Spiking neural networks (SNNs) have been proposed as energy-efficient alternatives to deep neural networks. Their training is non-convex and involves a non-differentiable spike function, so existing methods offer no guarantee of reaching a global optimum. We address this issue by formulating SNN optimization as an equivalent convex optimization problem. Specifically, we show that the weight-decay-regularized training problem for a class of threshold RNNs, including leaky integrate-and-fire SNNs, has zero duality gap and the same optimal value as a convex program. Since the resulting convex programs can have exponentially many variables in the amount of data, we introduce a practical randomized surrogate and establish a high-probability guarantee on its approximation quality. Empirically, the best convex variant outperforms surrogate-gradient training on addition and first-last-XOR, matches its performance on sequential MNIST, and trails a reservoir baseline on base-3 and base-5 addition.
This paper demonstrates how to construct ensembles of spiking neural networks producing state-of-the-art results, achieving classification accuracies of 98.71%, 100.0%, and 99.09%, on the MNIST, NMNIST and DVS Gesture datasets respectively. Furthermore, this performance is achieved using simplified individual models, with ensembles containing less than 50% of the parameters of published reference models. We provide comprehensive exploration on the effect of spike train interpretation methods, and derive the theoretical methodology for combining model predictions such that performance improveme…
StochEP: Stochastic Equilibrium Propagation for Spiking Convergent Recurrent Neural Networks
School of Electrical Engineering and Computer Science（电气工程与计算机科学学院）；The Pennsylvania State University（宾夕法尼亚州立大学）；University Park（大学公园）；PA 16802, USA（宾夕法尼亚州16802, 美国）
Indian Institute of Technology Bombay（印度理工学院孟买分校）；New Jersey Institute of Technology（新泽西理工学院）
Spiking neural networks (SNNs) have garnered a great amount of interest for supervised and unsupervised learning applications. This paper deals with the problem of training multi-layer feedforward SNNs. The non-linear integrate-and-fire dynamics employed by spiking neurons make it difficult to train SNNs to generate desired spike trains in response to a given input. To tackle this, first the problem of training a multi-layer SNN is formulated as an optimization problem such that its objective function is based on the deviation in membrane potential rather than the spike arrival instants. Then,…
Online Pseudo-Zeroth-Order Training of Neuromorphic Spiking Neural Networks
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）；Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；Peng Cheng Laboratory（鹏城实验室）
Hebbian Learning based Orthogonal Projection for Continual Learning of Spiking Neural Networks
School of Intelligence Science and Technology, Peking University（北京大学智能科学与技术学院）；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）；School of Public Health, Peking University（北京大学公共卫生学院）；Institute for Artificial Intelligence, Peking University（北京大学人工智能研究院）；Peng Cheng Laboratory（鹏城实验室）
The field of neuromorphic computing promises extremely low-power and low-latency sensing and processing. Challenges in transferring learning algorithms from traditional artificial neural networks (ANNs) to spiking neural networks (SNNs) have so far prevented their application to large-scale, complex regression tasks. Furthermore, realizing a truly asynchronous and fully neuromorphic pipeline that maximally attains the abovementioned benefits involves rethinking the way in which this pipeline takes in and accumulates information. In the case of perception, spikes would be passed as-is and one-b…
We study consistency properties of machine learning methods based on minimizing convex surrogates. We extend the recent framework of Osokin et al. (2017) for the quantitative analysis of consistency properties to the case of inconsistent surrogates. Our key technical contribution consists in a new lower bound on the calibration function for the quadratic surrogate, which is non-trivial (not always zero) for inconsistent cases. The new bound allows to quantify the level of inconsistency of the setting and shows how learning with inconsistent surrogates can have guarantees on sample complexity a…
Training neural networks is a challenging non-convex optimization problem, and backpropagation or gradient descent can get stuck in spurious local optima. We propose a novel algorithm based on tensor decomposition for guaranteed training of two-layer neural networks. We provide risk bounds for our proposed method, with a polynomial sample complexity in the relevant parameters, such as input dimension and number of neurons. While learning arbitrary target functions is NP-hard, we provide transparent conditions on the function and the input for learnability. Our training method is based on tenso…
Submodular continuous functions are a category of (generally) non-convex/non-concave functions with a wide spectrum of applications. We characterize these functions and demonstrate that they can be maximized efficiently with approximation guarantees. Specifically, i) We introduce the weak DR property that gives a unified characterization of submodularity for all set, integer-lattice and continuous functions; ii) for maximizing monotone DR-submodular continuous functions under general down-closed convex constraints, we propose a Frank-Wolfe variant with $(1-1/e)$ approximation guarantee, and su…
Technical University of Denmark（丹麦技术大学）；Syracuse University（雪城大学）；Georgia Institute of Technology（佐治亚理工学院）
This paper develops a novel differentially private framework to solve convex optimization problems with sensitive optimization data and complex physical or operational constraints. Unlike standard noise-additive algorithms, that act primarily on the problem data, objective or solution, and disregard the problem constraints, this framework requires the optimization variables to be a function of the noise and exploits a chance-constrained problem reformulation with formal feasibility guarantees. The noise is calibrated to provide differential privacy for identity and linear queries on the optimi…
Diminishing-returns (DR) submodular optimization is an important field with many real-world applications in machine learning, economics and communication systems. It captures a subclass of non-convex optimization that provides both practical and theoretical guarantees. In this paper, we study the fundamental problem of maximizing non-monotone DR-submodular functions over down-closed and general convex sets in both offline and online settings. First, we show that for offline maximizing non-monotone DR-submodular functions over a general convex set, the Frank-Wolfe algorithm achieves an approxim…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；University of California, Los Angeles（加州大学洛杉矶分校）
Recent years have seen increased interest in performance guarantees of gradient descent algorithms for non-convex optimization. A number of works have uncovered that gradient noise plays a critical role in the ability of gradient descent recursions to efficiently escape saddle-points and reach second-order stationary points. Most available works limit the gradient noise component to be bounded with probability one or sub-Gaussian and leverage concentration inequalities to arrive at high-probability results. We present an alternate approach, relying primarily on mean-square arguments and show t…
Distributionally robust optimization (DRO) has shown lot of promise in providing robustness in learning as well as sample based optimization problems. We endeavor to provide DRO solutions for a class of sum of fractionals, non-convex optimization which is used for decision making in prominent areas such as facility location and security games. In contrast to previous work, we find it more tractable to optimize the equivalent variance regularized form of DRO rather than the minimax form. We transform the variance regularized form to a mixed-integer second order cone program (MISOCP), which, whi…
We introduce an online convex optimization algorithm which utilizes projected subgradient descent with optimal adaptive learning rates. Our method provides second-order minimax-optimal dynamic regret guarantee (i.e. dependent on the sum of squared subgradient norms) for a sequence of general convex functions, which may not have strong convexity, smoothness, exp-concavity or even Lipschitz-continuity. The regret guarantee is against any comparator decision sequence with bounded path variation (i.e. sum of the distances between successive decisions). We generate the lower bound of the worst-case…
We consider the decision-making framework of online convex optimization with a very large number of experts. This setting is ubiquitous in contextual and reinforcement learning problems, where the size of the policy class renders enumeration and search within the policy class infeasible. Instead, we consider generalizing the methodology of online boosting. We define a weak learning algorithm as a mechanism that guarantees multiplicatively approximate regret against a base class of experts. In this access model, we give an efficient boosting algorithm that guarantees near-optimal regret against…
We consider the problem of finding critical points of functions that are non-convex and non-smooth. Studying a fairly broad class of such problems, we analyze the behavior of three gradient-based methods (gradient descent, proximal update, and Frank-Wolfe update). For each of these methods, we establish rates of convergence for general problems, and also prove faster rates for continuous sub-analytic functions. We also show that our algorithms can escape strict saddle points for a class of non-smooth functions, thereby generalizing known results for smooth functions. Our analysis leads to a si…
MPI for Intelligent Systems（智能系统马克斯·普朗克研究所）；ETH Zurich（苏黎世联邦理工学院）；EPFL（洛桑联邦理工学院）
Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical perfor…
Many popular learning-rate schedules for deep neural networks combine a decaying trend with local perturbations that attempt to escape saddle points and bad local minima. We derive convergence guarantees for bandwidth-based step-sizes, a general class of learning rates that are allowed to vary in a banded region. This framework includes many popular cyclic and non-monotonic step-sizes for which no theoretical guarantees were previously known. We provide worst-case guarantees for SGD on smooth non-convex problems under several bandwidth-based step sizes, including stagewise $1/\sqrt{t}$ and the…
Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）；University of California, Berkeley（加州大学伯克利分校）；University of California, Los Angeles（加州大学洛杉矶分校）
Although the optimization objectives for learning neural networks are highly non-convex, gradient-based methods have been wildly successful at learning neural networks in practice. This juxtaposition has led to a number of recent studies on provable guarantees for neural networks trained by gradient descent. Unfortunately, the techniques in these works are often highly specific to the particular setup in each problem, making it difficult to generalize across different settings. To address this drawback in the literature, we propose a unified non-convex optimization framework for the analysis o…
