We consider decentralized stochastic optimization with the objective function (e.g. data samples for machine learning task) being distributed over $n$ machines that can only communicate to their neighbors on a fixed communication graph. To reduce the communication bottleneck, the nodes compress (e.g. quantize or sparsify) their model updates. We cover both unbiased and biased compression operators with quality denoted by $\omega \leq 1$ ($\omega=1$ meaning no compression). We (i) propose a novel gossip-based stochastic gradient descent algorithm, CHOCO-SGD, that converges at rate $\mathcal{O}\left(1/(nT) + 1/(T \delta^2 \omega)^2\right)$ for strongly convex objectives, where $T$ denotes the number of iterations and $\delta$ the eigengap of the connectivity matrix. Despite compression quality and network connectivity affecting the higher order terms, the first term in the rate, $\mathcal{O}(1/(nT))$, is the same as for the centralized baseline with exact communication. We (ii) present a novel gossip algorithm, CHOCO-GOSSIP, for the average consensus problem that converges in time $\mathcal{O}(1/(\delta^2\omega) \log (1/\epsilon))$ for accuracy $\epsilon > 0$. This is (up to our knowledge) the first gossip algorithm that supports arbitrary compressed messages for $\omega > 0$ and still exhibits linear convergence. We (iii) show in experiments that both of our algorithms do outperform the respective state-of-the-art baselines and CHOCO-SGD can reduce communication by at least two orders of magnitudes.

Stochastic Variance-Reduced Cubic regularization (SVRC) algorithms have received increasing attention due to its improved gradient/Hessian complexities (i.e., number of queries to stochastic gradient/Hessian oracles) to find local minima for nonconvex finite-sum optimization. However, it is unclear whether existing SVRC algorithms can be further improved. Moreover, the semi-stochastic Hessian estimator adopted in existing SVRC algorithms prevents the use of Hessian-vector product-based fast cubic subproblem solvers, which makes SVRC algorithms computationally intractable for high-dimensional problems. In this paper, we first present a Stochastic Recursive Variance-Reduced Cubic regularization method (SRVRC) using a recursively updated semi-stochastic gradient and Hessian estimators. It enjoys improved gradient and Hessian complexities to find an $(\epsilon, \sqrt{\epsilon})$-approximate local minimum, and outperforms the state-of-the-art SVRC algorithms. Built upon SRVRC, we further propose a Hessian-free SRVRC algorithm, namely SRVRC$_{\text{free}}$, which only requires stochastic gradient and Hessian-vector product computations, and achieves $\tilde O(dn\epsilon^{-2} \land d\epsilon^{-3})$ runtime complexity, where $n$ is the number of component functions in the finite-sum structure, $d$ is the problem dimension, and $\epsilon$ is the optimization precision. This outperforms the best-known runtime complexity $\tilde O(d\epsilon^{-3.5})$ achieved by stochastic cubic regularization algorithm proposed in Tripuraneni et al. 2018.

Adaptive data analysis is frequently criticized for its pessimistic generalization guarantees. The source of these pessimistic bounds is a model that permits arbitrary, possibly adversarial analysts that optimally use information to bias results. While being a central issue in the field, still lacking are notions of natural analysts that allow for more optimistic bounds faithful to the reality that typical analysts aren't adversarial. In this work, we propose notions of natural analysts that smoothly interpolate between the optimal non-adaptive bounds and the best-known adaptive generalization bounds. To accomplish this, we model the analyst's knowledge as evolving according to the rules of an unknown dynamical system that takes in revealed information and outputs new statistical queries to the data. This allows us to restrict the analyst through different natural control-theoretic notions. One such notion corresponds to a recency bias, formalizing an inability to arbitrarily use distant information. Another complementary notion formalizes an anchoring bias, a tendency to weight initial information more strongly. Both notions come with quantitative parameters that smoothly interpolate between the non-adaptive case and the fully adaptive case, allowing for a rich spectrum of intermediate analysts that are neither non-adaptive nor adversarial. Natural not only from a cognitive perspective, we show that our notions also capture standard optimization methods, like gradient descent in various settings. This gives a new interpretation to the fact that gradient descent tends to overfit much less than its adaptive nature might suggest.

State space models (SSMs) provide a flexible framework for modeling complex time series via a latent stochastic process. Inference for nonlinear, non-Gaussian SSMs is often tackled with particle methods that do not scale well to long time series. The challenge is two-fold: not only do computations scale linearly with time, as in the linear case, but particle filters additionally suffer from increasing particle degeneracy with longer series. Stochastic gradient MCMC methods have been developed to scale inference for hidden Markov models (HMMs) and linear SSMs using buffered stochastic gradient estimates to account for temporal dependencies. We extend these stochastic gradient estimators to nonlinear SSMs using particle methods. We present error bounds that account for both buffering error and particle error in the case of nonlinear SSMs that are log-concave in the latent process. We evaluate our proposed particle buffered stochastic gradient using SGMCMC for inference on both long sequential synthetic and minute-resolution financial returns data, demonstrating the importance of this class of methods.

