In this paper, we study the performance of extremum estimators from the perspective of generalization ability (GA): the ability of a model to predict outcomes in new samples from the same population. By adapting the classical concentration inequalities, we derive upper bounds on the empirical out-of-sample prediction errors as a function of the in-sample errors, in-sample data size, heaviness in the tails of the error distribution, and model complexity. We show that the error bounds may be used for tuning key estimation hyper-parameters, such as the number of folds $K$ in cross-validation. We also show how $K$ affects the bias-variance trade-off for cross-validation. We demonstrate that the $\mathcal{L}_2$-norm difference between penalized and the corresponding un-penalized regression estimates is directly explained by the GA of the estimates and the GA of empirical moment conditions. Lastly, we prove that all penalized regression estimates are $L_2$-consistent for both the $n \geqslant p$ and the $n < p$ cases. Simulations are used to demonstrate key results. Keywords: generalization ability, upper bound of generalization error, penalized regression, cross-validation, bias-variance trade-off, $\mathcal{L}_2$ difference between penalized and unpenalized regression, lasso, high-dimensional data.

We apply a novel spectral graph technique, that of locally-biased semi-supervised eigenvectors, to study the diversity of galaxies. This technique permits us to characterize empirically the natural variations in observed spectra data, and we illustrate how this approach can be used in an exploratory manner to highlight both large-scale global as well as small-scale local structure in Sloan Digital Sky Survey (SDSS) data. We use this method in a way that simultaneously takes into account the measurements of spectral lines as well as the continuum shape. Unlike Principal Component Analysis, this method does not assume that the Euclidean distance between galaxy spectra is a good global measure of similarity between all spectra, but instead it only assumes that local difference information between similar spectra is reliable. Moreover, unlike other nonlinear dimensionality methods, this method can be used to characterize very finely both small-scale local as well as large-scale global properties of realistic noisy data. The power of the method is demonstrated on the SDSS Main Galaxy Sample by illustrating that the derived embeddings of spectra carry an unprecedented amount of information. By using a straightforward global or unsupervised variant, we observe that the main features correlate strongly with star formation rate and that they clearly separate active galactic nuclei. Computed parameters of the method can be used to describe line strengths and their interdependencies. By using a locally-biased or semi-supervised variant, we are able to focus on typical variations around specific objects of astronomical interest. We present several examples illustrating that this approach can enable new discoveries in the data as well as a detailed understanding of very fine local structure that would otherwise be overwhelmed by large-scale noise and global trends in the data.

We formulate and analyze a graphical model selection method for inferring the conditional independence graph of a high-dimensional nonstationary Gaussian random process (time series) from a finite-length observation. The observed process samples are assumed uncorrelated over time and having a time-varying marginal distribution. The selection method is based on testing conditional variances obtained for small subsets of process components. This allows to cope with the high-dimensional regime, where the sample size can be (drastically) smaller than the process dimension. We characterize the required sample size such that the proposed selection method is successful with high probability.

Information theoretic measures (e.g. the Kullback Liebler divergence and Shannon mutual information) have been used for exploring possibly nonlinear multivariate dependencies in high dimension. If these dependencies are assumed to follow a Markov factor graph model, this exploration process is called structure discovery. For discrete-valued samples, estimates of the information divergence over the parametric class of multinomial models lead to structure discovery methods whose mean squared error achieves parametric convergence rates as the sample size grows. However, a naive application of this method to continuous nonparametric multivariate models converges much more slowly. In this paper we introduce a new method for nonparametric structure discovery that uses weighted ensemble divergence estimators that achieve parametric convergence rates and obey an asymptotic central limit theorem that facilitates hypothesis testing and other types of statistical validation.

I don't think K-means clustering in R has any special meaning, whatever package you use the basic K-means algorithm remains the same.

The estimate of the gradient from just a mini-batch is usually good enough to point you in the right descent direction. It doesn't make sense to do the extra computation for a marginally better estimate. Plus, the inaccuracy or noise introduced by the mini-batch approximation can act as a regularizer. Here is an interesting paper that performs statistical tests during optimization: if the gradient is not statistically significant, more samples are added to the mini-batch.

Last week, we discussed Multi-class SVM loss; specifically, the hinge loss and squared hinge loss functions. A loss function, in the context of Machine Learning and Deep Learning, allows us to quantify how "good" or "bad" a given classification function (also called a "scoring function") is at correctly classifying data points in our dataset. In fact, if you have done previous work in Deep Learning, you have likely heard of this function before -- do the terms Softmax classifier and cross-entropy loss sound familiar? I'll go as far to say that if you do any work in Deep Learning (especially Convolutional Neural Networks) that you'll run into the term "Softmax": it's the final layer at the end of the network that yields your actual probability scores for each class label. To learn more about Softmax classifiers and the cross-entropy loss function, keep reading.

I happened to recently reread this article of mine from 2010, and I absolutely love it. I don't think it's been read by many people--it was published as one of three discussions of an article by Brad Efron in Statistical Science--so I wanted to share it with you again here. The information principle: the key to a good statistical method is not its underlying philosophy or mathematical reasoning, but rather what information the method allows us to use. Good methods make use of more information. The methodological attribution problem: the many useful contributions of a good statistical consultant, or collaborator, will often be attributed to the statistician's methods or philosophy rather than to the artful efforts of the statistician himself or herself.

The results are a little different to those obtained using RMSE as the objective function. The focus is still well and truly on the volatility indicators, but in this case the best cross validated performance occurred when selecting only 2 out of the 15 candidate variables. Here's a plot of the cross validated performance of the best feature set for various numbers of features: The model clearly performs better in terms of absolute return for a smaller number of predictors. Performance bottoms at 8 predictors and then improves, but never again achieves the performance obtained with 2-4 predictors. This is consistent with Aronson's assertion that we should stick with at most 3-4 variables otherwise overfitting is almost unavoidable.

We can see a few interesting things in this table. In a similar talk by Owen at ODSC Boston 2015 titled "Open Source Tools and Data Science Competitions", he again summarized common parameters he uses: We can see some minor differences that may be relevant. Finally, Abhishek Thakur, in his post titled "Approaching (Almost) Any Machine Learning Problem" provided a similar table listing out key XGBoost parameters and suggestions for tuning. The spreads do cover the general defaults suggested above and more. It is interesting to note that Abhishek does provides some suggestions for tuning the alpha and beta model penalization terms as well as row sampling. You can develop and evaluate XGBoost models in just a few lines of Python code.