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Deep neural networks have achieved impressive performance in many areas. Designing a fast and provable method for training neural networks is a fundamental question in machine learning. The classical training method requires paying $\Omega(mnd)$ cost for both forward computation and backward computation, where $m$ is the width of the neural network, and we are given $n$ training points in $d$-dimensional space. In this paper, we propose two novel preprocessing ideas to bypass this $\Omega(mnd)$ barrier: $\bullet$ First, by preprocessing the initial weights of the neural networks, we can train the neural network in $\widetilde{O}(m^{1-\Theta(1/d)} n d)$ cost per iteration. $\bullet$ Second, by preprocessing the input data points, we can train the neural network in $\widetilde{O} (m^{4/5} nd )$ cost per iteration. From the technical perspective, our result is a sophisticated combination of tools in different fields, greedy-type convergence analysis in optimization, sparsity observation in practical work, high-dimensional geometric search in data structure, concentration and anti-concentration in probability. Our results also provide theoretical insights for a large number of previously established fast training methods. In addition, our classical algorithm can be generalized to the Quantum computation model. Interestingly, we can get a similar sublinear cost per iteration but avoid preprocessing initial weights or input data points., Comment: NeurIPS 2021

We propose a simple and efficient approach for training the BERT model. Our approach exploits the special structure of BERT that contains a stack of repeated modules (i.e., transformer encoders). Our proposed approach first trains BERT with the weights shared across all the repeated modules till some point. This is for learning the commonly shared component of weights across all repeated layers. We then stop weight sharing and continue training until convergence. We present theoretic insights for training by sharing weights then unsharing with analysis for simplified models. Empirical experiments on the BERT model show that our method yields better performance of trained models, and significantly reduces the number of training iterations.

A more realistic object detection paradigm, Open-World Object Detection, has arisen increasing research interests in the community recently. A qualified open-world object detector can not only identify objects of known categories, but also discover unknown objects, and incrementally learn to categorize them when their annotations progressively arrive. Previous works rely on independent modules to recognize unknown categories and perform incremental learning, respectively. In this paper, we provide a unified perspective: Semantic Topology. During the life-long learning of an open-world object detector, all object instances from the same category are assigned to their corresponding pre-defined node in the semantic topology, including the `unknown' category. This constraint builds up discriminative feature representations and consistent relationships among objects, thus enabling the detector to distinguish unknown objects out of the known categories, as well as making learned features of known objects undistorted when learning new categories incrementally. Extensive experiments demonstrate that semantic topology, either randomly-generated or derived from a well-trained language model, could outperform the current state-of-the-art open-world object detectors by a large margin, e.g., the absolute open-set error is reduced from 7832 to 2546, exhibiting the inherent superiority of semantic topology on open-world object detection., Comment: ICLR 2022

We investigate the tensor network representations of fermionic crystalline symmetry-protected topological (SPT) phases on two-dimensional lattices. As a mapping from virtual indices to physical indices, projected entangled-pair state (PEPS) serves as a concrete way to construct the wavefunctions of 2D crystalline fermionic SPT (fSPT) phases protected by 17 wallpaper group symmetries, for both spinless and spin-1/2 fermions. Based on PEPS, the full classification of 2D crystalline fSPT phases with wallpaper groups can be obtained. Tensor network states provide a natural framework for studying 2D crystalline fSPT phases., Comment: 12 figures, 3 tables, comments are welcome

We present a general approach for controlling robotic systems that make and break contact with their environments. Contact-implicit model predictive control (CI-MPC) generalizes linear MPC to contact-rich settings by utilizing a bi-level planning formulation with lower-level contact dynamics formulated as time-varying linear complementarity problems (LCPs) computed using strategic Taylor approximations about a reference trajectory. These dynamics enable the upper-level planning problem to reason about contact timing and forces, and generate entirely new contact-mode sequences online. To achieve reliable and fast numerical convergence, we devise a structure-exploiting interior-point solver for these LCP contact dynamics and a custom trajectory optimizer for the tracking problem. We demonstrate real-time solution rates for CI-MPC and the ability to generate and track non-periodic behaviours in hardware experiments on a quadrupedal robot. We also show that the controller is robust to model mismatch and can respond to disturbances by discovering and exploiting new contact modes across a variety of robotic systems in simulation, including a pushbot, planar hopper, planar quadruped, and planar biped., Comment: submitted to Transactions on Robotics (T-RO), under review

