将MLP转化为闭式多项式，揭示其可解释性与演化规律
Converting MLPs into Polynomials in Closed Form
- 用任意阶多项式推导MLP和GLU的最小二乘最优闭式近似
- 训练后期二次近似仍能解释95%以上网络输出方差
- 可用于可视化分析、对抗样本生成及网络复杂度演化研究
近期研究表明，仅用二次函数即可替代Transformer中的MLP而性能无明显下降，并支持基于线性代数的可解释性方法。本文从理论上推导了前馈网络（多层感知机和门控线性单元）的闭式最小二乘最优多项式近似，阶数任意。当决定系数$R^2$较高时，可通过可视化其线性和二次近似系数的特征分解来解释MLP和GLU。此外，这些近似还可用于构建基于SVD的对抗样本。通过追踪训练过程中线性与二次近似的$R^2$变化，发现网络初始简单，随训练逐步变复杂。即使在训练结束时，二次近似仍能解释超过95%的网络输出方差。
原文摘要 · Abstract (English)
Recent work has shown that purely quadratic functions can replace MLPs in transformers with no significant loss in performance, while enabling new methods of interpretability based on linear algebra. In this work, we theoretically derive closed-form least-squares optimal approximations of feedforward networks (multilayer perceptrons and gated linear units) using polynomial functions of arbitrary degree. When the $R^2$ is high, this allows us to interpret MLPs and GLUs by visualizing the eigendecomposition of the coefficients of their linear and quadratic approximants. We also show that these approximants can be used to create SVD-based adversarial examples. By tracing the $R^2$ of linear and quadratic approximants across training time, we find new evidence that networks start out simple, and get progressively more complex. Even at the end of training, however, our quadratic approximants explain over 95% of the variance in network outputs.
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