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 differentiable by means of the implicit function theorem, and develop an implicit differentiation approach that improves the stability and tractability of training by decoupling the forward and backward passes. This connection enables us to apply advances in optimizing implicit layers to not only improve the optimization of the slot attention module in SLATE, a state-of-the-art method for learning entity representations, but do so with constant space and time complexity in backpropagation and only one additional line of code.
University of California, Berkeley（加州大学伯克利分校）；University of California, Davis（加州大学戴维斯分校）
We apply the technique of implicit differentiation to boost performance, reduce numerical error, and remove required user-tuning in the Marginal Unbiased Score Expansion (MUSE) algorithm for hierarchical Bayesian inference. We demonstrate these improvements on three representative inference problems: 1) an extended Neal's funnel 2) Bayesian neural networks, and 3) probabilistic principal component analysis. On our particular test cases, MUSE with implicit differentiation is faster than Hamiltonian Monte Carlo by factors of 155, 397, and 5, respectively, or factors of 65, 278, and 1 without imp…
A deep equilibrium model uses implicit layers, which are implicitly defined through an equilibrium point of an infinite sequence of computation. It avoids any explicit computation of the infinite sequence by finding an equilibrium point directly via root-finding and by computing gradients via implicit differentiation. In this paper, we analyze the gradient dynamics of deep equilibrium models with nonlinearity only on weight matrices and non-convex objective functions of weights for regression and classification. Despite non-convexity, convergence to global optimum at a linear rate is guarantee…
Using huge training datasets can be costly and inconvenient. This article explores various data distillation techniques that can reduce the amount of data required to successfully train deep networks. Inspired by recent ideas, we suggest new data distillation techniques based on generative teaching networks, gradient matching, and the Implicit Function Theorem. Experiments with the MNIST image classification problem show that the new methods are computationally more efficient than previous ones and allow to increase the performance of models trained on distilled data.
Université Laval（拉瓦尔大学）；University of Western Ontario（西安大略大学）；Mila（米拉人工智能研究所）
We consider a fair representation learning perspective, where optimal predictors, on top of the data representation, are ensured to be invariant with respect to different sub-groups. Specifically, we formulate this intuition as a bi-level optimization, where the representation is learned in the outer-loop, and invariant optimal group predictors are updated in the inner-loop. Moreover, the proposed bi-level objective is demonstrated to fulfill the sufficiency rule, which is desirable in various practical scenarios but was not commonly studied in the fair learning. Besides, to avoid the high com…
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…
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…
Panasonic AI Lab（松下人工智能实验室）；AIoli Labs（AIoli实验室）；Panasonic Technology Division（松下技术部门）
AutoAugment has sparked an interest in automated augmentation methods for deep learning models. These methods estimate image transformation policies for train data that improve generalization to test data. While recent papers evolved in the direction of decreasing policy search complexity, we show that those methods are not robust when applied to biased and noisy data. To overcome these limitations, we reformulate AutoAugment as a generalized automated dataset optimization (AutoDO) task that minimizes the distribution shift between test data and distorted train dataset. In our AutoDO model, we…
Mila（米拉人工智能研究所）；Université de Montréal（蒙特利尔大学）；Vector Institute（向量研究所）；University of Toronto（多伦多大学）；Google Research（谷歌研究院）；Facebook（脸书）
The shortcomings of maximum likelihood estimation in the context of model-based reinforcement learning have been highlighted by an increasing number of papers. When the model class is misspecified or has a limited representational capacity, model parameters with high likelihood might not necessarily result in high performance of the agent on a downstream control task. To alleviate this problem, we propose an end-to-end approach for model learning which directly optimizes the expected returns using implicit differentiation. We treat a value function that satisfies the Bellman optimality operato…
Training neural networks with auxiliary tasks is a common practice for improving the performance on a main task of interest. Two main challenges arise in this multi-task learning setting: (i) designing useful auxiliary tasks; and (ii) combining auxiliary tasks into a single coherent loss. Here, we propose a novel framework, AuxiLearn, that targets both challenges based on implicit differentiation. First, when useful auxiliaries are known, we propose learning a network that combines all losses into a single coherent objective function. This network can learn non-linear interactions between task…
Université Paris-Saclay（巴黎-萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Institut Mathématique de Bourgogne（勃艮第数学研究所）；Université de Bourgogne（勃艮第大学）；Google Research（谷歌研究院）；CNRS（法国国家科学研究中心）；IMAG（蒙彼利埃亚历山大·格罗滕迪克研究所）；Université de Montpellier（蒙彼利埃大学）
Setting regularization parameters for Lasso-type estimators is notoriously difficult, though crucial in practice. The most popular hyperparameter optimization approach is grid-search using held-out validation data. Grid-search however requires to choose a predefined grid for each parameter, which scales exponentially in the number of parameters. Another approach is to cast hyperparameter optimization as a bi-level optimization problem, one can solve by gradient descent. The key challenge for these methods is the estimation of the gradient with respect to the hyperparameters. Computing this gra…
