Convex optimizers have known many applications as differentiable layers within deep neural architectures. One application of these convex layers is to project points into a convex set. However, both forward and backward passes of these convex layers are significantly more expensive to compute than those of a typical neural network. We investigate in this paper whether an inexact, but cheaper projection, can drive a descent algorithm to an optimum. Specifically, we propose an interpolation-based projection that is computationally cheap and easy to compute given a convex, domain defining, function. We then propose an optimization algorithm that follows the gradient of the composition of the objective and the projection and prove its convergence for linear objectives and arbitrary convex and Lipschitz domain defining inequality constraints. In addition to the theoretical contributions, we demonstrate empirically the practical interest of the interpolation projection when used in conjunction with neural networks in a reinforcement learning and a supervised learning setting.
This work examines the problem of exact data interpolation via sparse (neuron count), infinitely wide, single hidden layer neural networks with leaky rectified linear unit activations. Using the atomic norm framework of [Chandrasekaran et al., 2012], we derive simple characterizations of the convex hulls of the corresponding atomic sets for this problem under several different constraints on the weights and biases of the network, thus obtaining equivalent convex formulations for these problems. A modest extension of our proposed framework to a binary classification problem is also presented. W…
LTCI, Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院电信学院LTCI研究所）；ENS Paris, CREST-ENSAE, CNRS（巴黎高等师范学院、CREST-ENSAE、法国国家科学研究中心）；Mila and DIRO, Université de Montréal（蒙特利尔大学Mila研究所与计算机科学与运筹学系）
Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Poly…
Mila（米拉人工智能研究所）；Université de Montréal（蒙特利尔大学）；University of British Columbia（不列颠哥伦比亚大学）；Element AI（元素人工智能公司）；QBit（1量子比特公司）；Amii（阿尔伯塔机器智能研究所）
Recent works have shown that stochastic gradient descent (SGD) achieves the fast convergence rates of full-batch gradient descent for over-parameterized models satisfying certain interpolation conditions. However, the step-size used in these works depends on unknown quantities and SGD's practical performance heavily relies on the choice of this step-size. We propose to use line-search techniques to automatically set the step-size when training models that can interpolate the data. In the interpolation setting, we prove that SGD with a stochastic variant of the classic Armijo line-search attain…
Technion - Israel Institute of Technology（以色列理工学院）
We revisit the challenge of designing online algorithms for the bandit convex optimization problem (BCO) which are also scalable to high dimensional problems. Hence, we consider algorithms that are \textit{projection-free}, i.e., based on the conditional gradient method whose only access to the feasible decision set, is through a linear optimization oracle (as opposed to other methods which require potentially much more computationally-expensive subprocedures, such as computing Euclidean projections). We present the first such algorithm that attains $O(T^{3/4})$ expected regret using only $O(T…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）
In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.
