Efficient Direct Density Ratio Estimation for Non-stationarity Adaptation and Outlier Detection Takafumi Kanamori Shohei Hido NIPS 2008

Outline Motivation Importance Estimation Direct Importance Estimation Approximation Algorithm Experiments Conclusions

Motivation Importance Sampling Covariate Shift Outlier Detection

Importance Sampling Rather than sampling from the distribution p, importance sampling is to reduce the variance of Ê[f(X)] by an appropriate choice of q, hence the name importance sampling, as samples from q can be more "important" for the estimation of the integral. Other reasons include difficulties to draw samples from distribution p or efficiency considerations. [2] R. Srinivasan, Importance sampling - Applications in communications and detection, Springer-Verlag, Berlin, [3]P. J.Smith, M.Shafi, and H. Gao, "Quick simulation: A review of importance sampling techniques in communication systems," IEEE J.Select.Areas Commun., vol. 15, pp , May 1997.

Covariate Shift Compensated by weighting the training samples according to the importance Distribution of input training and testing set changed, while the conditional distribution that output given input unchanged. Then, standard learning techniques such as MLE or CV are biased. [4]Jiayuan Huang, Alexander J. Smola,Arthur Gretton,et al. Correcting Sample Selection Bias by Unlabeled Data, NIPS 2006.

Outlier Detection The importance for regular samples are close to one, while those for outliers tend to be significantly deviated from one. The values of the importance could be used as an index of the degree of outlyingness.

Related Works Kernel Density Estimation Kernel Mean Matching a map into the feature space the expectation operator μ(Pr) := E x~Pr(x) [Φ(x)]. [4]Jiayuan Huang, Alexander J. Smola,Arthur Gretton,et al. Correcting Sample Selection Bias by Unlabeled Data, NIPS 2006.

Direct Importance Estimation

Least-square Approach Model w(x) with linear model Determine the parameter alpha so that the squared error on training samples is minimized:

Least Square Importance Fitting LSIF Empirical estimation Regularization term to avoid over-fitting

Model Selection for LSIF Model the parameter lambda, the basis function phi Model selection Cross Validation

Heuristics for Basic function Design Gaussian kernel centered at the test samples

Unconstrained Least-squares Approach (uLSIF) Ignore the non-negativity constraints Learned parameters could be negative To compensate for the approximation error, modify the solution

Efficient Computation of LOOCV Samples learned without the LOOCV score According to the Sherman-Woodbury-Morrison formula, the matrix inverse needs to be computed only once.

Experiments Importance Estimation p train is the d-dimensional normal distribution with mean zero and covariance identity. p test is the d-dimensional normal distribution with mean (1,0,…,0) T and covariance identity. Normalized mean squared error

Covariate Shift Adaptation in classification and regression Given the training samples, the test samples, and the outputs of the training samples The task is to predict the outputs for test samples

Experimental Description Divide the training samples into R disjoint subsets The function is learned using by IWRLS and its mean test error for the remaining samples is computed: Where

Covariate shift adaptation

Experiment Outlier Detection

Conclusions Application –Covariate shift adaptation –Outlier detection –Feature selection –Conditional distribution estimation –ICA –……

Reference [1]Takafumi Kanamori, Shohei Hido. Efficient direct density ratio estimation for non-stationarity adaptation and outlier detection, NIPS [2] R. Srinivasan, Importance sampling - Applications in communications and detection, Springer-Verlag, Berlin, [3]P. J.Smith, M.Shafi, and H. Gao, "Quick simulation: A review of importance sampling techniques in communication systems," IEEE J.Select.Areas Commun., vol. 15, pp , May [4]Jiayuan Huang, Alexander J. Smola,Arthur Gretton,et al. Correcting Sample Selection Bias by Unlabeled Data, NIPS [5] Jing Jiang. A Literature Survey on Domain Adaptation of Statistical Classifiers