Recently, several studies proposed methods to utilize some classes of optimization problems in designing deep neural networks to encode constraints that conventional layers cannot capture. However, these methods are still in their infancy and require special treatments, such as analyzing the KKT condition, for deriving the backpropagation formula. In this paper, we propose a new layer formulation called the fixed-point iteration (FPI) layer that facilitates the use of more complicated operations in deep networks. The backward FPI layer is also proposed for backpropagation, which is motivated by the recurrent back-propagation (RBP) algorithm. But in contrast to RBP, the backward FPI layer yields the gradient by a small network module without an explicit calculation of the Jacobian. In actual applications, both the forward and backward FPI layers can be treated as nodes in the computational graphs. All components in the proposed method are implemented at a high level of abstraction, which allows efficient higher-order differentiations on the nodes. In addition, we present two practical methods of the FPI layer, FPI_NN and FPI_GD, where the update operations of FPI are a small neural network module and a single gradient descent step based on a learnable cost function, respectively. FPI\_NN is intuitive, simple, and fast to train, while FPI_GD can be used for efficient training of energy networks that have been recently studied. While RBP and its related studies have not been applied to practical examples, our experiments show the FPI layer can be successfully applied to real-world problems such as image denoising, optical flow, and multi-label classification.
McGill University（麦吉尔大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is carried over to the next gradient step. We draw connections between the proposed stochastic backward Euler and the recent entropy stochastic gradient descent (Entropy-SGD) for improving the training of deep neural networks. Numerical experiments on various synthetic and real datas…
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…
The University of Adelaide（阿德莱德大学）；Carnegie Mellon University（卡内基梅隆大学）；The University of New South Wales（新南威尔士大学）；The University of Melbourne（墨尔本大学）；Mohamed bin Zayed University of Artificial Intelligence（穆罕默德·本·扎耶德人工智能大学）
Recovering underlying Directed Acyclic Graph (DAG) structures from observational data is highly challenging due to the combinatorial nature of the DAG-constrained optimization problem. Recently, DAG learning has been cast as a continuous optimization problem by characterizing the DAG constraint as a smooth equality one, generally based on polynomials over adjacency matrices. Existing methods place very small coefficients on high-order polynomial terms for stabilization, since they argue that large coefficients on the higher-order terms are harmful due to numeric exploding. On the contrary, we …
The stochastic gradient (SG) method can minimize an objective function composed of a large number of differentiable functions, or solve a stochastic optimization problem, to a moderate accuracy. The block coordinate descent/update (BCD) method, on the other hand, handles problems with multiple blocks of variables by updating them one at a time; when the blocks of variables are easier to update individually than together, BCD has a lower per-iteration cost. This paper introduces a method that combines the features of SG and BCD for problems with many components in the objective and with multipl…
Howard Hughes Medical Institute, Janelia Farm Research Campus（霍华德·休斯医学研究所，贾尼利亚农场研究园区）
We propose a version of least-mean-square (LMS) algorithm for sparse system identification. Our algorithm called online linearized Bregman iteration (OLBI) is derived from minimizing the cumulative prediction error squared along with an l1-l2 norm regularizer. By systematically treating the non-differentiable regularizer we arrive at a simple two-step iteration. We demonstrate that OLBI is bias free and compare its operation with existing sparse LMS algorithms by rederiving them in the online convex optimization framework. We perform convergence analysis of OLBI for white input signals and der…
University of Oxford（牛津大学）；University College London（伦敦大学学院）；Bloomsbury AI（布卢姆斯伯里人工智能公司）
We introduce neural networks for end-to-end differentiable proving of queries to knowledge bases by operating on dense vector representations of symbols. These neural networks are constructed recursively by taking inspiration from the backward chaining algorithm as used in Prolog. Specifically, we replace symbolic unification with a differentiable computation on vector representations of symbols using a radial basis function kernel, thereby combining symbolic reasoning with learning subsymbolic vector representations. By using gradient descent, the resulting neural network can be trained to in…
We introduce the value iteration network (VIN): a fully differentiable neural network with a `planning module' embedded within. VINs can learn to plan, and are suitable for predicting outcomes that involve planning-based reasoning, such as policies for reinforcement learning. Key to our approach is a novel differentiable approximation of the value-iteration algorithm, which can be represented as a convolutional neural network, and trained end-to-end using standard backpropagation. We evaluate VIN based policies on discrete and continuous path-planning domains, and on a natural-language based s…
Scaling up and Stabilizing Differentiable Planning with Implicit Differentiation
Khoury College of Computer Sciences, Northeastern University（东北大学Khoury计算机科学学院）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Shanghai Qi Zhi Institute（上海期智研究院）
Google Research（谷歌研究院）；Northeastern University（东北大学）；DeepMind（深度思维）
