Provable Training of a ReLU Gate with an Iterative Non-Gradient Algorithm

25 Pages Posted: 10 Mar 2021 Last revised: 4 Apr 2022

See all articles by Sayar Karmakar

Sayar Karmakar

University of Florida

Anirbit Mukherjee

Wharton (UPenn), Department of Statistics

Date Written: January 4, 2021


In this work, we demonstrate provable guarantees on the training of a single ReLU gate in hitherto unexplored regimes. We give a simple iterative stochastic algorithm that can train a ReLU gate in the realizable setting in linear time while using significantly milder conditions on the data distribution than previous such results. Leveraging certain additional moment assumptions, we also show a first-of-its-kind approximate recovery of the true label generating parameters under an (online) data-poisoning attack on the true labels, while training a ReLU gate by the same algorithm. Our guarantee is shown to be nearly optimal in the worst case and its accuracy of recovering the true weight degrades gracefully with increasing probability of attack and its magnitude. For both the realizable and the non-realizable cases as outlined above, our analysis allows for mini-batching and computes how the convergence time scales with the mini-batch size. We corroborate our theorems with simulation results which also bring to light a striking similarity in trajectories between our algorithm and the popular S.G.D. algorithm - for which similar guarantees as here are still unknown.

Keywords: neural nets, non-gradient iterative algorithms, stochastic algorithms, non-smooth non-convex optimization

Suggested Citation

Karmakar, Sayar and Mukherjee, Anirbit, Provable Training of a ReLU Gate with an Iterative Non-Gradient Algorithm (January 4, 2021). Available at SSRN: or

Sayar Karmakar (Contact Author)

University of Florida ( email )

PO Box 117165, 201 Stuzin Hall
Gainesville, FL 32610-0496
United States

Anirbit Mukherjee

Wharton (UPenn), Department of Statistics ( email )

Wharton School
Philadelphia, PA 19104
United States

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