Emergent Symbol-like Number Variables in Artificial Neural Networks

What types of numeric representations emerge in Neural Networks (NNs)? To what degree do NNs induce abstract, mutable, slot-like numeric variables, and in what situations do these representations emerge? How do these representations change over learning, and how can we understand the neural implementations in ways that are unified across different NNs? In this work, we approach these questions by first training sequence based neural systems using Next Token Prediction (NTP) objectives on numeric tasks. We then seek to understand the neural solutions through the lens of causal abstractions or symbolic algorithms. We use a combination of causal interventions and visualization methods to find that artificial neural models do indeed develop analogs of interchangeable, mutable, latent number variables purely from the NTP objective. We then ask how variations on the tasks and model architectures affect the models' learned solutions to find that these symbol-like numeric representations do not form for every variant of the task, and transformers solve the problem in a notably different way than their recurrent counterparts. We then show how the symbol-like variables change over the course of training to find a strong correlation between the models' task performance and the alignment of their symbol-like representations. Lastly, we show that in all cases, some degree of gradience exists in these neural symbols, highlighting the difficulty of finding simple, interpretable symbolic stories of how neural networks perform numeric tasks. Taken together, our results are consistent with the view that neural networks can approximate interpretable symbolic programs of number cognition, but the particular program they approximate and the extent to which they approximate it can vary widely, depending on the network architecture, training data, extent of training, and network size.

BoxingGym: Benchmarking Progress in Automated Experimental Design and Model Discovery

Understanding the world and explaining it with scientific theories is a central aspiration of artificial intelligence research. Proposing theories, designing experiments to test them, and then revising them based on data are fundamental to scientific discovery. Despite the significant promise of LLM-based scientific agents, no benchmarks systematically test LLM's ability to propose scientific models, collect experimental data, and revise them in light of new data. We introduce BoxingGym, a benchmark with 10 environments for systematically evaluating both experimental design (e.g. collecting data to test a scientific theory) and model discovery (e.g. proposing and revising scientific theories). To enable tractable and quantitative evaluation, we implement each environment as a generative probabilistic model with which a scientific agent can run interactive experiments. These probabilistic models are drawn from various real-world scientific domains ranging from psychology to ecology. To quantitatively evaluate a scientific agent's ability to collect informative experimental data, we compute the expected information gain (EIG), an information-theoretic quantity which measures how much an experiment reduces uncertainty about the parameters of a generative model. A good scientific theory is a concise and predictive explanation. Therefore, to quantitatively evaluate model discovery, we ask a scientific agent to explain their model and then assess whether this explanation enables another scientific agent to make reliable predictions about this environment. In addition to this explanation-based evaluation, we compute standard model evaluation metrics such as prediction errors. We find that current LLMs, such as GPT-4o, struggle with both experimental design and model discovery. We find that augmenting the LLM-based agent with an explicit statistical model does not reliably improve these results.

CriticAL: Critic Automation with Language Models

Understanding the world through models is a fundamental goal of scientific research. While large language model (LLM) based approaches show promise in automating scientific discovery, they often overlook the importance of criticizing scientific models. Criticizing models deepens scientific understanding and drives the development of more accurate models. Automating model criticism is difficult because it traditionally requires a human expert to define how to compare a model with data and evaluate if the discrepancies are significant--both rely heavily on understanding the modeling assumptions and domain. Although LLM-based critic approaches are appealing, they introduce new challenges: LLMs might hallucinate the critiques themselves. Motivated by this, we introduce CriticAL (Critic Automation with Language Models). CriticAL uses LLMs to generate summary statistics that capture discrepancies between model predictions and data, and applies hypothesis tests to evaluate their significance. We can view CriticAL as a verifier that validates models and their critiques by embedding them in a hypothesis testing framework. In experiments, we evaluate CriticAL across key quantitative and qualitative dimensions. In settings where we synthesize discrepancies between models and datasets, CriticAL reliably generates correct critiques without hallucinating incorrect ones. We show that both human and LLM judges consistently prefer CriticAL's critiques over alternative approaches in terms of transparency and actionability. Finally, we show that CriticAL's critiques enable an LLM scientist to improve upon human-designed models on real-world datasets.

Bayesian scaling laws for in-context learning

In-context learning (ICL) is a powerful technique for getting language models to perform complex tasks with no training updates. Prior work has established strong correlations between the number of in-context examples provided and the accuracy of the model's predictions. In this paper, we seek to explain this correlation by showing that ICL approximates a Bayesian learner. This perspective gives rise to a family of novel Bayesian scaling laws for ICL. In experiments with \mbox{GPT-2} models of different sizes, our scaling laws exceed or match existing scaling laws in accuracy while also offering interpretable terms for task priors, learning efficiency, and per-example probabilities. To illustrate the analytic power that such interpretable scaling laws provide, we report on controlled synthetic dataset experiments designed to inform real-world studies of safety alignment. In our experimental protocol, we use SFT to suppress an unwanted existing model capability and then use ICL to try to bring that capability back (many-shot jailbreaking). We then experiment on real-world instruction-tuned LLMs using capabilities benchmarks as well as a new many-shot jailbreaking dataset. In all cases, Bayesian scaling laws accurately predict the conditions under which ICL will cause the suppressed behavior to reemerge, which sheds light on the ineffectiveness of post-training at increasing LLM safety.

Human-like Affective Cognition in Foundation Models

Understanding emotions is fundamental to human interaction and experience. Humans easily infer emotions from situations or facial expressions, situations from emotions, and do a variety of other affective cognition. How adept is modern AI at these inferences? We introduce an evaluation framework for testing affective cognition in foundation models. Starting from psychological theory, we generate 1,280 diverse scenarios exploring relationships between appraisals, emotions, expressions, and outcomes. We evaluate the abilities of foundation models (GPT-4, Claude-3, Gemini-1.5-Pro) and humans (N = 567) across carefully selected conditions. Our results show foundation models tend to agree with human intuitions, matching or exceeding interparticipant agreement. In some conditions, models are ``superhuman'' -- they better predict modal human judgements than the average human. All models benefit from chain-of-thought reasoning. This suggests foundation models have acquired a human-like understanding of emotions and their influence on beliefs and behavior.

What Makes a Maze Look Like a Maze?

A unique aspect of human visual understanding is the ability to flexibly interpret abstract concepts: acquiring lifted rules explaining what they symbolize, grounding them across familiar and unfamiliar contexts, and making predictions or reasoning about them. While off-the-shelf vision-language models excel at making literal interpretations of images (e.g., recognizing object categories such as tree branches), they still struggle to make sense of such visual abstractions (e.g., how an arrangement of tree branches may form the walls of a maze). To address this challenge, we introduce Deep Schema Grounding (DSG), a framework that leverages explicit structured representations of visual abstractions for grounding and reasoning. At the core of DSG are schemas--dependency graph descriptions of abstract concepts that decompose them into more primitive-level symbols. DSG uses large language models to extract schemas, then hierarchically grounds concrete to abstract components of the schema onto images with vision-language models. The grounded schema is used to augment visual abstraction understanding. We systematically evaluate DSG and different methods in reasoning on our new Visual Abstractions Dataset, which consists of diverse, real-world images of abstract concepts and corresponding question-answer pairs labeled by humans. We show that DSG significantly improves the abstract visual reasoning performance of vision-language models, and is a step toward human-aligned understanding of visual abstractions.

Is Child-Directed Speech Effective Training Data for Language Models?

While high-performing language models are typically trained on hundreds of billions of words, human children become fluent language users with a much smaller amount of data. What are the features of the data they receive, and how do these features support language modeling objectives? To investigate this question, we train GPT-2 models on 29M words of English-language child-directed speech and a new matched, synthetic dataset (TinyDialogues), comparing to a heterogeneous blend of datasets from the BabyLM challenge. We evaluate both the syntactic and semantic knowledge of these models using developmentally-inspired evaluations. Through pretraining experiments, we test whether the global developmental ordering or the local discourse ordering of children's training data support high performance relative to other datasets. The local properties of the data affect model results, but somewhat surprisingly, global properties do not. Further, child language input is not uniquely valuable for training language models. These findings support the hypothesis that, rather than proceeding from better data, children's learning is instead substantially more efficient than current language modeling techniques.

