Deep Equilibrium Models (DEQs) are a class of implicit neural networks that solve for a fixed point of a neural network in their forward pass. Traditionally, DEQs take sequences as inputs, but have since been applied to a variety of data. In this work, we present Distributional Deep Equilibrium Models (DDEQs), extending DEQs to discrete measure inputs, such as sets or point clouds. We provide a theoretically grounded framework for DDEQs. Leveraging Wasserstein gradient flows, we show how the forward pass of the DEQ can be adapted to find fixed points of discrete measures under permutation-invariance, and derive adequate network architectures for DDEQs. In experiments, we show that they can compete with state-of-the-art models in tasks such as point cloud classification and point cloud completion, while being significantly more parameter-efficient.

In this paper, we explore the nature of sudden breakthroughs in language model performance at scale, which stands in contrast to smooth improvements governed by scaling laws. While advocates of "emergence" view abrupt performance gains as capabilities unlocking at specific scales, others have suggested that they are produced by thresholding effects and alleviated by continuous metrics. We propose that breakthroughs are instead driven by continuous changes in the probability distribution of training outcomes, particularly when performance is bimodally distributed across random seeds. In synthetic length generalization tasks, we show that different random seeds can produce either highly linear or emergent scaling trends. We reveal that sharp breakthroughs in metrics are produced by underlying continuous changes in their distribution across seeds. Furthermore, we provide a case study of inverse scaling and show that even as the probability of a successful run declines, the average performance of a successful run continues to increase monotonically. We validate our distributional scaling framework on realistic settings by measuring MMLU performance in LLM populations. These insights emphasize the role of random variation in the effect of scale on LLM capabilities.

Large language models ($\textbf{LLMs}$) have emerged as a powerful method for discovery. Instead of utilizing numerical data, LLMs utilize associated variable $\textit{semantic metadata}$ to predict variable relationships. Simultaneously, LLMs demonstrate impressive abilities to act as black-box optimizers when given an objective $f$ and sequence of trials. We study LLMs at the intersection of these two capabilities by applying LLMs to the task of $\textit{interactive graph discovery}$: given a ground truth graph $G^*$ capturing variable relationships and a budget of $I$ edge experiments over $R$ rounds, minimize the distance between the predicted graph $\hat{G}_R$ and $G^*$ at the end of the $R$-th round. To solve this task we propose $\textbf{IGDA}$, a LLM-based pipeline incorporating two key components: 1) an LLM uncertainty-driven method for edge experiment selection 2) a local graph update strategy utilizing binary feedback from experiments to improve predictions for unselected neighboring edges. Experiments on eight different real-world graphs show our approach often outperforms all baselines including a state-of-the-art numerical method for interactive graph discovery. Further, we conduct a rigorous series of ablations dissecting the impact of each pipeline component. Finally, to assess the impact of memorization, we apply our interactive graph discovery strategy to a complex, new (as of July 2024) causal graph on protein transcription factors, finding strong performance in a setting where memorization is impossible. Overall, our results show IGDA to be a powerful method for graph discovery complementary to existing numerically driven approaches.

Language models (LMs), like other neural networks, often favor shortcut heuristics based on surface-level patterns. Although LMs behave like n-gram models early in training, they must eventually learn hierarchical syntactic representations to correctly apply grammatical rules out-of-distribution (OOD). In this work, we use case studies of English grammar to explore how complex, diverse training data drives models to generalize OOD. We construct a framework that unifies our understanding of random variation with training dynamics, rule selection with memorization, and data diversity with complexity. We show that these factors are nuanced, and that intermediate levels of diversity and complexity lead to inconsistent behavior across random seeds and to unstable training dynamics. Our findings emphasize the critical role of training data in shaping generalization patterns and illuminate how competing model strategies lead to inconsistent generalization outcomes across random seeds.

Modern large language models use a fixed tokenizer to effectively compress text drawn from a source domain. However, applying the same tokenizer to a new target domain often leads to inferior compression, more costly inference, and reduced semantic alignment. To address this deficiency, we introduce Sparse Sinkhorn Token Translation (S2T2). S2T2 trains a tailored tokenizer for the target domain and learns to translate between target and source tokens, enabling more effective reuse of the pre-trained next-source-token predictor. In our experiments with finetuned English language models, S2T2 improves both the perplexity and the compression of out-of-domain protein sequences, outperforming direct finetuning with either the source or target tokenizer. In addition, we find that token translations learned for smaller, less expensive models can be directly transferred to larger, more powerful models to reap the benefits of S2T2 at lower cost.

