Achieving Small Test Error in Mildly Overparameterized Neural Networks

Recent theoretical works on over-parameterized neural nets have focused on two aspects: optimization and generalization. Many existing works that study optimization and generalization together are based on neural tangent kernel and require a very large width. In this work, we are interested in the following question: for a binary classification problem with two-layer mildly over-parameterized ReLU network, can we find a point with small test error in polynomial time? We first show that the landscape of loss functions with explicit regularization has the following property: all local minima and certain other points which are only stationary in certain directions achieve small test error. We then prove that for convolutional neural nets, there is an algorithm which finds one of these points in polynomial time (in the input dimension and the number of data points). In addition, we prove that for a fully connected neural net, with an additional assumption on the data distribution, there is a polynomial time algorithm.

On a Faster $R$-Linear Convergence Rate of the Barzilai-Borwein Method

The Barzilai-Borwein (BB) method has demonstrated great empirical success in nonlinear optimization. However, the convergence speed of BB method is not well understood, as the known convergence rate of BB method for quadratic problems is much worse than the steepest descent (SD) method. Therefore, there is a large discrepancy between theory and practice. To shrink this gap, we prove that the BB method converges $R$-linearly at a rate of $1-1/\kappa$, where $\kappa$ is the condition number, for strongly convex quadratic problems. In addition, an example with the theoretical rate of convergence is constructed, indicating the tightness of our bound.

Towards a Better Global Loss Landscape of GANs

Understanding of GAN training is still very limited. One major challenge is its non-convex-non-concave min-max objective, which may lead to sub-optimal local minima. In this work, we perform a global landscape analysis of the empirical loss of GANs. We prove that a class of separable-GAN, including the original JS-GAN, has exponentially many bad basins which are perceived as mode-collapse. We also study the relativistic pairing GAN (RpGAN) loss which couples the generated samples and the true samples. We prove that RpGAN has no bad basins. Experiments on synthetic data show that the predicted bad basin can indeed appear in training. We also perform experiments to support our theory that RpGAN has a better landscape than separable-GAN. For instance, we empirically show that RpGAN performs better than separable-GAN with relatively narrow neural nets. The code is available at https://github.com/AilsaF/RS-GAN.

Center-wise Local Image Mixture For Contrastive Representation Learning

Recent advances in unsupervised representation learning have experienced remarkable progress, especially with the achievements of contrastive learning, which regards each image as well its augmentations as a separate class, while does not consider the semantic similarity among images. This paper proposes a new kind of data augmentation, named Center-wise Local Image Mixture, to expand the neighborhood space of an image. CLIM encourages both local similarity and global aggregation while pulling similar images. This is achieved by searching local similar samples of an image, and only selecting images that are closer to the corresponding cluster center, which we denote as center-wise local selection. As a result, similar representations are progressively approaching the clusters, while do not break the local similarity. Furthermore, image mixture is used as a smoothing regularization to avoid overconfidence on the selected samples. Besides, we introduce multi-resolution augmentation, which enables the representation to be scale invariant. Integrating the two augmentations produces better feature representation on several unsupervised benchmarks. Notably, we reach 75.5% top-1 accuracy with linear evaluation over ResNet-50, and 59.3% top-1 accuracy when fine-tuned with only 1% labels, as well as consistently outperforming supervised pretraining on several downstream transfer tasks.

A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems

Nonconvex-concave min-max problem arises in many machine learning applications including minimizing a pointwise maximum of a set of nonconvex functions and robust adversarial training of neural networks. A popular approach to solve this problem is the gradient descent-ascent (GDA) algorithm which unfortunately can exhibit oscillation in case of nonconvexity. In this paper, we introduce a "smoothing" scheme which can be combined with GDA to stabilize the oscillation and ensure convergence to a stationary solution. We prove that the stabilized GDA algorithm can achieve an $O(1/\epsilon^2)$ iteration complexity for minimizing the pointwise maximum of a finite collection of nonconvex functions. Moreover, the smoothed GDA algorithm achieves an $O(1/\epsilon^4)$ iteration complexity for general nonconvex-concave problems. Extensions of this stabilized GDA algorithm to multi-block cases are presented. To the best of our knowledge, this is the first algorithm to achieve $O(1/\epsilon^2)$ for a class of nonconvex-concave problem. We illustrate the practical efficiency of the stabilized GDA algorithm on robust training.

