Regularized Gradient Clipping Provably Trains Wide and Deep Neural Networks

In this work, we instantiate a regularized form of the gradient clipping algorithm and prove that it can converge to the global minima of deep neural network loss functions provided that the net is of sufficient width. We present empirical evidence that our theoretically founded regularized gradient clipping algorithm is also competitive with the state-of-the-art … Read more

Predictive Low Rank Matrix Learning under Partial Observations: Mixed-Projection ADMM

\(\) We study the problem of learning a partially observed matrix under the low rank assumption in the presence of fully observed side information that depends linearly on the true underlying matrix. This problem consists of an important generalization of the Matrix Completion problem, a central problem in Statistics, Operations Research and Machine Learning, that … Read more

Distributionally and Adversarially Robust Logistic Regression via Intersecting Wasserstein Balls

Empirical risk minimization often fails to provide robustness against adversarial attacks in test data, causing poor out-of-sample performance. Adversarially robust optimization (ARO) has thus emerged as the de facto standard for obtaining models that hedge against such attacks. However, while these models are robust against adversarial attacks, they tend to suffer severely from overfitting. To … Read more

A Stochastic Objective-Function-Free Adaptive Regularization Method with Optimal Complexity

\(\) A fully stochastic second-order adaptive-regularization method for unconstrained nonconvex optimization is presented which never computes the objective-function value, but yet achieves the optimal $\mathcal{O}(\epsilon^{-3/2})$ complexity bound for finding first-order critical points. The method is noise-tolerant and the inexactness conditions required for convergence depend on the history of past steps. Applications to cases where derivative … Read more

Complexity of Adagrad and other first-order methods for nonconvex optimization problems with bounds and convex constraints

A parametric class of trust-region algorithms for constrained nonconvex optimization is analyzed, where the objective function is never computed. By defining appropriate first-order stationarity criteria, we are able to extend the Adagrad method to the newly considered problem and retrieve the standard complexity rate of the projected gradient method that uses both the gradient and … Read more

Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization

\(\) This paper considers the problem of minimizing the sum of a smooth function and the Schatten-\(p\) norm of the matrix. Our contribution involves proposing accelerated iteratively reweighted nuclear norm methods designed for solving the nonconvex low-rank minimization problem. Two major novelties characterize our approach. Firstly, the proposed method possesses a rank identification property, enabling … Read more

An Extended Validity Domain for Constraint Learning

We consider embedding a predictive machine-learning model within a prescriptive optimization problem. In this setting, called constraint learning, we study the concept of a validity domain, i.e., a constraint added to the feasible set, which keeps the optimization close to the training data, thus helping to ensure that the computed optimal solution exhibits less prediction … Read more

A mathematical introduction to SVMs with self-concordant kernel

A derivation of so-called “soft-margin Support Vector Machines with kernel” is presented which does not rely on concepts from functional analysis such as Mercer’s theorem that is frequently cited in this context, and that leads to a new analysis of the continuity properties of the kernel functions such as a new self-concordance condition for the … Read more

Statistical and Computational Guarantees of Kernel Max-Sliced Wasserstein Distances

Optimal transport has been very successful for various machine learning tasks; however, it is known to suffer from the curse of dimensionality. Hence, dimensionality reduction is desirable when applied to high-dimensional data with low-dimensional structures. The kernel max-sliced (KMS) Wasserstein distance is developed for this purpose by finding an optimal nonlinear mapping that reduces data … Read more