The Mixture-of-Experts (MoE) architecture enables a significant increase in the total number of model parameters with minimal computational overhead. However, it is not clear what performance tradeoffs, if any, exist between MoEs and standard dense transformers. In this paper, we show that as we increase the number of experts (while fixing the number of active parameters), the memorization performance consistently increases while the reasoning capabilities saturate. We begin by analyzing the theoretical limitations of MoEs at reasoning. We prove that there exist graph problems that cannot be solved by any number of experts of a certain width; however, the same task can be easily solved by a dense model with a slightly larger width. On the other hand, we find that on memory-intensive tasks, MoEs can effectively leverage a small number of active parameters with a large number of experts to memorize the data. Lastly, we pre-train a series of MoEs and dense transformers and evaluate them on commonly used benchmarks in math and natural language. We find that increasing the number of experts helps solve knowledge-intensive tasks, but fails to yield the same benefits for reasoning tasks. The explosion in capabilities of large language models in recent years has largely been enabled by scaling their size, as measured by the number of parameters in the model. In the standard Transformer architecture, scaling the number of parameters entails a proportional increase in computational cost, e.g. Mixture-of-Experts (MoE) were introduced as a solution for this problem (Shazeer et al., 2017; Lepikhin et al., 2020; Fedus et al., 2022). MoEs replace the single MLP in each Transformer block with multiple MLPs (called experts), where each token is routed to a few experts based on a linear routing function. The number of parameters in the MoE layer therefore increases with the total number of experts, while the compute increases only with the number of "active" experts (i.e., the number of experts to which the token is routed to). For this reason, MoEs have become very popular, and many frontier models today are based on the MoE architecture (Achiam et al., 2023; Databricks, 2023; Anil et al., 2023; Dai et al., 2024; Jiang et al., 2024; Yang et al., 2024).

Large language models (LLMs) have shown remarkable potential for problem solving, with open source models achieving increasingly impressive performance on benchmarks measuring areas from logical reasoning to mathematical ability. Ensembling models can further improve capabilities across a variety of domains. However, conventional methods of combining models at inference time such as shallow fusion necessitate a shared vocabulary and tokenization, and alternatives like fine-tuning for domain-specific performance are both time consuming and computationally expensive. We therefore present an inference-time ensembling algorithm aimed at "averaging" outputs from multiple LLMs and illustrate its improved performance across multiple domains compared to its constituent models alone. Character-wise ensemble decoding, CharED, finds the marginal distribution of each character for an individual model and performs a weighted average to generate an output, character by character. In coding, math, and toxicity benchmarks, we find our proposed model able to combine complimentary strengths of multiple LLMs, regardless of vocabulary, tokenization, or model size.

Data $\textit{quality}$ is a crucial factor in the performance of machine learning models, a principle that dataset distillation methods exploit by compressing training datasets into much smaller counterparts that maintain similar downstream performance. Understanding how and why data distillation methods work is vital not only for improving these methods but also for revealing fundamental characteristics of "good" training data. However, a major challenge in achieving this goal is the observation that distillation approaches, which rely on sophisticated but mostly disparate methods to generate synthetic data, have little in common with each other. In this work, we highlight a largely overlooked aspect common to most of these methods: the use of soft (probabilistic) labels. Through a series of ablation experiments, we study the role of soft labels in depth. Our results reveal that the main factor explaining the performance of state-of-the-art distillation methods is not the specific techniques used to generate synthetic data but rather the use of soft labels. Furthermore, we demonstrate that not all soft labels are created equal; they must contain $\textit{structured information}$ to be beneficial. We also provide empirical scaling laws that characterize the effectiveness of soft labels as a function of images-per-class in the distilled dataset and establish an empirical Pareto frontier for data-efficient learning. Combined, our findings challenge conventional wisdom in dataset distillation, underscore the importance of soft labels in learning, and suggest new directions for improving distillation methods. Code for all experiments is available at https://github.com/sunnytqin/no-distillation.

As large language models (LLMs) have gained popularity for a variety of use cases, making them adaptable and controllable has become increasingly important, especially for user-facing applications. While the existing literature on LLM adaptation primarily focuses on finding a model (or models) that optimizes a single predefined objective, here we focus on the challenging case where the model must dynamically adapt to diverse -- and often changing -- user preferences. For this, we leverage adaptation methods based on linear weight interpolation, casting them as continuous multi-domain interpolators that produce models with specific prescribed generation characteristics on-the-fly. Specifically, we use low-rank updates to fine-tune a base model to various different domains, yielding a set of anchor models with distinct generation profiles. Then, we use the weight updates of these anchor models to parametrize the entire (infinite) class of models contained within their convex hull. We empirically show that varying the interpolation weights yields predictable and consistent change in the model outputs with respect to all of the controlled attributes. We find that there is little entanglement between most attributes and identify and discuss the pairs of attributes for which this is not the case. Our results suggest that linearly interpolating between the weights of fine-tuned models facilitates predictable, fine-grained control of model outputs with respect to multiple stylistic characteristics simultaneously.

What does a neural network learn when training from a task-specific dataset? Synthesizing this knowledge is the central idea behind Dataset Distillation, which recent work has shown can be used to compress large datasets into a small set of input-label pairs ($\textit{prototypes}$) that capture essential aspects of the original dataset. In this paper, we make the key observation that existing methods distilling into explicit prototypes are very often suboptimal, incurring in unexpected storage cost from distilled labels. In response, we propose $\textit{Distributional Dataset Distillation}$ (D3), which encodes the data using minimal sufficient per-class statistics and paired with a decoder, we distill dataset into a compact distributional representation that is more memory-efficient compared to prototype-based methods. To scale up the process of learning these representations, we propose $\textit{Federated distillation}$, which decomposes the dataset into subsets, distills them in parallel using sub-task experts and then re-aggregates them. We thoroughly evaluate our algorithm on a three-dimensional metric and show that our method achieves state-of-the-art results on TinyImageNet and ImageNet-1K. Specifically, we outperform the prior art by $6.9\%$ on ImageNet-1K under the storage budget of 2 images per class.