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.
The credit scoring industry has a long tradition of using statistical tools for loan default probability prediction and domain specific standards have been established long before the hype of machine learning. Although several commercial software companies offer specific solutions for credit scorecard modelling in R explicit packages for this purpose have been missing long time. In the recent years this has changed and several packages have been developed which are dedicated to credit scoring. The aim of this paper is to give a structured overview on these packages. This may guide users to sel…
Dataloaders, in charge of moving data from storage into GPUs while training machine learning models, might hold the key to drastically improving the performance of training jobs. Recent advances have shown promise not only by considerably decreasing training time but also by offering new features such as loading data from remote storage like S3. In this paper, we are the first to distinguish the dataloader as a separate component in the Deep Learning (DL) workflow and to outline its structure and features. Finally, we offer a comprehensive comparison of the different dataloading libraries avai…
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep linear neural networks)}. In specific, we characterize the locations of stationary points and the null space of Hessian matrices \xl{of the objective function} via the lens of invariant groups\removed{for associated optimization problems, including low-rank matrix factorization, phase retrieval, and deep linear neural networks}. As a major motivating example, w…
For one-hidden-layer ReLU networks, we prove that all differentiable local minima are global inside differentiable regions. We give the locations and losses of differentiable local minima, and show that these local minima can be isolated points or continuous hyperplanes, depending on an interplay between data, activation pattern of hidden neurons and network size. Furthermore, we give necessary and sufficient conditions for the existence of saddle points as well as non-differentiable local minima, and their locations if they exist.
Imitation Learning is a sequential task where the learner tries to mimic an expert's action in order to achieve the best performance. Several algorithms have been proposed recently for this task. In this project, we aim at proposing a wide review of these algorithms, presenting their main features and comparing them on their performance and their regret bounds.
Center for Machine Learning, Capital One（美国Capital One公司机器学习中心）；Columbia University（哥伦比亚大学）
A barrier to the wider adoption of neural networks is their lack of interpretability. While local explanation methods exist for one prediction, most global attributions still reduce neural network decisions to a single set of features. In response, we present an approach for generating global attributions called GAM, which explains the landscape of neural network predictions across subpopulations. GAM augments global explanations with the proportion of samples that each attribution best explains and specifies which samples are described by each attribution. Global explanations also have tunabl…
University of California, Berkeley（加州大学伯克利分校）；International Computer Science Institute（国际计算机科学研究所）
Viewing neural network models in terms of their loss landscapes has a long history in the statistical mechanics approach to learning, and in recent years it has received attention within machine learning proper. Among other things, local metrics (such as the smoothness of the loss landscape) have been shown to correlate with global properties of the model (such as good generalization performance). Here, we perform a detailed empirical analysis of the loss landscape structure of thousands of neural network models, systematically varying learning tasks, model architectures, and/or quantity/quali…
Faculty of Electrical Engineering, Technion（以色列理工学院电气工程学院）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
While optimizing convex objective (loss) functions has been a powerhouse for machine learning for at least two decades, non-convex loss functions have attracted fast growing interests recently, due to many desirable properties such as superior robustness and classification accuracy, compared with their convex counterparts. The main obstacle for non-convex estimators is that it is in general intractable to find the optimal solution. In this paper, we study the computational issues for some non-convex M-estimators. In particular, we show that the stochastic variance reduction methods converge to…
Microsoft Research（微软研究院）；Google Research（谷歌研究院）；University of Denver（丹佛大学）；University of Michigan（密歇根大学）
When training deep neural networks for classification tasks, an intriguing empirical phenomenon has been widely observed in the last-layer classifiers and features, where (i) the class means and the last-layer classifiers all collapse to the vertices of a Simplex Equiangular Tight Frame (ETF) up to scaling, and (ii) cross-example within-class variability of last-layer activations collapses to zero. This phenomenon is called Neural Collapse (NC), which seems to take place regardless of the choice of loss functions. In this work, we justify NC under the mean squared error (MSE) loss, where recen…
The past decade has witnessed a successful application of deep learning to solving many challenging problems in machine learning and artificial intelligence. However, the loss functions of deep neural networks (especially nonlinear networks) are still far from being well understood from a theoretical aspect. In this paper, we enrich the current understanding of the landscape of the square loss functions for three types of neural networks. Specifically, when the parameter matrices are square, we provide an explicit characterization of the global minimizers for linear networks, linear residual n…
