Decreasing Step Size Research Articles

Training highly multiclass classifiers

Classification problems with thousands or more classes often have a large range of class-confusabilities, and we show that the more-confusable classes add more noise to the empirical loss that is minimized during training. We propose an online solution that reduces the effect of highly confusable classes in training the classifier parameters, and focuses the training on pairs of classes that are easier to differentiate at any given time in the training. We also show that the adagrad method, recently proposed for automatically decreasing step sizes for convex stochastic gradient descent, can also be profitably applied to the nonconvex joint training of supervised dimensionality reduction and linear classifiers as done in Wsabie. Experiments on ImageNet benchmark data sets and proprietary image recognition problems with 15,000 to 97,000 classes show substantial gains in classification accuracy compared to one-vs-all linear SVMs and Wsabie.

Approximate adjoint-based iterative learning control

This paper characterises stochastic convergence properties of adjoint-based (gradient-based) iterative learning control (ILC) applied to systems with load disturbances, when provided only with approximate gradient information and noisy measurements. Specifically, conditions are discussed under which the approximations will result in a scheme which converges to an optimal control input. Both the cases of time-invariant step sizes and cases of decreasing step sizes (as in stochastic approximation) are discussed. These theoretical results are supplemented with an application on a sequencing batch reactor for wastewater treatment plants, where approximate gradient information is available. It is found that for such case adjoint-based ILC outperforms inverse-based ILC and model-free P-type ILC, both in terms of convergence rate and measurement noise tolerance.

Performance of a Distributed Stochastic Approximation Algorithm

In this paper, a distributed stochastic approximation algorithm is studied. Applications of such algorithms include decentralized estimation, optimization, control or computing. The algorithm consists in two steps: a local step, where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size, and a gossip step, where a node computes a local weighted average between its estimates and those of its neighbors. Convergence of the estimates toward a consensus is established under weak assumptions. The approach relies on two main ingredients: the existence of a Lyapunov function for the mean field in the agreement subspace, and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace. A second-order analysis of the algorithm is also performed under the form of a central limit Theorem. The Polyak-averaged version of the algorithm is also considered.

Stochastic Approximations with Constant Step Size and Differential Inclusions

We consider stochastic approximation processes with constant step size whose associated deterministic system is an upper semicontinuous differential inclusion. We prove that over any finite time span, the sample paths of the stochastic process are closely approximated by a solution of the differential inclusion with high probability. We then analyze infinite horizon behavior, showing that if the process is Markov, its stationary measures must become concentrated on the Birkhoff center of the deterministic system. Our results extend those of Benaïm for settings in which the deterministic system is Lipschitz continuous and build on the work of Benaïm, Hofbauer, and Sorin for the case of decreasing step sizes. We apply our results to models of population dynamics in games, obtaining new conclusions about the medium and long run behavior of myopic optimizing agents.

Measurement of geometrically necessary dislocation density with high resolution electron backscatter diffraction: Effects of detector binning and step size

Recent advances using cross-correlation analysis of full resolution high quality electron backscatter diffraction (EBSD) patterns have provided a method for quantitatively mapping the stored dislocation density at high spatial resolution. Larger areas could be mapped with image binning or coarser step sizes. We have studied the effects of image binning and step size on the recovery of GND density. Our results suggest that: (i) the measured lower bound GND density noise floor broadly agrees with Wilkinson and Randman's 2009 prediction, where a decrease in step size or an increase in misorientation uncertainty increases the noise floor; (ii) increasing the step size results in a lower GND density being recovered as some dislocations are now considered as statistically stored dislocations (SSDs); (iii) in deformed samples the average GND density stays relatively constant as the degree of pattern binning is increased up to 8×8. Pattern binning thus provides a means of increasing the data acquisition and analysis rate without unduly degrading the data quality.

Stochastic approximation, cooperative dynamics and supermodular games

This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the nonconvergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of Hirsch [SIAM J. Math. Anal. 16 (1985) 423–439]. In particular, this allows us to extend significantly the main result of Hofbauer and Sandholm [Econometrica 70 (2002) 2265–2294] on the convergence of stochastic fictitious play in supermodular games.

