally, we provide theoretical insights linking our measure to differential entropy, unifying and extending previous sampling-based uncertainty measures such as the semantic entropy. Semantic Volume is shown to be a robust and interpretable approach to improving the reliability of LLMs by systematically detecting uncertainty in both user queries and model responses.

SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models

Generative Large Language Models (LLMs) such as GPT-3 are capable of generating highly fluent responses to a wide variety of user prompts. However, LLMs are known to hallucinate facts and make non-factual statements which can undermine trust in their output. Existing fact-checking approaches either require access to token-level output probability distribution (which may not be available for systems such as ChatGPT) or external databases that are interfaced via separate, often complex, modules. In this work, we propose "SelfCheckGPT", a simple sampling-based approach that can be used to fact-check black-box models in a zero-resource fashion, i.e. without an external database. SelfCheckGPT leverages the simple idea that if a LLM has knowledge of a given concept, sampled responses are likely to be similar and contain consistent facts. However, for hallucinated facts, stochastically sampled responses are likely to diverge and contradict one another. We investigate this approach by using GPT-3 to generate passages about individuals from the WikiBio dataset, and manually annotate the factuality of the generated passages. We demonstrate that SelfCheckGPT can: i) detect non-factual and factual sentences; and ii) rank passages in terms of factuality. We compare our approach to several existing baselines and show that in sentence hallucination detection, our approach has AUC-PR scores comparable to grey-box methods, while SelfCheckGPT is best at passage factuality assessment.

A Hierarchical Bayesian Model for Deep Few-Shot Meta Learning

We propose a novel hierarchical Bayesian model for learning with a large (possibly infinite) number of tasks/episodes, which suits well the few-shot meta learning problem. We consider episode-wise random variables to model episode-specific target generative processes, where these local random variables are governed by a higher-level global random variate. The global variable helps memorize the important information from historic episodes while controlling how much the model needs to be adapted to new episodes in a principled Bayesian manner. Within our model framework, the prediction on a novel episode/task can be seen as a Bayesian inference problem. However, a main obstacle in learning with a large/infinite number of local random variables in online nature, is that one is not allowed to store the posterior distribution of the current local random variable for frequent future updates, typical in conventional variational inference. We need to be able to treat each local variable as a one-time iterate in the optimization. We propose a Normal-Inverse-Wishart model, for which we show that this one-time iterate optimization becomes feasible due to the approximate closed-form solutions for the local posterior distributions. The resulting algorithm is more attractive than the MAML in that it is not required to maintain computational graphs for the whole gradient optimization steps per episode. Our approach is also different from existing Bayesian meta learning methods in that unlike dealing with a single random variable for the whole episodes, our approach has a hierarchical structure that allows one-time episodic optimization, desirable for principled Bayesian learning with many/infinite tasks. The code is available at https://github.com/minyoungkim21/niwmeta.

Chimera: A Lossless Decoding Method for Accelerating Large Language Models Inference by Fusing all Tokens

Large language models (LLMs) have demonstrated remarkable capabilities across various tasks. However, their widespread application is hindered by the resource-intensive decoding process. To address this challenge, current approaches have incorporated additional decoding heads to enable parallel prediction of multiple subsequent tokens, thereby achieving inference acceleration. Nevertheless, the accuracy of these decoding heads falls short of the auto-regressive decoding approach. In light of these limitations, we propose Chimera, a novel framework specifically designed for speculative sampling. Within this framework, we introduce a lightweight draft model that effectively utilizes previously generated tokens to predict subsequent words. To ensure both accuracy and efficiency, we present two strategies within the lightweight draft model. Firstly, we focus on capturing short-range dependencies at the bottom layer. Secondly, we leverage the readily available representations from the original LLM.Through empirical evaluation on the Vicuna and LlaMA-2 series, Chimera demonstrates impressive results, achieving an average latency speedup ratio of 2.7x compared to the vanilla auto-regressive decoding approach. This highlights the potential of our proposed framework in significantly improving the efficiency of large language models during the decoding process.

