In this paper, we propose Stochastic Block-ADMM as an approach to train deep neural networks in batch and online settings. Our method works by splitting neural networks into an arbitrary number of blocks and utilizes auxiliary variables to connect these blocks while optimizing with stochastic gradient descent. This allows training deep networks with non-differentiable constraints where conventional backpropagation is not applicable. An application of this is supervised feature disentangling, where our proposed DeepFacto inserts a non-negative matrix factorization (NMF) layer into the network. Since backpropagation only needs to be performed within each block, our approach alleviates vanishing gradients and provides potentials for parallelization. We prove the convergence of our proposed method and justify its capabilities through experiments in supervised and weakly-supervised settings.

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows that it provably converges at a linear rate independent of the condition number of the ground truth tensor for two canonical problems -- tensor completion and tensor regression -- as soon as the sample size is above the order of $n^{3/2}$ ignoring other dependencies, where $n$ is the dimension of the tensor. This leads to an extremely scalable approach to low-rank tensor estimation compared with prior art, which suffers from at least one of the following drawbacks: extreme sensitivity to ill-conditioning, high per-iteration costs in terms of memory and computation, or poor sample complexity guarantees. To the best of our knowledge, ScaledGD is the first algorithm that achieves near-optimal statistical and computational complexities simultaneously for low-rank tensor completion with the Tucker decomposition. Our algorithm highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the underlying symmetry in low-rank tensor factorization.

Many fundamental machine learning tasks can be formulated as a problem of learning with vector-valued functions, where we learn multiple scalar-valued functions together. Although there is some generalization analysis on different specific algorithms under the empirical risk minimization principle, a unifying analysis of vector-valued learning under a regularization framework is still lacking. In this paper, we initiate the generalization analysis of regularized vector-valued learning algorithms by presenting bounds with a mild dependency on the output dimension and a fast rate on the sample size. Our discussions relax the existing assumptions on the restrictive constraint of hypothesis spaces, smoothness of loss functions and low-noise condition. To understand the interaction between optimization and learning, we further use our results to derive the first generalization bounds for stochastic gradient descent with vector-valued functions. We apply our general results to multi-class classification and multi-label classification, which yield the first bounds with a logarithmic dependency on the output dimension for extreme multi-label classification with the Frobenius regularization. As a byproduct, we derive a Rademacher complexity bound for loss function classes defined in terms of a general strongly convex function.

Many reinforcement learning algorithms rely on value estimation. However, the most widely used algorithms -- namely temporal difference algorithms -- can diverge under both off-policy sampling and nonlinear function approximation. Many algorithms have been developed for off-policy value estimation which are sound under linear function approximation, based on the linear mean-squared projected Bellman error (PBE). Extending these methods to the non-linear case has been largely unsuccessful. Recently, several methods have been introduced that approximate a different objective, called the mean-squared Bellman error (BE), which naturally facilities nonlinear approximation. In this work, we build on these insights and introduce a new generalized PBE, that extends the linear PBE to the nonlinear setting. We show how this generalized objective unifies previous work, including previous theory, and obtain new bounds for the value error of the solutions of the generalized objective. We derive an easy-to-use, but sound, algorithm to minimize the generalized objective which is more stable across runs, is less sensitive to hyperparameters, and performs favorably across four control domains with neural network function approximation.

Federated averaging (FedAvg) is a communication efficient algorithm for the distributed training with an enormous number of clients. In FedAvg, clients keep their data locally for privacy protection; a central parameter server is used to communicate between clients. This central server distributes the parameters to each client and collects the updated parameters from clients. FedAvg is mostly studied in centralized fashions, which requires massive communication between server and clients in each communication. Moreover, attacking the central server can break the whole system's privacy. In this paper, we study the decentralized FedAvg with momentum (DFedAvgM), which is implemented on clients that are connected by an undirected graph. In DFedAvgM, all clients perform stochastic gradient descent with momentum and communicate with their neighbors only. To further reduce the communication cost, we also consider the quantized DFedAvgM. We prove convergence of the (quantized) DFedAvgM under trivial assumptions; the convergence rate can be improved when the loss function satisfies the P{\L} property. Finally, we numerically verify the efficacy of DFedAvgM.

