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Towards the Dynamics of a DNN Learning Symbolic Interactions

NeurIPS 2024 · Probabilistic and scientific modeling › Dynamical systems · approach: Latent dynamics models · 1 citations (OpenAlex)

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Proves analytically that a DNN's two-phase learning dynamics of interaction complexity (first removing medium/high-order interactions, then learning increasingly complex ones) follow from modeling training as regression under decreasing parameter noise.

Problem

Zhang et al. empirically observed a two-phase dynamics in how DNNs learn interactions of different complexities during training; this paper aims to mathematically prove that dynamics and explain how generalization power changes with training.

Approach

Rewrites the DNN output on masked inputs as a linear combination of interaction triggering functions, models training as linear regression under parameter noise whose variance grows exponentially with interaction order, and derives an analytic solution for interaction strengths at each noise level, decreasing noise explaining the second phase and large noise the first.

Key results

• The theoretical analytic solution's interaction-strength distribution matches the real distribution across training epochs for AlexNet, VGG, BERT-Tiny/Medium, and DGCNN on the listed datasets.
• The ratio of low-order to high-order interaction norms decreases monotonically as the noise level sigma^2 decreases, verifying the proposition for several n.
• When sigma=0 the analytic solution recovers the converged interactions exactly (w_hat = w*).

Limitations

• Relies on simplifying assumptions, e.g., independent noise terms with variance growing as 2^|T| sigma^2 and a regression toward the converged interactions.
• The noise level sigma^2 at each epoch is fitted to match real distributions rather than derived from training dynamics.
• Theory applies to interactions extracted from the converged DNN as the target, which may include over-fitted patterns.

Experimental setup

Datasets: MNIST CIFAR-10 CUB-200-2011 Tiny-ImageNet SST-2 ShapeNet

Benchmarks: MNIST CIFAR-10 CUB-200-2011 Tiny-ImageNet SST-2 ShapeNet

Metrics: normalized interaction strength by order (real vs theoretica relative strength ratio r^(k) of interaction norms

Builds on: AND-OR interactions Harsanyi interaction lottery ticket hypothesis Zhang et al. two-phase dynamics

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