Alternating Differentiation for Optimization Layers
ShanghaiTech University（上海科技大学）；University of Technology Sydney（悉尼科技大学）；Princeton University（普林斯顿大学）；The University of Sydney（悉尼大学）；JD Explore Academy（京东探索研究院）
Haixiang Sun, Ye Shi, Jingya Wang, Hoang Duong Tuan, H. Vincent Poor, Dacheng Tao
The idea of embedding optimization problems into deep neural networks as optimization layers to encode constraints and inductive priors has taken hold in recent years. Most existing methods focus on implicitly differentiating Karush-Kuhn-Tucker (KKT) conditions in a way that requires expensive computations on the Jacobian matrix, which can be slow and memory-intensive. In this paper, we developed a new framework, named Alternating Differentiation (Alt-Diff), that differentiates optimization problems (here, specifically in the form of convex optimization problems with polyhedral constraints) in a fast and recursive way. Alt-Diff decouples the differentiation procedure into a primal update and a dual update in an alternating way. Accordingly, Alt-Diff substantially decreases the dimensions of the Jacobian matrix especially for optimization with large-scale constraints and thus increases the computational speed of implicit differentiation. We show that the gradients obtained by Alt-Diff are consistent with those obtained by differentiating KKT conditions. In addition, we propose to truncate Alt-Diff to further accelerate the computational speed. Under some standard assumptions, we show that the truncation error of gradients is upper bounded by the same order of variables' estimation error. Therefore, Alt-Diff can be truncated to further increase computational speed without sacrificing much accuracy. A series of comprehensive experiments validate the superiority of Alt-Diff.
Toulouse School of Economics, University of Toulouse Capitole（图卢兹经济学院，图卢兹第一大学）；IRIT, CNRS, Université de Toulouse, ANITI（图卢兹信息学研究所，法国国家科学研究中心，图卢兹大学，图卢兹人工智能与自然智能研究所）
Differentiation along algorithms, i.e., piggyback propagation of derivatives, is now routinely used to differentiate iterative solvers in differentiable programming. Asymptotics is well understood for many smooth problems but the nondifferentiable case is hardly considered. Is there a limiting object for nonsmooth piggyback automatic differentiation (AD)? Does it have any variational meaning and can it be used effectively in machine learning? Is there a connection with classical derivative? All these questions are addressed under appropriate nonexpansivity conditions in the framework of conser…
Recent advances in neural-network architecture allow for seamless integration of convex optimization problems as differentiable layers in an end-to-end trainable neural network. Integrating medium and large scale quadratic programs into a deep neural network architecture, however, is challenging as solving quadratic programs exactly by interior-point methods has worst-case cubic complexity in the number of variables. In this paper, we present an alternative network layer architecture based on the alternating direction method of multipliers (ADMM) that is capable of scaling to problems with a m…
This work proposes a Neural Network model that can control its depth using an iterate-to-fixed-point operator. The architecture starts with a standard layered Network but with added connections from current later to earlier layers, along with a gate to make them inactive under most circumstances. These ``temporal wormhole'' connections create a shortcut that allows the Neural Network to use the information available at deeper layers and re-do earlier computations with modulated inputs. End-to-end training is accomplished by using appropriate calculations for a numeric iterate-to-fixed-point op…
SPO-VCS: An End-to-End Smart Predict-then-Optimize Framework with Alternating Differentiation Method for Relocation Problems in Large-Scale Vehicle Crowd Sensing
Civil and Environmental Engineering, The Hong Kong Polytechnic University（香港理工大学土木与环境工程学院）
In this work, we present a hardware compatible neural network training algorithm in which we used alternating direction method of multipliers (ADMM) and iterative least-square methods. The motive behind this approach was to conduct a method of training neural networks that is scalable and can be parallelised. These characteristics make this algorithm suitable for hardware implementation. We have achieved 6.9\% and 6.8\% better accuracy comparing to SGD and Adam respectively, with a four-layer neural network with hidden size of 28 on HIGGS dataset. Likewise, we could observe 21.0\% and 2.2\% ac…
Alternating minimization methods have recently been proposed as alternatives to the gradient descent for deep neural network optimization. Alternating minimization methods can typically decompose a deep neural network into layerwise subproblems, which can then be optimized in parallel. Despite the significant parallelism, alternating minimization methods are rarely explored in training deep neural networks because of the severe accuracy degradation. In this paper, we analyze the reason and propose to achieve a compelling trade-off between parallelism and accuracy by a reformulation called Tuna…
Syracuse University（雪城大学）；Florida International University（佛罗里达国际大学）；Northeastern University（东北大学）
Weight pruning methods for deep neural networks (DNNs) have been investigated recently, but prior work in this area is mainly heuristic, iterative pruning, thereby lacking guarantees on the weight reduction ratio and convergence time. To mitigate these limitations, we present a systematic weight pruning framework of DNNs using the alternating direction method of multipliers (ADMM). We first formulate the weight pruning problem of DNNs as a nonconvex optimization problem with combinatorial constraints specifying the sparsity requirements, and then adopt the ADMM framework for systematic weight …
