Sebastian J. Vollmer

Oscar Giles, Kasra Hosseini, Grigorios Mingas et al.

Synthetic datasets are often presented as a silver-bullet solution to the problem of privacy-preserving data publishing. However, for many applications, synthetic data has been shown to have limited utility when used to train predictive models. One promising potential application of these data is in the exploratory phase of the machine learning workflow, which involves understanding, engineering and selecting features. This phase often involves considerable time, and depends on the availability of data. There would be substantial value in synthetic data that permitted these steps to be carried out while, for example, data access was being negotiated, or with fewer information governance restrictions. This paper presents an empirical analysis of the agreement between the feature importance obtained from raw and from synthetic data, on a range of artificially generated and real-world datasets (where feature importance represents how useful each feature is when predicting a the outcome). We employ two differentially-private methods to produce synthetic data, and apply various utility measures to quantify the agreement in feature importance as this varies with the level of privacy. Our results indicate that synthetic data can sometimes preserve several representations of the ranking of feature importance in simple settings but their performance is not consistent and depends upon a number of factors. Particular caution should be exercised in more nuanced real-world settings, where synthetic data can lead to differences in ranked feature importance that could alter key modelling decisions. This work has important implications for developing synthetic versions of highly sensitive data sets in fields such as finance and healthcare.

Tobias Schröder, Zijing Ou, Jen Ning Lim et al.

Energy-based models are a simple yet powerful class of probabilistic models, but their widespread adoption has been limited by the computational burden of training them. We propose a novel loss function called Energy Discrepancy (ED) which does not rely on the computation of scores or expensive Markov chain Monte Carlo. We show that ED approaches the explicit score matching and negative log-likelihood loss under different limits, effectively interpolating between both. Consequently, minimum ED estimation overcomes the problem of nearsightedness encountered in score-based estimation methods, while also enjoying theoretical guarantees. Through numerical experiments, we demonstrate that ED learns low-dimensional data distributions faster and more accurately than explicit score matching or contrastive divergence. For high-dimensional image data, we describe how the manifold hypothesis puts limitations on our approach and demonstrate the effectiveness of energy discrepancy by training the energy-based model as a prior of a variational decoder model.

David Selby, Kai Spriestersbach, Yuichiro Iwashita et al.

Large language models (LLMs) have been extensively studied for their abilities to generate convincing natural language sequences, however their utility for quantitative information retrieval is less well understood. Here we explore the feasibility of LLMs as a mechanism for quantitative knowledge retrieval to aid two data analysis tasks: elicitation of prior distributions for Bayesian models and imputation of missing data. We introduce a framework that leverages LLMs to enhance Bayesian workflows by eliciting expert-like prior knowledge and imputing missing data. Tested on diverse datasets, this approach can improve predictive accuracy and reduce data requirements, offering significant potential in healthcare, environmental science and engineering applications. We discuss the implications and challenges of treating LLMs as 'experts'.

Gerrit Großmann, Larisa Ivanova, Sai Leela Poduru et al.

According to Yuval Noah Harari, large-scale human cooperation is driven by shared narratives that encode common beliefs and values. This study explores whether such narratives can similarly nudge LLM agents toward collaboration. We use a finitely repeated public goods game in which LLM agents choose either cooperative or egoistic spending strategies. We prime agents with stories highlighting teamwork to different degrees and test how this influences negotiation outcomes. Our experiments explore four questions:(1) How do narratives influence negotiation behavior? (2) What differs when agents share the same story versus different ones? (3) What happens when the agent numbers grow? (4) Are agents resilient against self-serving negotiators? We find that story-based priming significantly affects negotiation strategies and success rates. Common stories improve collaboration, benefiting each agent. By contrast, priming agents with different stories reverses this effect, and those agents primed toward self-interest prevail. We hypothesize that these results carry implications for multi-agent system design and AI alignment.

Yilong Zang, Lingfei Ren, Yue Li et al.

