Revisiting Gradient Clipping: Stochastic bias and tight convergence guarantees
This work addresses the theoretical limitations of gradient clipping for researchers and practitioners in machine learning, providing tight convergence guarantees that clarify its behavior in both deterministic and stochastic settings, which is incremental but precise.
The paper tackles the convergence guarantees of gradient clipping in stochastic gradient descent, showing that for deterministic gradients, clipping only affects higher-order terms, while in the stochastic setting, convergence to the true optimum cannot be guaranteed under standard noise assumptions, with matching upper and lower bounds provided.
Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value $c >0$. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of $c$ and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds $c$ and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.