Complexity Control Facilitates Reasoning-Based Compositional Generalization in Transformers

Transformers have demonstrated impressive capabilities across various tasks, yet their performance on compositional problems remains a subject of debate. In this study, we investigate the internal mechanisms underlying Transformers' behavior in compositional tasks. We find that complexity control strategies significantly influence whether the model learns primitive-level rules that generalize out-of-distribution (reasoning-based solutions) or relies solely on memorized mappings (memory-based solutions). By applying masking strategies to the model's information circuits and employing multiple complexity metrics, we reveal distinct internal working mechanisms associated with different solution types. Further analysis reveals that reasoning-based solutions exhibit a lower complexity bias, which aligns with the well-studied neuron condensation phenomenon. This lower complexity bias is hypothesized to be the key factor enabling these solutions to learn reasoning rules. We validate these conclusions across multiple real-world datasets, including image generation and natural language processing tasks, confirming the broad applicability of our findings.

Local Linear Recovery Guarantee of Deep Neural Networks at Overparameterization

Determining whether deep neural network (DNN) models can reliably recover target functions at overparameterization is a critical yet complex issue in the theory of deep learning. To advance understanding in this area, we introduce a concept we term "local linear recovery" (LLR), a weaker form of target function recovery that renders the problem more amenable to theoretical analysis. In the sense of LLR, we prove that functions expressible by narrower DNNs are guaranteed to be recoverable from fewer samples than model parameters. Specifically, we establish upper limits on the optimistic sample sizes, defined as the smallest sample size necessary to guarantee LLR, for functions in the space of a given DNN. Furthermore, we prove that these upper bounds are achieved in the case of two-layer tanh neural networks. Our research lays a solid groundwork for future investigations into the recovery capabilities of DNNs in overparameterized scenarios.

Towards Understanding How Transformer Perform Multi-step Reasoning with Matching Operation

Large language models have consistently struggled with complex reasoning tasks, such as mathematical problem-solving. Investigating the internal reasoning mechanisms of these models can help us design better model architectures and training strategies, ultimately enhancing their reasoning capabilities. In this study, we examine the matching mechanism employed by Transformer for multi-step reasoning on a constructed dataset. We investigate factors that influence the model's matching mechanism and discover that small initialization and post-LayerNorm can facilitate the formation of the matching mechanism, thereby enhancing the model's reasoning ability. Moreover, we propose a method to improve the model's reasoning capability by adding orthogonal noise. Finally, we investigate the parallel reasoning mechanism of Transformers and propose a conjecture on the upper bound of the model's reasoning ability based on this phenomenon. These insights contribute to a deeper understanding of the reasoning processes in large language models and guide designing more effective reasoning architectures and training strategies.

Initialization is Critical to Whether Transformers Fit Composite Functions by Inference or Memorizing

Transformers have shown impressive capabilities across various tasks, but their performance on compositional problems remains a topic of debate. In this work, we investigate the mechanisms of how transformers behave on unseen compositional tasks using anchor functions. We discover that the parameter initialization scale plays a critical role in determining whether the model learns inferential solutions, which capture the underlying compositional primitives, or symmetric solutions, which simply memorize mappings without understanding the compositional structure. By analyzing the information flow and vector representations within the model, we reveal the distinct mechanisms underlying these solution types. We further find that inferential solutions exhibit low complexity bias, which we hypothesize is a key factor enabling them to learn individual mappings for single anchors. Building upon our understanding of these mechanisms, we can predict the learning behavior of models with different initialization scales when faced with data of varying inferential complexity. Our findings provide valuable insights into the role of initialization scale in shaping the type of solution learned by transformers and their ability to learn and generalize compositional functions.

Loss Jump During Loss Switch in Solving PDEs with Neural Networks

Using neural networks to solve partial differential equations (PDEs) is gaining popularity as an alternative approach in the scientific computing community. Neural networks can integrate different types of information into the loss function. These include observation data, governing equations, and variational forms, etc. These loss functions can be broadly categorized into two types: observation data loss directly constrains and measures the model output, while other loss functions indirectly model the performance of the network, which can be classified as model loss. However, this alternative approach lacks a thorough understanding of its underlying mechanisms, including theoretical foundations and rigorous characterization of various phenomena. This work focuses on investigating how different loss functions impact the training of neural networks for solving PDEs. We discover a stable loss-jump phenomenon: when switching the loss function from the data loss to the model loss, which includes different orders of derivative information, the neural network solution significantly deviates from the exact solution immediately. Further experiments reveal that this phenomenon arises from the different frequency preferences of neural networks under different loss functions. We theoretically analyze the frequency preference of neural networks under model loss. This loss-jump phenomenon provides a valuable perspective for examining the underlying mechanisms of neural networks in solving PDEs.