Graph neural networks (GNNs) have shown promise in integrating protein-protein interaction (PPI) networks for identifying cancer genes in recent studies. However, due to the insufficient modeling of the biological information in PPI networks, more faithfully depiction of complex protein interaction patterns for cancer genes within the graph structure remains largely unexplored. This study takes a pioneering step toward bridging biological anomalies in protein interactions caused by cancer genes to statistical graph anomaly. We find a unique graph anomaly exhibited by cancer genes, namely weight heterogeneity, which manifests as significantly higher variance in edge weights of cancer gene nodes within the graph. Additionally, from the spectral perspective, we demonstrate that the weight heterogeneity could lead to the "flattening out" of spectral energy, with a concentration towards the extremes of the spectrum. Building on these insights, we propose the HIerarchical-Perspective Graph Neural Network (HIPGNN) that not only determines spectral energy distribution variations on the spectral perspective, but also perceives detailed protein interaction context on the spatial perspective. Extensive experiments are conducted on two reprocessed datasets STRINGdb and CPDB, and the experimental results demonstrate the superiority of HIPGNN.

Sumantrak Mukherjee, Serafima Lebedeva, Valentin Margraf et al.

We propose a novel Bayesian framework for efficient exploration in contextual multi-task multi-armed bandit settings, where the context is only observed partially and dependencies between reward distributions are induced by latent context variables. In order to exploit these structural dependencies, our approach integrates observations across all tasks and learns a global joint distribution, while still allowing personalised inference for new tasks. In this regard, we identify two key sources of epistemic uncertainty, namely structural uncertainty in the latent reward dependencies across arms and tasks, and user-specific uncertainty due to incomplete context and limited interaction history. To put our method into practice, we represent the joint distribution over tasks and rewards using a particle-based approximation of a log-density Gaussian process. This representation enables flexible, data-driven discovery of both inter-arm and inter-task dependencies without prior assumptions on the latent variables. Empirically, we demonstrate that our method outperforms baselines such as hierarchical model bandits, especially in settings with model misspecification or complex latent heterogeneity.

Dennis Wagner, Arjun Nair, Billy Joe Franks et al.

Undetected anomalies in time series can trigger catastrophic failures in safety-critical systems, such as chemical plant explosions or power grid outages. Although many detection methods have been proposed, their performance remains unclear because current metrics capture only narrow aspects of the task and often yield misleading results. We address this issue by introducing verifiable properties that formalize essential requirements for evaluating time-series anomaly detection. These properties enable a theoretical framework that supports principled evaluations and reliable comparisons. Analyzing 37 widely used metrics, we show that most satisfy only a few properties, and none satisfy all, explaining persistent inconsistencies in prior results. To close this gap, we propose LARM, a flexible metric that provably satisfies all properties, and extend it to ALARM, an advanced variant meeting stricter requirements.

Siddharth Gangwar, David A. Selby, Sebastian J. Vollmer

Making a good graphic that accurately and efficiently conveys the desired message to the audience is both an art and a science, typically not taught in the data science curriculum. Visualisation makeovers are exercises where the community exchange feedback to improve charts and data visualizations. Can multi-modal large language models (LLMs) emulate this task? Given a plot in the form of an image file, or the code used to generate it, an LLM, primed with a list of visualization best practices, is employed to semi-automatically generate constructive criticism to produce a better plot. Our system is centred around prompt engineering of a pre-trained model, relying on a combination of userspecified guidelines and any latent knowledge of data visualization practices that might lie within an LLMs training corpus. Unlike other works, the focus is not on generating valid visualization scripts from raw data or prompts, but on educating the user how to improve their existing data visualizations according to an interpretation of best practices. A quantitative evaluation is performed to measure the sensitivity of the LLM agent to various plotting issues across different chart types. We make the tool available as a simple self-hosted applet with an accessible Web interface.

Sumantrak Mukherjee, Mengyan Zhang, Seth Flaxman et al.

We study the problem of globally optimising a target variable of an unknown causal graph on which a sequence of soft or hard interventions can be performed. The problem of optimising the target variable associated with a causal graph is formalised as Causal Bayesian Optimisation (CBO). We study the CBO problem under the cumulative regret objective with unknown causal graphs for two settings, namely structural causal models with hard interventions and function networks with soft interventions. We propose Graph Agnostic Causal Bayesian Optimisation (GACBO), an algorithm that actively discovers the causal structure that contributes to achieving optimal rewards. GACBO seeks to balance exploiting the actions that give the best rewards against exploring the causal structures and functions. To the best of our knowledge, our work is the first to study causal Bayesian optimization with cumulative regret objectives in scenarios where the graph is unknown or partially known. We show our proposed algorithm outperforms baselines in simulated experiments and real-world applications.

