Graph-based machine learning methods are valuable tools for identifying and predicting variation in genetic data. In particular, understanding phenotypic effects at the cellular level is an accelerating area in pharmacogenomics. Insight into how drugs or disease affect bio-networks could aid drug development and precision medicine. This article proposes a novel graph-theoretic approach to infer a co-occurrence network from 16S microbiome data, designed specifically for smallsample datasets. Such datasets pose challenges due to sparsity, compositionality, and complex interactions. The methodology includes steps to enrich and statistically filter the inferred networks. The approach extracts informative, feature-rich, biologically meaningful, and statistically significant networks from limited data. While tailored for small datasets, it is broadly applicable and can be extended to multi-omics integration. The method is tested on data from chickens vaccinated and challenged with Eimeria tenella. Genetic reads are processed, and networks inferred to characterize intestinal ecosystems at three disease progression stages. Analysis of network features yields biologically intuitive conclusions using statistical methods. Notably, the distribution of node features evolves with disease progression, and distributions reveal mutualistic and parasitic species clusters. A sub-network consistently appears across all conditions, suggesting a 'persistent microbiome'. A clustering algorithm is also applied to demonstrate the methods utility for downstream analysis.

We analyze the prediction error of ridge regression in an asymptotic regime where the sample size and dimension go to infinity at a proportional rate. In particular, we consider the role played by the structure of the true regression parameter. We observe that the case of a general deterministic parameter can be reduced to the case of a random parameter from a structured prior. The latter assumption is a natural adaptation of classic smoothness assumptions in nonparametric regression, which are known as source conditions in the the context of regularization theory for inverse problems. Roughly speaking, we assume the large coefficients of the parameter are in correspondence to the principal components. In this setting a precise characterisation of the test error is obtained, depending on the inputs covariance and regression parameter structure. We illustrate this characterisation in a simplified setting to investigate the influence of the true parameter on optimal regularisation for overparameterized models. We show that interpolation (no regularisation) can be optimal even with bounded signal-to-noise ratio (SNR), provided that the parameter coefficients are larger on high-variance directions of the data, corresponding to a more regular function than posited by the regularization term. This contrasts with previous work considering ridge regression with isotropic prior, in which case interpolation is only optimal in the limit of infinite SNR.

We study the learning performance of gradient descent when the empirical risk is weakly convex, namely, the smallest negative eigenvalue of the empirical risk’s Hessian is bounded in magnitude. By showing that this eigenvalue can control the stability of gradient descent, generalisation error bounds are proven that hold under a wider range of step sizes compared to previous work. Out of sample guarantees are then achieved by decomposing the test error into generalisation, optimisation and approximation errors, each of which can be bounded and traded off with respect to algorithmic parameters, sample size and magnitude of this eigenvalue. In the case of a two layer neural network, we demonstrate that the empirical risk can satisfy a notion of local weak convexity, specifically, the Hessian’s smallest eigenvalue during training can be controlled by the normalisation of the layers, i.e., network scaling. This allows test error guarantees to then be achieved when the population risk minimiser satisfies a complexity assumption. By trading off the network complexity and scaling, insights are gained into the implicit bias of neural network scaling, which are further supported by experimental findings.

We revisit on-average algorithmic stability of Gradient Descent (GD) for training overparameterised shallow neural networks and prove new generalisation and excess risk bounds without the Neural Tangent Kernel (NTK) or Polyak-Łojasiewicz (PL) assumptions. In particular, we show oracle type bounds which reveal that the generalisation and excess risk of GD is controlled by an interpolating network with the shortest GD path from initialisation (in a sense, an interpolating network with the smallest relative norm). While this was known for kernelised interpolants, our proof applies directly to networks trained by GD without intermediate kernelisation. At the same time, by relaxing oracle inequalities developed here we recover existing NTK-based risk bounds in a straightforward way, which demonstrates that our analysis is tighter. Finally, unlike most of the NTK-based analyses we focus on regression with label noise and show that GD with early stopping is consistent

