Towards an Understanding of Stepwise Inference in Transformers: A Synthetic Graph Navigation Model

Stepwise inference protocols, such as scratchpads and chain-of-thought, help language models solve complex problems by decomposing them into a sequence of simpler subproblems. Despite the significant gain in performance achieved via these protocols, the underlying mechanisms of stepwise inference have remained elusive. To address this, we propose to study autoregressive Transformer models on a synthetic task that embodies the multi-step nature of problems where stepwise inference is generally most useful. Specifically, we define a graph navigation problem wherein a model is tasked with traversing a path from a start to a goal node on the graph. Despite is simplicity, we find we can empirically reproduce and analyze several phenomena observed at scale: (i) the stepwise inference reasoning gap, the cause of which we find in the structure of the training data; (ii) a diversity-accuracy tradeoff in model generations as sampling temperature varies; (iii) a simplicity bias in the model's output; and (iv) compositional generalization and a primacy bias with in-context exemplars. Overall, our work introduces a grounded, synthetic framework for studying stepwise inference and offers mechanistic hypotheses that can lay the foundation for a deeper understanding of this phenomenon.

How Capable Can a Transformer Become? A Study on Synthetic, Interpretable Tasks

Transformers trained on huge text corpora exhibit a remarkable set of capabilities, e.g., performing simple logical operations. Given the inherent compositional nature of language, one can expect the model to learn to compose these capabilities, potentially yielding a combinatorial explosion of what operations it can perform on an input. Motivated by the above, we aim to assess in this paper "how capable can a transformer become?". Specifically, we train autoregressive Transformer models on a data-generating process that involves compositions of a set of well-defined monolithic capabilities. Through a series of extensive and systematic experiments on this data-generating process, we show that: (1) autoregressive Transformers can learn compositional structures from the training data and generalize to exponentially or even combinatorially many functions; (2) composing functions by generating intermediate outputs is more effective at generalizing to unseen compositions, compared to generating no intermediate outputs; (3) the training data has a significant impact on the model's ability to compose unseen combinations of functions; and (4) the attention layers in the latter half of the model are critical to compositionality.

Mechanistically analyzing the effects of fine-tuning on procedurally defined tasks

Fine-tuning large pre-trained models has become the de facto strategy for developing both task-specific and general-purpose machine learning systems, including developing models that are safe to deploy. Despite its clear importance, there has been minimal work that explains how fine-tuning alters the underlying capabilities learned by a model during pretraining: does fine-tuning yield entirely novel capabilities or does it just modulate existing ones? We address this question empirically in synthetic, controlled settings where we can use mechanistic interpretability tools (e.g., network pruning and probing) to understand how the model's underlying capabilities are changing. We perform an extensive analysis of the effects of fine-tuning in these settings, and show that: (i) fine-tuning rarely alters the underlying model capabilities; (ii) a minimal transformation, which we call a 'wrapper', is typically learned on top of the underlying model capabilities, creating the illusion that they have been modified; and (iii) further fine-tuning on a task where such hidden capabilities are relevant leads to sample-efficient 'revival' of the capability, i.e., the model begins reusing these capability after only a few gradient steps. This indicates that practitioners can unintentionally remove a model's safety wrapper merely by fine-tuning it on a, e.g., superficially unrelated, downstream task. We additionally perform analysis on language models trained on the TinyStories dataset to support our claims in a more realistic setup.

In-Context Learning Dynamics with Random Binary Sequences

Large language models (LLMs) trained on huge corpora of text datasets demonstrate complex, emergent capabilities, achieving state-of-the-art performance on tasks they were not explicitly trained for. The precise nature of LLM capabilities is often mysterious, and different prompts can elicit different capabilities through in-context learning. We propose a Cognitive Interpretability framework that enables us to analyze in-context learning dynamics to understand latent concepts in LLMs underlying behavioral patterns. This provides a more nuanced understanding than success-or-failure evaluation benchmarks, but does not require observing internal activations as a mechanistic interpretation of circuits would. Inspired by the cognitive science of human randomness perception, we use random binary sequences as context and study dynamics of in-context learning by manipulating properties of context data, such as sequence length. In the latest GPT-3.5+ models, we find emergent abilities to generate pseudo-random numbers and learn basic formal languages, with striking in-context learning dynamics where model outputs transition sharply from pseudo-random behaviors to deterministic repetition.

