Zhangchen Zhou

Zhengan Chen, Yuqing Li, Tao Luo et al.

The phenomenon of distinct behaviors exhibited by neural networks under varying scales of initialization remains an enigma in deep learning research. In this paper, based on the earlier work by Luo et al.~\cite{luo2021phase}, we present a phase diagram of initial condensation for two-layer neural networks. Condensation is a phenomenon wherein the weight vectors of neural networks concentrate on isolated orientations during the training process, and it is a feature in non-linear learning process that enables neural networks to possess better generalization abilities. Our phase diagram serves to provide a comprehensive understanding of the dynamical regimes of neural networks and their dependence on the choice of hyperparameters related to initialization. Furthermore, we demonstrate in detail the underlying mechanisms by which small initialization leads to condensation at the initial training stage.

Zhiwei Bai, Jiajie Zhao, Zhangchen Zhou et al.

Adam is a widely used optimization algorithm in deep learning, yet the specific class of objective functions where it exhibits inherent advantages remains underexplored. Unlike prior studies requiring external schedulers and $β_2$ near 1 for convergence, this work investigates the "natural" auto-convergence properties of Adam. We identify a class of highly degenerate polynomials where Adam converges automatically without additional schedulers. Specifically, we derive theoretical conditions for local asymptotic stability on degenerate polynomials and demonstrate strong alignment between theoretical bounds and experimental results. We prove that Adam achieves local linear convergence on these degenerate functions, significantly outperforming the sub-linear convergence of Gradient Descent and Momentum. This acceleration stems from a decoupling mechanism between the second moment $v_t$ and squared gradient $g_t^2$, which exponentially amplifies the effective learning rate. Finally, we characterize Adam's hyperparameter phase diagram, identifying three distinct behavioral regimes: stable convergence, spikes, and SignGD-like oscillation.

Zhi-Qin John Xu, Yaoyu Zhang, Zhangchen Zhou

In this paper, we provide an overview of a common phenomenon, condensation, observed during the nonlinear training of neural networks: During the nonlinear training of neural networks, neurons in the same layer tend to condense into groups with similar outputs. Empirical observations suggest that the number of condensed clusters of neurons in the same layer typically increases monotonically as training progresses. Neural networks with small weight initializations or Dropout optimization can facilitate this condensation process. We also examine the underlying mechanisms of condensation from the perspectives of training dynamics and the structure of the loss landscape. The condensation phenomenon offers valuable insights into the generalization abilities of neural networks and correlates to stronger reasoning abilities in transformer-based language models.

Zhiwei Wang, Yunji Wang, Zhongwang Zhang et al.

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 capability. In this study, we constructed a symbolic multi-step reasoning task to investigate the information propagation mechanisms in Transformer models when solving the task through direct answering and Chain-of-Thought (CoT) reasoning. We introduced the concept of buffer mechanism: the model stores various information in distinct buffers and selectively extracts it through the query-key matrix. We proposed a random matrix-based algorithm to enhance the model's reasoning ability. This algorithm introduces only 132 trainable parameters, yet leads to significant performance improvements on 7 multi-step reasoning datasets, including PrOntoQA, LogicAsker, and LogicInference. These findings provide new insights into understanding the large language models.

Liangkai Hang, Junjie Yao, Zhiwei Bai et al.

The reasoning ability of large language models (LLMs) has been rapidly advancing in recent years, attracting interest in more fundamental approaches that can reliably enhance their generalizability. This work demonstrates that model complexity control, conveniently implementable by adjusting the initialization rate and weight decay coefficient, improves the scaling law of LLMs consistently over varying model sizes and data sizes. This gain is further illustrated by comparing the benchmark performance of 2.4B models pretrained on 1T tokens with different complexity hyperparameters. Instead of fixing the initialization std, we found that a constant initialization rate (the exponent of std) enables the scaling law to descend faster in both model and data sizes. These results indicate that complexity control is a promising direction for the continual advancement of LLMs.

Zhangchen Zhou, Yaoyu Zhang, Zhi-Qin John Xu

Grokking is the phenomenon where neural networks NNs initially fit the training data and later generalize to the test data during training. In this paper, we empirically provide a frequency perspective to explain the emergence of this phenomenon in NNs. The core insight is that the networks initially learn the less salient frequency components present in the test data. We observe this phenomenon across both synthetic and real datasets, offering a novel viewpoint for elucidating the grokking phenomenon by characterizing it through the lens of frequency dynamics during the training process. Our empirical frequency-based analysis sheds new light on understanding the grokking phenomenon and its underlying mechanisms.

Zhiwei Bai, Zhangchen Zhou, Jiajie Zhao et al.

Loss spikes emerge commonly during training across neural networks of varying architectures and scales when using the Adam optimizer. In this work, we investigate the underlying mechanism responsible for Adam spikes. While previous explanations attribute these phenomena to the lower-loss-as-sharper characteristics of the loss landscape, our analysis reveals that Adam's adaptive preconditioners themselves can trigger spikes. Specifically, we identify a critical regime where squared gradients become substantially smaller than the second-order moment estimates, causing the latter to undergo a $β_2$-exponential decay and to respond sluggishly to current gradient information. This mechanism can push the maximum eigenvalue of the preconditioned Hessian beyond the classical stability threshold $2/η$ for a sustained period, inducing instability. This instability further leads to an alignment between the gradient and the maximum eigendirection, and a loss spike occurs precisely when the gradient-directional curvature exceeds $2/η$. We verify this mechanism through extensive experiments on fully connected networks, convolutional networks, and Transformer architectures.

Zhongwang Zhang, Zhiwei Wang, Junjie Yao et al.

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.

Zhangchen Zhou, Hanxu Zhou, Yuqing Li et al.

Previous research has shown that fully-connected networks with small initialization and gradient-based training methods exhibit a phenomenon known as condensation during training. This phenomenon refers to the input weights of hidden neurons condensing into isolated orientations during training, revealing an implicit bias towards simple solutions in the parameter space. However, the impact of neural network structure on condensation has not been investigated yet. In this study, we focus on the investigation of convolutional neural networks (CNNs). Our experiments suggest that when subjected to small initialization and gradient-based training methods, kernel weights within the same CNN layer also cluster together during training, demonstrating a significant degree of condensation. Theoretically, we demonstrate that in a finite training period, kernels of a two-layer CNN with small initialization will converge to one or a few directions. This work represents a step towards a better understanding of the non-linear training behavior exhibited by neural networks with specialized structures.