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Gradient dynamics of single-neuron autoencoders on orthogonal data
Nikhil Ghosh · Spencer Frei · Wooseok Ha · Bin Yu

In this work we investigate the dynamics of (stochastic) gradient descent when training a single-neuron ReLU autoencoder on orthogonal inputs. We show that for this non-convex problem there exists a manifold of global minima all with the same maximum Hessian eigenvalue and that gradient descent reaches a particular global minimum when initialized randomly. Interestingly, which minimum is reached depends heavily on the batch-size. For full batch gradient descent, the directions of the neuron that are initially positively correlated with the data are merely rescaled uniformly, hence in high-dimensions the learned neuron is a near uniform mixture of these directions. On the other hand, with batch-size one the neuron exactly aligns with a single such direction, showing that when using a small batch-size a qualitatively different type of feature selection" occurs.

#### Author Information

##### Bin Yu (UC Berkeley)

Bin Yu is Chancellor’s Professor in the Departments of Statistics and of Electrical Engineering & Computer Sciences at the University of California at Berkeley and a former chair of Statistics at UC Berkeley. Her research focuses on practice, algorithm, and theory of statistical machine learning and causal inference. Her group is engaged in interdisciplinary research with scientists from genomics, neuroscience, and precision medicine. In order to augment empirical evidence for decision-making, they are investigating methods/algorithms (and associated statistical inference problems) such as dictionary learning, non-negative matrix factorization (NMF), EM and deep learning (CNNs and LSTMs), and heterogeneous effect estimation in randomized experiments (X-learner). Their recent algorithms include staNMF for unsupervised learning, iterative Random Forests (iRF) and signed iRF (s-iRF) for discovering predictive and stable high-order interactions in supervised learning, contextual decomposition (CD) and aggregated contextual decomposition (ACD) for phrase or patch importance extraction from an LSTM or a CNN. She is a member of the U.S. National Academy of Sciences and Fellow of the American Academy of Arts and Sciences. She was a Guggenheim Fellow in 2006, and the Tukey Memorial Lecturer of the Bernoulli Society in 2012. She was President of IMS (Institute of Mathematical Statistics) in 2013-2014 and the Rietz Lecturer of IMS in 2016. She received the E. L. Scott Award from COPSS (Committee of Presidents of Statistical Societies) in 2018. Moreover, Yu was a founding co-director of the Microsoft Research Asia (MSR) Lab at Peking Univeristy and is a member of the scientific advisory board at the UK Alan Turning Institute for data science and AI.