Align Forward, Adapt Backward: Closing the Discretization Gap in Logic Gate Networks

In neural network models, soft mixtures of fixed candidate components (e.g., logic gates and sub-networks) are often used during training for stable optimization, while hard selection is typically used at inference. This raises questions about training-inference mismatch. We analyze this gap by separating forward-pass computation (hard selection vs. soft mixture) from stochasticity (with vs. without Gumbel noise). Using logic gate networks as a testbed, we observe distinct behaviors across four methods: Hard-ST achieves zero selection gap by construction; Gumbel-ST achieves near-zero gap when training succeeds but suffers accuracy collapse at low temperatures; Soft-Mix achieves small gap only at low temperature via weight concentration; and Soft-Gumbel exhibits large gaps despite Gumbel noise, confirming that noise alone does not reduce the gap. We propose CAGE (Confidence-Adaptive Gradient Estimation) to maintain gradient flow while preserving forward alignment. On logic gate networks, Hard-ST and Gumbel-ST with CAGE both achieve over 98% accuracy on MNIST and ${\sim}58\%$ on CIFAR-10; Hard-ST with CAGE further maintains zero selection gap across all temperatures by construction, while Gumbel-ST without CAGE suffers a ${\sim}47$-point accuracy collapse.