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.

We introduce SODA, a self-supervised diffusion model, designed for representation learning. The model incorporates an image encoder, which distills a source view into a compact representation, that, in turn, guides the generation of related novel views. We show that by imposing a tight bottleneck between the encoder and a denoising decoder, and leveraging novel view synthesis as a self-supervised objective, we can turn diffusion models into strong representation learners, capable of capturing visual semantics in an unsupervised manner. To the best of our knowledge, SODA is the first diffusion model to succeed at ImageNet linear-probe classification, and, at the same time, it accomplishes reconstruction, editing and synthesis tasks across a wide range of datasets. Further investigation reveals the disentangled nature of its emergent latent space, that serves as an effective interface to control and manipulate the model's produced images. All in all, we aim to shed light on the exciting and promising potential of diffusion models, not only for image generation, but also for learning rich and robust representations.

Abstract reasoning is a key ability for an intelligent system. Large language models (LMs) achieve above-chance performance on abstract reasoning tasks, but exhibit many imperfections. However, human abstract reasoning is also imperfect. For example, human reasoning is affected by our real-world knowledge and beliefs, and shows notable "content effects"; humans reason more reliably when the semantic content of a problem supports the correct logical inferences. These content-entangled reasoning patterns play a central role in debates about the fundamental nature of human intelligence. Here, we investigate whether language models $\unicode{x2014}$ whose prior expectations capture some aspects of human knowledge $\unicode{x2014}$ similarly mix content into their answers to logical problems. We explored this question across three logical reasoning tasks: natural language inference, judging the logical validity of syllogisms, and the Wason selection task. We evaluate state of the art large language models, as well as humans, and find that the language models reflect many of the same patterns observed in humans across these tasks $\unicode{x2014}$ like humans, models answer more accurately when the semantic content of a task supports the logical inferences. These parallels are reflected both in answer patterns, and in lower-level features like the relationship between model answer distributions and human response times. Our findings have implications for understanding both these cognitive effects in humans, and the factors that contribute to language model performance.

Accounts of human language processing have long appealed to implicit ``situation models'' that enrich comprehension with relevant but unstated world knowledge. Here, we apply causal intervention techniques to recent transformer models to analyze performance on the Winograd Schema Challenge (WSC), where a single context cue shifts interpretation of an ambiguous pronoun. We identify a relatively small circuit of attention heads that are responsible for propagating information from the context word that guides which of the candidate noun phrases the pronoun ultimately attends to. We then compare how this circuit behaves in a closely matched ``syntactic'' control where the situation model is not strictly necessary. These analyses suggest distinct pathways through which implicit situation models are constructed to guide pronoun resolution.

Out-of-distribution generalization (OODG) is a longstanding challenge for neural networks. This challenge is quite apparent in tasks with well-defined variables and rules, where explicit use of the rules could solve problems independently of the particular values of the variables, but networks tend to be tied to the range of values sampled in their training data. Large transformer-based language models have pushed the boundaries on how well neural networks can solve previously unseen problems, but their complexity and lack of clarity about the relevant content in their training data obfuscates how they achieve such robustness. As a step toward understanding how transformer-based systems generalize, we explore the question of OODG in small scale transformers trained with examples from a known distribution. Using a reasoning task based on the puzzle Sudoku, we show that OODG can occur on a complex problem if the training set includes examples sampled from the whole distribution of simpler component tasks. Successful generalization depends on carefully managing positional alignment when absolute position encoding is used, but we find that suppressing sensitivity to absolute positions overcomes this limitation. Taken together our results represent a small step toward understanding and promoting systematic generalization in transformers.

Transformer networks have seen great success in natural language processing and machine vision, where task objectives such as next word prediction and image classification benefit from nuanced context sensitivity across high-dimensional inputs. However, there is an ongoing debate about how and when transformers can acquire highly structured behavior and achieve systematic generalization. Here, we explore how well a causal transformer can perform a set of algorithmic tasks, including copying, sorting, and hierarchical compositions of these operations. We demonstrate strong generalization to sequences longer than those used in training by replacing the standard positional encoding typically used in transformers with labels arbitrarily paired with items in the sequence. We search for the layer and head configuration sufficient to solve these tasks, then probe for signs of systematic processing in latent representations and attention patterns. We show that two-layer transformers learn reliable solutions to multi-level problems, develop signs of task decomposition, and encode input items in a way that encourages the exploitation of shared computation across related tasks. These results provide key insights into how attention layers support structured computation both within a task and across multiple tasks.