(Dynamic) Prompting might be all you need to repair Compressed LLMs

Large language models (LLMs), while transformative for NLP, come with significant computational demands, underlining the need for efficient, training-free compression. Notably, the reliability of perplexity as a benchmark for compressed model efficacy is in question, as our tests using LLaMA-7B and OPT-6.7b reveal a significant performance drop in several realistic downstream tasks, underscoring the disparity between perplexity as a performance indicator and real-world performance. Investigation into the trade-off between resource-intensive post-compression re-training highlights the prospect of prompt-driven recovery as a lightweight adaption tool. However, existing studies, confined mainly to perplexity evaluations and simple tasks, fail to offer unequivocal confidence in the scalability and generalizability of prompting. We tackle this uncertainty in two key ways. First, we uncover the vulnerability of naive prompts in LLM compression as an over-reliance on a singular prompt per input. In response, we propose inference-time dynamic prompting (IDP), a mechanism that autonomously chooses from a set of curated prompts based on the context of each individual input. Second, we delve into a scientific understanding of why ``prompting might be all you need post-LLM compression". Our findings suggest that compression doesn't irretrievably erase LLM model knowledge but displace it, necessitating a new inference path. IDP effectively redirects this path, enabling the model to tap into its inherent yet displaced knowledge and thereby recover performance. Empirical tests affirm the value of IDP, demonstrating an average performance improvement of 1.24% across nine varied tasks spanning multiple knowledge domains.

COLEP: Certifiably Robust Learning-Reasoning Conformal Prediction via Probabilistic Circuits

Conformal prediction has shown spurring performance in constructing statistically rigorous prediction sets for arbitrary black-box machine learning models, assuming the data is exchangeable. However, even small adversarial perturbations during the inference can violate the exchangeability assumption, challenge the coverage guarantees, and result in a subsequent decline in empirical coverage. In this work, we propose a certifiably robust learning-reasoning conformal prediction framework (COLEP) via probabilistic circuits, which comprise a data-driven learning component that trains statistical models to learn different semantic concepts, and a reasoning component that encodes knowledge and characterizes the relationships among the trained models for logic reasoning. To achieve exact and efficient reasoning, we employ probabilistic circuits (PCs) within the reasoning component. Theoretically, we provide end-to-end certification of prediction coverage for COLEP in the presence of bounded adversarial perturbations. We also provide certified coverage considering the finite size of the calibration set. Furthermore, we prove that COLEP achieves higher prediction coverage and accuracy over a single model as long as the utilities of knowledge models are non-trivial. Empirically, we show the validity and tightness of our certified coverage, demonstrating the robust conformal prediction of COLEP on various datasets, including GTSRB, CIFAR10, and AwA2. We show that COLEP achieves up to 12% improvement in certified coverage on GTSRB, 9% on CIFAR-10, and 14% on AwA2.

Adaptive Elicitation of Latent Information Using Natural Language

Eliciting information to reduce uncertainty about a latent entity is a critical task in many application domains, e.g., assessing individual student learning outcomes, diagnosing underlying diseases, or learning user preferences. Though natural language is a powerful medium for this purpose, large language models (LLMs) and existing fine-tuning algorithms lack mechanisms for strategically gathering information to refine their own understanding of the latent entity. To harness the generalization power and world knowledge of LLMs in developing effective information-gathering strategies, we propose an adaptive elicitation framework that actively reduces uncertainty on the latent entity. Since probabilistic modeling of an abstract latent entity is difficult, our framework adopts a predictive view of uncertainty, using a meta-learned language model to simulate future observations and enable scalable uncertainty quantification over complex natural language. Through autoregressive forward simulation, our model quantifies how new questions reduce epistemic uncertainty, enabling the development of sophisticated information-gathering strategies to choose the most informative next queries. In experiments on the 20 questions game, dynamic opinion polling, and adaptive student assessment, our method consistently outperforms baselines in identifying critical unknowns and improving downstream predictions, illustrating the promise of strategic information gathering in natural language settings.

