发现神经网络损失曲面中通往无穷的平坦通道,揭示了隐藏的计算机制。
Flat Channels to Infinity in Neural Loss Landscapes
- 通过梯度流分析损失曲面中的特殊平坦通道,发现参数发散但输出不变。
- 通道末端两神经元形成门控线性单元,实现非线性函数的高效表达。
- 该结构解释了优化器看似收敛却参数无限的现象,适用于多种回归任务。
神经网络的损失曲面包含可能相互连接或孤立的极小值点和鞍点。本文识别并刻画了一种特殊结构:沿特定通道损失极慢下降,同时至少两个神经元的输出权重 $a_i$、$a_j$ 趋向 $/pm$∞,其输入权重向量 $m{w_i}$、$m{w_j}$ 逐渐相等。在收敛时,这两个神经元实现一个门控线性单元:$a_iσ(m{w_i} m{x}) + a_jσ(m{w_j} m{x}) \rightarrow σ(m{w} m{x}) + (m{v} m{x}) σ'(\bm{w} m{x})$。几何上,这些通往无穷的通道渐近平行于由对称性诱导的临界点线。梯度流求解器及类似SGD、ADAM的优化方法在多种回归设置下以高概率抵达这些通道,但若不仔细检查,它们会误认为是参数有限的平坦局部极小值。本研究从梯度动力学、几何与功能角度全面揭示了这些准平坦区域的本质。通道末端出现的门控线性单元凸显了全连接层惊人的计算能力。
原文摘要 · Abstract (English)
The loss landscapes of neural networks contain minima and saddle points that may be connected in flat regions or appear in isolation. We identify and characterize a special structure in the loss landscape: channels along which the loss decreases extremely slowly, while the output weights of at least two neurons, $a_i$ and $a_j$, diverge to $\pm$infinity, and their input weight vectors, $\mathbf{w_i}$ and $\mathbf{w_j}$, become equal to each other. At convergence, the two neurons implement a gated linear unit: $a_iσ(\mathbf{w_i} \cdot \mathbf{x}) + a_jσ(\mathbf{w_j} \cdot \mathbf{x}) \rightarrow σ(\mathbf{w} \cdot \mathbf{x}) + (\mathbf{v} \cdot \mathbf{x}) σ'(\mathbf{w} \cdot \mathbf{x})$. Geometrically, these channels to infinity are asymptotically parallel to symmetry-induced lines of critical points. Gradient flow solvers, and related optimization methods like SGD or ADAM, reach the channels with high probability in diverse regression settings, but without careful inspection they look like flat local minima with finite parameter values. Our characterization provides a comprehensive picture of these quasi-flat regions in terms of gradient dynamics, geometry, and functional interpretation. The emergence of gated linear units at the end of the channels highlights a surprising aspect of the computational capabilities of fully connected layers.
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