Multi-modal distributions are commonly used to model clustered data in statistical learning tasks. In this paper, we consider the Mixed Linear Regression (MLR) problem. We propose an optimal transport-based framework for MLR problems, Wasserstein Mixed Linear Regression (WMLR), which minimizes the Wasserstein distance between the learned and target mixture regression models. Through a model-based duality analysis, WMLR reduces the underlying MLR task to a nonconvex-concave minimax optimization problem, which can be provably solved to find a minimax stationary point by the Gradient Descent Ascent (GDA) algorithm. In the special case of mixtures of two linear regression models, we show that WMLR enjoys global convergence and generalization guarantees. We prove that WMLR’s sample complexity grows linearly with the dimension of data. Finally, we discuss the application of WMLR to the federated learning task where the training samples are collected by multiple agents in a network. Unlike the Expectation-Maximization algorithm, WMLR directly extends to the distributed, federated learning setting. We support our theoretical results through several numerical experiments, which highlight our framework’s ability to handle the federated learning setting with mixture models.

Stochastic Neural Networks (SNNs) that inject noise into their hidden layers have recently been shown to achieve strong robustness against adversarial attacks. However, existing SNNs are usually heuristically motivated, and often rely on adversarial training, which is computationally costly. We propose a new SNN that achieves state-of-the-art performance without relying on adversarial training, and enjoys solid theoretical justification. Specifically, while existing SNNs inject learned or hand-tuned isotropic noise, our SNN learns an anisotropic noise distribution to optimize a learning-theoretic bound on adversarial robustness. We evaluate our method on a number of popular benchmarks, show that it can be applied to different architectures, and that it provides robustness to a variety of white-box and black-box attacks, while being simple and fast to train compared to existing alternatives.

Boundary discontinuity and its inconsistency to the final detection metric have been the bottleneck for rotating detection regression loss design. In this paper, we propose a novel regression loss based on Gaussian Wasserstein distance as a fundamental approach to solve the problem. Specifically, the rotated bounding box is converted to a 2-D Gaussian distribution, which enables to approximate the indifferentiable rotational IoU induced loss by the Gaussian Wasserstein distance (GWD) which can be learned efficiently by gradient back-propagation. GWD can still be informative for learning even there is no overlapping between two rotating bounding boxes which is often the case for small object detection. Thanks to its three unique properties, GWD can also elegantly solve the boundary discontinuity and square-like problem regardless how the bounding box is defined. Experiments on five datasets using different detectors show the effectiveness of our approach, and codes are available at https://github.com/yangxue0827/RotationDetection.

We study distributed computing of the truncated singular value decomposition (SVD). We develop an algorithm that we call \texttt{LocalPower} for improving communication efficiency. Specifically, we uniformly partition the dataset among $m$ nodes and alternate between multiple (precisely $p$) local power iterations and one global aggregation. In the aggregation, we propose to weight each local eigenvector matrix with orthogonal Procrustes transformation (OPT). As a practical surrogate of OPT, sign-fixing, which uses a diagonal matrix with $\pm 1$ entries as weights, has better computation complexity and stability in experiments. We theoretically show that under certain assumptions \texttt{LocalPower} lowers the required number of communications by a factor of $p$ to reach a constant accuracy. We also show that the strategy of periodically decaying $p$ helps obtain high-precision solutions. We conduct experiments to demonstrate the effectiveness of \texttt{LocalPower}.

The Wasserstein distance has become increasingly important in machine learning and deep learning. Despite its popularity, the Wasserstein distance is hard to approximate because of the curse of dimensionality. A recently proposed approach to alleviate the curse of dimensionality is to project the sampled data from the high dimensional probability distribution onto a lower-dimensional subspace, and then compute the Wasserstein distance between the projected data. However, this approach requires to solve a max-min problem over the Stiefel manifold, which is very challenging in practice. In this paper, we propose a Riemannian block coordinate descent (RBCD) method to solve this problem, which is based on a novel reformulation of the regularized max-min problem over the Stiefel manifold. We show that the complexity of arithmetic operations for RBCD to obtain an $\epsilon$-stationary point is $O(\epsilon^{-3})$, which is significantly better than the complexity of existing methods. Numerical results on both synthetic and real datasets demonstrate that our method is more efficient than existing methods, especially when the number of sampled data is very large.

We consider a best arm identification (BAI) problem for stochastic bandits with adversarial corruptions in the fixed-budget setting of T steps. We design a novel randomized algorithm, Probabilistic Sequential Shrinking(u) (PSS(u)), which is agnostic to the amount of corruptions. When the amount of corruptions per step (CPS) is below a threshold, PSS(u) identifies the best arm or item with probability tending to 1 as T→∞. Otherwise, the optimality gap of the identified item degrades gracefully with the CPS.We argue that such a bifurcation is necessary. In PSS(u), the parameter u serves to balance between the optimality gap and success probability. The injection of randomization is shown to be essential to mitigate the impact of corruptions. To demonstrate this, we design two attack strategies that are applicable to any algorithm. We apply one of them to a deterministic analogue of PSS(u) known as Successive Halving (SH) by Karnin et al. (2013). The attack strategy results in a high failure probability for SH, but PSS(u) remains robust. In the absence of corruptions, PSS(2)'s performance guarantee matches SH's. We show that when the CPS is sufficiently large, no algorithm can achieve a BAI probability tending to 1 as T→∞. Numerical experiments corroborate our theoretical findings.

Inductive program synthesis, or inferring programs from examples of desired behavior, offers a general paradigm for building interpretable, robust, andgeneralizable machine learning systems. Effective program synthesis depends on two key ingredients: a strong library of functions from which to build programs, and an efficient search strategy for finding programs that solve a given task. We introduce LAPS (Language for Abstraction and Program Search), a technique for using natural language annotations to guide joint learning of libraries and neurally-guided search models for synthesis. When integrated into a state-of-the-art library learning system (DreamCoder), LAPS produces higher-quality libraries and improves search efficiency and generalization on three domains – string editing, image composition, and abstract reasoning about scenes – even when no natural language hints are available at test time.