Searcharxiv⌕ Search

arXiv subjects

Gavin Brown

Publications and source records attributed to Gavin Brown.

At least 37 records · Page 2Linked to original sources
arXiv2023

Outlier detection of vital sign trajectories from COVID-19 patients

In this work, we present a novel trajectory comparison algorithm to identify abnormal vital sign trends, with the aim of improving recognition of deteriorating health. There is growing interest in continuous wearable vital sign sensors for monitoring patients remotely at home. These monitors are usually coupled to an alerting system, which is triggered when vital sign measurements fall outside a predefined normal range. Trends in vital signs, such as increasing heart rate, are often indicative of deteriorating health, but are rarely incorporated into alerting systems. We introduce a dynamic time warp distance-based measure to compare time series trajectories. We split each multi-variable sign time series into 180 minute, non-overlapping epochs. We then calculate the distance between all pairs of epochs. Each epoch is characterized by its mean pairwise distance (average link distance) to all other epochs, with clusters forming with nearby epochs. We demonstrate in synthetically generated data that this method can identify abnormal epochs and cluster epochs with similar trajectories. We then apply this method to a real-world data set of vital signs from 8 patients who had recently been discharged from hospital after contracting COVID-19. We show how outlier epochs correspond well with the abnormal vital signs and identify patients who were subsequently readmitted to hospital.

cs.LG↗
arXiv2022

Computation and data in the classification of Fano varieties

Fano varieties are 'atomic pieces' of algebraic varieties, the shapes that can be defined by polynomial equations. We describe the role of computation and database methods in the construction and classification of Fano varieties, with an emphasis on three-dimensional Fano varieties with mild singularities called Q-Fano threefolds. The classification of Q-Fano threefolds has been open for several decades, but there has been significant recent progress. These advances combine computational algebraic geometry and large-scale data analysis with new ideas that originated in theoretical physics.

math.AG↗
arXiv2022

Strong Memory Lower Bounds for Learning Natural Models

We give lower bounds on the amount of memory required by one-pass streaming algorithms for solving several natural learning problems. In a setting where examples lie in $\{0,1\}^d$ and the optimal classifier can be encoded using $κ$ bits, we show that algorithms which learn using a near-minimal number of examples, $\tilde O(κ)$, must use $\tilde Ω( dκ)$ bits of space. Our space bounds match the dimension of the ambient space of the problem's natural parametrization, even when it is quadratic in the size of examples and the final classifier. For instance, in the setting of $d$-sparse linear classifiers over degree-2 polynomial features, for which $κ=Θ(d\log d)$, our space lower bound is $\tildeΩ(d^2)$. Our bounds degrade gracefully with the stream length $N$, generally having the form $\tildeΩ\left(dκ\cdot \fracκ{N}\right)$. Bounds of the form $Ω(dκ)$ were known for learning parity and other problems defined over finite fields. Bounds that apply in a narrow range of sample sizes are also known for linear regression. Ours are the first such bounds for problems of the type commonly seen in recent learning applications that apply for a large range of input sizes.

cs.LG↗
arXiv2022

Bias-Variance Decompositions for Margin Losses

We introduce a novel bias-variance decomposition for a range of strictly convex margin losses, including the logistic loss (minimized by the classic LogitBoost algorithm), as well as the squared margin loss and canonical boosting loss. Furthermore, we show that, for all strictly convex margin losses, the expected risk decomposes into the risk of a "central" model and a term quantifying variation in the functional margin with respect to variations in the training data. These decompositions provide a diagnostic tool for practitioners to understand model overfitting/underfitting, and have implications for additive ensemble models -- for example, when our bias-variance decomposition holds, there is a corresponding "ambiguity" decomposition, which can be used to quantify model diversity.

stat.ML↗
arXiv2022

Performative Prediction in a Stateful World

Deployed supervised machine learning models make predictions that interact with and influence the world. This phenomenon is called performative prediction by Perdomo et al. (ICML 2020). It is an ongoing challenge to understand the influence of such predictions as well as design tools so as to control that influence. We propose a theoretical framework where the response of a target population to the deployed classifier is modeled as a function of the classifier and the current state (distribution) of the population. We show necessary and sufficient conditions for convergence to an equilibrium of two retraining algorithms, repeated risk minimization and a lazier variant. Furthermore, convergence is near an optimal classifier. We thus generalize results of Perdomo et al., whose performativity framework does not assume any dependence on the state of the target population. A particular phenomenon captured by our model is that of distinct groups that acquire information and resources at different rates to be able to respond to the latest deployed classifier. We study this phenomenon theoretically and empirically.

