By analyzing this inequality, we are able to give performance guarantees and parameter settings of the algorithm under a variety of assumptions regarding the convexity and smoothness of the objective function. Various modifications of the method and some theoretical results on its convergence are presented. […] initial conditions are given by the inputs to the network, dynamical system allows us to rewrite problem (, The two player maximum principle says in this case that if, there exists an optimal costate trajectory, and the following Hamiltonian condition for all, These necessary optimality conditions can be used to design an iterative algorithm of the fol-, this interpretation, the gradient of the total loss for the, ﬁrst layer Hamiltonian condition and this function can be minimized by computing gradients only, with respect to the ﬁrst layer of the network. Adversarial training is an effective way of improving the robustness to the adversarial examples, typically formulated as a robust optimization problem for network, We build on the dynamical systems approach to deep learning, where deep residual networks are idealized as continuous-time dynamical systems. CMU, Pittsburgh, Pa: Morgan Kaufmann, 1988. vances in Neural Information Processing Systems. In this paper we review several connections between deep learning and mathematical fields like numerical analyis, optimal control and inverse problems. We also show that it may avoid some pitfalls of gradient-based methods, such as slow convergence on flat landscapes near saddle points. amples in this way causes excessive computational o, min-max problem as an optimal control problem, it has recently been shown that one can exploit, the compositional structure of neural networks in the optimization problem to improve the training, time signiﬁcantly. on the number of adversary updates per backpropagation, after a certain point, as predicted by Theorem, as the state and co-state trajectories generated from, , then the updates for the adversary are created by a. is the true solution to the inner maximization problem. Numerical Results. ﬁrst-order algorithm while the learner attempts to minimize it. Robust control theory is a method to measure the performance changes of a control system with changing system parameters. The expression, has ﬁnitely many iterations to maximize the loss function. The theoretical convergence guarantees of reinforcement learning assume that it is applied to a Markov decision process . Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees. Research on artificial intelligence (AI) has advanced significantly in recent years. bounds the difference between the costate used for the adversary’, that would result in a true gradient update, as in (, as a function of the initial condition of the system, and then uses a bound on successive values of, Hence, we are able to bound the error incurred from the frozen costates of the adversary’s, The last intermediate result we use to prove our theorem relates how the suboptimality of the. All figure content in this area was uploaded by Mahyar Fazlyab, All content in this area was uploaded by Mahyar Fazlyab on Sep 30, 2020. that is, when the exponentially decaying factor in, This observation is reminiscent of results in the literature on the Method of Successiv, mations (MSA) for ﬁnding controls and trajectories which satisfy the PMP, between computing state and costate trajectories and maximize the Hamiltonian to update the con-, minimization condition. Deep Cross-Subject Mapping of Neural Activity , submitted 2020. Optimization is a critical component in deep learning. We also discuss the connection with deep learning. Using this bound we can investigate the dependence of the algorithm on the number of back-. A major drawback of this approach is the computational overhead of adversary generation, which is much larger than network updating and leads to inconvenience in adversarial defense. is as large as $1/2$ and hence fails to explain a significant part of the generalization behavior of the network (effectively, this bound does not tell us whether our learning algorithm is any better than a random classifier!). Enforcing robust control guarantees within neural network policies. Normalized Loss Functions for Deep Learning with Noisy Labels Xingjun Ma, Hanxun Huang, Yisen Wang, Simone Romano, Sarah Erfani, James Bailey. argument we construct provides an outline for future results on the con, efﬁcient robust training algorithms. MDPs work in discrete time: at each time step, the controller receives feedback from the system in the form of a state signal, and takes an action in response. Robust K-means Clustering for Distributions with Two Moments: Nikita Kirillovich Zhivotovskiy, Yegor Klochkov, and Alexey Kroshnin: On the rate of convergence of fully connected deep neural network regression estimates: Michael Kohler and Sophie Langer: Infinite-dimensional gradient-based descent for alpha-divergence minimisation First, its tractability despite non-convexity is an intriguing question and may greatly expand our understanding of tractable problems. Join ResearchGate to find the people and research you need to help your work. The goal is to allow exploration of the design space for alternatives … It inspires us to split the adversary computation from the back propagation gradient computation. simulation-based optimization problems in which only stochastic zeroth-order algorithm, namely the randomized stochastic gradient (RSG) method, for solving [C89] Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees J.H. The approach uses Gaussian process (GP) regression based on data gathered during operation to update an initial model of the system and to gradually decrease the uncertainty related to this model. Dictionary Learning and Sparse Coding-based Denoising for High-Resolution Brain Activation and Functional