This paper concerns a convex, stochastic zeroth-order optimization (S-ZOO) problem, where the objective is to minimize the expectation of a cost function and its gradient is not accessible directly. To solve this problem, traditional optimization techniques mostly yield query complexities that grow polynomially with dimensionality, i.e., the number of function evaluations is a polynomial function of the number of decision variables. Consequently, these methods may not perform well in solving massive-dimensional problems arising in many modern applications. Although more recent methods can be provably dimension-insensitive, almost all of them work with arguably more stringent conditions such as everywhere sparse or compressible gradient. Thus, prior to this research, it was unknown whether dimension-insensitive S-ZOO is possible without such conditions. In this paper, we give an affirmative answer to this question by proposing a sparsity-inducing stochastic gradient-free (SI-SGF) algorithm. It is proved to achieve dimension-insensitive query complexity in both convex and strongly convex cases when neither gradient sparsity nor gradient compressibility is satisfied. Our numerical results demonstrate the strong potential of the proposed SI-SGF compared with existing alternatives.

UMAP has supplanted t-SNE as state-of-the-art for visualizing high-dimensional datasets in many disciplines, but the reason for its success is not well understood. In this work, we investigate UMAP's sampling based optimization scheme in detail. We derive UMAP's effective loss function in closed form and find that it differs from the published one. As a consequence, we show that UMAP does not aim to reproduce its theoretically motivated high-dimensional UMAP similarities. Instead, it tries to reproduce similarities that only encode the shared $k$ nearest neighbor graph, thereby challenging the previous understanding of UMAP's effectiveness. Instead, we claim that the key to UMAP's success is its implicit balancing of attraction and repulsion resulting from negative sampling. This balancing in turn facilitates optimization via gradient descent. We corroborate our theoretical findings on toy and single cell RNA sequencing data.

Many machine learning problems optimize an objective that must be measured with noise. The primary method is a first order stochastic gradient descent using one or more Monte Carlo (MC) samples at each step. There are settings where ill-conditioning makes second order methods such as L-BFGS more effective. We study the use of randomized quasi-Monte Carlo (RQMC) sampling for such problems. When MC sampling has a root mean squared error (RMSE) of $O(n^{-1/2})$ then RQMC has an RMSE of $o(n^{-1/2})$ that can be close to $O(n^{-3/2})$ in favorable settings. We prove that improved sampling accuracy translates directly to improved optimization. In our empirical investigations for variational Bayes, using RQMC with stochastic L-BFGS greatly speeds up the optimization, and sometimes finds a better parameter value than MC does.

Modern deep learning models are powerful but brittle: standard stochastic gradient descent (SGD) training of deep neural networks can lead to remarkable performance as measured by the classification accuracy on the test set, but this performance rapidly degrades if the metric is instead adversarially robust accuracy. This brittleness is most apparent for image classification tasks (Szegedy et al., 2014; Goodfellow et al., 2015), where neural networks trained by gradient descent achieve state-of-the-art classification accuracy on a number of benchmark tasks, but where imperceptible (adversarial) perturbations of an image can force the neural network to get nearly all of its predictions incorrect. To formalize the above comment, let us define the robust error of a classifier.

Machine learning is vulnerable to a wide variety of different attacks. It is now well understood that by changing the underlying data distribution, an adversary can poison the model trained with it or introduce backdoors. In this paper we present a novel class of training-time attacks that require no changes to the underlying model dataset or architecture, but instead only change the order in which data are supplied to the model. In particular, an attacker can disrupt the integrity and availability of a model by simply reordering training batches, with no knowledge about either the model or the dataset. Indeed, the attacks presented here are not specific to the model or dataset, but rather target the stochastic nature of modern learning procedures. We extensively evaluate our attacks to find that the adversary can disrupt model training and even introduce backdoors. For integrity we find that the attacker can either stop the model from learning, or poison it to learn behaviours specified by the attacker. For availability we find that a single adversarially-ordered epoch can be enough to slow down model learning, or even to reset all of the learning progress. Such attacks have a long-term impact in that they decrease model performance hundreds of epochs after the attack took place. Reordering is a very powerful adversarial paradigm in that it removes the assumption that an adversary must inject adversarial data points or perturbations to perform training-time attacks. It reminds us that stochastic gradient descent relies on the assumption that data are sampled at random. If this randomness is compromised, then all bets are off.