Feature selection is the data analysis process that selects a smaller and curated subset of the original dataset by filtering out data (features) which are irrelevant or redundant. The most important features can be ranked and selected based on statistical measures, such as mutual information. Feature selection not only reduces the size of dataset as well as the execution time for training Machine Learning (ML) models, but it can also improve the accuracy of the inference. This paper analyses mutual-information-based feature selection for resource-constrained FPGAs and proposes FINESSD, a novel approach that can be deployed for near-storage acceleration. This paper highlights that the Mutual Information Maximization (MIM) algorithm does not require multiple passes over the data while being a good trade-off between accuracy and FPGA resources, when approximated appropriately. The new FPGA accelerator for MIM generated by FINESSD can fully utilize the NVMe bandwidth of a modern SSD and perform feature selection without requiring full dataset transfers onto the main processor. The evaluation using a Samsung SmartSSD over small, large and out-of-core datasets shows that, compared to the mainstream multiprocessing Python ML libraries and an optimized C library, FINESSD yields up to 35 × and 19 ×speedup respectively while being more than 70 × more energy efficient for large, out-of-core datasets.

We present a theory of ensemble diversity, explaining the nature of diversity for a wide range of supervised learning scenarios. This challenge has been referred to as the “holy grail” of ensemble learning, an open research issue for over 30 years. Our framework reveals that diversity is in fact a hidden dimension in the bias-variance decomposition of the ensemble loss. We prove a family of exact bias-variance-diversity decompositions, for a wide range of losses in both regression and classification, e.g., squared, cross-entropy, and Poisson losses. For losses where an additive bias-variance decomposition is not available (e.g., 0/1 loss) we present an alternative approach: quantifying the effects of diversity, which turn out to be dependent on the label distribution. Overall, we argue that diversity is a measure of model fit, in precisely the same sense as bias and variance, but accounting for statistical dependencies between ensemble members. Thus, we should not be ‘maximising diversity’ as so many works aim to do---instead, we have a bias/variance/diversity trade-off to manage.

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.

Information theoretic feature selection methods quantify the importance of each feature by estimating mutual information terms to capture: the relevancy, the redundancy and the complementarity. These terms are commonly estimated by maximum likelihood, while an under-explored area of research is how to use shrinkage methods instead. Our work suggests a novel shrinkage method for data-efficient estimation of information theoretic terms. The small sample behaviour makes it particularly suitable for estimation of discrete distributions with large number of categories (bins). Using our novel estimators we derive a framework for generating feature selection criteria that capture any high-order feature interaction for redundancy and complementarity. We perform a thorough empirical study across datasets from diverse sources and using various evaluation measures. Our first finding is that our shrinkage based methods achieve better results, while they keep the same computational cost as the simple maximum likelihood based methods. Furthermore, under our framework we derive efficient novel high-order criteria that outperform state-of-the-art methods in various tasks.

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 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. Unlike previous methods such as the UCBogram and Adaptively Binned Successive Elimination, the $k$-Nearest Neighbour UCB algorithm does not require prior knowledge of the intrinsic dimension of the marginal distribution. It is also naturally anytime, without resorting to the doubling trick. 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.