Rahul Sharma, Sergey Redyuk, Sumantrak Mukherjee et al.

Explainable AI (XAI) and interpretable machine learning methods help to build trust in model predictions and derived insights, yet also present a perverse incentive for analysts to manipulate XAI metrics to support pre-specified conclusions. This paper introduces the concept of X-hacking, a form of p-hacking applied to XAI metrics such as SHAP values. We show how easily an automated machine learning pipeline can be adapted to exploit model multiplicity at scale: searching a Rashomon set of 'defensible' models with similar predictive performance to find a desired explanation. We formulate the trade-off between explanation and accuracy as a multi-objective optimisation problem, and illustrate empirically on familiar real-world datasets that, on average, Bayesian optimisation accelerates X-hacking 3-fold for features susceptible to it, versus random sampling. We show the vulnerability of a dataset to X-hacking can be determined by information redundancy among features. Finally, we suggest possible methods for detection and prevention, and discuss ethical implications for the credibility and reproducibility of XAI.

Jen Ning Lim, Sebastian Vollmer, Lorenz Wolf et al.

Energy-Based Models (EBMs) have proven to be a highly effective approach for modelling densities on finite-dimensional spaces. Their ability to incorporate domain-specific choices and constraints into the structure of the model through composition make EBMs an appealing candidate for applications in physics, biology and computer vision and various other fields. Recently, Energy-Based Processes (EBP) for modelling stochastic processes was proposed for \textit{unconditional} exchangeable data (e.g., point clouds). In this work, we present a novel subclass of EBPs, called $\mathcal{F}$-EBM for \textit{conditional} exchangeable data, which is able to learn distributions of functions (such as curves or surfaces) from functional samples evaluated at finitely many points. Two unique challenges arise in the functional context. Firstly, training data is often not evaluated along a fixed set of points. Secondly, steps must be taken to control the behaviour of the model between evaluation points, to mitigate overfitting. The proposed model is an energy based model on function space that is decomposed spectrally, where a Gaussian Process path measure is used to reweight the distribution to capture smoothness properties of the underlying process being modelled. The resulting model has the ability to utilize irregularly sampled training data and can output predictions at any resolution, providing an effective approach to up-scaling functional data. We demonstrate the efficacy of our proposed approach for modelling a range of datasets, including data collected from Standard and Poor's 500 (S\&P) and UK National grid.

Sahra Ghalebikesabi, Harrison Wilde, Jack Jewson et al.

Increasing interest in privacy-preserving machine learning has led to new and evolved approaches for generating private synthetic data from undisclosed real data. However, mechanisms of privacy preservation can significantly reduce the utility of synthetic data, which in turn impacts downstream tasks such as learning predictive models or inference. We propose several re-weighting strategies using privatised likelihood ratios that not only mitigate statistical bias of downstream estimators but also have general applicability to differentially private generative models. Through large-scale empirical evaluation, we show that private importance weighting provides simple and effective privacy-compliant augmentation for general applications of synthetic data.

Anthony D. Blaom, Sebastian J. Vollmer

A graph-based protocol called `learning networks' which combine assorted machine learning models into meta-models is described. Learning networks are shown to overcome several limitations of model composition as implemented in the dominant machine learning platforms. After illustrating the protocol in simple examples, a concise syntax for specifying a learning network, implemented in the MLJ framework, is presented. Using the syntax, it is shown that learning networks are are sufficiently flexible to include Wolpert's model stacking, with out-of-sample predictions for the base learners.

Harrison Wilde, Jack Jewson, Sebastian Vollmer et al.

There is significant growth and interest in the use of synthetic data as an enabler for machine learning in environments where the release of real data is restricted due to privacy or availability constraints. Despite a large number of methods for synthetic data generation, there are comparatively few results on the statistical properties of models learnt on synthetic data, and fewer still for situations where a researcher wishes to augment real data with another party's synthesised data. We use a Bayesian paradigm to characterise the updating of model parameters when learning in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate consideration of the synthetic data generating process and learning task. Recent results from general Bayesian updating support a novel and robust approach to Bayesian synthetic-learning founded on decision theory that outperforms standard approaches across repeated experiments on supervised learning and inference problems.