Compositional Abilities Emerge Multiplicatively: Exploring Diffusion Models on a Synthetic Task

Modern generative models exhibit unprecedented capabilities to generate extremely realistic data. However, given the inherent compositionality of the real world, reliable use of these models in practical applications requires that they exhibit the capability to compose a novel set of concepts to generate outputs not seen in the training data set. Prior work demonstrates that recent diffusion models do exhibit intriguing compositional generalization abilities, but also fail unpredictably. Motivated by this, we perform a controlled study for understanding compositional generalization in conditional diffusion models in a synthetic setting, varying different attributes of the training data and measuring the model's ability to generate samples out-of-distribution. Our results show: (i) the order in which the ability to generate samples from a concept and compose them emerges is governed by the structure of the underlying data-generating process; (ii) performance on compositional tasks exhibits a sudden ``emergence'' due to multiplicative reliance on the performance of constituent tasks, partially explaining emergent phenomena seen in generative models; and (iii) composing concepts with lower frequency in the training data to generate out-of-distribution samples requires considerably more optimization steps compared to generating in-distribution samples. Overall, our study lays a foundation for understanding capabilities and compositionality in generative models from a data-centric perspective.

Mechanistic Mode Connectivity

Neural networks are known to be biased towards learning mechanisms that help identify $spurious\, attributes$, yielding features that do not generalize well under distribution shifts. To understand and address this limitation, we study the geometry of neural network loss landscapes through the lens of $mode\, connectivity$, the observation that minimizers of neural networks are connected via simple paths of low loss. Our work addresses two questions: (i) do minimizers that encode dissimilar mechanisms connect via simple paths of low loss? (ii) can fine-tuning a pretrained model help switch between such minimizers? We define a notion of $\textit{mechanistic similarity}$ and demonstrate that lack of linear connectivity between two minimizers implies the corresponding models use dissimilar mechanisms for making their predictions. This property helps us demonstrate that na$\"{i}$ve fine-tuning can fail to eliminate a model's reliance on spurious attributes. We thus propose a method for altering a model's mechanisms, named $connectivity$-$based$ $fine$-$tuning$, and validate its usefulness by inducing models invariant to spurious attributes.

What shapes the loss landscape of self-supervised learning?

Prevention of complete and dimensional collapse of representations has recently become a design principle for self-supervised learning (SSL). However, questions remain in our theoretical understanding: When do those collapses occur? What are the mechanisms and causes? We provide answers to these questions by thoroughly analyzing SSL loss landscapes for a linear model. We derive an analytically tractable theory of SSL landscape and show that it accurately captures an array of collapse phenomena and identifies their causes. Finally, we leverage the interpretability afforded by the analytical theory to understand how dimensional collapse can be beneficial and what affects the robustness of SSL against data imbalance.

Rethinking the limiting dynamics of SGD: modified loss, phase space oscillations, and anomalous diffusion

In this work we explore the limiting dynamics of deep neural networks trained with stochastic gradient descent (SGD). We find empirically that long after performance has converged, networks continue to move through parameter space by a process of anomalous diffusion in which distance travelled grows as a power law in the number of gradient updates with a nontrivial exponent. We reveal an intricate interaction between the hyperparameters of optimization, the structure in the gradient noise, and the Hessian matrix at the end of training that explains this anomalous diffusion. To build this understanding, we first derive a continuous-time model for SGD with finite learning rates and batch sizes as an underdamped Langevin equation. We study this equation in the setting of linear regression, where we can derive exact, analytic expressions for the phase space dynamics of the parameters and their instantaneous velocities from initialization to stationarity. Using the Fokker-Planck equation, we show that the key ingredient driving these dynamics is not the original training loss, but rather the combination of a modified loss, which implicitly regularizes the velocity, and probability currents, which cause oscillations in phase space. We identify qualitative and quantitative predictions of this theory in the dynamics of a ResNet-18 model trained on ImageNet. Through the lens of statistical physics, we uncover a mechanistic origin for the anomalous limiting dynamics of deep neural networks trained with SGD.

Beyond BatchNorm: Towards a General Understanding of Normalization in Deep Learning

Inspired by BatchNorm, there has been an explosion of normalization layers for deep neural networks (DNNs). However, these alternative normalization layers have seen minimal use, partially due to a lack of guiding principles that can help identify when these layers can serve as a replacement for BatchNorm. To address this problem, we take a theoretical approach, generalizing the known beneficial mechanisms of BatchNorm to several recently proposed normalization techniques. Our generalized theory leads to the following set of principles: (i) similar to BatchNorm, activations-based normalization layers can prevent exponential growth of activations in ResNets, but parametric layers require explicit remedies; (ii) use of GroupNorm can ensure informative forward propagation, with different samples being assigned dissimilar activations, but increasing group size results in increasingly indistinguishable activations for different samples, explaining slow convergence speed in models with LayerNorm; (iii) small group sizes result in large gradient norm in earlier layers, hence explaining training instability issues in Instance Normalization and illustrating a speed-stability tradeoff in GroupNorm. Overall, our analysis reveals a unified set of mechanisms that underpin the success of normalization methods in deep learning, providing us with a compass to systematically explore the vast design space of DNN normalization layers.