Learning from Label Proportions: Bootstrapping Supervised Learners via Belief Propagation

Learning from Label Proportions (LLP) is a learning problem where only aggregate level labels are available for groups of instances, called bags, during training, and the aim is to get the best performance at the instance-level on the test data. This setting arises in domains like advertising and medicine due to privacy considerations. We propose a novel algorithmic framework for this problem that iteratively performs two main steps. For the first step (Pseudo Labeling) in every iteration, we define a Gibbs distribution over binary instance labels that incorporates a) covariate information through the constraint that instances with similar covariates should have similar labels and b) the bag level aggregated label. We then use Belief Propagation (BP) to marginalize the Gibbs distribution to obtain pseudo labels. In the second step (Embedding Refinement), we use the pseudo labels to provide supervision for a learner that yields a better embedding. Further, we iterate on the two steps again by using the second step's embeddings as new covariates for the next iteration. In the final iteration, a classifier is trained using the pseudo labels. Our algorithm displays strong gains against several SOTA baselines (up to 15%) for the LLP Binary Classification problem on various dataset types - tabular and Image. We achieve these improvements with minimal computational overhead above standard supervised learning due to Belief Propagation, for large bag sizes, even for a million samples.

On the Provable Advantage of Unsupervised Pretraining

Unsupervised pretraining, which learns a useful representation using a large amount of unlabeled data to facilitate the learning of downstream tasks, is a critical component of modern large-scale machine learning systems. Despite its tremendous empirical success, the rigorous theoretical understanding of why unsupervised pretraining generally helps remains rather limited -- most existing results are restricted to particular methods or approaches for unsupervised pretraining with specialized structural assumptions. This paper studies a generic framework, where the unsupervised representation learning task is specified by an abstract class of latent variable models Phi and the downstream task is specified by a class of prediction functions Psi. We consider a natural approach of using Maximum Likelihood Estimation (MLE) for unsupervised pretraining and Empirical Risk Minimization (ERM) for learning downstream tasks. We prove that, under a mild ''informative'' condition, our algorithm achieves an excess risk of mathcal{O}(mathcal{C_Phi/m} + mathcal{C_Psi/n}) for downstream tasks, where C_Phi, C_Psi are complexity measures of function classes Phi, Psi, and m, n are the number of unlabeled and labeled data respectively. Comparing to the baseline of mathcal{O}(mathcal{C_{Phi circ Psi}/n}) achieved by performing supervised learning using only the labeled data, our result rigorously shows the benefit of unsupervised pretraining when m gg n and C_{Phicirc Psi} > C_Psi. This paper further shows that our generic framework covers a wide range of approaches for unsupervised pretraining, including factor models, Gaussian mixture models, and contrastive learning.

A Survey on Inference Optimization Techniques for Mixture of Experts Models

The emergence of large-scale Mixture of Experts (MoE) models has marked a significant advancement in artificial intelligence, offering enhanced model capacity and computational efficiency through conditional computation. However, the deployment and inference of these models present substantial challenges in terms of computational resources, latency, and energy efficiency. This comprehensive survey systematically analyzes the current landscape of inference optimization techniques for MoE models across the entire system stack. We first establish a taxonomical framework that categorizes optimization approaches into model-level, system-level, and hardware-level optimizations. At the model level, we examine architectural innovations including efficient expert design, attention mechanisms, various compression techniques such as pruning, quantization, and knowledge distillation, as well as algorithm improvement including dynamic routing strategies and expert merging methods. At the system level, we investigate distributed computing approaches, load balancing mechanisms, and efficient scheduling algorithms that enable scalable deployment. Furthermore, we delve into hardware-specific optimizations and co-design strategies that maximize throughput and energy efficiency. This survey not only provides a structured overview of existing solutions but also identifies key challenges and promising research directions in MoE inference optimization. Our comprehensive analysis serves as a valuable resource for researchers and practitioners working on large-scale deployment of MoE models in resource-constrained environments. To facilitate ongoing updates and the sharing of cutting-edge advances in MoE inference optimization research, we have established a repository accessible at https://github.com/MoE-Inf/awesome-moe-inference/.

