Pre-trained Large Language Models Use Fourier Features to Compute Addition

Pre-trained large language models (LLMs) exhibit impressive mathematical reasoning capabilities, yet how they compute basic arithmetic, such as addition, remains unclear. This paper shows that pre-trained LLMs add numbers using Fourier features -- dimensions in the hidden state that represent numbers via a set of features sparse in the frequency domain. Within the model, MLP and attention layers use Fourier features in complementary ways: MLP layers primarily approximate the magnitude of the answer using low-frequency features, while attention layers primarily perform modular addition (e.g., computing whether the answer is even or odd) using high-frequency features. Pre-training is crucial for this mechanism: models trained from scratch to add numbers only exploit low-frequency features, leading to lower accuracy. Introducing pre-trained token embeddings to a randomly initialized model rescues its performance. Overall, our analysis demonstrates that appropriate pre-trained representations (e.g., Fourier features) can unlock the ability of Transformers to learn precise mechanisms for algorithmic tasks.

IsoBench: Benchmarking Multimodal Foundation Models on Isomorphic Representations

Current foundation models exhibit impressive capabilities when prompted either with text only or with both image and text inputs. But do their capabilities change depending on the input modality? In this work, we propose $\textbf{IsoBench}$, a benchmark dataset containing problems from four major areas: math, science, algorithms, and games. Each example is presented with multiple $\textbf{isomorphic representations}$ of inputs, such as visual, textual, and mathematical presentations. IsoBench provides fine-grained feedback to diagnose performance gaps caused by the form of the representation. Across various foundation models, we observe that on the same problem, models have a consistent preference towards textual representations. Most prominently, when evaluated on all IsoBench problems, Claude-3 Opus performs 28.7 points worse when provided with images instead of text; similarly, GPT-4 Turbo is 18.7 points worse and Gemini Pro is 14.9 points worse. Finally, we present two prompting techniques, $\textit{IsoCombination}$ and $\textit{IsoScratchPad}$, which improve model performance by considering combinations of, and translations between, different input representations.

Simplicity Bias of Transformers to Learn Low Sensitivity Functions

Transformers achieve state-of-the-art accuracy and robustness across many tasks, but an understanding of the inductive biases that they have and how those biases are different from other neural network architectures remains elusive. Various neural network architectures such as fully connected networks have been found to have a simplicity bias towards simple functions of the data; one version of this simplicity bias is a spectral bias to learn simple functions in the Fourier space. In this work, we identify the notion of sensitivity of the model to random changes in the input as a notion of simplicity bias which provides a unified metric to explain the simplicity and spectral bias of transformers across different data modalities. We show that transformers have lower sensitivity than alternative architectures, such as LSTMs, MLPs and CNNs, across both vision and language tasks. We also show that low-sensitivity bias correlates with improved robustness; furthermore, it can also be used as an efficient intervention to further improve the robustness of transformers.

DeLLMa: A Framework for Decision Making Under Uncertainty with Large Language Models

Large language models (LLMs) are increasingly used across society, including in domains like business, engineering, and medicine. These fields often grapple with decision-making under uncertainty, a critical yet challenging task. In this paper, we show that directly prompting LLMs on these types of decision-making problems yields poor results, especially as the problem complexity increases. To overcome this limitation, we propose DeLLMa (Decision-making Large Language Model assistant), a framework designed to enhance decision-making accuracy in uncertain environments. DeLLMa involves a multi-step scaffolding procedure, drawing upon principles from decision theory and utility theory, to provide an optimal and human-auditable decision-making process. We validate our framework on decision-making environments involving real agriculture and finance data. Our results show that DeLLMa can significantly improve LLM decision-making performance, achieving up to a 40% increase in accuracy over competing methods.

DreamSync: Aligning Text-to-Image Generation with Image Understanding Feedback

Despite their wide-spread success, Text-to-Image models (T2I) still struggle to produce images that are both aesthetically pleasing and faithful to the user's input text. We introduce DreamSync, a model-agnostic training algorithm by design that improves T2I models to be faithful to the text input. DreamSync builds off a recent insight from TIFA's evaluation framework -- that large vision-language models (VLMs) can effectively identify the fine-grained discrepancies between generated images and the text inputs. DreamSync uses this insight to train T2I models without any labeled data; it improves T2I models using its own generations. First, it prompts the model to generate several candidate images for a given input text. Then, it uses two VLMs to select the best generation: a Visual Question Answering model that measures the alignment of generated images to the text, and another that measures the generation's aesthetic quality. After selection, we use LoRA to iteratively finetune the T2I model to guide its generation towards the selected best generations. DreamSync does not need any additional human annotation. model architecture changes, or reinforcement learning. Despite its simplicity, DreamSync improves both the semantic alignment and aesthetic appeal of two diffusion-based T2I models, evidenced by multiple benchmarks (+1.7% on TIFA, +2.9% on DSG1K, +3.4% on VILA aesthetic) and human evaluation.

