We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected 0 - 1 loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a \textit{neural tangent random feature} (NTRF) model. For data distributions that can be classified by NTRF model with sufficiently small error, our result yields a generalization error bound in the order of \tilde{\mathcal{O}}(n {-1/2}) that is independent of the network width. Our result is more general and sharper than many existing generalization error bounds for over-parameterized neural networks. In addition, we establish a strong connection between our generalization error bound and the neural tangent kernel (NTK) proposed in recent work.

In the light of the fact that the stochastic gradient descent (SGD) often finds a flat minimum valley in the training loss, we propose a novel directional pruning method which searches for a sparse minimizer in or close to that flat region. The proposed pruning method does not require retraining or the expert knowledge on the sparsity level. To overcome the computational formidability of estimating the flat directions, we propose to use a carefully tuned \ell_1 proximal gradient algorithm which can provably achieve the directional pruning with a small learning rate after sufficient training. The empirical results demonstrate the promising results of our solution in highly sparse regime (92% sparsity) among many existing pruning methods on the ResNet50 with the ImageNet, while using only a slightly higher wall time and memory footprint than the SGD. Using the VGG16 and the wide ResNet 28x10 on the CIFAR-10 and CIFAR-100, we demonstrate that our solution reaches the same minima valley as the SGD, and the minima found by our solution and the SGD do not deviate in directions that impact the training loss.

Deep neural networks achieve stellar generalisation even when they have enough parameters to easily fit all their training data. We study this phenomenon by analysing the dynamics and the performance of over-parameterised two-layer neural networks in the teacher-student setup, where one network, the student, is trained on data generated by another network, called the teacher. We show how the dynamics of stochastic gradient descent (SGD) is captured by a set of differential equations and prove that this description is asymptotically exact in the limit of large inputs. Using this framework, we calculate the final generalisation error of student networks that have more parameters than their teachers. We find that the final generalisation error of the student increases with network size when training only the first layer, but stays constant or even decreases with size when training both layers.

Partially monotone regression is a regression analysis in which the target values are monotonically increasing with respect to a subset of input features. The TensorFlow Lattice library is one of the standard machine learning libraries for partially monotone regression. It consists of several neural network layers, and its core component is the lattice layer. One of the problems of the lattice layer is that it requires the projected gradient descent algorithm with many constraints to train it. Another problem is that it cannot receive a high-dimensional input vector due to the memory consumption. We propose a novel neural network layer, the hierarchical lattice layer (HLL), as an extension of the lattice layer so that we can use a standard stochastic gradient descent algorithm to train HLL while satisfying monotonicity constraints and so that it can receive a high-dimensional input vector.

We introduce new algorithms and convergence guarantees for privacy-preserving non-convex Empirical Risk Minimization (ERM) on smooth d -dimensional objectives. We develop an improved sensitivity analysis of stochastic gradient descent on smooth objectives that exploits the recurrence of examples in different epochs.

Stochastic Gradient Hamiltonian Monte Carlo (SGHMC) algorithms have received increasing attention in both theory and practice. In this paper, we propose a Stochastic Recursive Variance-Reduced gradient HMC (SRVR-HMC) algorithm. It makes use of a semi-stochastic gradient estimator that recursively accumulates the gradient information to reduce the variance of the stochastic gradient. We provide a convergence analysis of SRVR-HMC for sampling from a class of non-log-concave distributions and show that SRVR-HMC converges faster than all existing HMC-type algorithms based on underdamped Langevin dynamics. Thorough experiments on synthetic and real-world datasets validate our theory and demonstrate the superiority of SRVR-HMC.

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with 1\le\alpha\le2 which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the average sample complexity of SCRN can be reduced to {\mathcal{O}}(\epsilon {-2}) for \alpha 1 using a variance reduction method with time-varying batch sizes.

The Federated Averaging (FedAvg) algorithm, which consists of alternating between a few local stochastic gradient updates at client nodes, followed by a model averaging update at the server, is perhaps the most commonly used method in Federated Learning. Notwithstanding its simplicity, several empirical studies have illustrated that the model output by FedAvg leads to a model that generalizes well to new unseen tasks after a few fine-tuning steps. This surprising performance of such a simple method, however, is not fully understood from a theoretical point of view. In this paper, we formally investigate this phenomenon in the multi-task linear regression setting. We show that the reason behind the generalizability of the FedAvg output is FedAvg's power in learning the common data representation among the clients' tasks, by leveraging the diversity among client data distributions via multiple local updates between communication rounds.

Deep Neural Networks and Reinforcement Learning methods have empirically shown great promise in tackling challenging combinatorial problems. In those methods a deep neural network is used as a solution generator which is then trained by gradient-based methods (e.g., policy gradient) to successively obtain better solution distributions.In this work we introduce a novel theoretical framework for analyzing the effectiveness of such methods. We ask whether there exist generative models that (i) are expressive enough to generate approximately optimal solutions; (ii) have a tractable, i.e, polynomial in the size of the input, number of parameters; (iii) their optimization landscape is benign in the sense that it does not contain sub-optimal stationary points. Our main contribution is a positive answer to this question. As a byproduct of our analysis we introduce a novel regularization process over vanilla gradient descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.

Stochastic gradient algorithm is a key ingredient of many machine learning methods, particularly appropriate for large-scale learning. However, a major caveat of large data is their incompleteness. We propose an averaged stochastic gradient algorithm handling missing values in linear models. This approach has the merit to be free from the need of any data distribution modeling and to account for heterogeneous missing proportion. In both streaming and finite-sample settings, we prove that this algorithm achieves convergence rate of \mathcal{O}(\frac{1}{n}) at the iteration n, the same as without missing values. We show the convergence behavior and the relevance of the algorithm not only on synthetic data but also on real data sets, including those collected from medical register.