InOrMo, momentum isincorporated into ASGD byorganizing the gradients in order based on their iteration indexes. We theoretically prove the convergence of OrMo with both constant and delay-adaptive learning rates for non-convexproblems.

The main question about the spiked Wigner model is: how large should the signal-to-noise ratio λ > 0 be in order to achieve constant correlation withx?

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model $f(\boldsymbol{x})=g(\boldsymbol{U}\boldsymbol{x})$ with hidden subspace $\boldsymbol{U}\in \mathbb{R}^{r\times d}$, which is the canonical setup to study representation learning. We prove that under generic non-degenerate assumptions on the link function, a standard two-layer neural network trained via layer-wise gradient descent can agnostically learn the target with $o_d(1)$ test error using $\widetilde{\mathcal{O}}(d)$ samples and $\widetilde{\mathcal{O}}(d^2)$ time. The sample and time complexity both align with the information-theoretic limit up to leading order and are therefore optimal. During the first stage of gradient descent learning, the proof proceeds via showing that the inner weights can perform a power-iteration process. This process implicitly mimics a spectral start for the whole span of the hidden subspace and eventually eliminates finite-sample noise and recovers this span. It surprisingly indicates that optimal results can only be achieved if the first layer is trained for more than $\mathcal{O}(1)$ steps. This work demonstrates the ability of neural networks to effectively learn hierarchical functions with respect to both sample and time efficiency.

Recent advancement in online optimization and control has provided novel tools to study online linear quadratic regulator (LQR) problems, where cost matrices are varying adversarially over time. However, the controller parameterization of existing works may not satisfy practical conditions like sparsity due to physical connections. In this work, we study online linear quadratic Gaussian problems with a given linear constraint imposed on the controller. Inspired by the recent work of [1] which proposed, for a linearly constrained policy optimization of an offline LQR, a second order method equipped with a Riemannian metric that emerges naturally in the context of optimal control problems, we propose online optimistic Newton on manifold (OONM) which provides an online controller based on the prediction on the first and second order information of the function sequence. To quantify the proposed algorithm, we leverage the notion of regret defined as the sub-optimality of its cumulative cost to that of a (locally) minimizing controller sequence and provide the regret bound in terms of the path-length of the minimizer sequence. Simulation results are also provided to verify the property of OONM.

Planning for many manipulation tasks, such as using tools or assembling parts, often requires both symbolic and geometric reasoning. Task and Motion Planning (TAMP) algorithms typically solve these problems by conducting a tree search over high-level task sequences while checking for kinematic and dynamic feasibility. This can be inefficient as the width of the tree can grow exponentially with the number of possible actions and objects. In this paper, we propose a novel approach to TAMP that relaxes discrete-and-continuous TAMP problems into inference problems on a continuous domain. Our method, Stein Task and Motion Planning (STAMP) subsequently solves this new problem using a gradient-based variational inference algorithm called Stein Variational Gradient Descent, by obtaining gradients from a parallelized differentiable physics simulator. By introducing relaxations to the discrete variables, leveraging parallelization, and approaching TAMP as an Bayesian inference problem, our method is able to efficiently find multiple diverse plans in a single optimization run. We demonstrate our method on two TAMP problems and benchmark them against existing TAMP baselines.

Perfect synchronization in distributed machine learning problems is inefficient and even impossible due to the existence of latency, package losses and stragglers. We propose a Robust Fully-Asynchronous Stochastic Gradient Tracking method (R-FAST), where each device performs local computation and communication at its own pace without any form of synchronization. Different from existing asynchronous distributed algorithms, R-FAST can eliminate the impact of data heterogeneity across devices and allow for packet losses by employing a robust gradient tracking strategy that relies on properly designed auxiliary variables for tracking and buffering the overall gradient vector. More importantly, the proposed method utilizes two spanning-tree graphs for communication so long as both share at least one common root, enabling flexible designs in communication architectures. We show that R-FAST converges in expectation to a neighborhood of the optimum with a geometric rate for smooth and strongly convex objectives; and to a stationary point with a sublinear rate for general non-convex settings. Extensive experiments demonstrate that R-FAST runs 1.5-2 times faster than synchronous benchmark algorithms, such as Ring-AllReduce and D-PSGD, while still achieving comparable accuracy, and outperforms existing asynchronous SOTA algorithms, such as AD-PSGD and OSGP, especially in the presence of stragglers.

We provide a new online learning algorithm which utilizes online passive-aggressive learning (PA) and total-error-rate minimization (TER) for binary classification. The PA learning establishes not only large margin training but also the capacity to handle non-separable data. The TER learning on the other hand minimizes an approximated classification error based objective function. We propose an online PATER algorithm which combines those useful properties. In addition, we also present a weighted PATER algorithm to improve the ability to cope with data imbalance problems. Experimental results demonstrate that the proposed PATER algorithms achieves better performances in terms of efficiency and effectiveness than the existing state-of-the-art online learning algorithms in real-world data sets.

We study a bad arm existing checking problem in which a player's task is to judge whether a positive arm exists or not among given K arms by drawing as small number of arms as possible. Here, an arm is positive if its expected loss suffered by drawing the arm is at least a given threshold. This problem is a formalization of diagnosis of disease or machine failure. An interesting structure of this problem is the asymmetry of positive and negative (non-positive) arms' roles; finding one positive arm is enough to judge existence while all the arms must be discriminated as negative to judge non-existence. We propose an algorithms with arm selection policy (policy to determine the next arm to draw) and stopping condition (condition to stop drawing arms) utilizing this asymmetric problem structure and prove its effectiveness theoretically and empirically.