Numerical Stability and Dispersion Analysis of the Precise-Integration Time-Domain Method in Lossy Media

In this paper, both the numerical stability condition and dispersion relation of the precise-integration time-domain (PITD) method in lossy media are presented. It is found that the time step size of the PITD method is limited by both the spatial step size of the PITD method and the ratio of permittivity to conductivity. In numerical dispersion investigations, it is shown that: the numerical loss error of the PITD method is always positive; the numerical phase error of the PITD method can be positive or negative; the numerical loss and phase errors can be made nearly independent of the time step size; and as the spatial step size decreases, the amplitudes of the numerical loss and phase errors decrease. In good conductors, the numerical phase velocity of the PITD method is closer to the physical value as compared with the finite-difference time-domain method. The numerical phase anisotropy of the PITD method can be positive or negative. The numerical anisotropies of the PITD method in the 3-D case are usually larger than those in the 2-D case. There is a conductivity giving zero numerical phase anisotropy. These theoretical observations are confirmed by numerical experiments.

Sign Data Derivative Recovery

Given only the signs of signal plus noise added repetitively or sign data, signal amplitudes can be recovered with minimal variance. However, discrete derivatives of the signal are recovered from sign data with a variance which approaches infinity with decreasing step size and increasing order. For industries such as the seismic industry, which exploits amplitude recovery from sign data, these results place constraints on processing, which includes differentiation of the data. While methods for smoothing noisy data for finite difference calculations are known, sign data requires noisy data. In this paper, we derive the expectation values of continuous and discrete sign data derivatives and we explicitly characterize the variance of discrete sign data derivatives.

Preliminary Empirical Models for Predicting Shrinkage, Part Geometry and Metallurgical Aspects of Ti-6Al-4V Shaped Metal Deposition Builds

Shaped Metal Deposition (SMD) is an additive manufacturing process which creates parts layer by layer by weld depositions. In this work, empirical models that predict part geometry (wall thickness and outer diameter) and some metallurgical aspects (i.e. surface texture, portion of finer Widmanstätten microstructure) for the SMD process were developed. The models are based on an orthogonal fractional factorial design of experiments with four factors at two levels. The factors considered were energy level (a relationship between heat source power and the rate of raw material input.), step size, programmed diameter and travel speed. The models were validated using previous builds; the prediction error for part geometry was under 11%. Several relationships between the factors and responses were identified. Current had a significant effect on wall thickness; thickness increases with increasing current. Programmed diameter had a significant effect on percentage of shrinkage; this decreased with increasing component size. Surface finish decreased with decreasing step size and current.

Adaptive Linear Turbo Equalization Over Doubly Selective Channels

Over the last decade, tremendous gains, leading to near-capacity achieving performance, have been shown for a variety of communication systems through the application of the turbo principle, i.e., the exchange of extrinsic information between constituent algorithms for tasks such as channel decoding, equalization, and multiple-input–multiple-output (MIMO) detection. In this paper, we study the practical application of such an iterative detection and decoding (IDD) framework to underwater acoustic communications. We explore complexity and performance tradeoffs of a variety of turbo equalization (TEQ)-based receiver architectures. First, we elaborate on two popular but suboptimal turbo equalization techniques: a channel-estimate-based minimum mean-square error TEQ (CE-based MMSE-TEQ) and a direct-adaptive TEQ (DA-TEQ). We study the behavior of both TEQ approaches in the presence of channel estimation errors and adaptive filter adjustment errors. We confirm that after a sufficient number of iterations, the performance gap between these two TEQ algorithms becomes small. Next, we demonstrate that an underwater receiver architecture built upon the least mean squares (LMS) DA-TEQ technique can leverage and dramatically improve the performance of the conventional implementation based on the decision-feedback equalizer at a feasible complexity. To maintain performance gains over time-varying channels, the slow convergence speed of the LMS algorithm has been improved via two methods: 1) repeating the weight update for the same set of data with decreasing step size and 2) reducing the dimensionality of the equalizer by capturing sparse channel structure. This receiver architecture was used to process collected data from the SPACE 08 experiment (Martha's Vineyard, MA). Receiver performance for different modulation orders, channel codes, and hydrophone configurations is examined at a variety of distance, up to 1 km from the transmitters. Experimental results show great promise for this approach, as data rates in excess of 15 kb/s could readily be achieved without error.

A Novel Variable Step-Size Algorithm for Implicit Integration

A new variable step-size numerical algorithm for implicit integration is discussed in this paper. The scheme for increase and decrease of step size is discussed according to the difference of output variable value. The next step size is calculated through the variable coefficient method and the limitation rules. The convergence and accuracy are testified by the simulation result. Compared with the common used ode45 algorithm, the algorithm has more computing efficiency with a certain calculation precision.

