Neural loss landscapes contain flat channels to infinity along which gradient flow leads pairs of neurons to implement gated linear units.
No bad local minima: Data independent training error guarantees for multilayer neural networks
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abstract
We use smoothed analysis techniques to provide guarantees on the training loss of Multilayer Neural Networks (MNNs) at differentiable local minima. Specifically, we examine MNNs with piecewise linear activation functions, quadratic loss and a single output, under mild over-parametrization. We prove that for a MNN with one hidden layer, the training error is zero at every differentiable local minimum, for almost every dataset and dropout-like noise realization. We then extend these results to the case of more than one hidden layer. Our theoretical guarantees assume essentially nothing on the training data, and are verified numerically. These results suggest why the highly non-convex loss of such MNNs can be easily optimized using local updates (e.g., stochastic gradient descent), as observed empirically.
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cs.LG 5representative citing papers
Permutation symmetries generate permutation saddles and equal-loss valleys linking equivalent global minima, yielding a lower bound on symmetry-induced critical points.
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
Presents an active-sampling method that approximates the weight subspace from Hessian finite differences, recovers the rank-1 tensors by robust nonlinear programming, and attributes layers with gradient descent, yielding stable recovery under a-posteriori verifiable conditions.
Large-batch methods converge to sharp minima causing a generalization gap, while small-batch methods reach flat minima due to inherent gradient noise.
citing papers explorer
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Flat Channels to Infinity in Neural Loss Landscapes
Neural loss landscapes contain flat channels to infinity along which gradient flow leads pairs of neurons to implement gated linear units.
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Weight-space symmetry in deep networks gives rise to permutation saddles, connected by equal-loss valleys across the loss landscape
Permutation symmetries generate permutation saddles and equal-loss valleys linking equivalent global minima, yielding a lower bound on symmetry-induced critical points.
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Self-Play Fine-Tuning Converts Weak Language Models to Strong Language Models
SPIN lets weak LLMs become strong by self-generating training data from previous model versions and training to prefer human-annotated responses over its own outputs, outperforming DPO even with extra GPT-4 data on benchmarks.
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Robust and Resource Efficient Identification of Two Hidden Layer Neural Networks
Presents an active-sampling method that approximates the weight subspace from Hessian finite differences, recovers the rank-1 tensors by robust nonlinear programming, and attributes layers with gradient descent, yielding stable recovery under a-posteriori verifiable conditions.
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On Large-Batch Training for Deep Learning: Generalization Gap and Sharp Minima
Large-batch methods converge to sharp minima causing a generalization gap, while small-batch methods reach flat minima due to inherent gradient noise.