We prove that finding all globally optimal two-layer ReLU neural networks can be performed by solving a convex optimization program with cone constraints. Our analysis is novel, characterizes all optimal solutions, and does not leverage duality-based analysis which was recently used to lift neural network training into convex spaces. Given the set of solutions of our convex optimization program, we show how to construct exactly the entire set of optimal neural networks. We provide a detailed characterization of this optimal set and its invariant transformations. As additional consequences of o…
Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that…
We consider the problem of learning a one-hidden-layer neural network: we assume the input $x\in \mathbb{R}^d$ is from Gaussian distribution and the label $y = a^\top σ(Bx) + ξ$, where $a$ is a nonnegative vector in $\mathbb{R}^m$ with $m\le d$, $B\in \mathbb{R}^{m\times d}$ is a full-rank weight matrix, and $ξ$ is a noise vector. We first give an analytic formula for the population risk of the standard squared loss and demonstrate that it implicitly attempts to decompose a sequence of low-rank tensors simultaneously. Inspired by the formula, we design a non-convex objective function $G(\cdot)…
The existence of local minima for one-hidden-layer ReLU networks has been investigated theoretically in [8]. Based on the theory, in this paper, we first analyze how big the probability of existing local minima is for 1D Gaussian data and how it varies in the whole weight space. We show that this probability is very low in most regions. We then design and implement a linear programming based approach to judge the existence of genuine local minima, and use it to predict whether bad local minima exist for the MNIST and CIFAR-10 datasets, and find that there are no bad differentiable local minima…
Deep learning models evolve through training to learn the manifold in which the data exists to satisfy an objective. It is well known that evolution leads to different final states which produce inconsistent predictions of the same test data points. This calls for techniques to be able to empirically quantify the difference in the trajectories and highlight problematic regions. While much focus is placed on discovering what models learn, the question of how a model learns is less studied beyond theoretical landscape characterizations and local geometric approximations near optimal conditions. …
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
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 …
We study the effects of mild over-parameterization on the optimization landscape of a simple ReLU neural network of the form $\mathbf{x}\mapsto\sum_{i=1}^k\max\{0,\mathbf{w}_i^{\top}\mathbf{x}\}$, in a well-studied teacher-student setting where the target values are generated by the same architecture, and when directly optimizing over the population squared loss with respect to Gaussian inputs. We prove that while the objective is strongly convex around the global minima when the teacher and student networks possess the same number of neurons, it is not even \emph{locally convex} after any amo…
Saarland University（萨尔大学）；University of Tübingen（图宾根大学）
We analyze the loss landscape and expressiveness of practical deep convolutional neural networks (CNNs) with shared weights and max pooling layers. We show that such CNNs produce linearly independent features at a "wide" layer which has more neurons than the number of training samples. This condition holds e.g. for the VGG network. Furthermore, we provide for such wide CNNs necessary and sufficient conditions for global minima with zero training error. For the case where the wide layer is followed by a fully connected layer we show that almost every critical point of the empirical loss is a gl…
An overview of condensation phenomenon in deep learning
School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC, Shanghai Jiao Tong University（上海交通大学数学科学学院、自然科学研究院、教育部科学与工程计算重点实验室）；School of Artificial Intelligence, Shanghai Jiao Tong University（上海交通大学人工智能学院）
Securing Reliability: A Brief Overview on Enhancing In-Context Learning for Foundation Models
State Key Lab of Novel Software Technology, Nanjing University（南京大学现代软件技术国家重点实验室）；National University of Singapore（新加坡国立大学）；Peking University（北京大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the number of observations are fewer than the number of parameters in the model. We show that with quadratic activations the optimization landscape of training such shallow neural networks has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input/output pairs. For differentiable…
We explore some mathematical features of the loss landscape of overparameterized neural networks. A priori one might imagine that the loss function looks like a typical function from $\mathbb{R}^n$ to $\mathbb{R}$ - in particular, nonconvex, with discrete global minima. In this paper, we prove that in at least one important way, the loss function of an overparameterized neural network does not look like a typical function. If a neural net has $n$ parameters and is trained on $d$ data points, with $n>d$, we show that the locus $M$ of global minima of $L$ is usually not discrete, but rather an $…
Knowledge of search-landscape features of BlackBox Optimization (BBO) problems offers valuable information in light of the Algorithm Selection and/or Configuration problems. Exploratory Landscape Analysis (ELA) models have gained success in identifying predefined human-derived features and in facilitating portfolio selectors to address those challenges. Unlike ELA approaches, the current study proposes to transform the identification problem into an image recognition problem, with a potential to detect conception-free, machine-driven landscape features. To this end, we introduce the notion of …
The loss landscape of deep linear neural networks: a second-order analysis
RWTH Aachen University（亚琛工业大学）；Université de Toulouse（图卢兹大学）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；CNRS（法国国家科学研究中心）；Institut de Recherche Technologique Saint Exupéry（圣埃克苏佩里技术研究院）
LossLens: Diagnostics for Machine Learning through Loss Landscape Visual Analytics
Arizona State University（亚利桑那州立大学）；Dartmouth College（达特茅斯学院）；University of California, Davis（加州大学戴维斯分校）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；ICSI（国际计算机科学研究所）；University of California at Berkeley（加州大学伯克利分校）
