Analyzing Monotonic Linear Interpolation in Neural Network Loss Landscapes

Abstract

Linear interpolation between initial neural network parameters and converged parameters after training with stochastic gradient descent (SGD) typically leads to a monotonic decrease in the training objective. This Monotonic Linear Interpolation (MLI) property, first observed by Goodfellow et al. (2014), persists in spite of the non-convex objectives and highly non-linear training dynamics of neural networks. Extending this work, we evaluate several hypotheses for this property that, to our knowledge, have not yet been explored. Using tools from differential geometry, we draw connections between the interpolated paths in function space and the monotonicity of the network -providing sufficient conditions for the MLI property under mean squared error. While the MLI property holds under various settings (e.g. network architectures and learning problems), we show in practice that networks violating the MLI property can be produced systematically, by encouraging the weights to move far from initialization. The MLI property raises important questions about the loss landscape geometry of neural networks and highlights the need to further study their global properties.

Insights into machine learning models from chemical physics: an energy landscapes approach (EL for ML)

“…These results received considerable attention in the eld, and the matter is likely to be more complicated. Lucas et al 27 are able to create counterexamples to the MLI property and others have observed different results to Goodfellow et al 26 when revisiting the work on more modern architectures and data sets. 28 Further doubt that 'Machine learning may just be simple' is cast in ref.…”

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“…In [7] it was observed that the loss decreases monotonically over the line between the initialization and the final convergence points. This observation was shown not to hold when using larger learning rates [21]. Swirszcz et al [31] also show that it is possible to create datasets which lead to a landscape containing local minima.…”