cs.LG↗
arXiv2022

Kawamata boundedness for Fano threefolds and the Graded Ring Database

We explain an effective Kawamata boundedness result for Mori-Fano 3-folds. In particular, we describe a list of 39,550 possible Hilbert series of semistable Mori-Fano 3-folds, with examples to explain its meaning, its relationship to known classifications and the wealth of more general Fano 3-folds it contains, as well as its application to the on-going classification of Fano 3-folds.

math.AG↗
arXiv2021

Ensembles of Spiking Neural Networks

This paper demonstrates how to construct ensembles of spiking neural networks producing state-of-the-art results, achieving classification accuracies of 98.71%, 100.0%, and 99.09%, on the MNIST, NMNIST and DVS Gesture datasets respectively. Furthermore, this performance is achieved using simplified individual models, with ensembles containing less than 50% of the parameters of published reference models. We provide comprehensive exploration on the effect of spike train interpretation methods, and derive the theoretical methodology for combining model predictions such that performance improvements are guaranteed for spiking ensembles. For this, we formalize spiking neural networks as GLM predictors, identifying a suitable representation for their target domain. Further, we show how the diversity of our spiking ensembles can be measured using the Ambiguity Decomposition. The work demonstrates how ensembling can overcome the challenges of producing individual SNN models which can compete with traditional deep neural networks, and creates systems with fewer trainable parameters and smaller memory footprints, opening the door to low-power edge applications, e.g. implemented on neuromorphic hardware.

cs.NE↗
arXiv2021

When is Memorization of Irrelevant Training Data Necessary for High-Accuracy Learning?

Modern machine learning models are complex and frequently encode surprising amounts of information about individual inputs. In extreme cases, complex models appear to memorize entire input examples, including seemingly irrelevant information (social security numbers from text, for example). In this paper, we aim to understand whether this sort of memorization is necessary for accurate learning. We describe natural prediction problems in which every sufficiently accurate training algorithm must encode, in the prediction model, essentially all the information about a large subset of its training examples. This remains true even when the examples are high-dimensional and have entropy much higher than the sample size, and even when most of that information is ultimately irrelevant to the task at hand. Further, our results do not depend on the training algorithm or the class of models used for learning. Our problems are simple and fairly natural variants of the next-symbol prediction and the cluster labeling tasks. These tasks can be seen as abstractions of text- and image-related prediction problems. To establish our results, we reduce from a family of one-way communication problems for which we prove new information complexity lower bounds. Additionally, we present synthetic-data experiments demonstrating successful attacks on logistic regression and neural network classifiers.

cs.LG↗
arXiv2020

To Ensemble or Not Ensemble: When does End-To-End Training Fail?

End-to-End training (E2E) is becoming more and more popular to train complex Deep Network architectures. An interesting question is whether this trend will continue-are there any clear failure cases for E2E training? We study this question in depth, for the specific case of E2E training an ensemble of networks. Our strategy is to blend the gradient smoothly in between two extremes: from independent training of the networks, up to to full E2E training. We find clear failure cases, where over-parameterized models cannot be trained E2E. A surprising result is that the optimum can sometimes lie in between the two, neither an ensemble or an E2E system. The work also uncovers links to Dropout, and raises questions around the nature of ensemble diversity and multi-branch networks.

stat.ML↗
arXiv2020

Margin Maximization as Lossless Maximal Compression

The ultimate goal of a supervised learning algorithm is to produce models constructed on the training data that can generalize well to new examples. In classification, functional margin maximization -- correctly classifying as many training examples as possible with maximal confidence --has been known to construct models with good generalization guarantees. This work gives an information-theoretic interpretation of a margin maximizing model on a noiseless training dataset as one that achieves lossless maximal compression of said dataset -- i.e. extracts from the features all the useful information for predicting the label and no more. The connection offers new insights on generalization in supervised machine learning, showing margin maximization as a special case (that of classification) of a more general principle and explains the success and potential limitations of popular learning algorithms like gradient boosting. We support our observations with theoretical arguments and empirical evidence and identify interesting directions for future work.

cs.LG↗
arXiv2020

Better Boosting with Bandits for Online Learning

Probability estimates generated by boosting ensembles are poorly calibrated because of the margin maximization nature of the algorithm. The outputs of the ensemble need to be properly calibrated before they can be used as probability estimates. In this work, we demonstrate that online boosting is also prone to producing distorted probability estimates. In batch learning, calibration is achieved by reserving part of the training data for training the calibrator function. In the online setting, a decision needs to be made on each round: shall the new example(s) be used to update the parameters of the ensemble or those of the calibrator. We proceed to resolve this decision with the aid of bandit optimization algorithms. We demonstrate superior performance to uncalibrated and naively-calibrated on-line boosting ensembles in terms of probability estimation. Our proposed mechanism can be easily adapted to other tasks(e.g. cost-sensitive classification) and is robust to the choice of hyperparameters of both the calibrator and the ensemble.