Motivation: The identification of biomarkers to support decision-making is central to personalized medicine, in both clinical and research scenarios. The challenge can be seen in two halves: identifying predictive markers, which guide the development/use of tailored therapies; and identifying prognostic markers, which guide other aspects of care and clinical trial planning, i.e. prognostic markers can be considered as covariates for stratification. Mistakenly assuming a biomarker to be predictive, when it is in fact largely prognostic (and vice-versa) is highly undesirable, and can result in financial, ethical and personal consequences. We present a framework for data-driven ranking of biomarkers on their prognostic/predictive strength, using a novel information theoretic method. This approach provides a natural algebra to discuss and quantify the individual predictive and prognostic strength, in a self-consistent mathematical framework. Results: Our contribution is a novel procedure, INFO+, which naturally distinguishes the prognostic versus predictive role of each biomarker and handles higher order interactions. In a comprehensive empirical evaluation INFO+ outperforms more complex methods, most notably when noise factors dominate, and biomarkers are likely to be falsely identified as predictive, when in fact they are just strongly prognostic. Furthermore, we show that our methods can be 1-3 orders of magnitude faster than competitors, making it useful for biomarker discovery in 'big data' scenarios. Finally, we apply our methods to identify predictive biomarkers on two real clinical trials, and introduce a new graphical representation that provides greater insight into the prognostic and predictive strength of each biomarker. Availability and implementation: R implementations of the suggested methods are available at https://github.com/sechidis. Supplementary information: Supplementary data are available at Bioinformatics online.

Bioinformatics, (2018) https://doi.org/10.1093/bioinformatics/bty357 The author wishes to apologize for a mistake in Figure 2 in the above manuscript. The figure appears correctly below: When biomarkers have both prognostic/predictive strength (M-1) VT achieves higher TPR, otherwise (M-2) the gains in TPR are vanishing. In terms of FNRProg.⁠, VT always has very high error rate on selecting solely prognostic biomarkers as predictive, and it performs worse than random selection. This is the average TPR/FNRProg. over 200 simulated datasets for three different values of the predictive strength θ: 1/5 means a strongly prognostic signal, 1 means equal strength between prognostic and predictive signals, and 5 means a strongly predictive signal. The sample size is 2000, and the dimensionality p = 30 biomarkers. Dashed lines show the TPR/FNRProg. if we were ranking the biomarkers at random. (a) M-1: Biomarkers can be both prognostic and predictive. (b) M-2: Biomarkers are solely either prognostic or predictive The paper has been corrected online.

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.

What is the simplest thing you can do to solve a problem? In the context of semi-supervised feature selection, we tackle exactly this—how much we can gain from two simple classifier-independent strategies. If we have some binary labelled data and some unlabelled, we could assume the unlabelled data are all positives, or assume them all negatives. These minimalist, seemingly naive, approaches have not previously been studied in depth. However, with theoretical and empirical studies, we show they provide powerful results for feature selection, via hypothesis testing and feature ranking. Combining them with some “soft” prior knowledge of the domain, we derive two novel algorithms ( Semi -JMI, Semi -IAMB) that outperform significantly more complex competing methods, showing particularly good performance when the labels are missing-not-at-random. We conclude that simple approaches to this problem can work surprisingly well, and in many situations we can provably recover the exact feature selection dynamics, as if we had labelled the entire dataset .

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.

Microarray data classification has been typically seen as a difficult challenge for machine learning researchers mainly due to its high dimension in features while sample size is small. Because of this particularity, feature selection is usually applied trying to reduce its high dimensionality. However, existing algorithms may not scale well when dealing with this amount of features, and a possible solution is to distribute the features into several nodes. In this work we explore the process of distribution on microarray data - which has recently gained attention - and we evaluate to what extent it is possible to obtain similar results as those obtained with the whole dataset. We performed experiments with different aggregation methods, feature rankers and also evaluated the effect of distributing the feature ranking process in the subsequent classification performance.

Under-reporting occurs in survey data when there is a reason for participants to give a false negative response to a question, e.g. maternal smoking in epidemiological studies. Failing to correct this misreporting introduces biases and it may lead to misinformed decision making. Our work provides methods of correcting for this bias, by reinterpreting it as a missing data problem, and particularly learning from positive and unlabelled data. Focusing on information theoretic approaches we have three key contributions: (1) we provide a method to perform valid independence tests with known power by incorporating prior knowledge over misreporting; (2) we derive corrections for point/interval estimates of the mutual information that capture both relevance and redundancy; and finally, (3) we derive different ways for ranking under-reported risk factors. Furthermore, we show how to use our results in real-world problems and machine learning tasks.

