Stephan Sloth Lorenzen

Yi-Shan Wu, Andrés R. Masegosa, Stephan S. Lorenzen et al.

We present a new second-order oracle bound for the expected risk of a weighted majority vote. The bound is based on a novel parametric form of the Chebyshev- Cantelli inequality (a.k.a. one-sided Chebyshev's), which is amenable to efficient minimization. The new form resolves the optimization challenge faced by prior oracle bounds based on the Chebyshev-Cantelli inequality, the C-bounds [Germain et al., 2015], and, at the same time, it improves on the oracle bound based on second order Markov's inequality introduced by Masegosa et al. [2020]. We also derive a new concentration of measure inequality, which we name PAC-Bayes-Bennett, since it combines PAC-Bayesian bounding with Bennett's inequality. We use it for empirical estimation of the oracle bound. The PAC-Bayes-Bennett inequality improves on the PAC-Bayes-Bernstein inequality of Seldin et al. [2012]. We provide an empirical evaluation demonstrating that the new bounds can improve on the work of Masegosa et al. [2020]. Both the parametric form of the Chebyshev-Cantelli inequality and the PAC-Bayes-Bennett inequality may be of independent interest for the study of concentration of measure in other domains.

Stephan Sloth Lorenzen, Christian Igel, Mads Nielsen

The information bottleneck (IB) principle has been suggested as a way to analyze deep neural networks. The learning dynamics are studied by inspecting the mutual information (MI) between the hidden layers and the input and output. Notably, separate fitting and compression phases during training have been reported. This led to some controversy including claims that the observations are not reproducible and strongly dependent on the type of activation function used as well as on the way the MI is estimated. Our study confirms that different ways of binning when computing the MI lead to qualitatively different results, either supporting or refusing IB conjectures. To resolve the controversy, we study the IB principle in settings where MI is non-trivial and can be computed exactly. We monitor the dynamics of quantized neural networks, that is, we discretize the whole deep learning system so that no approximation is required when computing the MI. This allows us to quantify the information flow without measurement errors. In this setting, we observed a fitting phase for all layers and a compression phase for the output layer in all experiments; the compression in the hidden layers was dependent on the type of activation function. Our study shows that the initial IB results were not artifacts of binning when computing the MI. However, the critical claim that the compression phase may not be observed for some networks also holds true.

Andrés R. Masegosa, Stephan S. Lorenzen, Christian Igel et al.

We present a novel analysis of the expected risk of weighted majority vote in multiclass classification. The analysis takes correlation of predictions by ensemble members into account and provides a bound that is amenable to efficient minimization, which yields improved weighting for the majority vote. We also provide a specialized version of our bound for binary classification, which allows to exploit additional unlabeled data for tighter risk estimation. In experiments, we apply the bound to improve weighting of trees in random forests and show that, in contrast to the commonly used first order bound, minimization of the new bound typically does not lead to degradation of the test error of the ensemble.

Stephan S. Lorenzen, Ninh Pham

Top-k maximum inner product search (MIPS) is a central task in many machine learning applications. This paper extends top-k MIPS with a budgeted setting, that asks for the best approximate top-k MIPS given a limit of B computational operations. We investigate recent advanced sampling algorithms, including wedge and diamond sampling to solve it. Though the design of these sampling schemes naturally supports budgeted top-k MIPS, they suffer from the linear cost from scanning all data points to retrieve top-k results and the performance degradation for handling negative inputs. This paper makes two main contributions. First, we show that diamond sampling is essentially a combination between wedge sampling and basic sampling for top-k MIPS. Our theoretical analysis and empirical evaluation show that wedge is competitive (often superior) to diamond on approximating top-k MIPS regarding both efficiency and accuracy. Second, we propose a series of algorithmic engineering techniques to deploy wedge sampling on budgeted top-k MIPS. Our novel deterministic wedge-based algorithm runs significantly faster than the state-of-the-art methods for budgeted and exact top-k MIPS while maintaining the top-5 precision at least 80% on standard recommender system data sets.

Stephan Lorenzen, Niklas Hjuler, Stephen Alstrup

Analysis of log data generated by online educational systems is an essential task to better the educational systems and increase our understanding of how students learn. In this study we investigate previously unseen data from Clio Online, the largest provider of digital learning content for primary schools in Denmark. We consider data for 14,810 students with 3 million sessions in the period 2015-2017. We analyze student activity in periods of one week. By using non-negative matrix factorization techniques, we obtain soft clusterings, revealing dependencies among time of day, subject, activity type, activity complexity (measured by Bloom's taxonomy), and performance. Furthermore, our method allows for tracking behavioral changes of individual students over time, as well as general behavioral changes in the educational system. Based on the results, we give suggestions for behavioral changes, in order to optimize the learning experience and improve performance.

Stephan Lorenzen, Niklas Hjuler, Stephen Alstrup

In this paper we do the first large scale analysis of writing style development among Danish high school students. More than 10K students with more than 100K essays are analyzed. Writing style itself is often studied in the natural language processing community, but usually with the goal of verifying authorship, assessing quality or popularity, or other kinds of predictions. In this work, we analyze writing style changes over time, with the goal of detecting global development trends among students, and identifying at-risk students. We train a Siamese neural network to compute the similarity between two texts. Using this similarity measure, a student's newer essays are compared to their first essays, and a writing style development profile is constructed for the student. We cluster these student profiles and analyze the resulting clusters in order to detect general development patterns. We evaluate clusters with respect to writing style quality indicators, and identify optimal clusters, showing significant improvement in writing style, while also observing suboptimal clusters, exhibiting periods of limited development and even setbacks. Furthermore, we identify general development trends between high school students, showing that as students progress through high school, their writing style deviates, leaving students less similar when they finish high school, than when they start.

Magnus Stavngaard, August Sørensen, Stephan Lorenzen et al.

Students hiring ghostwriters to write their assignments is an increasing problem in educational institutions all over the world, with companies selling these services as a product. In this work, we develop automatic techniques with special focus on detecting such ghostwriting in high school assignments. This is done by training deep neural networks on an unprecedented large amount of data supplied by the Danish company MaCom, which covers 90% of Danish high schools. We achieve an accuracy of 0.875 and a AUC score of 0.947 on an evenly split data set.

Stephan Sloth Lorenzen, Christian Igel, Yevgeny Seldin

Existing guarantees in terms of rigorous upper bounds on the generalization error for the original random forest algorithm, one of the most frequently used machine learning methods, are unsatisfying. We discuss and evaluate various PAC-Bayesian approaches to derive such bounds. The bounds do not require additional hold-out data, because the out-of-bag samples from the bagging in the training process can be exploited. A random forest predicts by taking a majority vote of an ensemble of decision trees. The first approach is to bound the error of the vote by twice the error of the corresponding Gibbs classifier (classifying with a single member of the ensemble selected at random). However, this approach does not take into account the effect of averaging out of errors of individual classifiers when taking the majority vote. This effect provides a significant boost in performance when the errors are independent or negatively correlated, but when the correlations are strong the advantage from taking the majority vote is small. The second approach based on PAC-Bayesian C-bounds takes dependencies between ensemble members into account, but it requires estimating correlations between the errors of the individual classifiers. When the correlations are high or the estimation is poor, the bounds degrade. In our experiments, we compute generalization bounds for random forests on various benchmark data sets. Because the individual decision trees already perform well, their predictions are highly correlated and the C-bounds do not lead to satisfactory results. For the same reason, the bounds based on the analysis of Gibbs classifiers are typically superior and often reasonably tight. Bounds based on a validation set coming at the cost of a smaller training set gave better performance guarantees, but worse performance in most experiments.