Mirostat: A Neural Text Decoding Algorithm that Directly Controls Perplexity

Neural text decoding is important for generating high-quality texts using language models. To generate high-quality text, popular decoding algorithms like top-k, top-p (nucleus), and temperature-based sampling truncate or distort the unreliable low probability tail of the language model. Though these methods generate high-quality text after parameter tuning, they are ad hoc. Not much is known about the control they provide over the statistics of the output, which is important since recent reports show text quality is highest for a specific range of likelihoods. Here, first we provide a theoretical analysis of perplexity in top-k, top-p, and temperature sampling, finding that cross-entropy behaves approximately linearly as a function of p in top-p sampling whereas it is a nonlinear function of k in top-k sampling, under Zipfian statistics. We use this analysis to design a feedback-based adaptive top-k text decoding algorithm called mirostat that generates text (of any length) with a predetermined value of perplexity, and thereby high-quality text without any tuning. Experiments show that for low values of k and p in top-k and top-p sampling, perplexity drops significantly with generated text length, which is also correlated with excessive repetitions in the text (the boredom trap). On the other hand, for large values of k and p, we find that perplexity increases with generated text length, which is correlated with incoherence in the text (confusion trap). Mirostat avoids both traps: experiments show that cross-entropy has a near-linear relation with repetition in generated text. This relation is almost independent of the sampling method but slightly dependent on the model used. Hence, for a given language model, control over perplexity also gives control over repetitions. Experiments with human raters for fluency, coherence, and quality further verify our findings.

Grammar-Aligned Decoding

Large Language Models (LLMs) struggle with reliably generating highly structured outputs, such as program code, mathematical formulas, or well-formed markup. Constrained decoding approaches mitigate this problem by greedily restricting what tokens an LLM can output at each step to guarantee that the output matches a given constraint. Specifically, in grammar-constrained decoding (GCD), the LLM's output must follow a given grammar. In this paper, we demonstrate that GCD techniques (and in general constrained decoding techniques) can distort the LLM's distribution, leading to outputs that are grammatical but appear with likelihoods that are not proportional to the ones given by the LLM, and so ultimately are low-quality. We call the problem of aligning sampling with a grammar constraint, grammar-aligned decoding (GAD), and propose adaptive sampling with approximate expected futures (ASAp), a decoding algorithm that guarantees the output to be grammatical while provably producing outputs that match the conditional probability of the LLM's distribution conditioned on the given grammar constraint. Our algorithm uses prior sample outputs to soundly overapproximate the future grammaticality of different output prefixes. Our evaluation on code generation and structured NLP tasks shows how ASAp often produces outputs with higher likelihood (according to the LLM's distribution) than existing GCD techniques, while still enforcing the desired grammatical constraints.

Robust Active Distillation

Distilling knowledge from a large teacher model to a lightweight one is a widely successful approach for generating compact, powerful models in the semi-supervised learning setting where a limited amount of labeled data is available. In large-scale applications, however, the teacher tends to provide a large number of incorrect soft-labels that impairs student performance. The sheer size of the teacher additionally constrains the number of soft-labels that can be queried due to prohibitive computational and/or financial costs. The difficulty in achieving simultaneous efficiency (i.e., minimizing soft-label queries) and robustness (i.e., avoiding student inaccuracies due to incorrect labels) hurts the widespread application of knowledge distillation to many modern tasks. In this paper, we present a parameter-free approach with provable guarantees to query the soft-labels of points that are simultaneously informative and correctly labeled by the teacher. At the core of our work lies a game-theoretic formulation that explicitly considers the inherent trade-off between the informativeness and correctness of input instances. We establish bounds on the expected performance of our approach that hold even in worst-case distillation instances. We present empirical evaluations on popular benchmarks that demonstrate the improved distillation performance enabled by our work relative to that of state-of-the-art active learning and active distillation methods.

Bridging the Training-Inference Gap in LLMs by Leveraging Self-Generated Tokens

Language models are often trained to maximize the likelihood of the next token given past tokens in the training dataset. However, during inference time, they are utilized differently, generating text sequentially and auto-regressively by using previously generated tokens as input to predict the next one. Marginal differences in predictions at each step can cascade over successive steps, resulting in different distributions from what the models were trained for and potentially leading to unpredictable behavior. This paper proposes two simple approaches based on model own generation to address this discrepancy between the training and inference time. Our first approach is Batch-Scheduled Sampling, where, during training, we stochastically choose between the ground-truth token from the dataset and the model's own generated token as input to predict the next token. This is done in an offline manner, modifying the context window by interleaving ground-truth tokens with those generated by the model. Our second approach is Reference-Answer-based Correction, where we explicitly incorporate a self-correction capability into the model during training. This enables the model to effectively self-correct the gaps between the generated sequences and the ground truth data without relying on an external oracle model. By incorporating our proposed strategies during training, we have observed an overall improvement in performance compared to baseline methods, as demonstrated by our extensive experiments using summarization, general question-answering, and math question-answering tasks.