Landscape of Sparse Linear Network: A Brief Investigation

Network pruning, or sparse network has a long history and practical significance in modern application. A major concern for neural network training is that the non-convexity of the associated loss functions may cause bad landscape. We focus on analyzing sparse linear network generated from weight pruning strategy. With no unrealistic assumption, we prove the following statements for the squared loss objective of sparse linear neural networks: 1) every local minimum is a global minimum for scalar output with any sparse structure, or non-intersect sparse first layer and dense other layers with whitened data; 2) sparse linear networks have sub-optimal local-min for only sparse first layer or three target dimensions.

The Global Landscape of Neural Networks: An Overview

One of the major concerns for neural network training is that the non-convexity of the associated loss functions may cause bad landscape. The recent success of neural networks suggests that their loss landscape is not too bad, but what specific results do we know about the landscape? In this article, we review recent findings and results on the global landscape of neural networks. First, we point out that wide neural nets may have sub-optimal local minima under certain assumptions. Second, we discuss a few rigorous results on the geometric properties of wide networks such as "no bad basin", and some modifications that eliminate sub-optimal local minima and/or decreasing paths to infinity. Third, we discuss visualization and empirical explorations of the landscape for practical neural nets. Finally, we briefly discuss some convergence results and their relation to landscape results.

Global Convergence and Induced Kernels of Gradient-Based Meta-Learning with Neural Nets

Gradient-based meta-learning (GBML) with deep neural nets (DNNs) has become a popular approach for few-shot learning. However, due to the non-convexity of DNNs and the complex bi-level optimization in GBML, the theoretical properties of GBML with DNNs remain largely unknown. In this paper, we first develop a novel theoretical analysis to answer the following questions: Does GBML with DNNs have global convergence guarantees? We provide a positive answer to this question by proving that GBML with over-parameterized DNNs is guaranteed to converge to global optima at a linear rate. The second question we aim to address is: How does GBML achieve fast adaption to new tasks with experience on past similar tasks? To answer it, we prove that GBML is equivalent to a functional gradient descent operation that explicitly propagates experience from the past tasks to new ones. Finally, inspired by our theoretical analysis, we develop a new kernel-based meta-learning approach. We show that the proposed approach outperforms GBML with standard DNNs on the Omniglot dataset when the number of past tasks for meta-training is small. The code is available at https://github.com/ AI-secure/Meta-Neural-Kernel .

Distilling Object Detectors with Task Adaptive Regularization

Current state-of-the-art object detectors are at the expense of high computational costs and are hard to deploy to low-end devices. Knowledge distillation, which aims at training a smaller student network by transferring knowledge from a larger teacher model, is one of the promising solutions for model miniaturization. In this paper, we investigate each module of a typical detector in depth, and propose a general distillation framework that adaptively transfers knowledge from teacher to student according to the task specific priors. The intuition is that simply distilling all information from teacher to student is not advisable, instead we should only borrow priors from the teacher model where the student cannot perform well. Towards this goal, we propose a region proposal sharing mechanism to interflow region responses between the teacher and student models. Based on this, we adaptively transfer knowledge at three levels, \emph{i.e.}, feature backbone, classification head, and bounding box regression head, according to which model performs more reasonably. Furthermore, considering that it would introduce optimization dilemma when minimizing distillation loss and detection loss simultaneously, we propose a distillation decay strategy to help improve model generalization via gradually reducing the distillation penalty. Experiments on widely used detection benchmarks demonstrate the effectiveness of our method. In particular, using Faster R-CNN with FPN as an instantiation, we achieve an accuracy of $39.0\%$ with Resnet-50 on COCO dataset, which surpasses the baseline $36.3\%$ by $2.7\%$ points, and even better than the teacher model with $38.5\%$ mAP.