Online tracking of multiple objects in videos requires strong capacity of modeling and matching object appearances. Previous methods for learning appearance embedding mostly rely on instance-level matching without considering the temporal continuity provided by videos. We design a new instance-to-track matching objective to learn appearance embedding that compares a candidate detection to the embedding of the tracks persisted in the tracker. It enables us to learn not only from videos labeled with complete tracks, but also unlabeled or partially labeled videos. We implement this learning objective in a unified form following the spirit of constrastive loss. Experiments on multiple object tracking datasets demonstrate that our method can effectively learning discriminative appearance embeddings in a semi-supervised fashion and outperform state of the art methods on representative benchmarks.

A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical analysis of this LDG method, focusing on the free boundary case. The problem consists of minimizing a fourth order bending energy subject to a nonlinear and nonconvex metric constraint. The energy is discretized using LDG and a discrete gradient flow is used for computing discrete minimizers. We first show $\Gamma$-convergence of the discrete energy to the continuous one. Then we prove that the discrete gradient flow decreases the energy at each step and computes discrete minimizers with control of the metric constraint defect. We also present a numerical scheme for initialization of the gradient flow, and discuss the conditional stability of it., Comment: 29 pages, no figure

In label-noise learning, estimating the transition matrix is a hot topic as the matrix plays an important role in building statistically consistent classifiers. Traditionally, the transition from clean labels to noisy labels (i.e., clean-label transition matrix (CLTM)) has been widely exploited to learn a clean label classifier by employing the noisy data. Motivated by that classifiers mostly output Bayes optimal labels for prediction, in this paper, we study to directly model the transition from Bayes optimal labels to noisy labels (i.e., Bayes-label transition matrix (BLTM)) and learn a classifier to predict Bayes optimal labels. Note that given only noisy data, it is ill-posed to estimate either the CLTM or the BLTM. But favorably, Bayes optimal labels have less uncertainty compared with the clean labels, i.e., the class posteriors of Bayes optimal labels are one-hot vectors while those of clean labels are not. This enables two advantages to estimate the BLTM, i.e., (a) a set of examples with theoretically guaranteed Bayes optimal labels can be collected out of noisy data; (b) the feasible solution space is much smaller. By exploiting the advantages, we estimate the BLTM parametrically by employing a deep neural network, leading to better generalization and superior classification performance., Comment: ICML 22 camera ready

We tackle the problem of visual search under resource constraints. Existing systems use the same embedding model to compute representations (embeddings) for the query and gallery images. Such systems inherently face a hard accuracy-efficiency trade-off: the embedding model needs to be large enough to ensure high accuracy, yet small enough to enable query-embedding computation on resource-constrained platforms. This trade-off could be mitigated if gallery embeddings are generated from a large model and query embeddings are extracted using a compact model. The key to building such a system is to ensure representation compatibility between the query and gallery models. In this paper, we address two forms of compatibility: One enforced by modifying the parameters of each model that computes the embeddings. The other by modifying the architectures that compute the embeddings, leading to compatibility-aware neural architecture search (CMP-NAS). We test CMP-NAS on challenging retrieval tasks for fashion images (DeepFashion2), and face images (IJB-C). Compared to ordinary (homogeneous) visual search using the largest embedding model (paragon), CMP-NAS achieves 80-fold and 23-fold cost reduction while maintaining accuracy within 0.3% and 1.6% of the paragon on DeepFashion2 and IJB-C respectively., Comment: Accepted at CVPR 2021