Transformers as Intrinsic Optimizers: Forward Inference through the Energy Principle
This work provides a theoretical lens for interpreting Transformers, which could aid in designing new attention mechanisms, but it is incremental as it builds on existing energy-based principles without demonstrating broad practical gains.
The paper tackles the problem of understanding Transformer mechanisms by proposing an energy-based framework that interprets attention as minimizing Helmholtz free energy, and it introduces modified attention structures inspired by gradient descent variants, with experiments showing preliminary support for these designs.
Transformers have demonstrated strong adaptability across a wide range of tasks and have become the backbone of modern Large Language Models (LLMs). However, their underlying mechanisms remain open for further exploration. The energy-based perspective has long provided a valuable principle for understanding neural computation. In this paper, we revisit the principle of energy as a lens to understand attention-based Transformer models. We present a unified energy-based framework which is composed of three key components: the global energy $F^*$, the energy function $E_i$ and the employed gradient descent (GD) form. Within this framework, standard softmax attention can be viewed as a special case of minimizing the Helmholtz free energy as $F^*$ using standard GD when $E_i$ takes the form of elastic potential energy, with residual connections ensuring that this optimization proceeds in an incremental manner. In addition, linear attentions can also be naturally incorporated into this framework by adjusting the corresponding energy forms. We also extend the above analysis to the multi-head setting, where the energy is defined across multiple low-dimensional subspaces. Building on this framework, we propose energy-based modifications of attention structures. Inspired by classical GD algorithms, we extend the original attention formulation based on standard GD to the momentum-based GD, Nesterov Accelerated Gradient (NAG), and Newton's method variants, each inducing a corresponding new attention structure. Our experiments provide preliminary support for the potential of the energy-based framework for designing attention mechanisms.