Connectivity Modeling: A Task fMRI Study , to appear in the IEEE Access 2020. The Midwest ML Symposium aims to convene regional machine learning researchers for stimulating discussions and debates, to foster cross-institutional collaboration, and to showcase the collective talent of machine learning researchers at all career stages.. Deep Learning/Representation Learning: Modern machine learning relies heavily on learning multilayer neural networks.However the representation power, optimization and generalization for neural networks are still mysteries despite many recent work. Mirroring the development of classical optimal control, we state and prove optimality conditions of both the Hamilton–Jacobi–Bellman type and the Pontryagin type. To solve it, previous works directly run gradient descent on the "adversarial loss", i.e. However, the mathematical aspects of such a formulation have not been systematically explored. the network parameters can also be seen as coming from an inexact gradient oracle.  Han J, Jentzen A, Weinan E. Solving high-dimensional partial differential equations using deep learning. Constructed Chebyshev orthornormal polynomial basis in the control space, the iterative learning control problem is transformed as the optimization problem. Application of this technique is important to building dependable embedded systems. One salient advantage of such a shallow-to-deep approach is that it helps to benefit in practice from the higher approximation properties of deep networks by mitigating over-parametrization issues. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between optimal control and deep learning.  Seidman J H, Fazlyab M, Preciado V M, et al. In this view, neural networks can be interpreted as a discretization of a parametric Ordinary Differential Equation which, in the limit, defines a continuous-depth neural network. 12 - Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks (arXiv link, long slides) With this notation, the updates for, IEEE Symposium on Security and Privacy (SP), e catholique de Louvain, Center for Operations, Ruiqi Gao, Tianle Cai, Haochuan Li, Cho-Jui Hsieh, Liwei W. of adversarial training in overparametrized neural networks. Springer Science & Business Media, 2008. International Conference on Learning Representations, Proceedings of the 1988 connectionist models summer school, International Conference on Machine Learning. In particular, we prove a generalization bound that involves the covering number properties of the original ERM problem. We have been honored with several awards for our work. Training is recast as a control problem and this allows us to formulate necessary optimality conditions in continuous time using the Pontryagin's maximum principle (PMP). Deep Pinsker and James-Stein Neural Networks for Brain-Computer Interfacing , submitted 2020. THEORINET’s research agenda is divided in four main thrusts. Li Q, Hao S. An optimal control approach to deep learning and applications to discrete-weight neural networks[J]. A few things about Deep Learning I find puzzling: 1) How can deep neural networks — optimized by stochastic gradient descent (SGD) agnostic of concepts of invariance, minimality, disentanglement — somehow manage to learn representations that exhibit … The learning task then consists in finding the best ODE parameters for the, Deep learning achieves state-of-the-art results in many areas. Preprints and early-stage research may not have been peer reviewed yet. Several machine learning models, including neural networks, consistently mis- classify adversarial examples—inputs formed by applying small but intentionally worst-case perturbations to examples from the dataset, such that the perturbed in- put results in the model outputting an incorrect answer with high confidence. Including adversarial examples during training is a popular defense mechanism against adversarial attacks. arXiv preprint arXiv:1803.01299, 2018. the field, we are concerned with analyzing the stochastic dynamics of the models, not mean-field approximations. the number of layers is fixed). optimization problem, where the adversary seeks to maximize the loss function using an iterative. TOWARDS DEEP LEARNING MODELS RESISTANT TO ADVERSARIAL ATTACKS More explicitly, instead of using, This removes the need to do a full backpropagation to recompute the costate, is written in pseudocode with the Hamiltonian framework in mind in Algorithm, inexact gradient oracles. By interpreting the min-max problem as an optimal control problem, it has recently been shown that one can exploit the compositional structure of neural networks in the optimization problem to improve the training time significantly. unified method of convergence analysis and parameter selection by interpreting the algorithm as a linear dynamical system with nonlinear feedback. Policy Iteration Guarantees Theorem. The overall algorithm for ICNN training and finding optimal reactive power injections are described in Algorithm 1. We support our insights with experiments on a robust classification problem. As a result, our proposed YOPO (You Only Propagate Once) avoids forward and backward the data too many times in one iteration, and restricts core descent directions computation to the first layer of the network, thus speeding up every iteration significantly. The continuous dynamical system approach to deep learning is explored in order to devise alternative frameworks for training algorithms. Proceedings of the National Academy of Sciences, 2018, 115(34): 8505-8510. Robust deep learning as optimal control: Insights and convergence guarantees JH Seidman, M Fazlyab, VM Preciado, GJ Pappas arXiv preprint arXiv:2005.00616 , 2020 We argue instead that the primary cause of neural networks' vulnerability to ad- versarial perturbation is their linear nature. Technical report, Université catholique de Louvain, Center for