Data Science Theory

A graph-structured distance for heterogeneous datasets with meta variables

  • Edward Hallé-Hannan
  • Charles Audet
  • Youssef Diouane
  • Sébastien Le Digabel
  • Paul Saves
    • Categories Data Science Theory, Optimization in Data Science Tags , , ,

    Heterogeneous datasets emerge in various machine learning or optimization applications that feature different data sources, various data types and complex relationships between variables. In practice, heterogeneous datasets are often partitioned into smaller well-behaved ones that are easier to process. However, some applications involve expensive-to-generate or limited size datasets, which motivates methods based on the whole … Read more

    Stochastic Aspects of Dynamical Low-Rank Approximation in the Context of Machine Learning

  • Arsen Hnatiuk
  • Lisa Kusch
  • Jonas Kusch
  • Nicolas R. Gauger
  • Andrea Walther
    • Categories Data Science Theory, Nonlinear Optimization, Optimization in Data Science Tags , , , ,

    The central challenges of today’s neural network architectures are the prohibitive memory footprint and the training costs associated with determining optimal weights and biases. A large portion of research in machine learning is therefore dedicated to constructing memory-efficient training methods. One promising approach is dynamical low-rank training (DLRT) which represents and trains parameters as a … Read more

    Expected Value of Matrix Quadratic Forms with Wishart distributed Random Matrices

  • Melinda Hagedorn
    • Categories Convex Optimization, Data Science Theory, Stochastic Approaches Tags , , , , ,

    To explore the limits of a stochastic gradient method, it may be useful to consider an example consisting of an infinite number of quadratic functions. In this context, it is appropriate to determine the expected value and the covariance matrix of the stochastic noise, i.e. the difference of the true gradient and the approximated gradient … Read more

    A polyhedral study of multivariate decision trees

  • Carla Michini
  • Zachary Zhou
    • Categories (Mixed) Integer Linear Programming, Data Science Theory, Polyhedra Tags , ,

    Decision trees are a widely used tool for interpretable machine learning. Multivariate decision trees employ hyperplanes at the branch nodes to route datapoints throughout the tree and yield more compact models than univariate trees. Recently, mixed-integer programming (MIP) has been applied to formulate the optimal decision tree problem. To strengthen MIP formulations, it is crucial … Read more

    Wasserstein Regularization for 0-1 Loss

  • Rui Gao
    • Categories Data Science Theory, Robust Optimization, Stochastic Programming

    Wasserstein distributionally robust optimization (DRO) finds robust solutions by hedging against data perturbation specified by distributions in a Wasserstein ball. The robustness is linked to the regularization effect, which has been studied for continuous losses in various settings. However, existing results cannot be simply applied to the 0-1 loss, which is frequently seen in uncertainty … Read more

    Sparse PCA With Multiple Components

  • Ryan Cory-Wright
  • Jean Pauphilet
    • Categories Data Science Theory, Data-Mining, Semi-definite Programming

    Sparse Principal Component Analysis (sPCA) is a cardinal technique for obtaining combinations of features, or principal components (PCs), that explain the variance of high-dimensional datasets in an interpretable manner. This involves solving a sparsity and orthogonality-constrained convex maximization problem, which is extremely computationally challenging. Most existing works address sparse PCA via methods—such as iteratively computing … Read more

    Optimized convergence of stochastic gradient descent by weighted averaging

  • Melinda Hagedorn
  • Florian Jarre
    • Categories Convex Optimization, Data Science Theory, Stochastic Approaches Tags , , , , ,

    Under mild assumptions stochastic gradient methods asymptotically achieve an optimal rate of convergence if the arithmetic mean of all iterates is returned as an approximate optimal solution. However, in the absence of stochastic noise, the arithmetic mean of all iterates converges considerably slower to the optimal solution than the iterates themselves. And also in the … Read more