We study a new technique for understanding convergence of learning agents under small modifications of data. We show that such convergence can be understood via an analogue of Fatou's lemma which yields gamma-convergence. We show it's relevance and applications in general machine learning tasks and domain adaptation transfer learning.
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One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly available and can be approximated conventionally by finite differences. However, the discrete approximations of time derivatives may result in poor estimations when state data are scarce and/or corrupted by noise, thus compromising the predictiveness of the learned dynamical models. To overcome this technical hurdle, we propose a new method that learns nonlinear dynamics through a Bayesian inference of characterizing model parameters. This method leverages a Gaussian process representation of states, and constructs a likelihood function using the correlation between state data and their derivatives, yet prevents explicit evaluations of time derivatives. Through a Bayesian scheme, a probabilistic estimate of the model parameters is given by the posterior distribution, and thus a quantification is facilitated for uncertainties from noisy state data and the learning process. Specifically, we will discuss the applicability of the proposed method to several typical scenarios for dynamical systems: identification and estimation with an affine parametrization, nonlinear parametric approximation without prior knowledge, and general parameter estimation for a given dynamical system.
The rise of machine learning has fueled the discovery of new materials and, especially, metamaterials--truss lattices being their most prominent class. While their tailorable properties have been explored extensively, the design of truss-based metamaterials has remained highly limited and often heuristic, due to the vast, discrete design space and the lack of a comprehensive parameterization. We here present a graph-based deep learning generative framework, which combines a variational autoencoder and a property predictor, to construct a reduced, continuous latent representation covering an enormous range of trusses. This unified latent space allows for the fast generation of new designs through simple operations (e.g., traversing the latent space or interpolating between structures). We further demonstrate an optimization framework for the inverse design of trusses with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, pentamode-like, and tailored nonlinear behaviors. This generative model can predict manufacturable (and counter-intuitive) designs with extreme target properties beyond the training domain.
In recent years, deep reinforcement learning has emerged as a technique to solve closed-loop flow control problems. Employing simulation-based environments in reinforcement learning enables a priori end-to-end optimization of the control system, provides a virtual testbed for safety-critical control applications, and allows to gain a deep understanding of the control mechanisms. While reinforcement learning has been applied successfully in a number of rather simple flow control benchmarks, a major bottleneck toward real-world applications is the high computational cost and turnaround time of flow simulations. In this contribution, we demonstrate the benefits of model-based reinforcement learning for flow control applications. Specifically, we optimize the policy by alternating between trajectories sampled from flow simulations and trajectories sampled from an ensemble of environment models. The model-based learning reduces the overall training time by up to $85\%$ for the fluidic pinball test case. Even larger savings are expected for more demanding flow simulations.
In this study, our main objective is to address the challenge of solving elliptic equations with quasiperiodic coefficients. To achieve accurate and efficient computation, we introduce the projection method, which enables the embedding of quasiperiodic systems into higher-dimensional periodic systems. To enhance the computational efficiency, we propose a compressed storage strategy for the stiffness matrix, reducing memory requirements while preserving accuracy. Furthermore, we design a diagonal preconditioner to efficiently solve the resulting high-dimensional linear system by reducing the condition number of the stiffness matrix. These techniques collectively contribute to the computational effectiveness of our proposed approach. We demonstrate the effectiveness and accuracy of our approach through a series of numerical examples. Moreover, we apply our method to achieve a highly accurate computation of the homogenized coefficients for a quasiperiodic multiscale elliptic equation.
In federated learning, data heterogeneity is a critical challenge. A straightforward solution is to shuffle the clients' data to homogenize the distribution. However, this may violate data access rights, and how and when shuffling can accelerate the convergence of a federated optimization algorithm is not theoretically well understood. In this paper, we establish a precise and quantifiable correspondence between data heterogeneity and parameters in the convergence rate when a fraction of data is shuffled across clients. We prove that shuffling can quadratically reduce the gradient dissimilarity with respect to the shuffling percentage, accelerating convergence. Inspired by the theory, we propose a practical approach that addresses the data access rights issue by shuffling locally generated synthetic data. The experimental results show that shuffling synthetic data improves the performance of multiple existing federated learning algorithms by a large margin.
We consider the computation of model-free bounds for multi-asset options in a setting that combines dependence uncertainty with additional information on the dependence structure. More specifically, we consider the setting where the marginal distributions are known and partial information, in the form of known prices for multi-asset options, is also available in the market. We provide a fundamental theorem of asset pricing in this setting, as well as a superhedging duality that allows to transform the maximization problem over probability measures in a more tractable minimization problem over trading strategies. The latter is solved using a penalization approach combined with a deep learning approximation using artificial neural networks. The numerical method is fast and the computational time scales linearly with respect to the number of traded assets. We finally examine the significance of various pieces of additional information. Empirical evidence suggests that "relevant" information, i.e. prices of derivatives with the same payoff structure as the target payoff, are more useful that other information, and should be prioritized in view of the trade-off between accuracy and computational efficiency.
We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.
An interesting case of the well-known Dataset Shift Problem is the classification of Electroencephalogram (EEG) signals in the context of Brain-Computer Interface (BCI). The non-stationarity of EEG signals can lead to poor generalisation performance in BCI classification systems used in different sessions, also from the same subject. In this paper, we start from the hypothesis that the Dataset Shift problem can be alleviated by exploiting suitable eXplainable Artificial Intelligence (XAI) methods to locate and transform the relevant characteristics of the input for the goal of classification. In particular, we focus on an experimental analysis of explanations produced by several XAI methods on an ML system trained on a typical EEG dataset for emotion recognition. Results show that many relevant components found by XAI methods are shared across the sessions and can be used to build a system able to generalise better. However, relevant components of the input signal also appear to be highly dependent on the input itself.
Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining appropriate loss functions is therefore critical to successfully solving problems in this field. We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.
In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.
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