Robustness, Weak Stability, and Stability in Distribution of Adaptive Filtering Algorithms under Model Mismatch

This work is concerned with robustness, convergence, and stability of adaptive filtering (AF) type algorithms in the presence of model mismatch. The algorithms under consideration are recursive and have inherent multiscale structure. They can be considered as dynamic systems, in which the “state” changes much more slowly than the perturbing noise. Beyond the existing results on adaptive algorithms, model mismatch significantly affects convergence properties of AF algorithms, raising issues of algorithm robustness. Weak convergence and weak stability (i.e., recurrence) under model mismatch are derived. Based on the limiting stochastic differential equations of suitably scaled iterates, stability in distribution is established. Then algorithms with decreasing step sizes and their convergence properties are examined. When input signals are large, identification bias due to model mismatch will become large and unacceptable. Methods for reducing such bias are introduced when the identified models are used in r...

High order local linearization methods: An approach for constructing A-stable explicit schemes for stochastic differential equations with additive noise

An approach for the construction of A-stable high order explicit strong schemes for stochastic differential equations (SDEs) with additive noise is proposed. We prove that such schemes also have the dynamical property that we call Random A-stability (RA-stability), which ensures that, for linear equations with stationary solutions, the numerical scheme has a random attractor that converges to the exact one as the step size decreases. Basically, the proposed integrators are obtained by splitting, at each time step, the solution of the original equation into two parts: the solution of a linear ordinary differential equation plus the solution of an auxiliary SDE. The first one is solved by the Local Linearization scheme in such a way that A-stability is guaranteed, while the second one is approximated by any extant scheme, preferably an explicit one that yields high order of convergence with low computational cost. Numerical integrators constructed in this way are called High Order Local Linearization (HOLL) methods. Various efficient HOLL schemes are elaborated in detail, and their performance is illustrated through computer simulations. Furthermore, mean-square convergence of the introduced methods is studied.

Sources of Error in a Simulation of Rigid Parts on a Vibrating Rigid Plate

We present a simulation study of an important rigid-body contact problem. The system in question is composed of a rigid plate and a single rigid body (or particle). The plate follows a prescribed periodic motion of small amplitude and high frequency, such that the net force applied to the part appears to be from a time-independent, position-dependent velocity field in the plane of the plate. Theoretical results obtained by Vose et al. were found to be in good agreement with simulation results obtained with the Stewart–Trinkle time-stepping method. In addition, simulations were found to agree with the qualitative experimental results of Vose et al. After such verification of the simulation method, additional numerical studies were done that would have been impossible to carry out analytically. Specifically, we were able to demonstrate the convergence of the method with decreasing step size (as predicted theoretically by Stewart). Further analytical and numerical studies will be carried out in the future to develop and select robust simulation methods that best satisfy the speed and accuracy requirements of different applications. With the accuracy of our time-stepper verified for this system, we were able to study the inverse problem of designing new plate motions to generate a desired part motion. This is done through an optimization framework, where a simulation of the part interacting with the plate (including the full dynamics of the system) is performed, and based on the results of the simulation the motion of the plate is modified. The learned (by simulation) plate motion was experimentally run on the device, and without any tuning (of the simulation parameters or device parameters) our learned plate motion produced the desired part motion.

Adaptive OFDM synchronization algorithm in frequency selective fading channels

With the development of mobile communication, adaptive algorithms are often chosen in real time process because of their high speed. However, existing adaptive algorithms are not suitable in frequency selective fading channels. In this paper, an improved adaptive synchronization algorithm is proposed for the application under frequency selective fading channels. A novel correlation function is proposed to reduce the effect of intersymbol interference, which only requires coarse channel length estimation. A gate value is also used to combine both the constant and decreasing step size methods; thus, the proposed algorithm is capable of distinguishing between tracking and convergent stages, and then choosing the right step size. Furthermore, an accelerating method is introduced to make the algorithm track faster, i.e. making the algorithm move more steps if it searches in the same direction as the last step. It is shown that the mean square errors of time and frequency offset estimates can be reduced to 0.9243 and 0.003 respectively even when the channel length estimate is not good, which totally satisfies Beek's requirement for synchronization. Simulation also shows that it takes only 10-20 symbols for the proposed algorithm to track the change of channel delay. Compared with the existing adaptive algorithm, the proposed algorithm has a higher accuracy with a higher tracking speed, and the proposed algorithm also has advantages over some nonadaptive algorithms with higher accuracy and lower computational complexity.

Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms and Asymptotic Behavior

This paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors' states. For consensus seeking, we propose stochastic approximation-type algorithms with a decreasing step size, and introduce the notions of mean square and strong consensus. Although the decreasing step size reduces the detrimental effect of the noise, it also reduces the ability of the algorithm to drive the individual states towards each other. The key technique is to ensure a trade-off for the decreasing rate of the step size. By following this strategy, we first develop a stochastic double array analysis in a two-agent model, which leads to both mean square and strong consensus, and extend the analysis to a class of well-studied symmetric models. Subsequently, we consider a general network topology, and introduce stochastic Lyapunov functions together with the so-called direction of invariance to establish mean square consensus. Finally, we apply the stochastic Lyapunov analysis to a leader following scenario.

Branching random walk with exponentially decreasing steps, and stochastically self-similar measures

We consider a Branching Random Walk on R whose step size decreases by a fixed factor, 0 1/2 the limit measure is almost surely (a.s.) absolutely continuous with respect to the Lebesgue measure, but for Pisot 1/λ it is a.s. singular; (2) for all A > (5 - 1)/2 the support of the measure is a.s. the closure of its interior; (3) for Pisot 1/λ the support of the measure is fractured: it is a.s. disconnected, and the components of the complement are not isolated on both sides.

A Unified Framework for Multipath Routing for Unicast and Multicast Traffic

We study the problem of load balancing the traffic from a set of unicast and multicast sessions. The problem is formulated as an optimization problem. However, we assume that the gradient of the network cost function is not available and needs to be estimated. Multiple paths are provided between a source and a destination using application-layer overlay. We propose a novel algorithm that is based on what is known as simultaneous perturbation stochastic approximation and utilizes only noisy measurements collected and reported to the sources, using an overlay architecture. We consider three network models that reflect different sets of assumptions regarding multicast capabilities of the network. Using an analytical model we first prove the almost sure convergence of the algorithm to a corresponding optimal solution under each network model considered in this paper with decreasing step sizes. Then, we establish the weak convergence (or convergence in distribution) with a fixed step size. In addition, we investigate the benefits acquired from implementing additional multicast capabilities by studying the relative performance of our algorithm under the three network models.

SU-GG-T-28: Physical Principles of Intensity Modulated Electronic Brachytherapy (IMEB)

Purpose: To describe a new collimator design and methodology for controlling the effective intensity of radiation produced by a microminiature x-ray source which allows the modulation of dose distributions for electronic brachytherapy applications. Method and Materials: A close fitting, two-section adjustable lead collimator consisting of a lower cylindrical band and an upper cap was constructed for a 50kV electronic brachytherapy source (Xoft Inc., Fremont, CA) to provide controllable beam hardening and attenuation. Micro-indexing the source through an indwelling catheter using variable step sizes, in combination with adjustment of the collimator gap width, modifies the resulting depth-doses. Results: Depth dose increases with wider collimator aperture settings and for decreasing step sizes. For a multi-stepped plan using step sizes of 1mm, the relative dose at 1cm depth was 0.50 for a gap of 0.5mm and 0.74 for a gap of 2.0mm. For a multi-stepped plan with a constant 2mm aperture size, the relative dose at 1cm depth increased from 0.70 for a step size of 2.5mm to 0.74 for a step size of 1.0mm. Conclusion: Appropriate depth dose adjustment by microindexing and variable collimation of an electronic brachytherapy source facilitates the development of IMEB plans that can be made to conform to irregularly shaped tumor cavities or those closely bounded by critical structures. Conflict of Interest: Partial financial support was provided by Xoft, Inc.

Objective Function Effect Based Pattern Search—An Implementation for 3D Component Layout

Generalized pattern search (GPS) algorithms have been used successfully to solve three-dimensional (3D) component layout problems. These algorithms use a set of patterns and successively decreasing step sizes of these patterns to explore the search space before converging to good local minima. A shortcoming of conventional GPS algorithms is the lack of recognition of the fact that patterns affect the objective function by different amounts and hence it might be efficient to introduce them into the search in a certain order rather than introduce all of them at the beginning of the search. To address this shortcoming, it has been shown by the authors in previous work that it is more efficient to schedule patterns in decreasing order of their effect on the objective function. The effect of the patterns on the objective function was estimated by the a priori expectation of the objective function change due to the patterns. However, computing the a priori expectation is expensive, and to practically implement the scheduling of patterns, an inexpensive estimate of the effect on the objective function is necessary. This paper introduces a metric for geometric layout called the sensitivity metric that is computationally inexpensive, to estimate the effect of pattern moves on the objective function. A new pattern search algorithm that uses the sensitivity metric to schedule patterns is shown to perform as well as the pattern search algorithm that used the a priori expectation of the objective function change. Though the sensitivity metric applies to the class of geometric layout or placement problems, the foundation and approach is useful for developing metrics for other optimization problems.