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The higher dimensional autoregressive models would describe some of the econometric processes relatively generically if they incorporate the heterogeneity in dependence on times. This paper analyzes the stationarity of an autoregressive process of dimension $k>1$ having a sequence of coefficients $\beta$ multiplied by successively increasing powers of $0<\delta<1$. The theorem gives the conditions of stationarity in simple relations between the coefficients and $k$ in terms of $\delta$. Computationally, the evidence of stationarity depends on the parameters. The choice of $\delta$ sets the bounds on $\beta$ and the number of time lags for prediction of the model.

A model of interacting agents, following plausible behavioral rules into a world where the Covid-19 epidemic is affecting the actions of everyone. The model works with (i) infected agents categorized as symptomatic or asymptomatic and (ii) the places of contagion specified in a detailed way. The infection transmission is related to three factors: the characteristics of both the infected person and the susceptible one, plus those of the space in which contact occurs. The model includes the structural data of Piedmont, an Italian region, but we can easily calibrate it for other areas. The micro-based structure of the model allows factual, counterfactual, and conditional simulations to investigate both the spontaneous or controlled development of the epidemic. The model is generative of complex epidemic dynamics emerging from the consequences of agents' actions and interactions, with high variability in outcomes and stunning realistic reproduction of the successive contagion waves in the reference region. There is also an inverse generative side of the model, coming from the idea of using genetic algorithms to construct a meta-agent to optimize the vaccine distribution. This agent takes into account groups' characteristics -- by age, fragility, work conditions -- to minimize the number of symptomatic people.

We consider multivariate centered Gaussian models for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup of the group of permutations on $\{1,\ldots, p\}$. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter $\Sigma$ and also the analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter $K=\Sigma^{-1}$. We can thus perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group, which we could also call complete RCOP models. We illustrate our results with a toy example of dimension $4$ and several examples for selection within cyclic groups, including a high dimensional example with $p=100$.

Process mining is a scientific discipline that analyzes event data, often collected in databases called event logs. Recently, uncertain event logs have become of interest, which contain non-deterministic and stochastic event attributes that may represent many possible real-life scenarios. In this paper, we present a method to reliably estimate the probability of each of such scenarios, allowing their analysis. Experiments show that the probabilities calculated with our method closely match the true chances of occurrence of specific outcomes, enabling more trustworthy analyses on uncertain data.

The vulnerability of deep neural networks against adversarial examples - inputs with small imperceptible perturbations - has gained a lot of attention in the research community recently. Simultaneously, the number of parameters of state-of-the-art deep learning models has been growing massively, with implications on the memory and computational resources required to train and deploy such models. One approach to control the size of neural networks is retrospectively reducing the number of parameters, so-called neural network pruning. Available research on the impact of neural network pruning on the adversarial robustness is fragmentary and often does not adhere to established principles of robustness evaluation. We close this gap by evaluating the robustness of pruned models against L-0, L-2 and L-infinity attacks for a wide range of attack strengths, several architectures, data sets, pruning methods, and compression rates. Our results confirm that neural network pruning and adversarial robustness are not mutually exclusive. Instead, sweet spots can be found that are favorable in terms of model size and adversarial robustness. Furthermore, we extend our analysis to situations that incorporate additional assumptions on the adversarial scenario and show that depending on the situation, different strategies are optimal.

A free-floating bike-sharing system (FFBSS) is a dockless rental system where an individual can borrow a bike and returns it anywhere, within the service area. To improve the rental service, available bikes should be distributed over the entire service area: a customer leaving from any position is then more likely to find a near bike and then to use the service. Moreover, spreading bikes among the entire service area increases urban spatial equity since the benefits of FFBSS are not a prerogative of just a few zones. For guaranteeing such distribution, the FFBSS operator can use vans to manually relocate bikes, but it incurs high economic and environmental costs. We propose a novel approach that exploits the existing bike flows generated by customers to distribute bikes. More specifically, by envisioning the problem as an Influence Maximization problem, we show that it is possible to position batches of bikes on a small number of zones, and then the daily use of FFBSS will efficiently spread these bikes on a large area. We show that detecting these zones is NP-complete, but there exists a simple and efficient $1-1/e$ approximation algorithm; our approach is then evaluated on a dataset of rides from the free-floating bike-sharing system of the city of Padova.

Scaling up deep neural networks has been proven effective in improving model quality, while it also brings ever-growing training challenges. This paper presents Whale, an automatic and hardware-aware distributed training framework for giant models. Whale generalizes the expression of parallelism with four primitives, which can define various parallel strategies, as well as flexible hybrid strategies including combination and nesting patterns. It allows users to build models at an arbitrary scale by adding a few annotations and automatically transforms the local model to a distributed implementation. Moreover, Whale is hardware-aware and highly efficient even when training on GPUs of mixed types, which meets the growing demand of heterogeneous training in industrial clusters. Whale sets a milestone for training the largest multimodal pretrained model M6. The success of M6 is achieved by Whale's design to decouple algorithm modeling from system implementations, i.e., algorithm developers can focus on model innovation, since it takes only three lines of code to scale the M6 model to trillions of parameters on a cluster of 480 GPUs.

Model complexity is a fundamental problem in deep learning. In this paper we conduct a systematic overview of the latest studies on model complexity in deep learning. Model complexity of deep learning can be categorized into expressive capacity and effective model complexity. We review the existing studies on those two categories along four important factors, including model framework, model size, optimization process and data complexity. We also discuss the applications of deep learning model complexity including understanding model generalization capability, model optimization, and model selection and design. We conclude by proposing several interesting future directions.

Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.