The ability of neural networks to provide `best in class' approximation across a wide range of applications is well-documented. Nevertheless, the powerful expressivity of neural networks comes to naught if one is unable to effectively train (choose) the parameters defining the network. In general, neural networks are trained by gradient descent type optimization methods, or a stochastic variant thereof. In practice, such methods result in the loss function decreases rapidly at the beginning of training but then, after a relatively small number of steps, significantly slow down. The loss may even appear to stagnate over the period of a large number of epochs, only to then suddenly start to decrease fast again for no apparent reason. This so-called plateau phenomenon manifests itself in many learning tasks. The present work aims to identify and quantify the root causes of plateau phenomenon. No assumptions are made on the number of neurons relative to the number of training data, and our results hold for both the lazy and adaptive regimes. The main findings are: plateaux correspond to periods during which activation patterns remain constant, where activation pattern refers to the number of data points that activate a given neuron; quantification of convergence of the gradient flow dynamics; and, characterization of stationary points in terms solutions of local least squares regression lines over subsets of the training data. Based on these conclusions, we propose a new iterative training method, the Active Neuron Least Squares (ANLS), characterised by the explicit adjustment of the activation pattern at each step, which is designed to enable a quick exit from a plateau. Illustrative numerical examples are included throughout.

In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (ie, the size of the hidden layer) $N \to +\infty$. Following a probabilistic approach, we show 'propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any particle to a solution of a mean-field McKean-Vlasov equation in the metric space endowed with the Wasserstein distance. In comparison to previous works on the subject, we consider settings in which the sequence of stepsizes in SGD can potentially depend on the number of neurons and the iterations. We then identify two regimes under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand. We perform various experiments on real datasets to validate our theoretical results, assessing the existence of these two regimes on classification problems and illustrating our convergence results.

We provide a detailed asymptotic study of gradient flow trajectories and their implicit optimization bias when minimizing the exponential loss over "diagonal linear networks". This is the simplest model displaying a transition between "kernel" and non-kernel ("rich" or "active") regimes. We show how the transition is controlled by the relationship between the initialization scale and how accurately we minimize the training loss. Our results indicate that some limit behaviors of gradient descent only kick in at ridiculous training accuracies (well beyond $10^{-100}$). Moreover, the implicit bias at reasonable initialization scales and training accuracies is more complex and not captured by these limits.

Our understanding of learning input-output relationships with neural nets has improved rapidly in recent years, but little is known about the convergence of the underlying representations, even in the simple case of linear autoencoders (LAEs). We show that when trained with proper regularization, LAEs can directly learn the optimal representation -- ordered, axis-aligned principal components. We analyze two such regularization schemes: non-uniform $\ell_2$ regularization and a deterministic variant of nested dropout [Rippel et al, ICML' 2014]. Though both regularization schemes converge to the optimal representation, we show that this convergence is slow due to ill-conditioning that worsens with increasing latent dimension. We show that the inefficiency of learning the optimal representation is not inevitable -- we present a simple modification to the gradient descent update that greatly speeds up convergence empirically.

Stochastic Gradient Descent (SGD) is the key learning algorithm for many machine learning tasks. Because of its computational costs, there is a growing interest in accelerating SGD on HPC resources like GPU clusters. However, the performance of parallel SGD is still bottlenecked by the high communication costs even with a fast connection among the machines. A simple approach to alleviating this problem, used in many existing efforts, is to perform communication every few iterations, using a constant averaging period. In this paper, we show that the optimal averaging period in terms of convergence and communication cost is not a constant, but instead varies over the course of the execution. Specifically, we observe that reducing the variance of model parameters among the computing nodes is critical to the convergence of periodic parameter averaging SGD. Given a fixed communication budget, we show that it is more beneficial to synchronize more frequently in early iterations to reduce the initial large variance and synchronize less frequently in the later phase of the training process. We propose a practical algorithm, named ADaptive Periodic parameter averaging SGD (ADPSGD), to achieve a smaller overall variance of model parameters, and thus better convergence compared with the Constant Periodic parameter averaging SGD (CPSGD). We evaluate our method with several image classification benchmarks and show that our ADPSGD indeed achieves smaller training losses and higher test accuracies with smaller communication compared with CPSGD. Compared with gradient-quantization SGD, we show that our algorithm achieves faster convergence with only half of the communication. Compared with full-communication SGD, our ADPSGD achieves 1:14x to 1:27x speedups with a 100Gbps connection among computing nodes, and the speedups increase to 1:46x ~ 1:95x with a 10Gbps connection.

