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1. How and When Random Feedback Works: A Case Study of Low-Rank Matrix Factorization

   Shivam Garg and Santosh S. Vempala ( January 2022 , AISTATS)

   The success of gradient descent in ML and especially for learning neural networks is remarkable and robust. In the context of how the brain learns, one aspect of gradient descent that appears biologically difficult to realize (if not implausible) is that its updates rely on feedback from later layers to earlier layers through the same connections. Such bidirected links are relatively few in brain networks, and even when reciprocal connections exist, they may not be equi-weighted. Random Feedback Alignment (Lillicrap et al., 2016), where the backward weights are random and fixed, has been proposed as a bio-plausible alternative and found to be effective empirically. We investigate how and when feedback alignment (FA) works, focusing on one of the most basic problems with layered structure n×m, the goal is to find a low rank factorization Zn×rWr×m that minimizes the error ∥ZW−Y∥F. Gradient descent solves this problem optimally. We show that FA finds the optimal solution when r≥rank(Y). We also shed light on how FA works. It is observed empirically that the forward weight matrices and (random) feedback matrices come closer during FA updates. Our analysis rigorously derives this phenomenon and shows how it facilitates convergence of FA*, a closely related variantmore »of FA. We also show that FA can be far from optimal when r« less
2. Learning Low-rank Deep Neural Networks via Singular Vector Orthogonality Regularization and Singular Value Sparsification

   Yang, Huanrui ; Tang, Minxue ; Yan, Feng ; Hu, Daniel ; Li, Ang ; Li, Hai ; Chen, Yiran ( June 2020 , Joint Workshop on Efficient Deep Learning in Computer Vision)

   Modern deep neural networks (DNNs) often require high memory consumption and large computational loads. In order to deploy DNN algorithms efficiently on edge or mobile devices, a series of DNN compression algorithms have been explored, including factorization methods. Factorization methods approximate the weight matrix of a DNN layer with the multiplication of two or multiple low-rank matrices. However, it is hard to measure the ranks of DNN layers during the training process. Previous works mainly induce low-rank through implicit approximations or via costly singular value decomposition (SVD) process on every training step. The former approach usually induces a high accuracy loss while the latter has a low efficiency. In this work, we propose SVD training, the first method to explicitly achieve low-rank DNNs during training without applying SVD on every step. SVD training first decomposes each layer into the form of its full-rank SVD, then performs training directly on the decomposed weights. We add orthogonality regularization to the singular vectors, which ensure the valid form of SVD and avoid gradient vanishing/exploding. Low-rank is encouraged by applying sparsity-inducing regularizers on the singular values of each layer. Singular value pruning is applied at the end to explicitly reach a low-rank model. Wemore »empirically show that SVD training can significantly reduce the rank of DNN layers and achieve higher reduction on computation load under the same accuracy, comparing to not only previous factorization methods but also state-of-the-art filter pruning methods.« less
3. Preconditioned Gradient Descent for Over-Parameterized Nonconvex Matrix Factorization

   Zhang, Jialun ; Fattahi, Salar ; Zhang, Richard Y ( December 2021 , Advances in Neural Information Processing Systems)

   In practical instances of nonconvex matrix factorization, the rank of the true solution r^{\star} is often unknown, so the rank rof the model can be over-specified as r>r^{\star}. This over-parameterized regime of matrix factorization significantly slows down the convergence of local search algorithms, from a linear rate with r=r^{\star} to a sublinear rate when r>r^{\star}. We propose an inexpensive preconditioner for the matrix sensing variant of nonconvex matrix factorization that restores the convergence rate of gradient descent back to linear, even in the over-parameterized case, while also making it agnostic to possible ill-conditioning in the ground truth. Classical gradient descent in a neighborhood of the solution slows down due to the need for the model matrix factor to become singular. Our key result is that this singularity can be corrected by \ell_{2} regularization with a specific range of values for the damping parameter. In fact, a good damping parameter can be inexpensively estimated from the current iterate. The resulting algorithm, which we call preconditioned gradient descent or PrecGD, is stable under noise, and converges linearly to an information theoretically optimal error bound. Our numerical experiments find that PrecGD works equally well in restoring the linear convergence of other variants ofmore »nonconvex matrix factorization in the over-parameterized regime.« less
4. Beyond Lazy Training for Over-parameterized Tensor Decomposition

   Wang, Xiang ; Wu, Chenwei ; Lee, Jason D ; Ma, Tengyu ; Ge, Rong ( January 2020 , NeurIPS)

   Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an l-th order tensor in (Rd)⊗l of rank r (where r≪d), can variants of gradient descent find a rank m decomposition where m>r? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least m=Ω(dl−1), while a variant of gradient descent can find an approximate tensor when m=O∗(r2.5llogd). Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
5. Tensor recovery from noisy and multi-level quantized measurements

   https://doi.org/10.1186/s13634-020-00698-z  

   Wang, Ren ; Wang, Meng ; Xiong, Jinjun ( December 2020 , EURASIP Journal on Advances in Signal Processing)

   Abstract Higher-order tensors can represent scores in a rating system, frames in a video, and images of the same subject. In practice, the measurements are often highly quantized due to the sampling strategies or the quality of devices. Existing works on tensor recovery have focused on data losses and random noises. Only a few works consider tensor recovery from quantized measurements but are restricted to binary measurements. This paper, for the first time, addresses the problem of tensor recovery from multi-level quantized measurements by leveraging the low CANDECOMP/PARAFAC (CP) rank property. We study the recovery of both general low-rank tensors and tensors that have tensor singular value decomposition (TSVD) by solving nonconvex optimization problems. We provide the theoretical upper bounds of the recovery error, which diminish to zero when the sizes of dimensions increase to infinity. We further characterize the fundamental limit of any recovery algorithm and show that our recovery error is nearly order-wise optimal. A tensor-based alternating proximal gradient descent algorithm with a convergence guarantee and a TSVD-based projected gradient descent algorithm are proposed to solve the nonconvex problems. Our recovery methods can also handle data losses and do not necessarily need the information of the quantization rule.more »The methods are validated on synthetic data, image datasets, and music recommender datasets.« less