When training neural networks with low-precision computation, rounding errors often cause stagnation or are detrimental to the convergence of the optimizers; in this paper we study the influence of rounding errors on the convergence of the gradient descent method for problems satisfying the Polyak-Lojasiewicz inequality. Within this context, we show that, in contrast, biased stochastic rounding errors may be beneficial since choosing a proper rounding strategy eliminates the vanishing gradient problem and forces the rounding bias in a descent direction. Furthermore, we obtain a bound on the convergence rate that is stricter than the one achieved by unbiased stochastic rounding. The theoretical analysis is validated by comparing the performances of various rounding strategies when optimizing several examples using low-precision fixed-point number formats.

Injecting noise within gradient descent has several desirable features, such as smoothing and regularizing properties. In this paper, we investigate the effects of injecting noise before computing a gradient step. We demonstrate that small perturbations can induce explicit regularization for simple models based on the L1-norm, group L1-norms, or nuclear norms. However, when applied to overparametrized neural networks with large widths, we show that the same perturbations can cause variance explosion. To overcome this, we propose using independent layer-wise perturbations, which provably allow for explicit regularization without variance explosion. Our empirical results show that these small perturbations lead to improved generalization performance compared to vanilla gradient descent.

In federated learning (FL), the communication constraint between the remote learners and the Parameter Server (PS) is a crucial bottleneck. For this reason, model updates must be compressed so as to minimize the loss in accuracy resulting from the communication constraint. This paper proposes ``\emph{${\bf M}$-magnitude weighted $L_{\bf 2}$ distortion + $\bf 2$ degrees of freedom''} (M22) algorithm, a rate-distortion inspired approach to gradient compression for federated training of deep neural networks (DNNs). In particular, we propose a family of distortion measures between the original gradient and the reconstruction we referred to as ``$M$-magnitude weighted $L_2$'' distortion, and we assume that gradient updates follow an i.i.d. distribution -- generalized normal or Weibull, which have two degrees of freedom. In both the distortion measure and the gradient, there is one free parameter for each that can be fitted as a function of the iteration number. Given a choice of gradient distribution and distortion measure, we design the quantizer minimizing the expected distortion in gradient reconstruction. To measure the gradient compression performance under a communication constraint, we define the \emph{per-bit accuracy} as the optimal improvement in accuracy that one bit of communication brings to the centralized model over the training period. Using this performance measure, we systematically benchmark the choice of gradient distribution and distortion measure. We provide substantial insights on the role of these choices and argue that significant performance improvements can be attained using such a rate-distortion inspired compressor.

Batch gradient descent is a type of gradient descent that update the parameters after forward and backward pass through the entire dataset. It is called "batch" gradient descent because it uses the entire dataset to compute the gradient of the loss function at each iteration. Where n is the number of samples in the entire dataset. One of the main disadvantages of batch gradient descent is that it can be computationally expensive when the dataset is very large, as it requires a forward and backward pass through the entire dataset at each iteration. In addition, if the dataset is noisy or has a lot of outliers, the loss function can oscillate and never converge to a minimum. In this case, a more sophisticated optimization algorithm such as stochastic gradient descent or mini-batch gradient descent may be more appropriate.

Abstract: With the fast development of big data, it has been easier than before to learn the optimal decision rule by updating the decision rule recursively and making online decisions. We study the online statistical inference of model parameters in a contextual bandit framework of sequential decision-making. We propose a general framework for online and adaptive data collection environment that can update decision rules via weighted stochastic gradient descent. We allow different weighting schemes of the stochastic gradient and establish the asymptotic normality of the parameter estimator. Our proposed estimator significantly improves the asymptotic efficiency over the previous averaged SGD approach via inverse probability weights.

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. We prove that the total sample complexity of SCRN in achieving $\epsilon$-global optimum is $\mathcal{O}(\epsilon^{-7/(2\alpha)+1})$ for $1\le\alpha< 3/2$ and $\mathcal{\tilde{O}}(\epsilon^{-2/(\alpha)})$ for $3/2\le\alpha\le 2$. 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. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.

In federated learning (FL), it is commonly assumed that all data are placed at clients in the beginning of machine learning (ML) optimization (i.e., offline learning). However, in many real-world applications, it is expected to proceed in an online fashion. To this end, online FL (OFL) has been introduced, which aims at learning a sequence of global models from decentralized streaming data such that the so-called cumulative regret is minimized. Combining online gradient descent and model averaging, in this framework, FedOGD is constructed as the counterpart of FedSGD in FL. While it can enjoy an optimal sublinear regret, FedOGD suffers from heavy communication costs. In this paper, we present a communication-efficient method (named OFedIQ) by means of intermittent transmission (enabled by client subsampling and periodic transmission) and quantization. For the first time, we derive the regret bound that captures the impact of data-heterogeneity and the communication-efficient techniques. Through this, we efficiently optimize the parameters of OFedIQ such as sampling rate, transmission period, and quantization levels. Also, it is proved that the optimized OFedIQ can asymptotically achieve the performance of FedOGD while reducing the communication costs by 99%. Via experiments with real datasets, we demonstrate the effectiveness of the optimized OFedIQ.

Abstract: In this letter, we focus on the problem of millimeter-Wave channels estimation in massive MIMO communication systems. Inspired by the sparsity of mmWave MIMO channel in the angular domain, we formulate the estimation problem as a sparse signal recovery problem. We propose a deep learning based trainable proximal gradient descent network (TPGD-Net) for mmWave channel estimation. Specifically, we unfold the iterative proximal gradient descent (PGD) algorithm into a layer-wise network. Different from the PGD algorithm, the gradient descent step size in TPGD-Net is set as a trainable parameter.

The Stochastic Gradient Langevin Dynamics (SGLD), first introduced by Welling and Teh [25], has attracted a lot of attention in various areas [18, 26, 4]. The SGLD algorithm and its variants have fantastic performance when dealing with many practical sampling or optimization tasks. Recent decades have witnessed great development of theoretical research for SGLD, where most researchers focus on its discretization error, namely, the "distance" between the SGLD algorithm and the corresponding Langevin diffusion in terms of the time step (or learning rate) η [12, 18, 26, 16]. The SGLD itself can be regarded as a stochastic process and the ergodicity is also of great importance. So far, the justification of the geometric ergodicity of SGLD mostly relies on the strong convexity conditions, namely, the strong log-concaveness of the target distribution. In [4], under strong convexity settings, the authors considered the Synchronous coupling and established the geometric ergodicity of SGLD and some other numerical schemes in terms of Wasserstein-2 distance. However, the strong convexity assumption seems to limit the applicability of the result, and the ergodicity of the SGLD algorithm in a general setting and the existence of an invariant measure are still unclear. In our work, we aim to study the geometric ergodicity under locally nonconvex setting in this paper. The main technique we apply is reflection coupling [8], which was originally designed earlier to study the contraction property of many continuous SDEs.

Motivated by recent applications requiring differential privacy over adaptive streams, we investigate the question of optimal instantiations of the matrix mechanism in this setting. We prove fundamental theoretical results on the applicability of matrix factorizations to adaptive streams, and provide a parameter-free fixed-point algorithm for computing optimal factorizations. We instantiate this framework with respect to concrete matrices which arise naturally in machine learning, and train user-level differentially private models with the resulting optimal mechanisms, yielding significant improvements in a notable problem in federated learning with user-level differential privacy.