TANet++: Triple Attention Network with Filtered Pointcloud on 3D Detection

TANet is one of state-of-the-art 3D object detection method on KITTI and JRDB benchmark, the network contains a Triple Attention module and Coarse-to-Fine Regression module to improve the robustness and accuracy of 3D Detection. However, since the original input data (point clouds) contains a lot of noise during collecting the data, which will further affect the training of the model. For example, the object is far from the robot, the sensor is difficult to obtain enough pointcloud. If the objects only contains few point clouds, and the samples are fed into model with the normal samples together during training, the detector will be difficult to distinguish the individual with few pointcloud belong to object or background. In this paper, we propose TANet++ to improve the performance on 3D Detection, which adopt a novel training strategy on training the TANet. In order to reduce the negative impact by the weak samples, the training strategy previously filtered the training data, and then the TANet++ is trained by the rest of data. The experimental results shows that AP score of TANet++ is 8.98\% higher than TANet on JRDB benchmark.

Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows that it provably converges at a linear rate independent of the condition number of the ground truth tensor for two canonical problems -- tensor completion and tensor regression -- as soon as the sample size is above the order of $n^{3/2}$ ignoring other dependencies, where $n$ is the dimension of the tensor. This leads to an extremely scalable approach to low-rank tensor estimation compared with prior art, which suffers from at least one of the following drawbacks: extreme sensitivity to ill-conditioning, high per-iteration costs in terms of memory and computation, or poor sample complexity guarantees. To the best of our knowledge, ScaledGD is the first algorithm that achieves near-optimal statistical and computational complexities simultaneously for low-rank tensor completion with the Tucker decomposition. Our algorithm highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the underlying symmetry in low-rank tensor factorization.

Bridging Offline Reinforcement Learning and Imitation Learning: A Tale of Pessimism

Offline (or batch) reinforcement learning (RL) algorithms seek to learn an optimal policy from a fixed dataset without active data collection. Based on the composition of the offline dataset, two main categories of methods are used: imitation learning which is suitable for expert datasets and vanilla offline RL which often requires uniform coverage datasets. From a practical standpoint, datasets often deviate from these two extremes and the exact data composition is usually unknown a priori. To bridge this gap, we present a new offline RL framework that smoothly interpolates between the two extremes of data composition, hence unifying imitation learning and vanilla offline RL. The new framework is centered around a weak version of the concentrability coefficient that measures the deviation from the behavior policy to the expert policy alone. Under this new framework, we further investigate the question on algorithm design: can one develop an algorithm that achieves a minimax optimal rate and also adapts to unknown data composition? To address this question, we consider a lower confidence bound (LCB) algorithm developed based on pessimism in the face of uncertainty in offline RL. We study finite-sample properties of LCB as well as information-theoretic limits in multi-armed bandits, contextual bandits, and Markov decision processes (MDPs). Our analysis reveals surprising facts about optimality rates. In particular, in all three settings, LCB achieves a faster rate of $1/N$ for nearly-expert datasets compared to the usual rate of $1/\sqrt{N}$ in offline RL, where $N$ is the number of samples in the batch dataset. In the case of contextual bandits with at least two contexts, we prove that LCB is adaptively optimal for the entire data composition range, achieving a smooth transition from imitation learning to offline RL. We further show that LCB is almost adaptively optimal in MDPs.

Minimax Off-Policy Evaluation for Multi-Armed Bandits

We study the problem of off-policy evaluation in the multi-armed bandit model with bounded rewards, and develop minimax rate-optimal procedures under three settings. First, when the behavior policy is known, we show that the Switch estimator, a method that alternates between the plug-in and importance sampling estimators, is minimax rate-optimal for all sample sizes. Second, when the behavior policy is unknown, we analyze performance in terms of the competitive ratio, thereby revealing a fundamental gap between the settings of known and unknown behavior policies. When the behavior policy is unknown, any estimator must have mean-squared error larger -- relative to the oracle estimator equipped with the knowledge of the behavior policy -- by a multiplicative factor proportional to the support size of the target policy. Moreover, we demonstrate that the plug-in approach achieves this worst-case competitive ratio up to a logarithmic factor. Third, we initiate the study of the partial knowledge setting in which it is assumed that the minimum probability taken by the behavior policy is known. We show that the plug-in estimator is optimal for relatively large values of the minimum probability, but is sub-optimal when the minimum probability is low. In order to remedy this gap, we propose a new estimator based on approximation by Chebyshev polynomials that provably achieves the optimal estimation error. Numerical experiments on both simulated and real data corroborate our theoretical findings.

