An official website of the United States government
In this paper we prove that Local (S)GD (or FedAvg) can optimize deep neural networks with Rectified Linear Unit (ReLU) activation function in polynomial time. Despite the established convergence theory of Local SGD on optimizing general smooth functions in communication-efficient distributed optimization, its convergence on non-smooth ReLU networks still eludes full theoretical understanding. The key property used in many Local SGD analysis on smooth function is gradient Lipschitzness, so that the gradient on local models will not drift far away from that on averaged model. However, this decent property does not hold in networks with non-smooth ReLU activation function. We show that, even though ReLU network does not admit gradient Lipschitzness property, the difference between gradients on local models and average model will not change too much, under the dynamics of Local SGD. We validate our theoretical results via extensive experiments. This work is the first to show the convergence of Local SGD on non-smooth functions, and will shed lights on the optimization theory of federated training of deep neural networks.
more »
« less
This content will become publicly available on May 1, 2025
ReLUs Are Sufficient for Learning Implicit Neural Representations
Motivated by the growing theoretical understanding of neural networks that employ the Rectified Linear Unit (ReLU) as their activation function, we revisit the use of ReLU activation functions for learning implicit neural representations (INRs). Inspired by second order B-spline wavelets, we incorporate a set of simple constraints to the ReLU neurons in each layer of a deep neural network (DNN) to remedy the spectral bias. This in turn enables its use for various INR tasks. Empirically, we demonstrate that, contrary to popular belief, one can learn state-of-the-art INRs based on a DNN composed of only ReLU neurons. Next, by leveraging recent theoretical works which characterize the kinds of functions ReLU neural networks learn, we provide a way to quantify the regularity of the learned function. This offers a principled approach to selecting the hyperparameters in INR architectures. We substantiate our claims through experiments in signal representation, super resolution, and computed tomography, demonstrating the versatility and effectiveness of our method. The code for all experiments can be found at https://github.com/joeshenouda/relu-inrs.
more »
« less
- Award ID(s):
- 2023239
- PAR ID:
- 10533083
- Publisher / Repository:
- International Conference on Machine Learning
- Date Published:
- Format(s):
- Medium: X
- Location:
- Vienna, Austria
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A wide variety of activation functions have been proposed for neural networks. The Rectified Linear Unit (ReLU) is especially popular today. There are many practical reasons that motivate the use of the ReLU. This paper provides new theoretical characterizations that support the use of the ReLU, its variants such as the leaky ReLU, as well as other activation functions in the case of univariate, single-hidden layer feedforward neural networks. Our results also explain the importance of commonly used strategies in the design and training of neural networks such as “weight decay” and “path-norm” regularization, and provide a new justification for the use of “skip connections” in network architectures. These new insights are obtained through the lens of spline theory. In particular, we show how neural network training problems are related to infinite-dimensional optimizations posed over Banach spaces of functions whose solutions are well-known to be fractional and polynomial splines, where the particular Banach space (which controls the order of the spline) depends on the choice of activation function.more » « less
-
null (Ed.)The nonlinearity of activation functions used in deep learning models is crucial for the success of predictive models. Several simple nonlinear functions, including Rectified Linear Unit (ReLU) and Leaky-ReLU (L-ReLU) are commonly used in neural networks to impose the nonlinearity. In practice, these functions remarkably enhance the model accuracy. However, there is limited insight into the effects of nonlinearity in neural networks on their performance. Here, we investigate the performance of neural network models as a function of nonlinearity using ReLU and L-ReLU activation functions in the context of different model architectures and data domains. We use entropy as a measurement of the randomness, to quantify the effects of nonlinearity in different architecture shapes on the performance of neural networks. We show that the ReLU nonliearity is a better choice for activation function mostly when the network has sufficient number of parameters. However, we found that the image classification models with transfer learning seem to perform well with L-ReLU in fully connected layers. We show that the entropy of hidden layer outputs in neural networks can fairly represent the fluctuations in information loss as a function of nonlinearity. Furthermore, we investigate the entropy profile of shallow neural networks as a way of representing their hidden layer dynamics.more » « less
-
Pérez, Guillermo A.; Raskin, Jean-François (Ed.)Deep neural networks (DNNs) are increasingly being deployed to perform safety-critical tasks. The opacity of DNNs, which prevents humans from reasoning about them, presents new safety and security challenges. To address these challenges, the verification community has begun developing techniques for rigorously analyzing DNNs, with numerous verification algorithms proposed in recent years. While a significant amount of work has gone into developing these verification algorithms, little work has been devoted to rigorously studying the computability and complexity of the underlying theoretical problems. Here, we seek to contribute to the bridging of this gap. We focus on two kinds of DNNs: those that employ piecewise-linear activation functions (e.g., ReLU), and those that employ piecewise-smooth activation functions (e.g., Sigmoids). We prove the two following theorems: 1. The decidability of verifying DNNs with piecewise-smooth activation functions is equivalent to a well-known, open problem formulated by Tarski; and 2. The DNN verification problem for any quantifier-free linear arithmetic specification can be reduced to the DNN reachability problem, whose approximation is NP-complete. These results answer two fundamental questions about the computability and complexity of DNN verification, and the ways it is affected by the network’s activation functions and error tolerance; and could help guide future efforts in developing DNN verification tools.more » « less
-
null (Ed.)We study how neural networks trained by gradient descent extrapolate, i.e., what they learn outside the support of the training distribution. Previous works report mixed empirical results when extrapolating with neural networks: while feedforward neural networks, a.k.a. multilayer perceptrons (MLPs), do not extrapolate well in certain simple tasks, Graph Neural Networks (GNNs) – structured networks with MLP modules – have shown some success in more complex tasks. Working towards a theoretical explanation, we identify conditions under which MLPs and GNNs extrapolate well. First, we quantify the observation that ReLU MLPs quickly converge to linear functions along any direction from the origin, which implies that ReLU MLPs do not extrapolate most nonlinear functions. But, they can provably learn a linear target function when the training distribution is sufficiently “diverse”. Second, in connection to analyzing the successes and limitations of GNNs, these results suggest a hypothesis for which we provide theoretical and empirical evidence: the success of GNNs in extrapolating algorithmic tasks to new data (e.g., larger graphs or edge weights) relies on encoding task-specific non-linearities in the architecture or features. Our theoretical analysis builds on a connection of over-parameterized networks to the neural tangent kernel. Empirically, our theory holds across different training settings.more » « less