Psychometric Alignment: Capturing Human Knowledge Distributions via Language Models

Language models (LMs) are increasingly used to simulate human-like responses in scenarios where accurately mimicking a population's behavior can guide decision-making, such as in developing educational materials and designing public policies. The objective of these simulations is for LMs to capture the variations in human responses, rather than merely providing the expected correct answers. Prior work has shown that LMs often generate unrealistically accurate responses, but there are no established metrics to quantify how closely the knowledge distribution of LMs aligns with that of humans. To address this, we introduce "psychometric alignment," a metric that measures the extent to which LMs reflect human knowledge distribution. Assessing this alignment involves collecting responses from both LMs and humans to the same set of test items and using Item Response Theory to analyze the differences in item functioning between the groups. We demonstrate that our metric can capture important variations in populations that traditional metrics, like differences in accuracy, fail to capture. We apply this metric to assess existing LMs for their alignment with human knowledge distributions across three real-world domains. We find significant misalignment between LMs and human populations, though using persona-based prompts can improve alignment. Interestingly, smaller LMs tend to achieve greater psychometric alignment than larger LMs. Further, training LMs on human response data from the target distribution enhances their psychometric alignment on unseen test items, but the effectiveness of such training varies across domains.

On scalable oversight with weak LLMs judging strong LLMs

Scalable oversight protocols aim to enable humans to accurately supervise superhuman AI. In this paper we study debate, where two AI's compete to convince a judge; consultancy, where a single AI tries to convince a judge that asks questions; and compare to a baseline of direct question-answering, where the judge just answers outright without the AI. We use large language models (LLMs) as both AI agents and as stand-ins for human judges, taking the judge models to be weaker than agent models. We benchmark on a diverse range of asymmetries between judges and agents, extending previous work on a single extractive QA task with information asymmetry, to also include mathematics, coding, logic and multimodal reasoning asymmetries. We find that debate outperforms consultancy across all tasks when the consultant is randomly assigned to argue for the correct/incorrect answer. Comparing debate to direct question answering, the results depend on the type of task: in extractive QA tasks with information asymmetry debate outperforms direct question answering, but in other tasks without information asymmetry the results are mixed. Previous work assigned debaters/consultants an answer to argue for. When we allow them to instead choose which answer to argue for, we find judges are less frequently convinced by the wrong answer in debate than in consultancy. Further, we find that stronger debater models increase judge accuracy, though more modestly than in previous studies.

MathCAMPS: Fine-grained Synthesis of Mathematical Problems From Human Curricula

Mathematical problem solving is an important skill for Large Language Models (LLMs), both as an important capability and a proxy for a range of reasoning abilities. Existing benchmarks probe a diverse set of skills, but they yield aggregate accuracy metrics, obscuring specific abilities or weaknesses. Furthermore, they are difficult to extend with new problems, risking data contamination over time. To address these challenges, we propose MathCAMPS: a method to synthesize high-quality mathematical problems at scale, grounded on 44 fine-grained "standards" from the Mathematics Common Core (CC) Standard for K-8 grades. We encode each standard in a formal grammar, allowing us to sample diverse symbolic problems and their answers. We then use LLMs to realize the symbolic problems into word problems. We propose a cycle-consistency method for validating problem faithfulness. Finally, we derive follow-up questions from symbolic structures and convert them into follow-up word problems - a novel task of mathematical dialogue that probes for robustness in understanding. Experiments on 23 LLMs show surprising failures even in the strongest models (in particular when asked simple follow-up questions). Moreover, we evaluate training checkpoints of Pythia 12B on MathCAMPS, allowing us to analyze when particular mathematical skills develop during its training. Our framework enables the community to reproduce and extend our pipeline for a fraction of the typical cost of building new high-quality datasets.