Ashrya Agrawal, Florian Pfisterer, Bernd Bischl et al.

We present an empirical study of debiasing methods for classifiers, showing that debiasers often fail in practice to generalize out-of-sample, and can in fact make fairness worse rather than better. A rigorous evaluation of the debiasing treatment effect requires extensive cross-validation beyond what is usually done. We demonstrate that this phenomenon can be explained as a consequence of bias-variance trade-off, with an increase in variance necessitated by imposing a fairness constraint. Follow-up experiments validate the theoretical prediction that the estimation variance depends strongly on the base rates of the protected class. Considering fairness--performance trade-offs justifies the counterintuitive notion that partial debiasing can actually yield better results in practice on out-of-sample data.

James Liley, Samuel R Emerson, Bilal A Mateen et al.

Machine learning is increasingly being used to generate prediction models for use in a number of real-world settings, from credit risk assessment to clinical decision support. Recent discussions have highlighted potential problems in the updating of a predictive score for a binary outcome when an existing predictive score forms part of the standard workflow, driving interventions. In this setting, the existing score induces an additional causative pathway which leads to miscalibration when the original score is replaced. We propose a general causal framework to describe and address this problem, and demonstrate an equivalent formulation as a partially observed Markov decision process. We use this model to demonstrate the impact of such `naive updating' when performed repeatedly. Namely, we show that successive predictive scores may converge to a point where they predict their own effect, or may eventually tend toward a stable oscillation between two values, and we argue that neither outcome is desirable. Furthermore, we demonstrate that even if model-fitting procedures improve, actual performance may worsen. We complement these findings with a discussion of several potential routes to overcome these issues.

Harrison Wilde, Lucia Lushi Chen, Austin Nguyen et al.

Rough sleeping is a chronic problem faced by some of the most disadvantaged people in modern society. This paper describes work carried out in partnership with Homeless Link, a UK-based charity, in developing a data-driven approach to assess the quality of incoming alerts from members of the public aimed at connecting people sleeping rough on the streets with outreach service providers. Alerts are prioritised based on the predicted likelihood of successfully connecting with the rough sleeper, helping to address capacity limitations and to quickly, effectively, and equitably process all of the alerts that they receive. Initial evaluation concludes that our approach increases the rate at which rough sleepers are found following a referral by at least 15\% based on labelled data, implying a greater overall increase when the alerts with unknown outcomes are considered, and suggesting the benefit in a trial taking place over a longer period to assess the models in practice. The discussion and modelling process is done with careful considerations of ethics, transparency and explainability due to the sensitive nature of the data in this context and the vulnerability of the people that are affected.

Diego Arenas, Jon Atkins, Claire Austin et al.

We present a policy and process framework for secure environments for productive data science research projects at scale, by combining prevailing data security threat and risk profiles into five sensitivity tiers, and, at each tier, specifying recommended policies for data classification, data ingress, software ingress, data egress, user access, user device control, and analysis environments. By presenting design patterns for security choices for each tier, and using software defined infrastructure so that a different, independent, secure research environment can be instantiated for each project appropriate to its classification, we hope to maximise researcher productivity and minimise risk, allowing research organisations to operate with confidence.

Sebastian Vollmer, Bilal A. Mateen, Gergo Bohner et al.

Machine learning (ML), artificial intelligence (AI) and other modern statistical methods are providing new opportunities to operationalize previously untapped and rapidly growing sources of data for patient benefit. Whilst there is a lot of promising research currently being undertaken, the literature as a whole lacks: transparency; clear reporting to facilitate replicability; exploration for potential ethical concerns; and, clear demonstrations of effectiveness. There are many reasons for why these issues exist, but one of the most important that we provide a preliminary solution for here is the current lack of ML/AI- specific best practice guidance. Although there is no consensus on what best practice looks in this field, we believe that interdisciplinary groups pursuing research and impact projects in the ML/AI for health domain would benefit from answering a series of questions based on the important issues that exist when undertaking work of this nature. Here we present 20 questions that span the entire project life cycle, from inception, data analysis, and model evaluation, to implementation, as a means to facilitate project planning and post-hoc (structured) independent evaluation. By beginning to answer these questions in different settings, we can start to understand what constitutes a good answer, and we expect that the resulting discussion will be central to developing an international consensus framework for transparent, replicable, ethical and effective research in artificial intelligence (AI-TREE) for health.