cs.LG↗
arXiv2018

Tutorial on Tom and Jerry: the two smoothings of the anticanonical cone over PP(1,2,3)

This is a first introduction to unprojection methods, and more specifically to Tom and Jerry unprojections. These two harmless tricks deserve to be better known, since they answer many practical questions about constructing codimension 4 Gorenstein subschemes. In particular, we discuss here the two smoothing components of the anticanonical cone over PP(1,2,3).

math.AG↗
arXiv2018

Is feature selection secure against training data poisoning?

Learning in adversarial settings is becoming an important task for application domains where attackers may inject malicious data into the training set to subvert normal operation of data-driven technologies. Feature selection has been widely used in machine learning for security applications to improve generalization and computational efficiency, although it is not clear whether its use may be beneficial or even counterproductive when training data are poisoned by intelligent attackers. In this work, we shed light on this issue by providing a framework to investigate the robustness of popular feature selection methods, including LASSO, ridge regression and the elastic net. Our results on malware detection show that feature selection methods can be significantly compromised under attack (we can reduce LASSO to almost random choices of feature sets by careful insertion of less than 5% poisoned training samples), highlighting the need for specific countermeasures.

cs.LG↗
arXiv2018

Minimax rates for cost-sensitive learning on manifolds with approximate nearest neighbours

We study the approximate nearest neighbour method for cost-sensitive classification on low-dimensional manifolds embedded within a high-dimensional feature space. We determine the minimax learning rates for distributions on a smooth manifold, in a cost-sensitive setting. This generalises a classic result of Audibert and Tsybakov. Building upon recent work of Chaudhuri and Dasgupta we prove that these minimax rates are attained by the approximate nearest neighbour algorithm, where neighbours are computed in a randomly projected low-dimensional space. In addition, we give a bound on the number of dimensions required for the projection which depends solely upon the reach and dimension of the manifold, combined with the regularity of the marginal.

cs.LG↗
arXiv2018

Diversity and degrees of freedom in regression ensembles

Ensemble methods are a cornerstone of modern machine learning. The performance of an ensemble depends crucially upon the level of diversity between its constituent learners. This paper establishes a connection between diversity and degrees of freedom (i.e. the capacity of the model), showing that diversity may be viewed as a form of inverse regularisation. This is achieved by focusing on a previously published algorithm Negative Correlation Learning (NCL), in which model diversity is explicitly encouraged through a diversity penalty term in the loss function. We provide an exact formula for the effective degrees of freedom in an NCL ensemble with fixed basis functions, showing that it is a continuous, convex and monotonically increasing function of the diversity parameter. We demonstrate a connection to Tikhonov regularisation and show that, with an appropriately chosen diversity parameter, an NCL ensemble can always outperform the unregularised ensemble in the presence of noise. We demonstrate the practical utility of our approach by deriving a method to efficiently tune the diversity parameter. Finally, we use a Monte-Carlo estimator to extend the connection between diversity and degrees of freedom to ensembles of deep neural networks.

cs.LG↗
arXiv2018

The K-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates

In this paper we propose and explore the k-Nearest Neighbour UCB algorithm for multi-armed bandits with covariates. We focus on a setting where the covariates are supported on a metric space of low intrinsic dimension, such as a manifold embedded within a high dimensional ambient feature space. The algorithm is conceptually simple and straightforward to implement. The k-Nearest Neighbour UCB algorithm does not require prior knowledge of the either the intrinsic dimension of the marginal distribution or the time horizon. We prove a regret bound for the k-Nearest Neighbour UCB algorithm which is minimax optimal up to logarithmic factors. In particular, the algorithm automatically takes advantage of both low intrinsic dimensionality of the marginal distribution over the covariates and low noise in the data, expressed as a margin condition. In addition, focusing on the case of bounded rewards, we give corresponding regret bounds for the k-Nearest Neighbour KL-UCB algorithm, which is an analogue of the KL-UCB algorithm adapted to the setting of multi-armed bandits with covariates. Finally, we present empirical results which demonstrate the ability of both the k-Nearest Neighbour UCB and k-Nearest Neighbour KL-UCB to take advantage of situations where the data is supported on an unknown sub-manifold of a high-dimensional feature space.

cs.LG↗
arXiv2017

Gopakumar-Vafa invariants do not determine flops

Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new, and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar-Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analytic classification of 3-fold flops via contraction algebras.

math.AG↗