Horizontal Federated learning (FL) handles multi-client data that share the same set of features, and vertical FL trains a better predictor that combine all the features from different clients. This paper targets solving vertical FL in an asynchronous fashion, and develops a simple FL method. The new method allows each client to run stochastic gradient algorithms without coordination with other clients, so it is suitable for intermittent connectivity of clients. This method further uses a new technique of perturbed local embedding to ensure data privacy and improve communication efficiency. Theoretically, we present the convergence rate and privacy level of our method for strongly convex, nonconvex and even nonsmooth objectives separately. Empirically, we apply our method to FL on various image and healthcare datasets. The results compare favorably to centralized and synchronous FL methods.

In this paper, we introduce a discrete variant of the meta-learning framework. Meta-learning aims at exploiting prior experience and data to improve performance on future tasks. By now, there exist numerous formulations for meta-learning in the continuous domain. Notably, the Model-Agnostic Meta-Learning (MAML) formulation views each task as a continuous optimization problem and based on prior data learns a suitable initialization that can be adapted to new, unseen tasks after a few simple gradient updates. Motivated by this terminology, we propose a novel meta-learning framework in the discrete domain where each task is equivalent to maximizing a set function under a cardinality constraint. Our approach aims at using prior data, i.e., previously visited tasks, to train a proper initial solution set that can be quickly adapted to a new task at a relatively low computational cost. This approach leads to (i) a personalized solution for each individual task, and (ii) significantly reduced computational cost at test time compared to the case where the solution is fully optimized once the new task is revealed. The training procedure is performed by solving a challenging discrete optimization problem for which we present deterministic and randomized algorithms. In the case where the tasks are monotone and submodular, we show strong theoretical guarantees for our proposed methods even though the training objective may not be submodular. We also demonstrate the effectiveness of our framework on two real-world problem instances where we observe that our methods lead to a significant reduction in computational complexity in solving the new tasks while incurring a small performance loss compared to when the tasks are fully optimized.

The existence of noisy labels in the dataset causes significant performance degradation for deep neural networks (DNNs). To address this problem, we propose a Meta Soft Label Generation algorithm called MSLG, which can jointly generate soft labels using meta-learning techniques and learn DNN parameters in an end-to-end fashion. Our approach adapts the meta-learning paradigm to estimate optimal label distribution by checking gradient directions on both noisy training data and noise-free meta-data. In order to iteratively update soft labels, meta-gradient descent step is performed on estimated labels, which would minimize the loss of noise-free meta samples. In each iteration, the base classifier is trained on estimated meta labels. MSLG is model-agnostic and can be added on top of any existing model at hand with ease. We performed extensive experiments on CIFAR10, Clothing1M and Food101N datasets. Results show that our approach outperforms other state-of-the-art methods by a large margin.

Reinforcement learning with function approximation can be unstable and even divergent, especially when combined with off-policy learning and Bellman updates. In deep reinforcement learning, these issues have been dealt with empirically by adapting and regularizing the representation, in particular with auxiliary tasks. This suggests that representation learning may provide a means to guarantee stability. In this paper, we formally show that there are indeed nontrivial state representations under which the canonical TD algorithm is stable, even when learning off-policy. We analyze representation learning schemes that are based on the transition matrix of a policy, such as proto-value functions, along three axes: approximation error, stability, and ease of estimation. In the most general case, we show that a Schur basis provides convergence guarantees, but is difficult to estimate from samples. For a fixed reward function, we find that an orthogonal basis of the corresponding Krylov subspace is an even better choice. We conclude by empirically demonstrating that these stable representations can be learned using stochastic gradient descent, opening the door to improved techniques for representation learning with deep networks.

When training the parameters of a linear dynamical model, the gradient descent algorithm is likely to fail to converge if the squared-error loss is used as the training loss function. Restricting the parameter space to a smaller subset and running the gradient descent algorithm within this subset can allow learning stable dynamical systems, but this strategy does not work for unstable systems. In this work, we look into the dynamics of the gradient descent algorithm and pinpoint what causes the difficulty of learning unstable systems. We show that observations taken at different times from the system to be learned influence the dynamics of the gradient descent algorithm in substantially different degrees. We introduce a time-weighted logarithmic loss function to fix this imbalance and demonstrate its effectiveness in learning unstable systems.