Weight decay induces low-rank attention layers (2410.23819v1)

Abstract: The effect of regularizers such as weight decay when training deep neural networks is not well understood. We study the influence of weight decay as well as $L2$-regularization when training neural network models in which parameter matrices interact multiplicatively. This combination is of particular interest as this parametrization is common in attention layers, the workhorse of transformers. Here, key-query, as well as value-projection parameter matrices, are multiplied directly with each other: $W_K^TW_Q$ and $PW_V$. We extend previous results and show on one hand that any local minimum of a $L2$-regularized loss of the form $L(AB^\top) + \lambda (|A|^2 + |B|^2)$ coincides with a minimum of the nuclear norm-regularized loss $L(AB^\top) + \lambda|AB^\top|_*$, and on the other hand that the 2 losses become identical exponentially quickly during training. We thus complement existing works linking $L2$-regularization with low-rank regularization, and in particular, explain why such regularization on the matrix product affects early stages of training. Based on these theoretical insights, we verify empirically that the key-query and value-projection matrix products $W_K^TW_Q, PW_V$ within attention layers, when optimized with weight decay, as usually done in vision tasks and LLMling, indeed induce a significant reduction in the rank of $W_K^TW_Q$ and $PW_V$, even in fully online training. We find that, in accordance with existing work, inducing low rank in attention matrix products can damage LLM performance, and observe advantages when decoupling weight decay in attention layers from the rest of the parameters.

• The paper demonstrates that weight decay and L2-regularization encourage low-rank solutions via factorized parametrization in attention layers.
• The methodology links L2-regularized losses with nuclear norm regularization, showing an exponential reduction in their discrepancy during optimization.
• Empirical results reveal that strong weight decay reduces the rank of projection matrices, which can compromise performance in language models.

Insights into Weight Decay and Low-Rank Induction in Attention Layers

The paper "Weight decay induces low-rank attention layers" provides a comprehensive analysis of the effects of weight decay (WD) and $L2$-regularization on neural networks, particularly focusing on models with parameter matrix products, such as transformers. The paper's theoretical contributions delve into the optimization landscape of $L2$-regularized losses and elucidate how these regularizations influence the rank of attention layers within transformer architectures.

Theoretical Contributions

The paper introduces a robust theoretical framework to understand how weight decay and $L2$-regularization can affect rank minimization in matrices by employing a factorized parametrization. Central to the investigation is the consideration of neural network models where parameters are represented as products of matrices, denoted by $W = AB^\top$. This is especially pertinent in the context of transformers, where weight matrices interact multiplicatively within attention layers.

A key theoretical result demonstrates that any local minimum of the $L2$-regularized loss $L(AB^\top) + \lambda (\|A\|^2 + \|B\|^2)$ aligns with a local minimum of its nuclear norm-regularized counterpart, $L(AB^\top) + \lambda\|AB^\top\|_*$. This theoretical insight is significant because it establishes a relationship between $L2$-regularization and low-rank regularization, which was not fully explicated in prior literature. Furthermore, the authors reveal that during the optimization process, the discrepancy between the two regularizations diminishes exponentially.

Implications of this are profound; the nuclear norm is well-known for its rank minimization properties. Therefore, this paper suggests that the application of $L2$-regularization inherently applies pressure towards low-rank solutions, even in early training stages. The analysis is complemented by empirical evidence which shows the inductive biases introduced by factorized parameterizations, revealing their potential detrimental impact on certain tasks.

Empirical Findings and Validation

Empirical validation solidifies the theoretical claims through experiments demonstrating low-rank induction on key-query and value-projection products within attention layers under weight decay. This work effectively corroborates the hypothesis that training configurations traditionally deploying high-strength weight decay can indeed instigate significant rank reductions in component matrices such as $W_K^\top W_Q$ and $PW_V$. Furthermore, the paper underscores scenarios where this observational phenomenon leads to compromised performance in LLMs, despite being consistent with weight decay strategies reported for influential models like GPT-3, LLaMa, and ViT.

Practical and Theoretical Implications

These findings are instructive for designing better neural network optimizers and architectures. The implications pin down an unrecognized trade-off between inducing low-rank behavior and maintaining performance, prevalent in large-scale pre-trained models. The paper challenges existing practices by suggesting potential advantages from decoupling weight decay applications in attention layers from other model parameters. This avenue opens up methodologies for more nuanced application of regularization techniques, possibly leading to improved model adaptability across varied tasks.

Future Directions

This research paves the way for further exploration into layer-specific optimization strategies and the dynamic between $L2$-regularization and model expressivity. Future studies could investigate the optimal balance of rank-inducing regularization, particularly in transformers’ attention components, uncovering ways to harness, rather than exacerbate, this effect. Another future endeavor could address examining the combined use of regularization and advanced initialization strategies to mitigate adverse impacts on model robustness.

In summary, this paper provides a thorough theoretical and empirical exploration of weight decay-induced rank reduction in attention layers. By bridging previous theoretical gaps concerning regularization and low-rank induction, it furnishes a platform for future research aimed at refining deep learning model training practices, especially in transformer-based architectures.