We provide a unifying perspective for two decades of work on cost-sensitive Boosting algorithms. When analyzing the literature 1997–2016, we find 15 distinct cost-sensitive variants of the original algorithm; each of these has its own motivation and claims to superiority—so who should we believe? In this work we critique the Boosting literature using four theoretical frameworks: Bayesian decision theory, the functional gradient descent view, margin theory, and probabilistic modelling. Our finding is that only three algorithms are fully supported—and the probabilistic model view suggests that all require their outputs to be calibrated for best performance. Experiments on 18 datasets across 21 degrees of imbalance support the hypothesis—showing that once calibrated, they perform equivalently, and outperform all others. Our final recommendation—based on simplicity, flexibility and performance—is to use the original Adaboost algorithm with a shifted decision threshold and calibrated probability estimates.

Current parallelizing compilers can tackle applications exercising regular access patterns on arrays or affine indices, where data dependencies can be expressed in a linear form. Unfortunately, there are cases that independence between statements of code cannot be guaranteed and thus the compiler conservatively produces sequential code. Programs that involve extensive pointer use, irregular access patterns, and loops with unknown number of iterations are examples of such cases. This limits the extraction of parallelism in cases where dependencies are rarely or never triggered at runtime. Speculative parallelism refers to methods employed during program execution that aim to produce a valid parallel execution schedule for programs immune to static parallelization. The motivation for this article is to review recent developments in the area of compiler-driven software speculation for thread-level parallelism and how they came about. The article is divided into two parts. In the first part the fundamentals of speculative parallelization for thread-level parallelism are explained along with a design choice categorization for implementing such systems. Design choices include the ways speculative data is handled, how data dependence violations are detected and resolved, how the correct data are made visible to other threads, or how speculative threads are scheduled. The second part is structured around those design choices providing the advances and trends in the literature with reference to key developments in the area. Although the focus of the article is in software speculative parallelization, a section is dedicated for providing the interested reader with pointers and references for exploring similar topics such as hardware thread-level speculation, transactional memory, and automatic parallelization.

With the growth of high dimensional data, feature selection is a vital component of machine learning as well as an important stand alone data analytics tool. Without it, the computation cost of big data analytics can become unmanageable and spurious correlations and noise can reduce the accuracy of any results. Feature selection removes irrelevant and redundant information leading to faster, more reliable data analysis. Feature selection techniques based on information theory are among the fastest known and the Manchester AnalyticS Toolkit (MAST) provides an efficient, parallel and scalable implementation of these methods. This paper considers a number of data structures for storing the frequency counters that underpin MAST. We show that preprocessing the data to reduce the number of zero-valued counters in an array structure results in an order of magnitude reduction in both memory usage and execution time compared to state of the art structures that use explicit mappings to avoid zero-valued counters. We also describe a number of parallel processing techniques that enable MAST to scale linearly with the number of processors even on NUMA architectures. MAST targets scale-up servers rather than scale-out clusters and we show that it performs orders of magnitude faster than existing tools. Moreover, we show that MAST is 3.5 times faster than a scale-out solution built for Spark running on the same server. As an example of the performance of MAST, we were able to process a dataset of 100 million examples and 100,000 features in under 10 minutes on a four socket server which each socket containing an 8-core Intel Xeon E5-4620 processor.

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.

We study a dichotomy of scientific styles, unifying and diversifying , as proposed by Freeman J. Dyson. We discuss the extent to which the dichotomy transfers from the natural sciences (where Dyson proposed it) to the field of Pattern Recognition . To address this we must firstly ask what it means to be a “unifier” or “diversifier” in a field, and what are the relative merits of each style of thinking. Secondly, given that Dyson applied this to the sciences , does it also apply in a field known to be a blend of science and engineering? Parallels are drawn to Platonic/Aristotelian views, and to Cartesian/Baconian science, and questions are asked on what drives the Kuhnian paradigm shifts of our field. This article is intended not to marginalise individuals into categories (unifier/diversifier) but instead to demonstrate the utility of philosophical reflection on our field, showing the depth and complexities a seemingly simple idea can unearth.

