Robust and differentially private stochastic linear bandits

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.

Communication-efficient distributed eigenspace estimation with arbitrary node failures

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.

Communication-efficient distributed eigenspace estimation

Distributed computing is a standard way to scale up machine learning and data science algorithms to process large amounts of data. In such settings, avoiding communication amongst machines is paramount for achieving high performance. Rather than distribute the computation of existing algorithms, a common practice for avoiding communication is to compute local solutions or parameter estimates on each machine and then combine the results; in many convex optimization problems, even simple averaging of local solutions can work well. However, these schemes do not work when the local solutions are not unique. Spectral methods are a collection of such problems, where solutions are orthonormal bases of the leading invariant subspace of an associated data matrix, which are only unique up to rotation and reflections. Here, we develop a communication-efficient distributed algorithm for computing the leading invariant subspace of a data matrix. Our algorithm uses a novel alignment scheme that minimizes the Procrustean distance between local solutions and a reference solution, and only requires a single round of communication. For the important case of principal component analysis (PCA), we show that our algorithm achieves a similar error rate to that of a centralized estimator. We present numerical experiments demonstrating the efficacy of our proposed algorithm for distributed PCA, as well as other problems where solutions exhibit rotational symmetry, such as node embeddings for graph data and spectral initialization for quadratic sensing.

Entrywise convergence of iterative methods for eigenproblems

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 $\ell_{2 \to \infty}$ norm, but does not consider the actual computation of eigenvectors. Here we address the convergence of subspace iteration when distances are measured in the $\ell_{2 \to \infty}$ 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.

Stochastic algorithms with geometric step decay converge linearly on sharp functions

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay} and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of \emph{sharp} convex functions. In this work, we ask whether geometric step decay similarly improves stochastic algorithms for the class of sharp nonconvex problems. Such losses feature in modern statistical recovery problems and lead to a new challenge not present in the convex setting: the region of convergence is local, so one must bound the probability of escape. Our main result shows that for a large class of stochastic, sharp, nonsmooth, and nonconvex problems a geometric step decay schedule endows well-known algorithms with a local linear rate of convergence to global minimizers. This guarantee applies to the stochastic projected subgradient, proximal point, and prox-linear algorithms. As an application of our main result, we analyze two statistical recovery tasks---phase retrieval and blind deconvolution---and match the best known guarantees under Gaussian measurement models and establish new guarantees under heavy-tailed distributions.

Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence

The task of recovering a low-rank matrix from its noisy linear measurements plays a central role in computational science. Smooth formulations of the problem often exhibit an undesirable phenomenon: the condition number, classically defined, scales poorly with the dimension of the ambient space. In contrast, we here show that in a variety of concrete circumstances, nonsmooth penalty formulations do not suffer from the same type of ill-conditioning. Consequently, standard algorithms for nonsmooth optimization, such as subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. Moreover, nonsmooth formulations are naturally robust against outliers. Our framework subsumes such important computational tasks as phase retrieval, blind deconvolution, quadratic sensing, matrix completion, and robust PCA. Numerical experiments on these problems illustrate the benefits of the proposed approach.

Composite optimization for robust blind deconvolution

The blind deconvolution problem seeks to recover a pair of vectors from a set of rank one bilinear measurements. We consider a natural nonsmooth formulation of the problem and show that under standard statistical assumptions, its moduli of weak convexity, sharpness, and Lipschitz continuity are all dimension independent. This phenomenon persists even when up to half of the measurements are corrupted by noise. Consequently, standard algorithms, such as the subgradient and prox-linear methods, converge at a rapid dimension-independent rate when initialized within constant relative error of the solution. We then complete the paper with a new initialization strategy, complementing the local search algorithms. The initialization procedure is both provably efficient and robust to outlying measurements. Numerical experiments, on both simulated and real data, illustrate the developed theory and methods.

A Tropical Approach to Neural Networks with Piecewise Linear Activations

We present a new, unifying approach following some recent developments on the complexity of neural networks with piecewise linear activations. We treat neural network layers with piecewise linear activations, such as Maxout or ReLU units, as polynomials in the $(\max, +)$ (or so-called tropical) algebra. Following up on the work of Montufar et al. (arXiv:1402.1869), this approach enables us to improve their upper bound on linear regions of layers with ReLU or leaky ReLU activations to $\min \left\{ 2^m, 2 \cdot \sum_{j=0}^n \binom{m - 1}{j} \right\}$, where $n, m$ are the number of inputs and outputs, respectively. Additionally, we recover their upper bounds on maxout layers. Our work is parallel to the improvements reported in (arXiv:1711.02114, arXiv:1611.01491), though exclusively under the lens of tropical geometry.