Stochastic gradient descent (SGD) has achieved great success in training deep neural network, where the gradient is computed through back-propagation. However, the back-propagated values of different layers vary dramatically. This inconsistence of gradient magnitude across different layers renders optimization of deep neural network with a single learning rate problematic. We introduce the back-matching propagation which computes the backward values on the layer's parameter and the input by matching backward values on the layer's output. This leads to solving a bunch of least-squares problems,…
In view of training increasingly complex learning architectures, we establish a nonsmooth implicit function theorem with an operational calculus. Our result applies to most practical problems (i.e., definable problems) provided that a nonsmooth form of the classical invertibility condition is fulfilled. This approach allows for formal subdifferentiation: for instance, replacing derivatives by Clarke Jacobians in the usual differentiation formulas is fully justified for a wide class of nonsmooth problems. Moreover this calculus is entirely compatible with algorithmic differentiation (e.g., back…
Exhaustive Circuit Mapping of a Single-Cell Foundation Model Reveals Massive Redundancy, Heavy-Tailed Hub Architecture, and Layer-Dependent Differentiation Control
Department of Computer Science, University of Tübingen（图宾根大学计算机科学系）
The University of Sydney（悉尼大学）；Renmin University of China（中国人民大学）
Syndrome differentiation in Traditional Chinese Medicine (TCM) is the process of understanding and reasoning body condition, which is the essential step and premise of effective treatments. However, due to its complexity and lack of standardization, it is challenging to achieve. In this study, we consider each patient's record as a one-dimensional image and symptoms as pixels, in which missing and negative values are represented by zero pixels. The objective is to find relevant symptoms first and then map them to proper syndromes, that is similar to the object detection problem in computer vis…
The paper considers a Mixture Multilayer Stochastic Block Model (MMLSBM), where layers can be partitioned into groups of similar networks, and networks in each group are equipped with a distinct Stochastic Block Model. The goal is to partition the multilayer network into clusters of similar layers, and to identify communities in those layers. Jing et al. (2020) introduced the MMLSBM and developed a clustering methodology, TWIST, based on regularized tensor decomposition. The present paper proposes a different technique, an alternating minimization algorithm (ALMA), that aims at simultaneous re…
Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning. However, so far, implicit differentiation remained difficult to use for practitioners, as it often required case-by-case tedious mathematical deriv…
Laboratory of Physics of the \'Ecole Normale Sup\'erieure, CNRS UMR 8023 \& PSL Research, Sorbonne Universit\'e, 24 rue Lhomond, 75005 Paris, France
Restricted Boltzmann Machines (RBM) are bi-layer neural networks used for the unsupervised learning of model distributions from data. The bipartite architecture of RBM naturally defines an elegant sampling procedure, called Alternating Gibbs Sampling (AGS), where the configurations of the latent-variable layer are sampled conditional to the data-variable layer, and vice versa. We study here the performance of AGS on several analytically tractable models borrowed from statistical mechanics. We show that standard AGS is not more efficient than classical Metropolis-Hastings (MH) sampling of the e…
University of California, Berkeley（加州大学伯克利分校）；Princeton University（普林斯顿大学）
Iterative refinement -- start with a random guess, then iteratively improve the guess -- is a useful paradigm for representation learning because it offers a way to break symmetries among equally plausible explanations for the data. This property enables the application of such methods to infer representations of sets of entities, such as objects in physical scenes, structurally resembling clustering algorithms in latent space. However, most prior works differentiate through the unrolled refinement process, which can make optimization challenging. We observe that such methods can be made diffe…
Decoupled Structure-Feature Alignment via Alternating Optimization for Graph Learning
College of Artificial Intelligence, Shaoxing Institute of Technology（绍兴文理学院人工智能学院）；School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convolutional and fully-connected layers often cannot capture. We explore the foundations for such an architecture: we show how techniques from sensitivity analysis, bilevel optimization, and implicit differentiation can be used to exactly differentiate through these layers and with respe…
Despite a lack of theoretical understanding, deep neural networks have achieved unparalleled performance in a wide range of applications. On the other hand, shallow representation learning with component analysis is associated with rich intuition and theory, but smaller capacity often limits its usefulness. To bridge this gap, we introduce Deep Component Analysis (DeepCA), an expressive multilayer model formulation that enforces hierarchical structure through constraints on latent variables in each layer. For inference, we propose a differentiable optimization algorithm implemented using recur…
