[论文解读] Estimating Information Flow in Deep Neural Networks

Ziv Goldfeld, E. van den Berg+5 more|arXiv (Cornell University)|Oct 12, 2018

Adversarial Robustness in Machine Learning参考文献 22被引用 20

一句话总结

本文提出了一种基于矩逼近和协方差结构的高斯混合模型在深度神经网络中微分熵的上下界估计方法，用于量化信息流。核心贡献是提出了一种新颖的分析框架，通过分量均值、方差和混合权重导出的熵界来量化信息传播。

ABSTRACT

We study the flow of information and the evolution of internal representations during deep neural network (DNN) training, aiming to demystify the compression aspect of the information bottleneck theory. The theory suggests that DNN training comprises a rapid fitting phase followed by a slower compression phase, in which the mutual information $I(X;T)$ between the input $X$ and internal representations $T$ decreases. Several papers observe compression of estimated mutual information on different DNN models, but the true $I(X;T)$ over these networks is provably either constant (discrete $X$) or infinite (continuous $X$). This work explains the discrepancy between theory and experiments, and clarifies what was actually measured by these past works. To this end, we introduce an auxiliary (noisy) DNN framework for which $I(X;T)$ is a meaningful quantity that depends on the network's parameters. This noisy framework is shown to be a good proxy for the original (deterministic) DNN both in terms of performance and the learned representations. We then develop a rigorous estimator for $I(X;T)$ in noisy DNNs and observe compression in various models. By relating $I(X;T)$ in the noisy DNN to an information-theoretic communication problem, we show that compression is driven by the progressive clustering of hidden representations of inputs from the same class. Several methods to directly monitor clustering of hidden representations, both in noisy and deterministic DNNs, are used to show that meaningful clusters form in the $T$ space. Finally, we return to the estimator of $I(X;T)$ employed in past works, and demonstrate that while it fails to capture the true (vacuous) mutual information, it does serve as a measure for clustering. This clarifies the past observations of compression and isolates the geometric clustering of hidden representations as the true phenomenon of interest.

研究动机与目标

• 为深度神经网络中的高斯混合模型微分熵建立分析界。
• 通过基于矩的近似方法估计激活的熵，以量化深度网络中的信息流。
• 提供依赖于混合分量均值、方差和权重的可计算上下界。
• 在不依赖经验估计的前提下，实现对深度学习中信息传播的理论分析。

提出的方法

• 利用混合分量权重和分量均值之间成对距离推导微分熵的上界。
• 使用由 $\beta = \sum_{i\in[n]} c_i \mu_i \mu_i^\top - \mu\mu^\top + \beta^2 I_d $ 定义的协方差矩阵结构，其中 $\mu = \sum_{i\in[n]} c_i \mu_i $。
• 应用信息论不等式，通过对数行列式项和混合分量熵来界定熵。
• 建立结合加权熵项和分量均值之间几何距离的边界 (5a)、(5b) 和 (5c)。
• 利用高斯混合模型假设来近似深度网络中激活的分布。
• 引入基于矩的近似方法，在无需完整密度估计的前提下估计熵。

实验结果

研究问题

• RQ1 如何仅通过矩和混合权重对深度网络中高斯混合模型的微分熵进行界定？
• RQ2 激活协方差结构与深度神经网络中信息流之间有何关系？
• RQ3 能否在不依赖蒙特卡洛或密度估计的前提下推导出紧致的分析熵界？
• RQ4 分量均值之间的成对距离如何影响整体混合分布的熵？
• RQ5 加权协方差矩阵 $\beta$ 在塑造熵界方面起到什么作用？

主要发现

• 上界 (5a) 提供了基于混合分量权重及其方差的微分熵闭式表达式。
• 边界 (5b) 通过指数衰减项引入分量均值之间的成对距离，从而改进估计精度。
• 边界 (5c) 将熵表示为协方差矩阵 $\beta$ 的对数行列式，将其与混合分布的整体扩散程度联系起来。
• 所提出的边界在计算上是可行的，且仅依赖于混合分量的一阶和二阶矩。
• 该框架通过基于熵界的不确定性量化，实现了对信息流的理论分析。
• 这些边界适用于任意高斯混合模型，可应用于分析深度网络中跨层的信息传播。