Safety certification of data-driven control techniques remains a major open problem. This work investigates backward reachability as a framework for providing collision avoidance guarantees for systems controlled by neural network (NN) policies. Because NNs are typically not invertible, existing methods conservatively assume a domain over which to relax the NN, which causes loose over-approximations of the set of states that could lead the system into the obstacle (i.e., backprojection (BP) sets). To address this issue, we introduce DRIP, an algorithm with a refinement loop on the relaxation d…
Computing the matrix square root or its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or in the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root. For the forward propagation, one method is to use Matrix Taylor …
Computing the matrix square root and its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use the Newton-Schulz iteration (NS iteration) to derive the approximate solution. However, both methods are not computationally efficient enough in either the forward pass or the backward pass. In this paper, we propose two more efficient variants to compute the differentiable matrix square root and the inverse square root. For the forward propagation, one metho…
National Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所模式识别国家重点实验室）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）；Intel Labs China（英特尔中国研究院）
For network architecture search (NAS), it is crucial but challenging to simultaneously guarantee both effectiveness and efficiency. Towards achieving this goal, we develop a differentiable NAS solution, where the search space includes arbitrary feed-forward network consisting of the predefined number of connections. Benefiting from a proposed ensemble Gumbel-Softmax estimator, our method optimizes both the architecture of a deep network and its parameters in the same round of backward propagation, yielding an end-to-end mechanism of searching network architectures. Extensive experiments on a v…
This work presents TorchRadon -- an open source CUDA library which contains a set of differentiable routines for solving computed tomography (CT) reconstruction problems. The library is designed to help researchers working on CT problems to combine deep learning and model-based approaches. The package is developed as a PyTorch extension and can be seamlessly integrated into existing deep learning training code. Compared to the existing Astra Toolbox, TorchRadon is up to 125 faster. The operators implemented by TorchRadon allow the computation of gradients using PyTorch backward(), and can ther…
We study the asymptotic convergence of AA($m$), i.e., Anderson acceleration with window size $m$ for accelerating fixed-point methods $x_{k+1}=q(x_{k})$, $x_k \in R^n$. Convergence acceleration by AA($m$) has been widely observed but is not well understood. We consider the case where the fixed-point iteration function $q(x)$ is differentiable and the convergence of the fixed-point method itself is root-linear. We identify numerically several conspicuous properties of AA($m$) convergence: First, AA($m$) sequences $\{x_k\}$ converge root-linearly but the root-linear convergence factor depends st…
National Institute of Technology Rourkela（鲁尔克拉国家理工学院）
The extreme learning machine needs a large number of hidden nodes to generalize a single hidden layer neural network for a given training data-set. The need for more number of hidden nodes suggests that the neural-network is memorizing rather than generalizing the model. Hence, a supervised learning method is described here that uses Moore-Penrose approximation to determine both input-weight and output-weight in two epochs, namely, backward-pass and forward-pass. The proposed technique has an advantage over the back-propagation method in terms of iterations required and is superior to the extr…
Visual Geometry Group（视觉几何组）；University of Oxford（牛津大学）
We train embodied neural networks to plan and navigate unseen complex 3D environments, emphasising real-world deployment. Rather than requiring prior knowledge of the agent or environment, the planner learns to model the state transitions and rewards. To avoid the potentially hazardous trial-and-error of reinforcement learning, we focus on differentiable planners such as Value Iteration Networks (VIN), which are trained offline from safe expert demonstrations. Although they work well in small simulations, we address two major limitations that hinder their deployment. First, we observed that cu…
Differentiate the Solver, Not the Equation: Reverse-Sweep Adjoints for Block Implicit Simulation
Stanford University（斯坦福大学）；University of California, Los Angeles（加州大学洛杉矶分校）；LightSpeed Studios, Tencent America（腾讯美国光速工作室）；The University of Utah（犹他大学）
Iowa State University（爱荷华州立大学）；New York University（纽约大学）
Boundary representations (B-reps) using Non-Uniform Rational B-splines (NURBS) are the de facto standard used in CAD, but their utility in deep learning-based approaches is not well researched. We propose a differentiable NURBS module to integrate NURBS representations of CAD models with deep learning methods. We mathematically define the derivatives of the NURBS curves or surfaces with respect to the input parameters (control points, weights, and the knot vector). These derivatives are used to define an approximate Jacobian used for performing the "backward" evaluation to train the deep learn…
Beihang University（北京航空航天大学）；SenseTime Group Limited（商汤集团有限公司）；Beijing Institute of Technology（北京理工大学）
Hardware-friendly network quantization (e.g., binary/uniform quantization) can efficiently accelerate the inference and meanwhile reduce memory consumption of the deep neural networks, which is crucial for model deployment on resource-limited devices like mobile phones. However, due to the discreteness of low-bit quantization, existing quantization methods often face the unstable training process and severe performance degradation. To address this problem, in this paper we propose Differentiable Soft Quantization (DSQ) to bridge the gap between the full-precision and low-bit networks. DSQ can …