Large language models have recently shown promising progress in mathematical reasoning when fine-tuned with human-generated sequences walking through a sequence of solution steps. However, the solution sequences are not formally structured and the resulting model-generated sequences may not reflect the kind of systematic reasoning we might expect an expert human to produce. In this paper, we study how to build stronger reasoning capability in language models using the idea of relational abstractions. We introduce new types of sequences that more explicitly provide an abstract characterization of the transitions through intermediate solution steps to the goal state. We find that models that are supplied with such sequences as prompts can solve tasks with a significantly higher accuracy, and models that are trained to produce such sequences solve problems better than those that are trained with previously used human-generated sequences and other baselines. Our work thus takes several steps toward elucidating and improving how language models perform on tasks requiring multi-step mathematical reasoning.

Explanations play a considerable role in human learning, especially in areas that remain major challenges for AI -- forming abstractions, and learning about the relational and causal structure of the world. Here, we explore whether reinforcement learning agents might likewise benefit from explanations. We outline a family of relational tasks that involve selecting an object that is the odd one out in a set (i.e., unique along one of many possible feature dimensions). Odd-one-out tasks require agents to reason over multi-dimensional relationships among a set of objects. We show that agents do not learn these tasks well from reward alone, but achieve >90% performance when they are also trained to generate language explaining object properties or why a choice is correct or incorrect. In further experiments, we show how predicting explanations enables agents to generalize appropriately from ambiguous, causally-confounded training, and even to meta-learn to perform experimental interventions to identify causal structure. We show that explanations help overcome the tendency of agents to fixate on simple features, and explore which aspects of explanations make them most beneficial. Our results suggest that learning from explanations is a powerful principle that could offer a promising path towards training more robust and general machine learning systems.

Despite the groundbreaking successes of neural networks, contemporary models require extensive training with massive datasets and exhibit poor out-of-sample generalization. One proposed solution is to build systematicity and domain-specific constraints into the model, echoing the tenets of classical, symbolic cognitive architectures. In this paper, we consider the limitations of this approach by examining human adults' ability to learn an abstract reasoning task from a brief instructional tutorial and explanatory feedback for incorrect responses, demonstrating that human learning dynamics and ability to generalize outside the range of the training examples differ drastically from those of a representative neural network model, and that the model is brittle to changes in features not anticipated by its authors. We present further evidence from human data that the ability to consistently solve the puzzles was associated with education, particularly basic mathematics education, and with the ability to provide a reliably identifiable, valid description of the strategy used. We propose that rapid learning and systematic generalization in humans may depend on a gradual, experience-dependent process of learning-to-learn using instructions and explanations to guide the construction of explicit abstract rules that support generalizable inferences.

An important aspect of intelligence is the ability to adapt to a novel task without any direct experience (zero-shot), based on its relationship to previous tasks. Humans can exhibit this cognitive flexibility. By contrast, deep-learning models that achieve superhuman performance in specific tasks generally fail to adapt to even slight task alterations. To address this, we propose a general computational framework for adapting to novel tasks based on their relationship to prior tasks. We begin by learning vector representations of tasks. To adapt to new tasks, we propose meta-mappings, higher-order tasks that transform basic task representations. We demonstrate this framework across a wide variety of tasks and computational paradigms, ranging from regression to image classification and reinforcement learning. We compare to both human adaptability, and language-based approaches to zero-shot learning. Across these domains, meta-mapping is successful, often achieving 80-90% performance, without any data, on a novel task that directly contradicts its prior experience. We further show that using meta-mapping as a starting point can dramatically accelerate later learning on a new task, and reduce learning time and cumulative error substantially. Our results provide insight into a possible computational basis of intelligent adaptability, and offer a possible framework for modeling cognitive flexibility and building more flexible artificial intelligence.