Global Optimization for Cardinality-constrained Minimum Sum-of-Squares Clustering via Semidefinite Programming

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, has been recently extended to exploit prior knowledge on the cardinality of each cluster. Such knowledge is used to increase performance as well as solution quality. In this paper, we propose a global optimization approach based on the branch-and-cut technique to solve the cardinality-constrained MSSC. For the lower bound routine, we use the semidefinite programming (SDP) relaxation recently proposed by Rujeerapaiboon et al. [SIAM J. Optim. 29(2), 1211-1239, (2019)]. However, this relaxation can be used in a branch-and-cut method only for small-size instances. Therefore, we derive a new SDP relaxation that scales better with the instance size and the number of clusters. In both cases, we strengthen the bound by adding polyhedral cuts. Benefiting from a tailored branching strategy which enforces pairwise constraints, we reduce the complexity of the problems arising in the children nodes. For the upper bound, instead, we present a local search procedure that exploits the solution of the SDP relaxation solved at each node. Computational results show that the proposed algorithm globally solves, for the first time, real-world instances of size 10 times larger than those solved by state-of-the-art exact methods

A machine learning approach for forecasting hierarchical time series

In this paper, we propose a machine learning approach for forecasting hierarchical time series. When dealing with hierarchical time series, apart from generating accurate forecasts, one needs to select a suitable method for producing reconciled forecasts. Forecast reconciliation is the process of adjusting forecasts to make them coherent across the hierarchy. In literature, coherence is often enforced by using a post-processing technique on the base forecasts produced by suitable time series forecasting methods. On the contrary, our idea is to use a deep neural network to directly produce accurate and reconciled forecasts. We exploit the ability of a deep neural network to extract information capturing the structure of the hierarchy. We impose the reconciliation at training time by minimizing a customized loss function. In many practical applications, besides time series data, hierarchical time series include explanatory variables that are beneficial for increasing the forecasting accuracy. Exploiting this further information, our approach links the relationship between time series features extracted at any level of the hierarchy and the explanatory variables into an end-to-end neural network providing accurate and reconciled point forecasts. The effectiveness of the approach is validated on three real-world datasets, where our method outperforms state-of-the-art competitors in hierarchical forecasting

Supervised Feature Compression based on Counterfactual Analysis

Counterfactual Explanations are becoming a de-facto standard in post-hoc interpretable machine learning. For a given classifier and an instance classified in an undesired class, its counterfactual explanation corresponds to small perturbations of that instance that allows changing the classification outcome. This work aims to leverage Counterfactual Explanations to detect the important decision boundaries of a pre-trained black-box model. This information is used to build a supervised discretization of the features in the dataset with a tunable granularity. Using the discretized dataset, a smaller, therefore more interpretable Decision Tree can be trained, which, in addition, enhances the stability and robustness of the baseline Decision Tree. Numerical results on real-world datasets show the effectiveness of the approach in terms of accuracy and sparsity compared to the baseline Decision Tree.Comment: 29 pages, 12 figure

An Exact Algorithm for Semi-supervised Minimum Sum-of-Squares Clustering

The minimum sum-of-squares clustering (MSSC), or k-means type clustering, is traditionally considered an unsupervised learning task. In recent years, the use of background knowledge to improve the cluster quality and promote interpretability of the clustering process has become a hot research topic at the intersection of mathematical optimization and machine learning research. The problem of taking advantage of background information in data clustering is called semi-supervised or constrained clustering. In this paper, we present a branch-and-cut algorithm for semi-supervised MSSC, where background knowledge is incorporated as pairwise must-link and cannot-link constraints. For the lower bound procedure, we solve the semidefinite programming relaxation of the MSSC discrete optimization model, and we use a cutting-plane procedure for strengthening the bound. For the upper bound, instead, by using integer programming tools, we use an adaptation of the k-means algorithm to the constrained case. For the first time, the proposed global optimization algorithm efficiently manages to solve real-world instances up to 800 data points with different combinations of must-link and cannot-link constraints and with a generic number of features. This problem size is about four times larger than the one of the instances solved by state-of-the-art exact algorithms

Optimized Collaborative Brain-Computer Interfaces for Enhancing Face Recognition

: The aim of this study is to maximize group decision performance by optimally adapting EEG confidence decoders to the group composition. We train linear support vector machines to estimate the decision confidence of human participants from their EEG activity. We then simulate groups of different size and membership by combining individual decisions using a weighted majority rule. The weights assigned to each participant in the group are chosen solving a small-dimension, mixed, integer linear programming problem, where we maximize the group performance on the training set. We therefore introduce optimized collaborative brain-computer interfaces (BCIs), where the decisions of each team member are weighted according to both the individual neural activity and the group composition. We validate this approach on a face recognition task undertaken by 10 human participants. The results show that optimal collaborative BCIs significantly enhance team performance over other BCIs, while improving fairness within the group. This research paves the way for practical applications of collaborative BCIs to realistic scenarios characterized by stable teams, where optimizing the decision policy of a single group may lead to significant long-term benefits of team dynamics

Improving P300 Speller performance by means of optimization and machine learning

Brain-Computer Interfaces (BCIs) are systems allowing people to interact with the environment bypassing the natural neuromuscular and hormonal outputs of the peripheral nervous system (PNS). These interfaces record a user's brain activity and translate it into control commands for external devices, thus providing the PNS with additional artificial outputs. In this framework, the BCIs based on the P300 Event-Related Potentials (ERP), which represent the electrical responses recorded from the brain after specific events or stimuli, have proven to be particularly successful and robust. The presence or the absence of a P300 evoked potential within the EEG features is determined through a classification algorithm. Linear classifiers such as SWLDA and SVM are the most used for ERPs' classification. Due to the low signal-to-noise ratio of the EEG signals, multiple stimulation sequences (a.k.a. iterations) are carried out and then averaged before the signals being classified. However, while augmenting the number of iterations improves the Signal-to-Noise Ratio (SNR), it also slows down the process. In the early studies, the number of iterations was fixed (no stopping), but recently, several early stopping strategies have been proposed in the literature to dynamically interrupt the stimulation sequence when a certain criterion is met to enhance the communication rate. In this work, we explore how to improve the classification performances in P300 based BCIs by combining optimization and machine learning. First, we propose a new decision function that aims at improving classification performances in terms of accuracy and Information Transfer Rate both in a no stopping and early stopping environment. Then, we propose a new SVM training problem that aims to facilitate the target-detection process. Our approach proves to be effective on several publicly available datasets.Comment: 32 pages, research articl

A game-theoretic approach to computation offloading in mobile cloud computing

We consider a three-tier architecture for mobile and pervasive computing scenarios, consisting of a local tier ofmobile nodes, a middle tier (cloudlets) of nearby computing nodes, typically located at the mobile nodes access points but characterized by a limited amount of resources, and a remote tier of distant cloud servers, which have practically infinite resources. This architecture has been proposed to get the benefits of computation offloading from mobile nodes to external servers while limiting the use of distant servers whose higher latency could negatively impact the user experience. For this architecture, we consider a usage scenario where no central authority exists and multiple non-cooperative mobile users share the limited computing resources of a close-by cloudlet and can selfishly decide to send their computations to any of the three tiers. We define a model to capture the users interaction and to investigate the effects of computation offloading on the users’ perceived performance. We formulate the problem as a generalized Nash equilibrium problem and show existence of an equilibrium.We present a distributed algorithm for the computation of an equilibrium which is tailored to the problem structure and is based on an in-depth analysis of the underlying equilibrium problem. Through numerical examples, we illustrate its behavior and the characteristics of the achieved equilibria