Provable Training of a ReLU Gate with an Iterative Non-Gradient Algorithm
This work addresses the challenge of reliable training for neural network components, offering incremental improvements in theoretical guarantees for specific scenarios like adversarial robustness.
The paper tackles the problem of provably training a single ReLU gate with a simple iterative stochastic algorithm, achieving linear time convergence in the realizable setting under milder data conditions and demonstrating approximate recovery of true parameters under data-poisoning attacks with graceful degradation in accuracy.
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.