In this paper, we present a novel ensemble method random projection random discretization ensembles(RPRDE) to create ensembles of linear multivariate decision trees by using a univariate decision tree algorithm. The present method combines the better computational complexity of a univariate decision tree algorithm with the better representational power of linear multivariate decision trees. We develop random discretization (RD) method that creates random discretized features from continuous features. Random projection (RP) is used to create new features that are linear combinations of original features. A new dataset is created by augmenting discretized features (created by using RD) with features created by using RP. Each decision tree of a RPRD ensemble is trained on one dataset from the pool of these datasets by using a univariate decision tree algorithm. As these multivariate decision trees (because of features created by RP) have more representational power than univariate decision trees, we expect accurate decision trees in the ensemble. Diverse training datasets ensure diverse decision trees in the ensemble. We study the performance of RPRDE against other popular ensemble techniques using C4.5 tree as the base classifier. RPRDE matches or outperforms other popular ensemble methods. Experiments results also suggest that the proposed method is quite robust to the class noise.

As data sets become ever larger it becomes increasingly complex to apply traditional machine learning techniques to them. Feature selection can greatly reduce the computational requirements of machine learning but it too can be memory intensive. In this paper we explore the use of succinct data structures called sketches for probability estimation as a component of information theoretic feature selection. These data structures are sublinear in the number of items but were designed only for estimating the frequency of the most frequent items. To the best of our knowledge this is the first time they have been examined for estimating the frequency of all items and we find that often some information theoretic measures can be estimated to within a few percent of the correct values.

Thread-Level Speculation (TLS) overcomes limitations intrinsic with conservative compile-time auto-parallelizing tools by extracting parallel threads optimistically and only ensuring absence of data dependence violations at runtime. A significant barrier for adopting TLS (implemented in software) is the overheads associated with maintaining speculative state. Based on previous TLS limit studies, we observe that on future multicore systems we will likely have more cores idle than those which traditional TLS would be able to harness. This implies that a TLS system should focus on optimizing for small number of cores and find efficient ways to take advantage of the idle cores. Furthermore, research on optimistic systems has covered two important implementation design points: eager vs. lazy version management. With this knowledge, we propose new simple and effective techniques to reduce the execution time overheads for both of these design points. This article describes a novel compact version management data structure optimized for space overhead when using a small number of TLS threads. Furthermore, we describe two novel software runtime parallelization systems that utilize this compact data structure. The first software TLS system, MiniTLS, relies on eager memory data management (in-place updates) and, thus, when a misspeculation occurs a rollback process is required. MiniTLS takes advantage of the novel compact version management representation to parallelize the rollback process and is able to recover from misspeculation faster than existing software eager TLS systems. The second one, Lector (Lazy inspECTOR) is based on lazy version management. Since we have idle cores, the question is whether we can create “helper” tasks to determine whether speculation is actually needed without stopping or damaging the speculative execution. In Lector, for each conventional TLS thread running speculatively with lazy version management, there is associated with it a lightweight inspector . The inspector threads execute alongside to verify quickly whether data dependencies will occur. Inspector threads are generated by standard techniques for inspector/executor parallelization. We have applied both TLS systems to seven Java sequential benchmarks, including three benchmarks from SPECjvm2008. Two out of the seven benchmarks exhibit misspeculations. MiniTLS experiments report average speedups of 1.8x for 4 threads increasing close to 7x speedups with 32 threads. Facilitated by our novel compact representation, MiniTLS reduces the space overhead over state-of-the-art software TLS systems between 96% on 2 threads and 40% on 32 threads. The experiments for Lector, report average speedups of 1.7x for 2 threads (that is 1 TLS + 1 Inspector threads) increasing close to 8.2x speedups with 32 threads (16 + 16 threads). Compared to a well established software TLS baseline, Lector performs on average 1.7x faster for 32 threads and in no case ( x TLS + x Inspector threads) Lector delivers worse performance than the baseline TLS with the equivalent number of TLS threads (i.e. x TLS threads) nor doubling the equivalent number of TLS threads (i.e., x + x TLS threads).

