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We analyze the loss landscape of undercomplete linear autoencoders, revealing that finite-stepsize SGD breaks scale symmetries by favoring larger decoder weights on the PCA solution manifold. This directed scale drift, occurring on a slow timescale and analytically tractable, eventually encounters a stability boundary, leading to solutions with sharper Hessians in one sense but not universally. The study illustrates how loss geometry can transform residual gradient noise into directed movement across functionally equivalent solutions.
Undercomplete linear autoencoders exhibit a directional bias in scale symmetry breaking, with SGD consistently favoring larger decoder weights on the PCA manifold.
Neural network loss landscapes have many symmetries, which are preserved by gradient flow but broken by finite-stepsize stochastic gradient descent (SGD). A canonical example of such a symmetry is scale in homogeneous networks: one can scale up the parameters in one layer and down in the next without changing the network output. Previous work has documented cases in which SGD breaks this symmetry in favor of balancing gradient noise or minimizing fluctuations. Here, we show that the solution geometry of undercomplete linear autoencoders instead selects a preferred sign for scale drift: on the PCA solution manifold, SGD favors large decoder weights. This directed scale drift occurs on a slow timescale, and its dynamics admit an analytically-tractable effective description. However, it cannot continue indefinitely: increasing scale eventually drives the dynamics towards a finite-stepsize stability boundary. The resulting solutions are sharper than a balanced baseline in the sense of the maximum eigenvalue of the loss Hessian, but different sharpness measures can move in opposing directions. Thus, undercomplete autoencoders give a concrete illustration of how loss geometry can convert residual gradient noise into directed motion along a manifold of functionally-equivalent solutions.