Reasoning to Learn from Latent Thoughts

Compute scaling for language model (LM) pretraining has outpaced the growth of human-written texts, leading to concerns that data will become the bottleneck to LM scaling. To continue scaling pretraining in this data-constrained regime, we propose that explicitly modeling and inferring the latent thoughts that underlie the text generation process can significantly improve pretraining data efficiency. Intuitively, our approach views web text as the compressed final outcome of a verbose human thought process and that the latent thoughts contain important contextual knowledge and reasoning steps that are critical to data-efficient learning. We empirically demonstrate the effectiveness of our approach through data-constrained continued pretraining for math. We first show that synthetic data approaches to inferring latent thoughts significantly improve data efficiency, outperforming training on the same amount of raw data (5.7\% rightarrow 25.4\% on MATH). Furthermore, we demonstrate latent thought inference without a strong teacher, where an LM bootstraps its own performance by using an EM algorithm to iteratively improve the capability of the trained LM and the quality of thought-augmented pretraining data. We show that a 1B LM can bootstrap its performance across at least three iterations and significantly outperform baselines trained on raw data, with increasing gains from additional inference compute when performing the E-step. The gains from inference scaling and EM iterations suggest new opportunities for scaling data-constrained pretraining.

From Word Models to World Models: Translating from Natural Language to the Probabilistic Language of Thought

How does language inform our downstream thinking? In particular, how do humans make meaning from language -- and how can we leverage a theory of linguistic meaning to build machines that think in more human-like ways? In this paper, we propose rational meaning construction, a computational framework for language-informed thinking that combines neural models of language with probabilistic models for rational inference. We frame linguistic meaning as a context-sensitive mapping from natural language into a probabilistic language of thought (PLoT) -- a general-purpose symbolic substrate for probabilistic, generative world modeling. Our architecture integrates two powerful computational tools that have not previously come together: we model thinking with probabilistic programs, an expressive representation for flexible commonsense reasoning; and we model meaning construction with large language models (LLMs), which support broad-coverage translation from natural language utterances to code expressions in a probabilistic programming language. We illustrate our framework in action through examples covering four core domains from cognitive science: probabilistic reasoning, logical and relational reasoning, visual and physical reasoning, and social reasoning about agents and their plans. In each, we show that LLMs can generate context-sensitive translations that capture pragmatically-appropriate linguistic meanings, while Bayesian inference with the generated programs supports coherent and robust commonsense reasoning. We extend our framework to integrate cognitively-motivated symbolic modules to provide a unified commonsense thinking interface from language. Finally, we explore how language can drive the construction of world models themselves.

SpecTr: Fast Speculative Decoding via Optimal Transport

Autoregressive sampling from large language models has led to state-of-the-art results in several natural language tasks. However, autoregressive sampling generates tokens one at a time making it slow, and even prohibitive in certain tasks. One way to speed up sampling is speculative decoding: use a small model to sample a draft (block or sequence of tokens), and then score all tokens in the draft by the large language model in parallel. A subset of the tokens in the draft are accepted (and the rest rejected) based on a statistical method to guarantee that the final output follows the distribution of the large model. In this work, we provide a principled understanding of speculative decoding through the lens of optimal transport (OT) with membership cost. This framework can be viewed as an extension of the well-known maximal-coupling problem. This new formulation enables us to generalize the speculative decoding method to allow for a set of k candidates at the token-level, which leads to an improved optimal membership cost. We show that the optimal draft selection algorithm (transport plan) can be computed via linear programming, whose best-known runtime is exponential in k. We then propose a valid draft selection algorithm whose acceptance probability is (1-1/e)-optimal multiplicatively. Moreover, it can be computed in time almost linear with size of domain of a single token. Using this new draft selection algorithm, we develop a new autoregressive sampling algorithm called SpecTr, which provides speedup in decoding while ensuring that there is no quality degradation in the decoded output. We experimentally demonstrate that for state-of-the-art large language models, the proposed approach achieves a wall clock speedup of 2.13X, a further 1.37X speedup over speculative decoding on standard benchmarks.