School of Information Science and Technology, ShanghaiTech University（上海科技大学信息科学与技术学院）；Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences（中国科学院上海微系统与信息技术研究所）；University of Washington（华盛顿大学）；Tongji University（同济大学）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we propose a novel primal-dual inexact gradient projection method for nonlinear optimization problems with convex-set constraint. This method only needs inexact computation of the projections onto the convex set for each iteration, consequently reducing the computational cost for projections per iteration. This feature is attractive especially for solving problems where the projections are computationally not easy to calculate. Global convergence guarantee and O(1/k) ergodic convergence rate of the optimality residual are provided under loose assumptions. We apply our proposed s…
Yahoo Research（雅虎研究院）；The University of Iowa（爱荷华大学）；Nanjing University（南京大学）
We consider stochastic strongly convex optimization with a complex inequality constraint. This complex inequality constraint may lead to computationally expensive projections in algorithmic iterations of the stochastic gradient descent~(SGD) methods. To reduce the computation costs pertaining to the projections, we propose an Epoch-Projection Stochastic Gradient Descent~(Epro-SGD) method. The proposed Epro-SGD method consists of a sequence of epochs; it applies SGD to an augmented objective function at each iteration within the epoch, and then performs a projection at the end of each epoch. Gi…
Manifold learning has been successfully applied to a variety of medical imaging problems. Its use in real-time applications requires fast projection onto the low-dimensional space. To this end, out-of-sample extensions are applied by constructing an interpolation function that maps from the input space to the low-dimensional manifold. Commonly used approaches such as the Nyström extension and kernel ridge regression require using all training points. We propose an interpolation function that only depends on a small subset of the input training data. Consequently, in the testing phase each new …
University of Michigan, Ann Arbor（密歇根大学安娜堡分校）；Georgia Institute of Technology（佐治亚理工学院）
We revisit the Frank-Wolfe (FW) optimization under strongly convex constraint sets. We provide a faster convergence rate for FW without line search, showing that a previously overlooked variant of FW is indeed faster than the standard variant. With line search, we show that FW can converge to the global optimum, even for smooth functions that are not convex, but are quasi-convex and locally-Lipschitz. We also show that, for the general case of (smooth) non-convex functions, FW with line search converges with high probability to a stationary point at a rate of $O\left(\frac{1}{t}\right)$, as lo…
Michigan State University（密歇根州立大学）；GE Global Research（通用电气全球研究中心）；Zhejiang University（浙江大学）
Traditional algorithms for stochastic optimization require projecting the solution at each iteration into a given domain to ensure its feasibility. When facing complex domains, such as positive semi-definite cones, the projection operation can be expensive, leading to a high computational cost per iteration. In this paper, we present a novel algorithm that aims to reduce the number of projections for stochastic optimization. The proposed algorithm combines the strength of several recent developments in stochastic optimization, including mini-batch, extra-gradient, and epoch gradient descent, i…
Differentiable Projection-based Learn to Optimize in Wireless Network-Part I: Convex Constrained (Non-)Convex Programming
State Key Laboratory of ISN（ISN国家重点实验室）；School of Telecommunications Engineering, Xidian University（西安电子科技大学通信工程学院）；School of Cyber Engineering, Xidian University（西安电子科技大学网络与信息安全学院）
UC San Diego（加州大学圣地亚哥分校）；SPAWAR（SPAWAR（太空与海战系统司令部））
We present a neural network architecture based upon the Autoencoder (AE) and Generative Adversarial Network (GAN) that promotes a convex latent distribution by training adversarially on latent space interpolations. By using an AE as both the generator and discriminator of a GAN, we pass a pixel-wise error function across the discriminator, yielding an AE which produces non-blurry samples that match both high- and low-level features of the original images. Interpolations between images in this space remain within the latent-space distribution of real images as trained by the discriminator, and …
University of Iowa（爱荷华大学）；DAMO Academy, Alibaba Group（阿里巴巴集团达摩院）；Nanjing University（南京大学）；The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））
In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem include difference of convex (DC) functions and a family of bi-convex functions as special cases. We develop stochastic algorithms and establish their first-order convergence for finding a (nearly) stationary solution of the target non-convex function under different conditions of the component functions. To the best of our knowledge, this is the first work that comprehen…
Counterfactual explanations represent the minimal change to a data sample that alters its predicted classification, typically from an unfavorable initial class to a desired target class. Counterfactuals help answer questions such as "what needs to change for this application to get accepted for a loan?". A number of recently proposed approaches to counterfactual generation give varying definitions of "plausible" counterfactuals and methods to generate them. However, many of these methods are computationally intensive and provide unconvincing explanations. Here we introduce SharpShooter, a meth…
Technion - Israel Institute of Technology（以色列理工学院）
Projection-free optimization algorithms, which are mostly based on the classical Frank-Wolfe method, have gained significant interest in the machine learning community in recent years due to their ability to handle convex constraints that are popular in many applications, but for which computing projections is often computationally impractical in high-dimensional settings, and hence prohibit the use of most standard projection-based methods. In particular, a significant research effort was put on projection-free methods for online learning. In this paper we revisit the Online Frank-Wolfe (OFW)…
Our study focuses on determining the best weight windows for a weighted moving average smoother under squared loss. We show that there exists an optimal weight window that is symmetrical around its center. We study the class of tapered weight windows, which decrease in weight as they move away from the center. We formulate the corresponding least squares problem as a quadratic program and finally as a projection of the origin onto a convex polytope. Additionally, we provide some analytical solutions to the best window when some conditions are met on the input data.