Technion - Israel Institute of Technology（以色列理工学院）
We present new efficient \textit{projection-free} algorithms for online convex optimization (OCO), where by projection-free we refer to algorithms that avoid computing orthogonal projections onto the feasible set, and instead relay on different and potentially much more efficient oracles. While most state-of-the-art projection-free algorithms are based on the \textit{follow-the-leader} framework, our algorithms are fundamentally different and are based on the \textit{online gradient descent} algorithm with a novel and efficient approach to computing so-called \textit{infeasible projections}. A…
We consider the minimization of non-convex functions that typically arise in machine learning. Specifically, we focus our attention on a variant of trust region methods known as cubic regularization. This approach is particularly attractive because it escapes strict saddle points and it provides stronger convergence guarantees than first- and second-order as well as classical trust region methods. However, it suffers from a high computational complexity that makes it impractical for large-scale learning. Here, we propose a novel method that uses sub-sampling to lower this computational cost. B…
University of California, San Diego（加州大学圣地亚哥分校）；Google Research（谷歌研究院）；The Hong Kong University of Science and Technology（香港科技大学）
This paper considers the generalization performance of differentially private convex learning. We demonstrate that the convergence analysis of Langevin algorithms can be used to obtain new generalization bounds with differential privacy guarantees for DP-SGD. More specifically, by using some recently obtained dimension-independent convergence results for stochastic Langevin algorithms with convex objective functions, we obtain $O(n^{-1/4})$ privacy guarantees for DP-SGD with the optimal excess generalization error of $\tilde{O}(n^{-1/2})$ for certain classes of overparameterized smooth convex …
Boston University College of Engineering（波士顿大学工程学院）；University of Science and Technology of China（中国科学技术大学）
Recently, Stochastic Gradient Descent (SGD) and its variants have become the dominant methods in the large-scale optimization of machine learning (ML) problems. A variety of strategies have been proposed for tuning the step sizes, ranging from adaptive step sizes to heuristic methods to change the step size in each iteration. Also, momentum has been widely employed in ML tasks to accelerate the training process. Yet, there is a gap in our theoretical understanding of them. In this work, we start to close this gap by providing formal guarantees to a few heuristic optimization methods and propos…
We analyze the meta-learning of the initialization and step-size of learning algorithms for piecewise-Lipschitz functions, a non-convex setting with applications to both machine learning and algorithms. Starting from recent regret bounds for the exponential forecaster on losses with dispersed discontinuities, we generalize them to be initialization-dependent and then use this result to propose a practical meta-learning procedure that learns both the initialization and the step-size of the algorithm from multiple online learning tasks. Asymptotically, we guarantee that the average regret across…
In this paper, we study fundamental problems of maximizing DR-submodular continuous functions that have real-world applications in the domain of machine learning, economics, operations research and communication systems. It captures a subclass of non-convex optimization that provides both theoretical and practical guarantees. Here, we focus on minimizing regret for online arriving non-monotone DR-submodular functions over different types of convex sets: hypercube, down-closed and general convex sets. First, we present an online algorithm that achieves a $1/e$-approximation ratio with the regre…
Typal LLC（Typal有限责任公司）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Alibaba US, DAMO Academy（阿里巴巴美国达摩院）
Applications abound in which optimization problems must be repeatedly solved, each time with new (but similar) data. Analytic optimization algorithms can be hand-designed to provably solve these problems in an iterative fashion. On one hand, data-driven algorithms can "learn to optimize" (L2O) with much fewer iterations and similar cost per iteration as general-purpose optimization algorithms. On the other hand, unfortunately, many L2O algorithms lack converge guarantees. To fuse the advantages of these approaches, we present a Safe-L2O framework. Safe-L2O updates incorporate a safeguard to gu…
University of California, Irvine（加州大学欧文分校）；Duke University（杜克大学）
Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points such that the first and second order derivatives cannot distinguish them with local optima. In this paper we use higher order derivatives to escape these saddle points: we design the first efficient algorithm guaranteed to converge to a third order local optimum (while existing tec…
INRIA/ENS（法国国家信息与自动化研究所/巴黎高等师范学院）；HSE（高等经济学院）；MILA（米拉魁北克人工智能研究所）；DIRO（计算机科学与运筹学系）；Université de Montréal（蒙特利尔大学）
We provide novel theoretical insights on structured prediction in the context of efficient convex surrogate loss minimization with consistency guarantees. For any task loss, we construct a convex surrogate that can be optimized via stochastic gradient descent and we prove tight bounds on the so-called "calibration function" relating the excess surrogate risk to the actual risk. In contrast to prior related work, we carefully monitor the effect of the exponential number of classes in the learning guarantees as well as on the optimization complexity. As an interesting consequence, we formalize t…