Tigran Nagapetyan, Andrew B. Duncan, Leonard Hasenclever et al.

The problem of posterior inference is central to Bayesian statistics and a wealth of Markov Chain Monte Carlo (MCMC) methods have been proposed to obtain asymptotically correct samples from the posterior. As datasets in applications grow larger and larger, scalability has emerged as a central problem for MCMC methods. Stochastic Gradient Langevin Dynamics (SGLD) and related stochastic gradient Markov Chain Monte Carlo methods offer scalability by using stochastic gradients in each step of the simulated dynamics. While these methods are asymptotically unbiased if the stepsizes are reduced in an appropriate fashion, in practice constant stepsizes are used. This introduces a bias that is often ignored. In this paper we study the mean squared error of Lipschitz functionals in strongly log- concave models with i.i.d. data of growing data set size and show that, given a batchsize, to control the bias of SGLD the stepsize has to be chosen so small that the computational cost of reaching a target accuracy is roughly the same for all batchsizes. Using a control variate approach, the cost can be reduced dramatically. The analysis is performed by considering the algorithms as noisy discretisations of the Langevin SDE which correspond to the Euler method if the full data set is used. An important observation is that the 1scale of the step size is determined by the stability criterion if the accuracy is required for consistent credible intervals. Experimental results confirm our theoretical findings.

Jackson Gorham, Andrew B. Duncan, Sebastian J. Vollmer et al.

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targets have been analyzed. We introduce a new class of characterizing operators based on Ito diffusions and develop explicit multivariate Stein factor bounds for any target with a fast-coupling Ito diffusion. As example applications, we develop computable and convergence-determining diffusion Stein discrepancies for log-concave, heavy-tailed, and multimodal targets and use these quality measures to select the hyperparameters of biased Markov chain Monte Carlo (MCMC) samplers, compare random and deterministic quadrature rules, and quantify bias-variance tradeoffs in approximate MCMC. Our results establish a near-linear relationship between diffusion Stein discrepancies and Wasserstein distances, improving upon past work even for strongly log-concave targets. The exposed relationship between Stein factors and Markov process coupling may be of independent interest.

Mike Giles, Tigran Nagapetyan, Lukasz Szpruch et al.

Markov chain Monte Carlo (MCMC) algorithms are ubiquitous in Bayesian computations. However, they need to access the full data set in order to evaluate the posterior density at every step of the algorithm. This results in a great computational burden in big data applications. In contrast to MCMC methods, Stochastic Gradient MCMC (SGMCMC) algorithms such as the Stochastic Gradient Langevin Dynamics (SGLD) only require access to a batch of the data set at every step. This drastically improves the computational performance and scales well to large data sets. However, the difficulty with SGMCMC algorithms comes from the sensitivity to its parameters which are notoriously difficult to tune. Moreover, the Root Mean Square Error (RMSE) scales as $\mathcal{O}(c^{-\frac{1}{3}})$ as opposed to standard MCMC $\mathcal{O}(c^{-\frac{1}{2}})$ where $c$ is the computational cost. We introduce a new class of Multilevel Stochastic Gradient Markov chain Monte Carlo algorithms that are able to mitigate the problem of tuning the step size and more importantly of recovering the $\mathcal{O}(c^{-\frac{1}{2}})$ convergence of standard Markov Chain Monte Carlo methods without the need to introduce Metropolis-Hasting steps. A further advantage of this new class of algorithms is that it can easily be parallelised over a heterogeneous computer architecture. We illustrate our methodology using Bayesian logistic regression and provide numerical evidence that for a prescribed relative RMSE the computational cost is sublinear in the number of data items.

Xiaoyu Lu, Valerio Perrone, Leonard Hasenclever et al.

