Prevalidated ridge regression is a highly-efficient drop-in replacement for logistic regression for high-dimensional data

Logistic regression is a ubiquitous method for probabilistic classification. However, the effectiveness of logistic regression depends upon careful and relatively computationally expensive tuning, especially for the regularisation hyperparameter, and especially in the context of high-dimensional data. We present a prevalidated ridge regression model that closely matches logistic regression in terms of classification error and log-loss, particularly for high-dimensional data, while being significantly more computationally efficient and having effectively no hyperparameters beyond regularisation. We scale the coefficients of the model so as to minimise log-loss for a set of prevalidated predictions derived from the estimated leave-one-out cross-validation error. This exploits quantities already computed in the course of fitting the ridge regression model in order to find the scaling parameter with nominal additional computational expense.

Bayes beats Cross Validation: Efficient and Accurate Ridge Regression via Expectation Maximization

We present a novel method for tuning the regularization hyper-parameter, $\lambda$, of a ridge regression that is faster to compute than leave-one-out cross-validation (LOOCV) while yielding estimates of the regression parameters of equal, or particularly in the setting of sparse covariates, superior quality to those obtained by minimising the LOOCV risk. The LOOCV risk can suffer from multiple and bad local minima for finite $n$ and thus requires the specification of a set of candidate $\lambda$, which can fail to provide good solutions. In contrast, we show that the proposed method is guaranteed to find a unique optimal solution for large enough $n$, under relatively mild conditions, without requiring the specification of any difficult to determine hyper-parameters. This is based on a Bayesian formulation of ridge regression that we prove to have a unimodal posterior for large enough $n$, allowing for both the optimal $\lambda$ and the regression coefficients to be jointly learned within an iterative expectation maximization (EM) procedure. Importantly, we show that by utilizing an appropriate preprocessing step, a single iteration of the main EM loop can be implemented in $O(\min(n, p))$ operations, for input data with $n$ rows and $p$ columns. In contrast, evaluating a single value of $\lambda$ using fast LOOCV costs $O(n \min(n, p))$ operations when using the same preprocessing. This advantage amounts to an asymptotic improvement of a factor of $l$ for $l$ candidate values for $\lambda$ (in the regime $q, p \in O(\sqrt{n})$ where $q$ is the number of regression targets).

Scalable Probabilistic Forecasting in Retail with Gradient Boosted Trees: A Practitioner's Approach

The recent M5 competition has advanced the state-of-the-art in retail forecasting. However, we notice important differences between the competition challenge and the challenges we face in a large e-commerce company. The datasets in our scenario are larger (hundreds of thousands of time series), and e-commerce can afford to have a larger assortment than brick-and-mortar retailers, leading to more intermittent data. To scale to larger dataset sizes with feasible computational effort, firstly, we investigate a two-layer hierarchy and propose a top-down approach to forecasting at an aggregated level with less amount of series and intermittency, and then disaggregating to obtain the decision-level forecasts. Probabilistic forecasts are generated under distributional assumptions. Secondly, direct training at the lower level with subsamples can also be an alternative way of scaling. Performance of modelling with subsets is evaluated with the main dataset. Apart from a proprietary dataset, the proposed scalable methods are evaluated using the Favorita dataset and the M5 dataset. We are able to show the differences in characteristics of the e-commerce and brick-and-mortar retail datasets. Notably, our top-down forecasting framework enters the top 50 of the original M5 competition, even with models trained at a higher level under a much simpler setting.

Computing Marginal and Conditional Divergences between Decomposable Models with Applications

The ability to compute the exact divergence between two high-dimensional distributions is useful in many applications but doing so naively is intractable. Computing the alpha-beta divergence -- a family of divergences that includes the Kullback-Leibler divergence and Hellinger distance -- between the joint distribution of two decomposable models, i.e chordal Markov networks, can be done in time exponential in the treewidth of these models. However, reducing the dissimilarity between two high-dimensional objects to a single scalar value can be uninformative. Furthermore, in applications such as supervised learning, the divergence over a conditional distribution might be of more interest. Therefore, we propose an approach to compute the exact alpha-beta divergence between any marginal or conditional distribution of two decomposable models. Doing so tractably is non-trivial as we need to decompose the divergence between these distributions and therefore, require a decomposition over the marginal and conditional distributions of these models. Consequently, we provide such a decomposition and also extend existing work to compute the marginal and conditional alpha-beta divergence between these decompositions. We then show how our method can be used to analyze distributional changes by first applying it to a benchmark image dataset. Finally, based on our framework, we propose a novel way to quantify the error in contemporary superconducting quantum computers. Code for all experiments is available at: https://lklee.dev/pub/2023-icdm/code