Motivated by distributed machine learning settings such as Federated Learning, we consider the problem of fitting a statistical model across a distributed collection of heterogeneous data sets whose similarity structure is encoded by a graph topology. Precisely, we analyse the case where each node is associated with fitting a sparse linear model, and edges join two nodes if the difference of their solutions is also sparse. We propose a method based on Basis Pursuit Denoising with a total variation penalty, and provide finite sample guarantees for sub-Gaussian design matrices. Taking the root of the tree as a reference node, we show that if the sparsity of the differences across nodes is smaller than the sparsity at the root, then recovery is successful with fewer samples than by solving the problems independently, or by using methods that rely on a large overlap in the signal supports, such as the group Lasso. We consider both the noiseless and noisy setting, and numerically investigate the performance of distributed methods based on Distributed Alternating Direction Methods of Multipliers (ADMM) and hyperspectral unmixing.

We investigate the generalisation performance of Distributed Gradient Descent with implicit regularisation and random features in the homogenous setting where a network of agents are given data sampled independently from the same unknown distribution. Along with reducing the memory footprint, random features are particularly convenient in this setting as they provide a common parameterisation across agents that allows to overcome previous difficulties in implementing decentralised kernel regression. Under standard source and capacity assumptions, we establish high probability bounds on the predictive performance for each agent as a function of the step size, number of iterations, inverse spectral gap of the communication matrix and number of random features. By tuning these parameters, we obtain statistical rates that are minimax optimal with respect to the total number of samples in the network. The algorithm provides a linear improvement over single-machine gradient descent in memory cost and, when agents hold enough data with respect to the network size and inverse spectral gap, a linear speed up in computational run-time for any network topology. We present simulations that show how the number of random features, iterations and samples impact predictive performance.

We propose graph-dependent implicit regularisation strategies for synchronised distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, single-machine serial statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding constraints to the empirical risk minimisation rule. Particularly for distributed methods, the use of implicit regularisation allows the algorithm to remain simple, without projections or dual methods. To prove our results, we establish graph-independent generalisation bounds for Distributed SGD that match the single-machine serial SGD setting (using algorithmic stability), and we establish graph-dependent optimisation bounds that are of independent interest. We present numerical experiments to show that the qualitative nature of the upper bounds we derive can be representative of real behaviours.

We analyse the learning performance of Distributed Gradient Descent in the context of multi-agent decentralised non-parametric regression with the square loss function when i.i.d. samples are assigned to agents. We show that if agents hold sufficiently many samples with respect to the network size, then Distributed Gradient Descent achieves optimal statistical rates with a number of iterations that scales, up to a threshold, with the inverse of the spectral gap of the gossip matrix divided by the number of samples owned by each agent raised to a problem-dependent power. The presence of the threshold comes from statistics. It encodes the existence of a "big data" regime where the number of required iterations does not depend on the network topology. In this regime, Distributed Gradient Descent achieves optimal statistical rates with the same order of iterations as gradient descent run with all the samples in the network. Provided the communication delay is sufficiently small, the distributed protocol yields a linear speed-up in runtime compared to the single-machine protocol. This is in contrast to decentralised optimisation algorithms that do not exploit statistics and only yield a linear speed-up in graphs where the spectral gap is bounded away from zero. Our results exploit the statistical concentration of quantities held by agents and shed new light on the interplay between statistics and communication in decentralised methods. Bounds are given in the standard non-parametric setting with source/capacity assumptions.

SpiNNaker is a biologically-inspired massively-parallel computer designed to model up to a billion spiking neurons in real-time. A full-fledged implementation of a SpiNNaker system will comprise more than 105 integrated circuits (half of which are SDRAMs and half multi-core systems-on-chip). Given this scale, it is unavoidable that some components fail and, in consequence, fault-tolerance is a foundation of the system design. Although the target application can tolerate a certain, low level of failures, important efforts have been devoted to incorporate different techniques for fault tolerance. This paper is devoted to discussing how hardware and software mechanisms collaborate to make SpiNNaker operate properly even in the very likely scenario of component failures and how it can tolerate system-degradation levels well above those expected.