The essential difficulty of gradient-based bilevel optimization using implicit differentiation is to estimate the inverse Hessian vector product with respect to neural network parameters. This paper proposes to tackle this problem by the Nystrom method and the Woodbury matrix identity, exploiting the low-rankness of the Hessian. Compared to existing methods using iterative approximation, such as conjugate gradient and the Neumann series approximation, the proposed method avoids numerical instability and can be efficiently computed in matrix operations without iterations. As a result, the propo…
Samsung - SAIT AI Lab（三星SAIT人工智能实验室）；Montreal University（蒙特利尔大学）；University of Amsterdam（阿姆斯特丹大学）；Mila（米拉人工智能研究所）；Université de Montreal（蒙特利尔大学）；TNO（荷兰应用科学研究院）
Most set prediction models in deep learning use set-equivariant operations, but they actually operate on multisets. We show that set-equivariant functions cannot represent certain functions on multisets, so we introduce the more appropriate notion of multiset-equivariance. We identify that the existing Deep Set Prediction Network (DSPN) can be multiset-equivariant without being hindered by set-equivariance and improve it with approximate implicit differentiation, allowing for better optimization while being faster and saving memory. In a range of toy experiments, we show that the perspective o…
We study a class of algorithms for solving bilevel optimization problems in both stochastic and deterministic settings when the inner-level objective is strongly convex. Specifically, we consider algorithms based on inexact implicit differentiation and we exploit a warm-start strategy to amortize the estimation of the exact gradient. We then introduce a unified theoretical framework inspired by the study of singularly perturbed systems (Habets, 1974) to analyze such amortized algorithms. By using this framework, our analysis shows these algorithms to match the computational complexity of oracl…
University of Toronto（多伦多大学）；Vector Institute（向量研究所）
We propose an algorithm for inexpensive gradient-based hyperparameter optimization that combines the implicit function theorem (IFT) with efficient inverse Hessian approximations. We present results about the relationship between the IFT and differentiating through optimization, motivating our algorithm. We use the proposed approach to train modern network architectures with millions of weights and millions of hyper-parameters. For example, we learn a data-augmentation network - where every weight is a hyperparameter tuned for validation performance - outputting augmented training examples. Jo…
Université Paris-Saclay（巴黎萨克雷大学）；Inria（法国国家信息与自动化研究所）；CEA（法国原子能和替代能源委员会）；Université de Bourgogne（勃艮第大学）；Institut Mathématique de Bourgogne（勃艮第数学研究所）；Università degli Studi di Genova（热那亚大学）；Google Research（谷歌研究院）；CNRS（法国国家科学研究中心）；Université de Montpellier（蒙彼利埃大学）
Finding the optimal hyperparameters of a model can be cast as a bilevel optimization problem, typically solved using zero-order techniques. In this work we study first-order methods when the inner optimization problem is convex but non-smooth. We show that the forward-mode differentiation of proximal gradient descent and proximal coordinate descent yield sequences of Jacobians converging toward the exact Jacobian. Using implicit differentiation, we show it is possible to leverage the non-smoothness of the inner problem to speed up the computation. Finally, we provide a bound on the error made …
Peking University（北京大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；Center for Data Science, Academy for Advanced Interdisciplinary Studies, Peking University（北京大学前沿交叉学科研究院数据科学研究中心）；Pazhou Lab（琶洲实验室）
Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. However, the supervised training of SNNs remains a hard problem due to the discontinuity of the spiking neuron model. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks, and use surrogate derivatives or compute gradients with respect to the spiking time to deal with the problem. These approaches either accumulate approximation errors or only propagate information limitedly through existing spikes, an…
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）
Toulouse School of Economics, University of Toulouse Capitole（图卢兹第一大学图卢兹经济学院）；IRIT（图卢兹计算机科学研究所）；CNRS（法国国家科学研究中心）；Université de Toulouse（图卢兹大学）；Institut Universitaire de France (IUF)（法国大学研究院）；Université Côte d’Azur（蔚蓝海岸大学）；Laboratoire J. A. Dieudonné（J. A. 迪厄多内实验室）
University of Utah（犹他大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We present a differentiable soft-body physics simulator that can be composed with neural networks as a differentiable layer. In contrast to other differentiable physics approaches that use explicit forward models to define state transitions, we focus on implicit state transitions defined via function minimization. Implicit state transitions appear in implicit numerical integration methods, which offer the benefits of large time steps and excellent numerical stability, but require a special treatment to achieve differentiability due to the absence of an explicit differentiable forward pass. In …
University of Cambridge（剑桥大学）；Alan Turing Institute（艾伦·图灵研究所）
Machine learning training methods depend plentifully and intricately on hyperparameters, motivating automated strategies for their optimisation. Many existing algorithms restart training for each new hyperparameter choice, at considerable computational cost. Some hypergradient-based one-pass methods exist, but these either cannot be applied to arbitrary optimiser hyperparameters (such as learning rates and momenta) or take several times longer to train than their base models. We extend these existing methods to develop an approximate hypergradient-based hyperparameter optimiser which is applic…
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）