NTT Communication Science Laboratories（NTT通信科学实验室）
We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth loss function. As we show, adding a projection as output layer provably makes the loss smaller. We identify the marginal polytope, the output space's convex hull, as the best convex set on which to project. However, because the projection onto the marginal polytope can sometimes b…
Video Latent Flow Matching: Optimal Polynomial Projections for Video Interpolation and Extrapolation
Wyoming Seminary（怀俄明神学院）；Simons Institute for the Theory of Computing, University of California, Berkeley（加州大学伯克利分校西蒙斯理论计算研究所）；Sun Yat-sen University（中山大学）
本文提出 Video Latent Flow Matching (VLFM)，利用预训练图像生成模型和 HiPPO 最优多项式投影建模潜在块流，实现任意帧率的视频插值与外推生成，并具备理论逼近误差和时间尺度鲁棒性优势。
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We present and analyze an algorithm for optimizing smooth and convex or strongly convex objectives using minibatch stochastic gradient estimates. The algorithm is optimal with respect to its dependence on both the minibatch size and minimum expected loss simultaneously. This improves over the optimal method of Lan (2012), which is insensitive to the minimum expected loss; over the optimistic acceleration of Cotter et al. (2011), which has suboptimal dependence on the minibatch size; and over the algorithm of Liu and Belkin (2018), which is limited to least squares problems and is also similarl…
Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the $\ell_1$-norm. However, several important learning applications cannot benefit from this approach as they feature these convex norms as constraints in addition to the non-convex rank and sparsity constraints. In this setting, we derive efficient sparse projections onto the simplex and its extension, and illustrate how to use them to solve high-dimensional learning problems in quantum tomography, sparse density estimation and portfolio selection with non-convex…
Centre d’Analyse et de Mathématique Sociales（社会分析与数学中心）；École des Hautes Études en Sciences Sociales（社会科学高等学院）
There has been a long history of works showing that neural networks have hard time extrapolating beyond the training set. A recent study by Balestriero et al. (2021) challenges this view: defining interpolation as the state of belonging to the convex hull of the training set, they show that the test set, either in input or neural space, cannot lie for the most part in this convex hull, due to the high dimensionality of the data, invoking the well known curse of dimensionality. Neural networks are then assumed to necessarily work in extrapolative mode. We here study the neural activities of the…
MIT（麻省理工学院）；University of Toronto（多伦多大学）；Toyota Research Institute（丰田研究院）
Experience replay plays a crucial role in improving the sample efficiency of deep reinforcement learning agents. Recent advances in experience replay propose using Mixup (Zhang et al., 2018) to further improve sample efficiency via synthetic sample generation. We build upon this technique with Neighborhood Mixup Experience Replay (NMER), a geometrically-grounded replay buffer that interpolates transitions with their closest neighbors in state-action space. NMER preserves a locally linear approximation of the transition manifold by only applying Mixup between transitions with vicinal state-acti…
The notion of interpolation and extrapolation is fundamental in various fields from deep learning to function approximation. Interpolation occurs for a sample $x$ whenever this sample falls inside or on the boundary of the given dataset's convex hull. Extrapolation occurs when $x$ falls outside of that convex hull. One fundamental (mis)conception is that state-of-the-art algorithms work so well because of their ability to correctly interpolate training data. A second (mis)conception is that interpolation happens throughout tasks and datasets, in fact, many intuitions and theories rely on that …
