Zero-mean Distribution Research Articles

Symmetric Q-learning: Reducing Skewness of Bellman Error in Online Reinforcement Learning

In deep reinforcement learning, estimating the value function to evaluate the quality of states and actions is essential. The value function is often trained using the least squares method, which implicitly assumes a Gaussian error distribution. However, a recent study suggested that the error distribution for training the value function is often skewed because of the properties of the Bellman operator, and violates the implicit assumption of normal error distribution in the least squares method. To address this, we proposed a method called Symmetric Q-learning, in which the synthetic noise generated from a zero-mean distribution is added to the target values to generate a Gaussian error distribution. We evaluated the proposed method on continuous control benchmark tasks in MuJoCo. It improved the sample efficiency of a state-of-the-art reinforcement learning method by reducing the skewness of the error distribution.

Compressed Image Restoration via Artifacts-Free PCA Basis Learning and Adaptive Sparse Modeling

Visually unpleasant compression artifacts frequently appear in block-based transform coding, especially at low bit rates. This paper presents a new artifact reduction scheme based on Bayesian sparse modeling and artifacts-free PCA basis learning. To avoid the effect of blocking artifacts, we propose to learn artifacts-free PCA basis from clean images. We concatenate the clean patches and their compressed counterparts to learn paired distribution prior via the Gaussian Mixture Model (GMM). By this way, the GMM characterizes the mapping between the clean image and its compressed version. To restore a compressed patch, the best matched GMM component is assigned using the patch in the compressed image subspace. The artifacts-free PCA basis is obtained according to the mapping learned by the paired GMM. In practice, the statistical distributions of different sparse coefficients in different patches may dramatically vary with image contents. Instead of using a global zero-mean distribution for all coefficients, we propose to adaptively model the prior of each band in a Bayesian framework. The expectation and variance of each band are adaptively learned from the similar patches within the image. Thus, different transform bands are regularized unequally according to the learned priors. Experimental results show that the proposed scheme outperforms most of the compared schemes in terms of both objective quality and perceptual quality.

An Application of Reliability Estimation in Longitudinal Designs Through Modeling Item-Specific Error Variance.

The purpose of the present study was to apply the methodology developed by Raykov on modeling item-specific variance for the measurement of internal consistency reliability with longitudinal data. Participants were a randomly selected sample of 500 individuals who took on a professional qualifications test in Saudi Arabia over four different occasions. Data were analyzed by use of confirmatory factor analysis, and item error variance was corrected for item specificity. The estimation of reliability involved composite index omega. Results indicated that the initially low and unacceptable levels of internal consistency reliability approached acceptable levels after accounting for item-specific variance. Findings were verified by testing whether the difference estimates of internal consistency reliability deviated from a zero-mean distribution using 10,000 replicated samples assuming a known (symmetric) or unknown (asymmetric) population distribution of the difference reliability coefficients. Percentage improvement reliability estimates indices were also estimated along with their 95% confidence intervals. Two appendices provide annotated Mplus syntax files for future use.

Optimal two-value zero-mean disintegration of zero-mean random variables

For any continuous zero-mean random variable $X$, a reciprocating function $r$ is constructed, based only on the distribution of $X$, such that the conditional distribution of $X$ given the (at-most-)two-point set $\{X,r(X)\}$ is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations - of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) - go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.

Autoregressive spectral analysis based on statistical autocorrelation

The autoregressive method of spectral analysis is widely used in diverse areas for its solid theoretical foundation. Various aspects of its statistical performance have been investigated. People assume the times series x n to be samples from a zero-mean distribution whose variance remains constant in time. However, it is not available in fact. In this paper, by formulating the resolution event in the framework of statistical autocorrelation theory and directly determining its value from its center-autocorrelation function and statistical autocorrelation function, we obtain a power spectral density formula for the statistical resolution. On this basis, we determine the limiting resolving behavior of the sample autoregressive spectrum and develop the corresponding statistical insight in the time series. Simulation results are also presented to confirm and illustrate the effectiveness of the theory.

A distributed-µ implementation of the LMS algorithm

This paper describes the effects of component nonidealities upon the performance of analog and sampled data adaptive noise cancellers. It is shown that the effects of analog implementation differ significantly from those encountered in ideal cases or those in which the added circuit noise is assumed to have a zero-mean distribution. It is also shown that the effects of nonzero mean errors, such as offset voltages and nonlinearities of the input multipliers contribute an excess mean-square error (MSE) which is inversely proportional to the parameter which controls the stability and the rate of convergence of the algorithm. An implementation configuration is presented which minimizes these deleterious effects by distributing the gain of the adaptive system around the signal path.

A distributed-<tex>mu</tex>implementation of the LMS algorithm

This paper describes the effects of component nonidealities upon the performance of analog and sampled data adaptive noise cancellers. It is shown that the effects of analog implementation differ significantly from those encountered in ideal cases or those in which the added circuit noise is assumed to have a zero-mean distribution. It is also shown that the effects of nonzero mean errors, such as offset voltages and nonlinearities of the input multipliers contribute an excess mean-square error (MSE) which is inversely proportional to the parameter which controls the stability and the rate of convergence of the algorithm. An implementation configuration is presented which minimizes these deleterious effects by distributing the gain of the adaptive system around the signal path.

The Hackenbush number system for compression of numerical data

The Hackenbush number system is a method of representing numbers by expressing the integer part in unary and the fractional part in binary. The representation absorbs the sign bit and binary point into the string of binary digits which represents the number. Conversion between the Hackenbush representation and the more conventional fixed and floating point representations requires approximately the same small amount of computation as is required to convert between the fixed and floating representations. If the numbers to be represented are selected from a known zero-mean distribution which has bounded height and tails that fall off exponentially or faster, the m-bit Hackenbush representation is typically more accurate than any floating point representation. For example, if the numbers are selected from the normal Gaussian distribution, the mean-square error of the m-bit Hackenbush representation is less than the mean-square error of any (arbitrarily complex) (m − 1)-bit representation, and it is comparable to the mean square quantization error of the (m + n)-bit normalized floating point representation which has 1 bit of sign, n + 1 bits of exponent, and (m − 2) bits of characteristic, for all n.