In recent years there have been efforts to develop a probabilistic framework to explain the workings of a Learning Classifier System. This direction of research has met with limited success due to the intractability of complicated heuristic training rules used by the learning classifier systems. In this paper, we derive a learning classifier system from a mixture of experts that is similar to a sUpervised Classifier System (UCS) in terms of its training and prediction routines. We start by framing the learning model as a mixture of experts which uses an Expectation Maximisation (EM) procedure to learn its parameters. The batch updates of the EM is then converted into online updates and finally into a GA based sampled online update thus ending up with a classifier system similar to a sUpervised Classifier System. In this paper, we show the effectiveness of such a system as compared to UCS through a series of comparative studies on test datasets.

UCS is a a Learning Classifier System (LCS) which evolves condition-action rules for supervised classification tasks. In UCS the fitness of a rule is based on its accuracy raised to a power $\nu$, and this fitness is used in both the search for good rules (via a genetic algorithm) and in a classification vote. We trace the origin of the UCS fitness function through three successive versions of the XCS accuracy function, for which we present previously unpublished details and rationales. Through numerical examples and empirical studies we demonstrate that $\nu$ tunes both selective pressure in genetic search and the voting margin in classification, and demonstrate that $\nu$ (or some alternative) is necessary for both. We appeal to margin theory to explain the effect on classification and so connect the LCS field with ensemble systems, and we suggest $\nu$ might be useful as a noise-correction parameter. We argue that the design of fitness functions has always been the central difficulty for Michigan LCSs and that better understanding can help both parameterisation of existing algorithms and development of new ones.

MapReduce has been widely accepted as a simple programming pattern that can form the basis for efficient, large-scale, distributed data processing. The success of the MapReduce pattern has led to a variety of implementations for different computational scenarios. In this paper we present MRJ, a MapReduce Java framework for multi-core architectures. We evaluate its scalability on a four-core, hyperthreaded Intel Core i7 processor, using a set of standard MapReduce benchmarks. We investigate the significant impact that Java runtime garbage collection has on the performance and scalability of MRJ. We propose the use of memory management auto-tuning techniques based on machine learning. With our auto-tuning approach, we are able to achieve MRJ performance within 10% of optimal on 75% of our benchmark tests.

This paper argues that economic theory can improve our understanding of memory management. We introduce the allocation curve, as an analogue of the demand curve from microeconomics. An allocation curve for a program characterises how the amount of garbage collection activity required during its execution varies in relation to the heap size associated with that program. The standard treatment of microeconomic demand curves (shifts and elasticity) can be applied directly and intuitively to our new allocation curves. As an application of this new theory, we show how allocation elasticity can be used to control the heap growth rate for variable sized heaps in Jikes RVM.

We present a probabilistic formulation of UCS (a sUpervised Classifier System). UCS is shown to be a special case of mixture of experts where the experts are learned independently and later combined during prediction. In this work, we develop the links between the constituent components of UCS and a mixture of experts, thus lending UCS a strong analytical background. We find during our analysis that mixture of experts is a more generic formulation of UCS and possesses more generalization capability and flexibility than UCS, which is also verified using empirical evaluations. This is the first time that a simple probabilistic model has been proposed for UCS and we believe that this work will form a useful tool to analyse Learning Classifier Systems and gain useful insights into their working.

