In this paper, we study the stochastic linear bandit problem under the additional requirements of differential privacy, robustness and batched observations. In particular, we assume an adversary randomly chooses a constant fraction of the observed rewards in each batch, replacing them with arbitrary numbers. We present differentially private and robust variants of the arm elimination algorithm using logarithmic batch queries under two privacy models and provide regret bounds in both settings. In the first model, every reward in each round is reported by a potentially different client, which reduces to standard local differential privacy (LDP). In the second model, every action is "owned" by a different client, who may aggregate the rewards over multiple queries and privatize the aggregate response instead. To the best of our knowledge, our algorithms are the first simultaneously providing differential privacy and adversarial robustness in the stochastic linear bandits problem.

We develop an eigenspace estimation algorithm for distributed environments with arbitrary node failures, where a subset of computing nodes can return structurally valid but otherwise arbitrarily chosen responses. Notably, this setting encompasses several important scenarios that arise in distributed computing and data-collection environments such as silent/soft errors, outliers or corrupted data at certain nodes, and adversarial responses. Our estimator builds upon and matches the performance of a recently proposed non-robust estimator up to an additive $\tilde{O}(\sigma \sqrt{\alpha})$ error, where $\sigma^2$ is the variance of the existing estimator and $\alpha$ is the fraction of corrupted nodes.

Tropical geometry is a relatively recent field in mathematics and computer science, combining elements of algebraic geometry and polyhedral geometry. The scalar arithmetic of its analytic part preexisted in the form of max-plus and min-plus semiring arithmetic used in finite automata, nonlinear image processing, convex analysis, nonlinear control, optimization, and idempotent mathematics. Tropical geometry recently emerged in the analysis and extension of several classes of problems and systems in both classical machine learning and deep learning. Three such areas include: 1) deep neural networks with piecewise linear (PWL) activation functions; 2) probabilistic graphical models; and 3) nonlinear regression with PWL functions. In this article, we first summarize introductory ideas and objects of tropical geometry, providing a theoretical framework for both the max-plus algebra that underlies tropical geometry and its extensions to general max algebras. This unifies scalar and vector/signal operations over a class of nonlinear spaces, called weighted lattices, and allows us to provide optimal solutions for algebraic equations used in tropical geometry and generalize tropical geometric objects. Then, we survey the state of the art and recent progress in the aforementioned areas. First, we illustrate a purely geometric approach for studying the representation power of neural networks with PWL activations. Then, we review the tropical geometric analysis of parametric statistical models, such as HMMs; later, we focus on the Viterbi algorithm and related methods for weighted finite-state transducers and provide compact and elegant representations via their formal tropical modeling. Finally, we provide optimal solutions and an efficient algorithm for the convex regression problem, using concepts and tools from tropical geometry and max-plus algebra. Throughout this article, we also outline problems and future directions in machine learning that can benefit from the tropical-geometric point of view.

Several problems in machine learning, statistics, and other fields rely on computing eigenvectors. For large scale problems, the computation of these eigenvectors is typically performed via iterative schemes such as subspace iteration or Krylov methods. While there is classical and comprehensive analysis for subspace convergence guarantees with respect to the spectral norm, in many modern applications other notions of subspace distance are more appropriate. Recent theoretical work has focused on perturbations of subspaces measured in the ℓ2→∞ norm, but does not consider the actual computation of eigenvectors. Here we address the convergence of subspace iteration when distances are measured in the ℓ2→∞ norm and provide deterministic bounds. We complement our analysis with a practical stopping criterion and demonstrate its applicability via numerical experiments. Our results show that one can get comparable performance on downstream tasks while requiring fewer iterations, thereby saving substantial computational time.