PrivSpike: Employing Homomorphic Encryption for Private Inference of Deep Spiking Neural Networks
STAM Center, Ira A. Fulton Schools of Engineering, Arizona State University（STAM中心、伊拉·A·福鲁顿工程学院、亚利桑那州立大学）；Faculty of Technical Sciences, University of Novi Sad（技术科学学院、诺维萨德大学）
Cornell University（康奈尔大学）；University of Washington（华盛顿大学）
We consider a family of algorithms that successively sample and minimize simple stochastic models of the objective function. We show that under reasonable conditions on approximation quality and regularity of the models, any such algorithm drives a natural stationarity measure to zero at the rate $O(k^{-1/4})$. As a consequence, we obtain the first complexity guarantees for the stochastic proximal point, proximal subgradient, and regularized Gauss-Newton methods for minimizing compositions of convex functions with smooth maps. The guiding principle, underlying the complexity guarantees, is tha…
Université Rouen Normandie（鲁昂诺曼底大学）；Criteo AI Lab（Criteo人工智能实验室）；INSA Rouen Normandie（鲁昂国立应用科学学院）；Universitéé de Montpellier（蒙彼利埃大学）
Leveraging on the convexity of the Lasso problem , screening rules help in accelerating solvers by discarding irrelevant variables, during the optimization process. However, because they provide better theoretical guarantees in identifying relevant variables, several non-convex regularizers for the Lasso have been proposed in the literature. This work is the first that introduces a screening rule strategy into a non-convex Lasso solver. The approach we propose is based on a iterative majorization-minimization (MM) strategy that includes a screening rule in the inner solver and a condition for …
Policy gradients methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by standard dynamic programming techniques, policy gradient algorithms face non-convex optimization problems and are widely understood to converge only to a stationary point. This work identifies structural properties -- shared by several classic control problems -- that ensure the policy gradient objective function has no suboptimal stationary points despite being non-convex. Whe…
State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothness parameters is necessary. As a special case, we show Nesterov's accelerated cubic regularization method to be nearly tight.
Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully hand-picked stepsize for fast convergence, which is notoriously tedious and time-consuming to tune. Over the last several years, a plethora of adaptive gradient-based algorithms have emerged to ameliorate this problem. They have proved efficient in reducing the labor of tuning in practice, but many of them lack theoretic guarantees even in the convex setting. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic opti…
Université Paris Seine（巴黎塞纳大学）；Université Cergy-Pontoise（塞吉-蓬图瓦兹大学）；ENSEA（法国国立高等电子及应用学院）；CNRS（法国国家科学研究中心）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；LIG（格勒诺布尔信息学实验室）；ESIEE Paris（巴黎高等电子与电工技术工程师学院）；Université Paris-Est（巴黎东部大学）；University of Pisa（比萨大学）；CentraleSupélec（法国中央理工-高等电力学院）；Université Paris-Saclay（巴黎萨克雷大学）
Spurred by the enthusiasm surrounding the "Big Data" paradigm, the mathematical and algorithmic tools of online optimization have found widespread use in problems where the trade-off between data exploration and exploitation plays a predominant role. This trade-off is of particular importance to several branches and applications of signal processing, such as data mining, statistical inference, multimedia indexing and wireless communications (to name but a few). With this in mind, the aim of this tutorial paper is to provide a gentle introduction to online optimization and learning algorithms t…
Recently, both industry and academia have proposed several different neuromorphic systems to execute machine learning applications that are designed using Spiking Neural Networks (SNNs). With the growing complexity on design and technology fronts, programming such systems to admit and execute a machine learning application is becoming increasingly challenging. Additionally, neuromorphic systems are required to guarantee real-time performance, consume lower energy, and provide tolerance to logic and memory failures. Consequently, there is a clear need for system software frameworks that can imp…
University of California, Santa Barbara（加利福尼亚大学圣巴巴拉分校）
As an important class of spiking neural networks (SNNs), recurrent spiking neural networks (RSNNs) possess great computational power and have been widely used for processing sequential data like audio and text. However, most RSNNs suffer from two problems. 1. Due to a lack of architectural guidance, random recurrent connectivity is often adopted, which does not guarantee good performance. 2. Training of RSNNs is in general challenging, bottlenecking achievable model accuracy. To address these problems, we propose a new type of RSNNs called Skip-Connected Self-Recurrent SNNs (ScSr-SNNs). Recurr…
Spiking Neural Networks (SNN) are an emerging computation model, which uses event-driven activation and bio-inspired learning algorithms. SNN-based machine-learning programs are typically executed on tile- based neuromorphic hardware platforms, where each tile consists of a computation unit called crossbar, which maps neurons and synapses of the program. However, synthesizing such programs on an off-the-shelf neuromorphic hardware is challenging. This is because of the inherent resource and latency limitations of the hardware, which impact both model performance, e.g., accuracy, and hardware p…
