Xi-Lin Li

Xi-Lin Li

This paper proposes a family of online second order methods for possibly non-convex stochastic optimizations based on the theory of preconditioned stochastic gradient descent (PSGD), which can be regarded as an enhance stochastic Newton method with the ability to handle gradient noise and non-convexity simultaneously. We have improved the implementations of the original PSGD in several ways, e.g., new forms of preconditioners, more accurate Hessian vector product calculations, and better numerical stability with vanishing or ill-conditioned Hessian, etc.. We also have unrevealed the relationship between feature normalization and PSGD with Kronecker product preconditioners, which explains the excellent performance of Kronecker product preconditioners in deep neural network learning. A software package (https://github.com/lixilinx/psgd_tf) implemented in Tensorflow is provided to compare variations of stochastic gradient descent (SGD) and PSGD with five different preconditioners on a wide range of benchmark problems with commonly used neural network architectures, e.g., convolutional and recurrent neural networks. Experimental results clearly demonstrate the advantages of PSGD in terms of generalization performance and convergence speed.

Omead Pooladzandi, Xi-Lin Li

We present a novel approach to accelerate stochastic gradient descent (SGD) by utilizing curvature information obtained from Hessian-vector products or finite differences of parameters and gradients, similar to the BFGS algorithm. Our approach involves two preconditioners: a matrix-free preconditioner and a low-rank approximation preconditioner. We update both preconditioners online using a criterion that is robust to stochastic gradient noise and does not require line search or damping. To preserve the corresponding symmetry or invariance, our preconditioners are constrained to certain connected Lie groups. The Lie group's equivariance property simplifies the preconditioner fitting process, while its invariance property eliminates the need for damping, which is commonly required in second-order optimizers. As a result, the learning rate for parameter updating and the step size for preconditioner fitting are naturally normalized, and their default values work well in most scenarios. Our proposed approach offers a promising direction for improving the convergence of SGD with low computational overhead. We demonstrate that Preconditioned SGD (PSGD) outperforms SoTA on Vision, NLP, and RL tasks across multiple modern deep-learning architectures. We have provided code for reproducing toy and large scale experiments in this paper.

Xi-Lin Li

This report investigates the fitting of the Hessian or its inverse for stochastic optimizations using a Hessian fitting criterion derived from the preconditioned stochastic gradient descent (PSGD) method. This criterion is closely related to many widely used second-order and adaptive gradient optimization methods, including BFGS, the Gauss-Newton algorithm, natural gradient descent, and AdaGrad. Our analyses reveal the efficiency and reliability differences of a broad range of preconditioner fitting methods, ranging from closed-form to iterative approaches, using Hessian-vector products or stochastic gradients only, with Hessian fittings across various geometric settings (the Euclidean space, the manifold of symmetric positive definite (SPD) matrices, and a variety of Lie groups). The most intriguing finding is that the Hessian fitting problem is strongly convex under mild conditions in certain general Lie groups. This result turns the Hessian fitting into a well-behaved Lie group optimization problem and facilitates the design of highly efficient and elegant Lie group sparse preconditioner fitting methods for large-scale stochastic optimizations.

Xi-Lin Li

This paper studies the density priors for independent vector analysis (IVA) with convolutive speech mixture separation as the exemplary application. Most existing source priors for IVA are too simplified to capture the fine structures of speeches. Here, we first time show that it is possible to efficiently estimate the derivative of speech density with universal approximators like deep neural networks (DNN) by optimizing certain proxy separation related performance indices. Experimental results suggest that the resultant neural network density priors consistently outperform previous ones in convergence speed for online implementation and signal-to-interference ratio (SIR) for batch implementation.

Xi-Lin Li

We report a triangular neural network implementation of neural autoregressive flow (NAF). Unlike many universal autoregressive density models, our design is highly modular, parameter economy, computationally efficient, and applicable to density estimation of data with high dimensions. It achieves state-of-the-art bits-per-dimension indices on MNIST and CIFAR-10 (about 1.1 and 3.7, respectively) in the category of general-purpose density estimators.

Xi-Lin Li

We study a multiclass multiple instance learning (MIL) problem where the labels only suggest whether any instance of a class exists or does not exist in a training sample or example. No further information, e.g., the number of instances of each class, relative locations or orders of all instances in a training sample, is exploited. Such a weak supervision learning problem can be exactly solved by maximizing the model likelihood fitting given observations, and finds applications to tasks like multiple object detection and localization for image understanding. We discuss its relationship to the classic classification problem, the traditional MIL, and connectionist temporal classification (CTC). We use image recognition as the example task to develop our method, although it is applicable to data with higher or lower dimensions without much modification. Experimental results show that our method can be used to learn all convolutional neural networks for solving real-world multiple object detection and localization tasks with weak annotations, e.g., transcribing house number sequences from the Google street view imagery dataset.

Xi-Lin Li

We study two types of preconditioners and preconditioned stochastic gradient descent (SGD) methods in a unified framework. We call the first one the Newton type due to its close relationship to the Newton method, and the second one the Fisher type as its preconditioner is closely related to the inverse of Fisher information matrix. Both preconditioners can be derived from one framework, and efficiently estimated on any matrix Lie groups designated by the user using natural or relative gradient descent minimizing certain preconditioner estimation criteria. Many existing preconditioners and methods, e.g., RMSProp, Adam, KFAC, equilibrated SGD, batch normalization, etc., are special cases of or closely related to either the Newton type or the Fisher type ones. Experimental results on relatively large scale machine learning problems are reported for performance study.

Xi-Lin Li

This paper studies the performance of a recently proposed preconditioned stochastic gradient descent (PSGD) algorithm on recurrent neural network (RNN) training. PSGD adaptively estimates a preconditioner to accelerate gradient descent, and is designed to be simple, general and easy to use, as stochastic gradient descent (SGD). RNNs, especially the ones requiring extremely long term memories, are difficult to train. We have tested PSGD on a set of synthetic pathological RNN learning problems and the real world MNIST handwritten digit recognition task. Experimental results suggest that PSGD is able to achieve highly competitive performance without using any trick like preprocessing, pretraining or parameter tweaking.

Xi-Lin Li

Stochastic gradient descent (SGD) still is the workhorse for many practical problems. However, it converges slow, and can be difficult to tune. It is possible to precondition SGD to accelerate its convergence remarkably. But many attempts in this direction either aim at solving specialized problems, or result in significantly more complicated methods than SGD. This paper proposes a new method to estimate a preconditioner such that the amplitudes of perturbations of preconditioned stochastic gradient match that of the perturbations of parameters to be optimized in a way comparable to Newton method for deterministic optimization. Unlike the preconditioners based on secant equation fitting as done in deterministic quasi-Newton methods, which assume positive definite Hessian and approximate its inverse, the new preconditioner works equally well for both convex and non-convex optimizations with exact or noisy gradients. When stochastic gradient is used, it can naturally damp the gradient noise to stabilize SGD. Efficient preconditioner estimation methods are developed, and with reasonable simplifications, they are applicable to large scaled problems. Experimental results demonstrate that equipped with the new preconditioner, without any tuning effort, preconditioned SGD can efficiently solve many challenging problems like the training of a deep neural network or a recurrent neural network requiring extremely long term memories.