Transformers Learn Higher-Order Optimization Methods for In-Context Learning: A Study with Linear Models

Transformers are remarkably good at in-context learning (ICL) -- learning from demonstrations without parameter updates -- but how they perform ICL remains a mystery. Recent work suggests that Transformers may learn in-context by internally running Gradient Descent, a first-order optimization method. In this paper, we instead demonstrate that Transformers learn to implement higher-order optimization methods to perform ICL. Focusing on in-context linear regression, we show that Transformers learn to implement an algorithm very similar to Iterative Newton's Method, a higher-order optimization method, rather than Gradient Descent. Empirically, we show that predictions from successive Transformer layers closely match different iterations of Newton's Method linearly, with each middle layer roughly computing 3 iterations. In contrast, exponentially more Gradient Descent steps are needed to match an additional Transformers layer; this suggests that Transformers have an comparable rate of convergence with high-order methods such as Iterative Newton, which are exponentially faster than Gradient Descent. We also show that Transformers can learn in-context on ill-conditioned data, a setting where Gradient Descent struggles but Iterative Newton succeeds. Finally, we show theoretical results which support our empirical findings and have a close correspondence with them: we prove that Transformers can implement $k$ iterations of Newton's method with $\mathcal{O}(k)$ layers.

SCENE: Self-Labeled Counterfactuals for Extrapolating to Negative Examples

Detecting negatives (such as non-entailment relationships, unanswerable questions, and false claims) is an important and challenging aspect of many natural language understanding tasks. Though manually collecting challenging negative examples can help models detect them, it is both costly and domain-specific. In this work, we propose Self-labeled Counterfactuals for Extrapolating to Negative Examples (SCENE), an automatic method for synthesizing training data that greatly improves models' ability to detect challenging negative examples. In contrast with standard data augmentation, which synthesizes new examples for existing labels, SCENE can synthesize negative examples zero-shot from only positive ones. Given a positive example, SCENE perturbs it with a mask infilling model, then determines whether the resulting example is negative based on a self-training heuristic. With access to only answerable training examples, SCENE can close 69.6% of the performance gap on SQuAD 2.0, a dataset where half of the evaluation examples are unanswerable, compared to a model trained on SQuAD 2.0. Our method also extends to boolean question answering and recognizing textual entailment, and improves generalization from SQuAD to ACE-whQA, an out-of-domain extractive QA benchmark.

Topological Regularization for Dense Prediction

Dense prediction tasks such as depth perception and semantic segmentation are important applications in computer vision that have a concrete topological description in terms of partitioning an image into connected components or estimating a function with a small number of local extrema corresponding to objects in the image. We develop a form of topological regularization based on persistent homology that can be used in dense prediction tasks with these topological descriptions. Experimental results show that the output topology can also appear in the internal activations of trained neural networks which allows for a novel use of topological regularization to the internal states of neural networks during training, reducing the computational cost of the regularization. We demonstrate that this topological regularization of internal activations leads to improved convergence and test benchmarks on several problems and architectures.

Harnessing the Conditioning Sensorium for Improved Image Translation

Multi-modal domain translation typically refers to synthesizing a novel image that inherits certain localized attributes from a 'content' image (e.g. layout, semantics, or geometry), and inherits everything else (e.g. texture, lighting, sometimes even semantics) from a 'style' image. The dominant approach to this task is attempting to learn disentangled 'content' and 'style' representations from scratch. However, this is not only challenging, but ill-posed, as what users wish to preserve during translation varies depending on their goals. Motivated by this inherent ambiguity, we define 'content' based on conditioning information extracted by off-the-shelf pre-trained models. We then train our style extractor and image decoder with an easy to optimize set of reconstruction objectives. The wide variety of high-quality pre-trained models available and simple training procedure makes our approach straightforward to apply across numerous domains and definitions of 'content'. Additionally it offers intuitive control over which aspects of 'content' are preserved across domains. We evaluate our method on traditional, well-aligned, datasets such as CelebA-HQ, and propose two novel datasets for evaluation on more complex scenes: ClassicTV and FFHQ-Wild. Our approach, Sensorium, enables higher quality domain translation for more complex scenes.