Beyond Procrustes: Balancing-Free Gradient Descent for Asymmetric Low-Rank Matrix Sensing

Low-rank matrix estimation plays a central role in various applications across science and engineering. Recently, nonconvex formulations based on matrix factorization are provably solved by simple gradient descent algorithms with strong computational and statistical guarantees. However, when the low-rank matrices are asymmetric, existing approaches rely on adding a regularization term to balance the scale of the two matrix factors which in practice can be removed safely without hurting the performance when initialized via the spectral method. In this paper, we provide a theoretical justification to this for the matrix sensing problem, which aims to recover a low-rank matrix from a small number of linear measurements. As long as the measurement ensemble satisfies the restricted isometry property, gradient descent -- in conjunction with spectral initialization -- converges linearly without the need of explicitly promoting balancedness of the factors; in fact, the factors stay balanced automatically throughout the execution of the algorithm. Our analysis is based on analyzing the evolution of a new distance metric that directly accounts for the ambiguity due to invertible transforms, and might be of independent interest.

Spectral Methods for Data Science: A Statistical Perspective

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, data science, and signal processing. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to initialize other more sophisticated algorithms to improve performance. While the studies of spectral methods can be traced back to classical matrix perturbation theory and methods of moments, the past decade has witnessed tremendous theoretical advances in demystifying their efficacy through the lens of statistical modeling, with the aid of non-asymptotic random matrix theory. This monograph aims to present a systematic, comprehensive, yet accessible introduction to spectral methods from a modern statistical perspective, highlighting their algorithmic implications in diverse large-scale applications. In particular, our exposition gravitates around several central questions that span various applications: how to characterize the sample efficiency of spectral methods in reaching a target level of statistical accuracy, and how to assess their stability in the face of random noise, missing data, and adversarial corruptions? In addition to conventional $\ell_2$ perturbation analysis, we present a systematic $\ell_{\infty}$ and $\ell_{2,\infty}$ perturbation theory for eigenspace and singular subspaces, which has only recently become available owing to a powerful "leave-one-out" analysis framework.

Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number

Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are optimized via first-order methods over a smooth loss function, such as the residual sum of squares. While tremendous progresses have been made in recent years, the natural smooth formulation suffers from two sources of ill-conditioning, where the iteration complexity of gradient descent scales poorly both with the dimension as well as the condition number of the low-rank matrix. Moreover, the smooth formulation is not robust to corruptions. In this paper, we propose scaled subgradient methods to minimize a family of nonsmooth and nonconvex formulations -- in particular, the residual sum of absolute errors -- which is guaranteed to converge at a fast rate that is almost dimension-free and independent of the condition number, even in the presence of corruptions. We illustrate the effectiveness of our approach when the observation operator satisfies certain mixed-norm restricted isometry properties, and derive state-of-the-art performance guarantees for a variety of problems such as robust low-rank matrix sensing and quadratic sampling.

Learning Mixtures of Low-Rank Models

We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings -- low-rank matrix sensing and mixed linear regression -- by bringing latent variables (i.e. unknown labels) and structural priors (i.e. low-rank structures) into consideration. To cope with the non-convexity issues arising from unlabelled heterogeneous data and low-complexity structure, we develop a three-stage meta-algorithm that is guaranteed to recover the unknown matrices with near-optimal sample and computational complexities under Gaussian designs. In addition, the proposed algorithm is provably stable against random noise. We complement the theoretical studies with empirical evidence that confirms the efficacy of our algorithm.

Accelerating Ill-Conditioned Low-Rank Matrix Estimation via Scaled Gradient Descent

Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then seek to optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approaches can still be computationally expensive especially for ill-conditioned matrices: the convergence rate of gradient descent depends linearly on the condition number of the low-rank matrix, while the per-iteration cost of alternating minimization is often prohibitive for large matrices. The goal of this paper is to set forth a new algorithmic approach dubbed Scaled Gradient Descent (ScaledGD) which can be viewed as pre-conditioned or diagonally-scaled gradient descent, where the pre-conditioners are adaptive and iteration-varying with a minimal computational overhead. For low-rank matrix sensing and robust principal component analysis, we theoretically show that ScaledGD achieves the best of both worlds: it converges linearly at a rate independent of the condition number similar as alternating minimization, while maintaining the low per-iteration cost of gradient descent. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties. At the core of our analysis is the introduction of a new distance function that takes account of the pre-conditioners when measuring the distance between the iterates and the ground truth.

Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data

This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-a-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the $\ell_{\infty}$ loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper.