We study the performance of a wide class of convex optimization-based estimators for recovering a signal from corrupted one-bit measurements in high-dimensions. Our general result predicts sharply the performance of such estimators in the linear asymptotic regime when the measurement vectors have entries IID Gaussian. This includes, as a special case, the previously studied least-squares estimator and various novel results for other popular estimators such as least-absolute deviations, hinge-loss and logistic-loss. Importantly, we exploit the fact that our analysis holds for generic convex loss functions to prove a bound on the best achievable performance across the entire class of estimators. Numerical simulations corroborate our theoretical findings and suggest they are accurate even for relatively small problem dimensions.
Binary networks are extremely efficient as they use only two symbols to define the network: $\{+1,-1\}$. One can make the prior distribution of these symbols a design choice. The recent IR-Net of Qin et al. argues that imposing a Bernoulli distribution with equal priors (equal bit ratios) over the binary weights leads to maximum entropy and thus minimizes information loss. However, prior work cannot precisely control the binary weight distribution during training, and therefore cannot guarantee maximum entropy. Here, we show that quantizing using optimal transport can guarantee any bit ratio, …
The massive size of modern neural networks has motivated substantial recent interest in neural network quantization. We introduce Stochastic Markov Gradient Descent (SMGD), a discrete optimization method applicable to training quantized neural networks. The SMGD algorithm is designed for settings where memory is highly constrained during training. We provide theoretical guarantees of algorithm performance as well as encouraging numerical results.
The population recovery problem is a basic problem in noisy unsupervised learning that has attracted significant research attention in recent years [WY12,DRWY12, MS13, BIMP13, LZ15,DST16]. A number of different variants of this problem have been studied, often under assumptions on the unknown distribution (such as that it has restricted support size). In this work we study the sample complexity and algorithmic complexity of the most general version of the problem, under both bit-flip noise and erasure noise model. We give essentially matching upper and lower sample complexity bounds for both n…
Network quantization is a dominant paradigm of model compression. However, the abrupt changes in quantized weights during training often lead to severe loss fluctuations and result in a sharp loss landscape, making the gradients unstable and thus degrading the performance. Recently, Sharpness-Aware Minimization (SAM) has been proposed to smooth the loss landscape and improve the generalization performance of the models. Nevertheless, directly applying SAM to the quantized models can lead to perturbation mismatch or diminishment issues, resulting in suboptimal performance. In this paper, we pro…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Facebook AI Research（Facebook AI 研究院）
Adaptive gradient methods such as AdaGrad and its variants update the stepsize in stochastic gradient descent on the fly according to the gradients received along the way; such methods have gained widespread use in large-scale optimization for their ability to converge robustly, without the need to fine-tune the stepsize schedule. Yet, the theoretical guarantees to date for AdaGrad are for online and convex optimization. We bridge this gap by providing theoretical guarantees for the convergence of AdaGrad for smooth, nonconvex functions. We show that the norm version of AdaGrad (AdaGrad-Norm) …
We study the asynchronous stochastic gradient descent algorithm for distributed training over $n$ workers which have varying computation and communication frequency over time. In this algorithm, workers compute stochastic gradients in parallel at their own pace and return those to the server without any synchronization. Existing convergence rates of this algorithm for non-convex smooth objectives depend on the maximum gradient delay $τ_{\max}$ and show that an $ε$-stationary point is reached after $\mathcal{O}\!\left(σ^2ε^{-2}+ τ_{\max}ε^{-1}\right)$ iterations, where $σ$ denotes the variance …
Why Adam Can Beat SGD: Second-Moment Normalization Yields Sharper Tails
Department of Electrical and Computer Engineering, The Ohio State University（俄亥俄州立大学电气与计算机工程系）；School of Electrical, Computer and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）
We consider the matrix completion problem where the aim is to esti-mate a large data matrix for which only a relatively small random subset of its entries is observed. Quite popular approaches to matrix completion problem are iterative thresholding methods. In spite of their empirical success, the theoretical guarantees of such iterative thresholding methods are poorly understood. The goal of this paper is to provide strong theo-retical guarantees, similar to those obtained for nuclear-norm penalization methods and one step thresholding methods, for an iterative thresholding algorithm which is…
We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample-splitting. Cruc…
Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these are often guaranteed to output a stationary point of the optimization problem. Oracle inequalities are an important theoretical instrument to asses the statistical performance of an estimator. Oracle results have focused on the theoretical properties of the uncomputable (global) minimum or maximum. In the present work a general framework used for convex optimization problems to derive oracle inequalities for stationary points is extended. A main new ingredient of these oracle inequalities …