Anchor function: a type of benchmark functions for studying language models

Understanding transformer-based language models is becoming increasingly crucial, particularly as they play pivotal roles in advancing towards artificial general intelligence. However, language model research faces significant challenges, especially for academic research groups with constrained resources. These challenges include complex data structures, unknown target functions, high computational costs and memory requirements, and a lack of interpretability in the inference process, etc. Drawing a parallel to the use of simple models in scientific research, we propose the concept of an anchor function. This is a type of benchmark function designed for studying language models in learning tasks that follow an "anchor-key" pattern. By utilizing the concept of an anchor function, we can construct a series of functions to simulate various language tasks. The anchor function plays a role analogous to that of mice in diabetes research, particularly suitable for academic research. We demonstrate the utility of the anchor function with an example, revealing two basic operations by attention structures in language models: shifting tokens and broadcasting one token from one position to many positions. These operations are also commonly observed in large language models. The anchor function framework, therefore, opens up a series of valuable and accessible research questions for further exploration, especially for theoretical study.

Optimistic Estimate Uncovers the Potential of Nonlinear Models

We propose an optimistic estimate to evaluate the best possible fitting performance of nonlinear models. It yields an optimistic sample size that quantifies the smallest possible sample size to fit/recover a target function using a nonlinear model. We estimate the optimistic sample sizes for matrix factorization models, deep models, and deep neural networks (DNNs) with fully-connected or convolutional architecture. For each nonlinear model, our estimates predict a specific subset of targets that can be fitted at overparameterization, which are confirmed by our experiments. Our optimistic estimate reveals two special properties of the DNN models -- free expressiveness in width and costly expressiveness in connection. These properties suggest the following architecture design principles of DNNs: (i) feel free to add neurons/kernels; (ii) restrain from connecting neurons. Overall, our optimistic estimate theoretically unveils the vast potential of nonlinear models in fitting at overparameterization. Based on this framework, we anticipate gaining a deeper understanding of how and why numerous nonlinear models such as DNNs can effectively realize their potential in practice in the near future.

Stochastic Modified Equations and Dynamics of Dropout Algorithm

Dropout is a widely utilized regularization technique in the training of neural networks, nevertheless, its underlying mechanism and its impact on achieving good generalization abilities remain poorly understood. In this work, we derive the stochastic modified equations for analyzing the dynamics of dropout, where its discrete iteration process is approximated by a class of stochastic differential equations. In order to investigate the underlying mechanism by which dropout facilitates the identification of flatter minima, we study the noise structure of the derived stochastic modified equation for dropout. By drawing upon the structural resemblance between the Hessian and covariance through several intuitive approximations, we empirically demonstrate the universal presence of the inverse variance-flatness relation and the Hessian-variance relation, throughout the training process of dropout. These theoretical and empirical findings make a substantial contribution to our understanding of the inherent tendency of dropout to locate flatter minima.

Loss Spike in Training Neural Networks

In this work, we study the mechanism underlying loss spikes observed during neural network training. When the training enters a region, which has a smaller-loss-as-sharper (SLAS) structure, the training becomes unstable and loss exponentially increases once it is too sharp, i.e., the rapid ascent of the loss spike. The training becomes stable when it finds a flat region. The deviation in the first eigen direction (with maximum eigenvalue of the loss Hessian ($\lambda_{\mathrm{max}}$) is found to be dominated by low-frequency. Since low-frequency is captured very fast (frequency principle), the rapid descent is then observed. Inspired by our analysis of loss spikes, we revisit the link between $\lambda_{\mathrm{max}}$ flatness and generalization. For real datasets, low-frequency is often dominant and well-captured by both the training data and the test data. Then, a solution with good generalization and a solution with bad generalization can both learn low-frequency well, thus, they have little difference in the sharpest direction. Therefore, although $\lambda_{\mathrm{max}}$ can indicate the sharpness of the loss landscape, deviation in its corresponding eigen direction is not responsible for the generalization difference. We also find that loss spikes can facilitate condensation, i.e., input weights evolve towards the same, which may be the underlying mechanism for why the loss spike improves generalization, rather than simply controlling the value of $\lambda_{\mathrm{max}}$.

Linear Stability Hypothesis and Rank Stratification for Nonlinear Models

Models with nonlinear architectures/parameterizations such as deep neural networks (DNNs) are well known for their mysteriously good generalization performance at overparameterization. In this work, we tackle this mystery from a novel perspective focusing on the transition of the target recovery/fitting accuracy as a function of the training data size. We propose a rank stratification for general nonlinear models to uncover a model rank as an "effective size of parameters" for each function in the function space of the corresponding model. Moreover, we establish a linear stability theory proving that a target function almost surely becomes linearly stable when the training data size equals its model rank. Supported by our experiments, we propose a linear stability hypothesis that linearly stable functions are preferred by nonlinear training. By these results, model rank of a target function predicts a minimal training data size for its successful recovery. Specifically for the matrix factorization model and DNNs of fully-connected or convolutional architectures, our rank stratification shows that the model rank for specific target functions can be much lower than the size of model parameters. This result predicts the target recovery capability even at heavy overparameterization for these nonlinear models as demonstrated quantitatively by our experiments. Overall, our work provides a unified framework with quantitative prediction power to understand the mysterious target recovery behavior at overparameterization for general nonlinear models.