QUANT: A Minimalist Interval Method for Time Series Classification

We show that it is possible to achieve the same accuracy, on average, as the most accurate existing interval methods for time series classification on a standard set of benchmark datasets using a single type of feature (quantiles), fixed intervals, and an 'off the shelf' classifier. This distillation of interval-based approaches represents a fast and accurate method for time series classification, achieving state-of-the-art accuracy on the expanded set of 142 datasets in the UCR archive with a total compute time (training and inference) of less than 15 minutes using a single CPU core.

An Approach to Multiple Comparison Benchmark Evaluations that is Stable Under Manipulation of the Comparate Set

The measurement of progress using benchmarks evaluations is ubiquitous in computer science and machine learning. However, common approaches to analyzing and presenting the results of benchmark comparisons of multiple algorithms over multiple datasets, such as the critical difference diagram introduced by Dem\v{s}ar (2006), have important shortcomings and, we show, are open to both inadvertent and intentional manipulation. To address these issues, we propose a new approach to presenting the results of benchmark comparisons, the Multiple Comparison Matrix (MCM), that prioritizes pairwise comparisons and precludes the means of manipulating experimental results in existing approaches. MCM can be used to show the results of an all-pairs comparison, or to show the results of a comparison between one or more selected algorithms and the state of the art. MCM is implemented in Python and is publicly available.

Sparse Horseshoe Estimation via Expectation-Maximisation

The horseshoe prior is known to possess many desirable properties for Bayesian estimation of sparse parameter vectors, yet its density function lacks an analytic form. As such, it is challenging to find a closed-form solution for the posterior mode. Conventional horseshoe estimators use the posterior mean to estimate the parameters, but these estimates are not sparse. We propose a novel expectation-maximisation (EM) procedure for computing the MAP estimates of the parameters in the case of the standard linear model. A particular strength of our approach is that the M-step depends only on the form of the prior and it is independent of the form of the likelihood. We introduce several simple modifications of this EM procedure that allow for straightforward extension to generalised linear models. In experiments performed on simulated and real data, our approach performs comparable, or superior to, state-of-the-art sparse estimation methods in terms of statistical performance and computational cost.

HYDRA: Competing convolutional kernels for fast and accurate time series classification

We demonstrate a simple connection between dictionary methods for time series classification, which involve extracting and counting symbolic patterns in time series, and methods based on transforming input time series using convolutional kernels, namely ROCKET and its variants. We show that by adjusting a single hyperparameter it is possible to move by degrees between models resembling dictionary methods and models resembling ROCKET. We present HYDRA, a simple, fast, and accurate dictionary method for time series classification using competing convolutional kernels, combining key aspects of both ROCKET and conventional dictionary methods. HYDRA is faster and more accurate than the most accurate existing dictionary methods, and can be combined with ROCKET and its variants to further improve the accuracy of these methods.

MINIROCKET: A Very Fast (Almost) Deterministic Transform for Time Series Classification

Until recently, the most accurate methods for time series classification were limited by high computational complexity. ROCKET achieves state-of-the-art accuracy with a fraction of the computational expense of most existing methods by transforming input time series using random convolutional kernels, and using the transformed features to train a linear classifier. We reformulate ROCKET into a new method, MINIROCKET, making it up to 75 times faster on larger datasets, and making it almost deterministic (and optionally, with additional computational expense, fully deterministic), while maintaining essentially the same accuracy. Using this method, it is possible to train and test a classifier on all of 109 datasets from the UCR archive to state-of-the-art accuracy in less than 10 minutes. MINIROCKET is significantly faster than any other method of comparable accuracy (including ROCKET), and significantly more accurate than any other method of even roughly-similar computational expense. As such, we suggest that MINIROCKET should now be considered and used as the default variant of ROCKET.