Peking University（北京大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；Peng Cheng Laboratory（鹏城实验室）
Spiking neural networks (SNNs) with event-based computation are promising brain-inspired models for energy-efficient applications on neuromorphic hardware. However, most supervised SNN training methods, such as conversion from artificial neural networks or direct training with surrogate gradients, require complex computation rather than spike-based operations of spiking neurons during training. In this paper, we study spike-based implicit differentiation on the equilibrium state (SPIDE) that extends the recently proposed training method, implicit differentiation on the equilibrium state (IDE),…
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…
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）
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta-learning, continual learning, and reinforcement learning. Conventional BO methods need to differentiate through the low-level optimization process with implicit differentiation, which requires expensive calculations related to the Hessian matrix. There has been a recent quest for first-order methods for BO, but the methods proposed to date tend to be complicated and impractical for large-scale deep learning applications. In this work, w…
Istituto Italiano di Tecnologia（意大利技术研究院）；University College London（伦敦大学学院）
We study a general class of bilevel problems, consisting in the minimization of an upper-level objective which depends on the solution to a parametric fixed-point equation. Important instances arising in machine learning include hyperparameter optimization, meta-learning, and certain graph and recurrent neural networks. Typically the gradient of the upper-level objective (hypergradient) is hard or even impossible to compute exactly, which has raised the interest in approximation methods. We investigate some popular approaches to compute the hypergradient, based on reverse mode iterative differ…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；University of Tübingen（图宾根大学）；Amazon（亚马逊）；ETAS GmbH, Bosch Group（博世集团ETAS有限公司）
Learning-based 3D reconstruction methods have shown impressive results. However, most methods require 3D supervision which is often hard to obtain for real-world datasets. Recently, several works have proposed differentiable rendering techniques to train reconstruction models from RGB images. Unfortunately, these approaches are currently restricted to voxel- and mesh-based representations, suffering from discretization or low resolution. In this work, we propose a differentiable rendering formulation for implicit shape and texture representations. Implicit representations have recently gained …
Implicit models are a general class of learning models that forgo the hierarchical layer structure typical in neural networks and instead define the internal states based on an ``equilibrium'' equation, offering competitive performance and reduced memory consumption. However, training such models usually relies on expensive implicit differentiation for backward propagation. In this work, we present a new approach to training implicit models, called State-driven Implicit Modeling (SIM), where we constrain the internal states and outputs to match that of a baseline model, circumventing costly ba…
Zhejiang Lab（之江实验室）；Key Lab. of Machine Perception, School of AI, Peking University（北京大学人工智能学院机器感知重点实验室）；Carnegie Mellon University（卡内基梅隆大学）；Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；Pazhou Lab（琶洲实验室）
This paper focuses on training implicit models of infinite layers. Specifically, previous works employ implicit differentiation and solve the exact gradient for the backward propagation. However, is it necessary to compute such an exact but expensive gradient for training? In this work, we propose a novel gradient estimate for implicit models, named phantom gradient, that 1) forgoes the costly computation of the exact gradient; and 2) provides an update direction empirically preferable to the implicit model training. We theoretically analyze the condition under which an ascent direction of the…
chemtrain: Learning Deep Potential Models via Automatic Differentiation and Statistical Physics
Multiscale Modeling of Fluid Materials, Department of Engineering Physics and Computation, TUM School of Engineering and Design, Technical University of Munich, Germany（多尺度流体材料建模，工程物理与计算系，技术大学慕尼黑工程与设计学院，技术大学慕尼黑，德国）；Valence Labs, Montreal, QC, Canada（Valence实验室，加拿大蒙特利尔，魁北克省）；Atomistic Modeling Center, Munich Data Science Institute, Technical University of Munich, Germany（原子建模中心，慕尼黑数据科学研究所，技术大学慕尼黑，德国）
Online learning to rank is a core problem in machine learning. In Lattimore et al. (2018), a novel online learning algorithm was proposed based on topological sorting. In the paper they provided a set of self-normalized inequalities (a) in the algorithm as a criterion in iterations and (b) to provide an upper bound for cumulative regret, which is a measure of algorithm performance. In this work, we utilized method of mixtures and asymptotic expansions of certain implicit function to provide a tighter, iterated-log-like boundary for the inequalities, and as a consequence improve both the algori…
Heidelberg University（海德堡大学）；Université catholique de Louvain（鲁汶天主教大学）；Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）；University of California, Berkeley（加州大学伯克利分校）
Deep generative models are becoming widely used across science and industry for a variety of purposes. A common challenge is achieving a precise implicit or explicit representation of the data probability density. Recent proposals have suggested using classifier weights to refine the learned density of deep generative models. We extend this idea to all types of generative models and show how latent space refinement via iterated generative modeling can circumvent topological obstructions and improve precision. This methodology also applies to cases were the target model is non-differentiable an…
