We propose an approach to reduce the bias of ridge regression and regularization kernel network. When applied to a single data set the new algorithms have comparable learning performance with the original ones. When applied to incremental learning with block wise streaming data the new algorithms are more efficient due to bias reduction. Both theoretical characterizations and simulation studies are used to verify the effectiveness of these new algorithms.
The idea of bias correction has long history in statistics. For instance, bias correction to maximum likelihood estimation dates at least back to 1950s @cite_39 and a variety method were proposed later on; see e.g. @cite_29 @cite_5 @cite_19 @cite_23 . Bias reduction to kernel density estimators was studied in @cite_35 @cite_37 @cite_42 @cite_7 . Bias correction to nonparametric estimation was studied in @cite_38 @cite_26 @cite_24 .
@cite_35 @cite_37 @cite_38 @cite_26 @cite_7 @cite_29 @cite_42 @cite_39 @cite_24 @cite_19 @cite_23 @cite_5
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The bias correction to the maximum likelihood extimates of the parameters for logistic discrimination is examined under mixture and separate sampling schemes. An existing adjustment developed under mixture sampling and based on higher derivatives of the log likelihood is modified slightly for use under separate sampling. The effect of this bias correction on the sampling properties of the misclassification rates and of the estimated posterior probabilities is discussed. This paper presents a theoretical analysis of sample selection bias correction. The sample bias correction technique commonly used in machine learning consists of reweighting the cost of an error on each training point of a biased sample to more closely reflect the unbiased distribution. This relies on weights derived by various estimation techniques based on finite samples. We analyze the effect of an error in that estimation on the accuracy of the hypothesis returned by the learning algorithm for two estimation techniques: a cluster-based estimation technique and kernel mean matching. We also report the results of sample bias correction experiments with several data sets using these techniques. Our analysis is based on the novel concept of distributional stability which generalizes the existing concept of point-based stability. Much of our work and proof techniques can be used to analyze other importance weighting techniques and their effect on accuracy when using a distributionally stable algorithm. We present some new results on the nonparametric estimation of entropy and mutual information. First, we use an exact local expansion of the entropy function to prove almost sure consistency and central limit theorems for three of the most commonly used discretized information estimators. The setup is related to Grenander's method of sieves and places no assumptions on the underlying probability measure generating the data. Second, we prove a converse to these consistency theorems, demonstrating that a misapplication of the most common estimation techniques leads to an arbitrarily poor estimate of the true information, even given unlimited data. This "inconsistency" theorem leads to an analytical approximation of the bias, valid in surprisingly small sample regimes and more accurate than the usual 1 N formula of Miller and Madow over a large region of parameter space. The two most practical implications of these results are negative: (1) information estimates in a certain data regime are likely contaminated by bias, even if "bias-corrected" estimators are used, and (2) confidence intervals calculated by standard techniques drastically underestimate the error of the most common estimation methods.Finally, we note a very useful connection between the bias of entropy estimators and a certain polynomial approximation problem. By casting bias calculation problems in this approximation theory framework, we obtain the best possible generalization of known asymptotic bias results. More interesting, this framework leads to an estimator with some nice properties: the estimator comes equipped with rigorous bounds on the maximum error over all possible underlying probability distributions, and this maximum error turns out to be surprisingly small. We demonstrate the application of this new estimator on both real and simulated data. We study the problem of maximum entropy density estimation in the presence of known sample selection bias. We propose three bias correction approaches. The first one takes advantage of unbiased sufficient statistics which can be obtained from biased samples. The second one estimates the biased distribution and then factors the bias out. The third one approximates the second by only using samples from the sampling distribution. We provide guarantees for the first two approaches and evaluate the performance of all three approaches in synthetic experiments and on real data from species habitat modeling, where maxent has been successfully applied and where sample selection bias is a significant problem.