Adaptive Momentum for Neural Network Optimization

In this thesis, we develop a novel and efficient algorithm for optimizing neural networks inspired by a recently proposed geodesic optimization algorithm. Our algorithm, which we call Stochastic Geodesic Optimization (SGeO), utilizes an adaptive coefficient on top of Polyaks Heavy Ball method effectively controlling the amount of weight put on the previous update to the parameters based on the change of direction in the optimization path. Experimental results on strongly convex functions with Lipschitz gradients and deep Autoencoder benchmarks show that SGeO reaches lower errors than established first-order methods and competes well with lower or similar errors to a recent second-order method called K-FAC (Kronecker-Factored Approximate Curvature). We also incorporate Nesterov style lookahead gradient into our algorithm (SGeO-N) and observe notable improvements. We believe that our research will open up new directions for high-dimensional neural network optimization where combining the efficiency of first-order methods and the effectiveness of second-order methods proves a promising avenue to explore

High-Order Stochastic Gradient Thermostats for Bayesian Learning of Deep Models

Learning in deep models using Bayesian methods has generated significant attention recently. This is largely because of the feasibility of modern Bayesian methods to yield scalable learning and inference, while maintaining a measure of uncertainty in the model parameters. Stochastic gradient MCMC algorithms (SG-MCMC) are a family of diffusion-based sampling methods for large-scale Bayesian learning. In SG-MCMC, multivariate stochastic gradient thermostats (mSGNHT) augment each parameter of interest, with a momentum and a thermostat variable to maintain stationary distributions as target posterior distributions. As the number of variables in a continuous-time diffusion increases, its numerical approximation error becomes a practical bottleneck, so better use of a numerical integrator is desirable. To this end, we propose use of an efficient symmetric splitting integrator in mSGNHT, instead of the traditional Euler integrator. We demonstrate that the proposed scheme is more accurate, robust, and converges faster. These properties are demonstrated to be desirable in Bayesian deep learning. Extensive experiments on two canonical models and their deep extensions demonstrate that the proposed scheme improves general Bayesian posterior sampling, particularly for deep models.Comment: AAAI 201

Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks

Effective training of deep neural networks suffers from two main issues. The first is that the parameter spaces of these models exhibit pathological curvature. Recent methods address this problem by using adaptive preconditioning for Stochastic Gradient Descent (SGD). These methods improve convergence by adapting to the local geometry of parameter space. A second issue is overfitting, which is typically addressed by early stopping. However, recent work has demonstrated that Bayesian model averaging mitigates this problem. The posterior can be sampled by using Stochastic Gradient Langevin Dynamics (SGLD). However, the rapidly changing curvature renders default SGLD methods inefficient. Here, we propose combining adaptive preconditioners with SGLD. In support of this idea, we give theoretical properties on asymptotic convergence and predictive risk. We also provide empirical results for Logistic Regression, Feedforward Neural Nets, and Convolutional Neural Nets, demonstrating that our preconditioned SGLD method gives state-of-the-art performance on these models.Comment: AAAI 201

Second-Order Optimization for Non-Convex Machine Learning: An Empirical Study

While first-order optimization methods such as stochastic gradient descent (SGD) are popular in machine learning (ML), they come with well-known deficiencies, including relatively-slow convergence, sensitivity to the settings of hyper-parameters such as learning rate, stagnation at high training errors, and difficulty in escaping flat regions and saddle points. These issues are particularly acute in highly non-convex settings such as those arising in neural networks. Motivated by this, there has been recent interest in second-order methods that aim to alleviate these shortcomings by capturing curvature information. In this paper, we report detailed empirical evaluations of a class of Newton-type methods, namely sub-sampled variants of trust region (TR) and adaptive regularization with cubics (ARC) algorithms, for non-convex ML problems. In doing so, we demonstrate that these methods not only can be computationally competitive with hand-tuned SGD with momentum, obtaining comparable or better generalization performance, but also they are highly robust to hyper-parameter settings. Further, in contrast to SGD with momentum, we show that the manner in which these Newton-type methods employ curvature information allows them to seamlessly escape flat regions and saddle points.Comment: 21 pages, 11 figures. Restructure the paper and add experiment