The increasing interest in user privacy is leading to new privacy preserving machine learning paradigms. In the Federated Learning paradigm, a master machine learning model is distributed to user clients, the clients use their locally stored data and model for both inference and calculating model updates. The model updates are sent back and aggregated on the server to update the master model then redistributed to the clients. In this paradigm, the user data never leaves the client, greatly enhancing the user' privacy, in contrast to the traditional paradigm of collecting, storing and processing user data on a backend server beyond the user's control. In this paper we introduce, as far as we are aware, the first federated implementation of a Collaborative Filter. The federated updates to the model are based on a stochastic gradient approach. As a classical case study in machine learning, we explore a personalized recommendation system based on users' implicit feedback and demonstrate the method's applicability to both the MovieLens and an in-house dataset. Empirical validation confirms a collaborative filter can be federated without a loss of accuracy compared to a standard implementation, hence enhancing the user's privacy in a widely used recommender application while maintaining recommender performance.

Deep learning neural networks are trained using the stochastic gradient descent optimization algorithm. The learning rate is a hyperparameter that controls how much to change the model in response to the estimated error each time the model weights are updated. Choosing the learning rate is challenging as a value too small may result in a long training process that could get stuck, whereas a value too large may result in learning a sub-optimal set of weights too fast or an unstable training process. The learning rate may be the most important hyperparameter when configuring your neural network. Therefore it is vital to know how to investigate the effects of the learning rate on model performance and to build an intuition about the dynamics of the learning rate on model behavior. In this tutorial, you will discover the effects of the learning rate, learning rate schedules, and adaptive learning rates on model performance. Deep learning neural networks are trained using the stochastic gradient descent algorithm. Stochastic gradient descent is an optimization algorithm that estimates the error gradient for the current state of the model using examples from the training dataset, then updates the weights of the model using the back-propagation of errors algorithm, referred to as simply backpropagation.

We present ErasureHead, a new approach for distributed gradient descent (GD) that mitigates system delays by employing approximate gradient coding. Gradient coded distributed GD uses redundancy to exactly recover the gradient at each iteration from a subset of compute nodes. ErasureHead instead uses approximate gradient codes to recover an inexact gradient at each iteration, but with higher delay tolerance. Unlike prior work on gradient coding, we provide a performance analysis that combines both delay and convergence guarantees. We establish that down to a small noise floor, ErasureHead converges as quickly as distributed GD and has faster overall runtime under a probabilistic delay model. We conduct extensive experiments on real world datasets and distributed clusters and demonstrate that our method can lead to significant speedups over both standard and gradient coded GD.

This work is a part of ICLR Reproducibility Challenge 2019, we try to reproduce the results in the conference submission PADAM: Closing The Generalization Gap of Adaptive Gradient Methods In Training Deep Neural Networks. Adaptive gradient methods proposed in past demonstrate a degraded generalization performance than the stochastic gradient descent (SGD) with momentum. The authors try to address this problem by designing a new optimization algorithm that bridges the gap between the space of Adaptive Gradient algorithms and SGD with momentum. With this method a new tunable hyperparameter called partially adaptive parameter p is introduced that varies between [0, 0.5]. We build the proposed optimizer and use it to mirror the experiments performed by the authors. We review and comment on the empirical analysis performed by the authors. Finally, we also propose a future direction for further study of Padam. Our code is available at: https://github.com/yashkant/Padam-Tensorflow

We propose a novel diminishing learning rate scheme, coined Decreasing-Trend-Nature (DTN), which allows us to prove fast convergence of the Stochastic Gradient Descent (SGD) algorithm to a first-order stationary point for smooth general convex and some class of nonconvex including neural network applications for classification problems. We are the first to prove that SGD with diminishing learning rate achieves a convergence rate of $\mathcal{O}(1/t)$ for these problems. Our theory applies to neural network applications for classification problems in a straightforward way.

We present a novel method for learning the weights in multinomial logistic regression based on the alternating direction method of multipliers (ADMM). In each iteration, our algorithm decomposes the training into three steps; a linear least-squares problem for the weights, a global variable update involving a separable cross-entropy loss function, and a trivial dual variable update The least-squares problem can be factorized in the off-line phase, and the separability in the global variable update allows for efficient parallelization, leading to faster convergence. We compare our method with stochastic gradient descent for linear classification as well as for transfer learning and show that the proposed ADMM-Softmax leads to improved generalization and convergence.