Hamiltonian Monte Carlo (HMC) is a popular Markov chain Monte Carlo (MCMC) algorithm that generates proposals for a Metropolis-Hastings algorithm by simulating the dynamics of a Hamiltonian system. However, HMC is sensitive to large time discretizations and performs poorly if there is a mismatch between the spatial geometry of the target distribution and the scales of the momentum distribution. In particular the mass matrix of HMC is hard to tune well. In order to alleviate these problems we propose relativistic Hamiltonian Monte Carlo, a version of HMC based on relativistic dynamics that introduce a maximum velocity on particles. We also derive stochastic gradient versions of the algorithm and show that the resulting algorithms bear interesting relationships to gradient clipping, RMSprop, Adagrad and Adam, popular optimisation methods in deep learning. Based on this, we develop relativistic stochastic gradient descent by taking the zero-temperature limit of relativistic stochastic gradient Hamiltonian Monte Carlo. In experiments we show that the relativistic algorithms perform better than classical Newtonian variants and Adam.

Leonard Hasenclever, Stefan Webb, Thibaut Lienart et al.

This paper makes two contributions to Bayesian machine learning algorithms. Firstly, we propose stochastic natural gradient expectation propagation (SNEP), a novel alternative to expectation propagation (EP), a popular variational inference algorithm. SNEP is a black box variational algorithm, in that it does not require any simplifying assumptions on the distribution of interest, beyond the existence of some Monte Carlo sampler for estimating the moments of the EP tilted distributions. Further, as opposed to EP which has no guarantee of convergence, SNEP can be shown to be convergent, even when using Monte Carlo moment estimates. Secondly, we propose a novel architecture for distributed Bayesian learning which we call the posterior server. The posterior server allows scalable and robust Bayesian learning in cases where a data set is stored in a distributed manner across a cluster, with each compute node containing a disjoint subset of data. An independent Monte Carlo sampler is run on each compute node, with direct access only to the local data subset, but which targets an approximation to the global posterior distribution given all data across the whole cluster. This is achieved by using a distributed asynchronous implementation of SNEP to pass messages across the cluster. We demonstrate SNEP and the posterior server on distributed Bayesian learning of logistic regression and neural networks. Keywords: Distributed Learning, Large Scale Learning, Deep Learning, Bayesian Learn- ing, Variational Inference, Expectation Propagation, Stochastic Approximation, Natural Gradient, Markov chain Monte Carlo, Parameter Server, Posterior Server.

Sebastian J. Vollmer, Konstantinos C. Zygalakis, and Yee Whye Teh

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings accept-reject step, and it uses sequences of decreasing step sizes. In \cite{TehThierryVollmerSGLD2014}, we provided the mathematical foundations for the decreasing step size SGLD, including consistency and a central limit theorem. However, in practice the SGLD is run for a relatively small number of iterations, and its step size is not decreased to zero. The present article investigates the behaviour of the SGLD with fixed step size. In particular we characterise the asymptotic bias explicitly, along with its dependence on the step size and the variance of the stochastic gradient. On that basis a modified SGLD which removes the asymptotic bias due to the variance of the stochastic gradients up to first order in the step size is derived. Moreover, we are able to obtain bounds on the finite-time bias, variance and mean squared error (MSE). The theory is illustrated with a Gaussian toy model for which the bias and the MSE for the estimation of moments can be obtained explicitly. For this toy model we study the gain of the SGLD over the standard Euler method in the limit of large data sets.

Yee Whye Teh, Alexandre Thiéry, Sebastian Vollmer

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally expensive. Both the calculation of the acceptance probability and the creation of informed proposals usually require an iteration through the whole data set. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem by generating proposals which are only based on a subset of the data, by skipping the accept-reject step and by using decreasing step-sizes sequence $(δ_m)_{m \geq 0}$. %Under appropriate Lyapunov conditions, We provide in this article a rigorous mathematical framework for analysing this algorithm. We prove that, under verifiable assumptions, the algorithm is consistent, satisfies a central limit theorem (CLT) and its asymptotic bias-variance decomposition can be characterized by an explicit functional of the step-sizes sequence $(δ_m)_{m \geq 0}$. We leverage this analysis to give practical recommendations for the notoriously difficult tuning of this algorithm: it is asymptotically optimal to use a step-size sequence of the type $δ_m \asymp m^{-1/3}$, leading to an algorithm whose mean squared error (MSE) decreases at rate $\mathcal{O}(m^{-1/3})$