Credit assignment is a fundamental issue for the Learning Classifier Systems literature. We engage in a detailed investigation of credit assignment in one recent system called UCS, and in the process uncover two previously undocumented features. We draw on techniques from the classical pattern recognition literature, showing how to analytically derive an optimal credit assignment system, given certain assumptions. Our primary aim is not to improve accuracy, but to better understand the system and put it on a more solid theoreticalfoundation. Nonetheless, empirical results on benign data demonstrate our new system, called UCSpv (UCS with principled voting), can match or exceed the original UCS. Further, its fitness function is principled, and, unlike that of UCS, requires no tuning. However, on more difficult data it seems UCSpv does need some form of tuning or correction. We believe the framework we adopt offers a promising new direction for LCS research, providing principled methods for action selection and bringing LCS closer to the mainstream pattern recognition literature.

Java program execution times vary greatly with different garbage collection algorithms. Until now, it has not been possible to determine the best GC algorithm for aparticular program without exhaustively profiling that program for all available GC algorithms. This paper presents a new approach. We use machine learning techniques to build a prediction model that, given asingle profile run of a previously unseen Java program,can predict a good GC algorithm for that program. We implement this technique in Jikes RVM and test it onseveral standard benchmark suites. Our techniqueachieves 5% speedup in overall execution time (averagedacross all test programs for all heap sizes) compared with selecting the default GC algorithm in every trial. We present further experiments to show that an oracle predictor could achieve an average 17% speedup on the same experiments. In addition, we provide evidence to suggest that GC behaviour is sometimes independent of program inputs. These observations lead us to propose that intelligent selection of GC algorithms is suitably straight forward, efficient and effective to merit further exploration regarding its potential inclusion in the general Java software deployment process.

A variant of a sparse distributed memory (SDM) is shown to have the capability of storing and recalling patterns containing rank-order information. These are patterns where information is encoded not only in the subset of neuron outputs that fire, but also in the order in which that subset fires. This is an interesting companion to several recent works in the neuroscience literature, showing that human memories may be stored in terms of neural spike timings. In our model, the ordering is stored in static synaptic weights using a Hebbian single-shot learning algorithm, and can be reliably recovered whenever the associated input is supplied. It is shown that the memory can operate using only unipolar binary connections throughout. The behavior of the memory under noisy input conditions is also investigated. It is shown that the memory is capable of improving the quality of the data that passes through it. That is, under appropriate conditions the output retrieved from the memory is less noisy than the input used to retrieve it. Thus, this memory architecture could be used as a component in a complex system with stable noise properties and, we argue, it can be implemented using spiking neurons.

Ensembles are a widely used and effective technique in machine learning---their success is commonly attributed to the degree of disagreement, or 'diversity', within the ensemble. For ensembles where the individual estimators output crisp class labels, this 'diversity' is not well understood and remains an open research issue. For ensembles of regression estimators, the diversity can be exactly formulated in terms of the covariance between individual estimator outputs, and the optimum level is expressed in terms of a bias-variance-covariance trade-off. Despite this, most approaches to learning ensembles use heuristics to encourage the right degree of diversity. In this work we show how to explicitly control diversity through the error function. The first contribution of this paper is to show that by taking the combination mechanism for the ensemble into account we can derive an error function for each individual that balances ensemble diversity with individual accuracy. We show the relationship between this error function and an existing algorithm called negative correlation learning, which uses a heuristic penalty term added to the mean squared error function. It is demonstrated that these methods control the bias-variance-covariance trade-off systematically, and can be utilised with any estimator capable of minimising a quadratic error function, for example MLPs, or RBF networks. As a second contribution, we derive a strict upper bound on the coefficient of the penalty term, which holds for any estimator that can be cast in a generalised linear regression framework, with mild assumptions on the basis functions. Finally we present the results of an empirical study, showing significant improvements over simple ensemble learning, and finding that this technique is competitive with a variety of methods, including boosting, bagging, mixtures of experts, and Gaussian processes, on a number of tasks.