Innuo

First I would like to mention that, since my last post , I came across the paper from 2005 on Robust Supervised Learning by J. Andrew Bagnell that proposed almost exactly the same regularized minimax algorithm as the one I derived. He motivates the problem slightly differently and weights each example separately and not based on types, but the details are essentially identical. Experiments on Synthetic Data I tried the algorithm on some synthetic data and a linear logistic regression model. The results are shown in the figures below. In both examples, there are examples from two classes (red and blue). Each class is a drawn from a  mixture of two normal distributions (i.e., there are two types per class). The types are shown as red squares and red circles, and blue diamonds and blue triangles. Class-conditionally the types have a skewed distribution. There are 9 times as many red squares as red circles, and 9 times as many blue diamonds as triangles. We would expect a plain logistic ...

This post is about using minimax estimation for robust learning when the test data distribution is expected to be different from the training data distribution, i.e learning that is robust to data drift. Cost Sensitive Loss Functions Given a training data set $latex D = \{x_i, y_i\}_{i=1,\ldots,N}$, most learning algorithms learn a classifier $latex \phi$ that is parametrized by a vector $latex w$ by minimizing a loss function where $latex l(x_i, y_i, w)$ is the loss on example $latex i$ and $latex f(w)$ is some function that penalizes complexity. For example for logistic regression the loss function looks like for some $latex \lambda > 0$. If, in addition, the examples came with costs $latex c_i$ (that somehow specify the importance of minimizing the loss on that particular example), we can perform cost sensitive learning by over/under-sampling the training data or minimize a cost-weighted loss function (see this paper by Zadrozny et. al. ) We further constrain $latex \sum_i^N c_i...

In a previous post , I talked about an idea for sparsifying kernel logistic regression by using random prototypes. I also showed how the prototypes themselves (as well as the kernel parameters) can be updated. (Update Apr 2010. Slides for a tutorial on this stuff.) (As a brief aside, I note that an essentially identical approach was used to sparsify Gaussian Process Regression by Snelson and Gharahmani . For GPR they use gradient ascent on the log-likelihood to learn the prototypes and labels, which is akin to learning the prototypes and betas for logistic regression. The set of prototypes and labels generated by their algorithm can be thought of as a pseudo training set.) I recently (with the help of my super-competent Java developer colleague Hiroko Bretz) implemented the sparse kernel logistic regression algorithm. The learning is done in an online fashion (i.e., using stochastic gradient descent). It seems to perform reasonably well on large datasets. Below I'll show its behav...

I show how the Burrows-Wheeler Transform can be used to compute the similarity between two strings. We submitted results from this method (along with results from the Context-Chain metric developed by my colleagues Frank Schilder and Ravi Kondadadi) for the Automatically Evaluating the Summaries of Peers (AESOP) task of the TAC 2009 conference. The task was to produce an automatic metric to evaluate machine generated summaries (i.e., system summaries) against human generated summaries for the TAC '09 Update Summarization Task. Clearly the automatic metric is just some function that produces a similarity score between the system summary and the human generated (the so-called model ) summary. The  proposed metrics were evaluated by comparing their rankings of the system summaries from different peers to that of the ranking produced by human judges. Similarity Metric We use an estimate of the conditional "compressibility" of the model summary given the system summary as the...

The Burrows-Wheeler transform (BWT), which is the main step in the bzip2 compression algorithm, is a permutation transform on a string over an ordered alphabet. It is a clever idea and can be useful for some string processing for natural language processing.  I will present one such use. BWT massages the original string into being more amenable to compression. Of course the transform doesn't alter the compressibility (entropy rate) of the original string. All it does is make the string more compressible by algorithms we know. The reason string permutation by BWT (as opposed to say sorting the string, which makes it really compressible) is useful is that the reverse transform (undoing the permutation) can be done with very little additional information. Mark Nelson wrote a nice introduction to the transform.  Moreover, the BWT essentially involves the construction of the suffix array for the string, and therefore can be done in time and space linear in the length of the string. ...

One of the problems with using complex kernels with support vector machines is that they tend to produce classification boundaries that are odd, like the ones below. (I generated them using a java SVM applet from here , whose reliability I cannot swear to, but have no reason to doubt.) Both SVM boundaries are with Gaussian RBF kernels: the first with $latex \sigma = 1$ and the second with $latex \sigma = 10$ on two different data sets. Note the segments of the boundary to the east of the blue examples in the bottom figure, and those to the south and to the north-east of the blue examples in the top figure. They seem to violate intuition. The reason for these anomalous boundaries is of course the large complexity of the function class induced by the RBF kernel with large $latex \sigma$, which gives the classifier a propensity to make subtle distinctions even in regions of  somewhat low example density. A possible solution: using complex kernels only where they are needed We propose to b...

As opposed to the traditional supervised learning setting where the labeled training data is generated (we hope) independently and identically, in active learning the learner is allowed to select points for which labels are requested. Because it is often impossible to construct the equivalent real-world object from its feature values, almost universally, active learning is pool-based . That is we start with a large pool of unlabeled data and the learner (usually sequentially) picks the objects from the pool for which the labels are requested. One unavoidable effect of active learning is that we end up with a biased training data set. If the true data distribution is $latex P(x,y)$, we have data drawn from some distribution $latex \hat{P}(x,y)$ (as always $latex x$ is the feature vector and $latex y$ is the class label). We would like to correct for this bias so it does not lead to learning an incorrect classifier. And furthermore we want to use this biased data set to accurately evalu...