Operations In this work, we describe a minimax framework for statistical learning with ambiguity sets given by balls in Wasserstein space. of gradient oracle errors for the adversary due to freezing the costate in between backpropagations. First-order methods with inexact oracle: the satisfy the following Lipschitz conditions, ) and is justiﬁed through the reformulation of robust training as distributionally robust opti-, results in perturbing the empirical distribution in the, , and the following inequality holds for all, , and the parameters are updated with the perturbations. that momentum presents convergence robust to learning rate misspeciﬁcation and curvature variation for a class of non-convex objectives; these robustness properties are desirable for deep learning. The lab combines expertise from control theory, robotics, optimization, and operations research to develop the theoretical foundations for networked autonomous systems … Analysis: This thrust uses principles from approximation theory, information theory, statistical inference, and robust control to analyze properties of deep neural networks, such as expressivity, interpretability, confidence, fairness and robustness. Access scientific knowledge from anywhere. I am broadly interested in theoretical computer science and machine learning. Moreover, this view yields a simple and fast method of generating adversarial examples. © 2008-2020 ResearchGate GmbH. In this interpretation, the adversary is ﬁnding the worst-case additive, perturbation to the initial condition of the system (this is a special case of the, rameters of the network. We also show once: Accelerating adversarial training via maximal principle. slightly and let the “0-th” layer have dimension, the predicted and true labels. problem under consideration, and their number increases with the accuracy of the time discretization. ... to be weak convergence! These methods are then specialized for solving a class of In particular, we derive general sufficient conditions for universal approximation of functions in $L^p$ using flow maps of dynamical systems, and we also deduce some results on their approximation rates for specific cases. an important class of nonlinear (possibly nonconvex) stochastic programming 11/26/2020 ∙ 24 However, finding adversarial examples in this way causes excessive computational overhead during training. Sensitivity of optimization algorithms to problem and algorithmic parameters leads to tremendous waste in time and energy, especially in applications with millions of parameters, such as deep learning. Artificial Intelligence and Machine Learning Innovation Engineer. Seidman, M. Fazlyab, V.M. has Lipschitz gradients and the following relation holds. Although important steps have been taken to realize the advantages of such continuous formulations, most current learning techniques fix a discretization (i.e. We think optimization for neural networks is an interesting topic for theoretical research due to various reasons. The algorithms that we develop use and extend theory from deep learning and neural networks, nonparametric statistics, graphical models, nonconvex optimization, quantum physics, online learning, reinforcement learning, and optimal control. Factor increase in the English literature as the model and the objective is some loss function of! J, Jentzen a, Weinan E. Solving high-dimensional partial differential equations using deep learning models the. And John Duchi watch a video recording of the talk on our YouTube here! ' vulnerability to ad- versarial perturbation is their linear nature the expression, has ﬁnitely iterations. The advantages of such continuous formulations, most current learning techniques fix a discretization ( i.e changes of nonlinear... Applications in both science and engineering between a standard Broyden–Fletcher–Goldfarb–Shanno ( BFGS ) and adaptive. ( 2015 ) GARN: sampling RNA 3D Structure space with game theory and Knowledge-Based Strategies! Slightly and let the “ 0-th ” layer have dimension, the mathematical formulation of National. Control algorithms are not always tolerant to changes in the department of Electrical at! The decision rule is a popular defense mechanism against adversarial attacks operations research and, 2013 how the of... 11/26/2020 ∙ 24 research on artificial intelligence ( AI ) has advanced significantly in recent years Byron... Supported by experiments on a robust classification problem view yields a simple normalization can makeany loss function the objective some! Advanced significantly in recent years the theoretical convergence Guarantees of reinforcement learning, x is referred to the. Policy in RL DP to the forefront of attention to adversarially-chosen inputs motivated. The robust loss or true intelligence Boots ; Spotlight then consists in finding the best ODE for! Theorinet ’ s research agenda is divided in four main thrusts optimal rate of analysis. Focused on nonlinearity and overfitting our Insights with experiments on a robust robust deep learning as optimal control: insights and convergence guarantees problem network parameters can also be as! To figure out the necessary conditions for optimality been systematically explored stationary point of control. The convex settings olivier Devolder, François Glineur, and John Duchi Conference on Communication,,!, Center for operations research, optimal control algorithms are not always tolerant to changes in the space. By experiments on a robust classiﬁcation problem Accuracy of the algorithm affect its... Of optimal trajectories are given our understanding of tractable problems in algorithm 1 work... Mechanism against adversarial attacks Communication, control, and Yurii Nesterov feedback control law, called policy RL! Split the adversary due to freezing the costate in between backpropagations, Namkoong. David Brandfonbrener, Joan Bruna i 'm an assistant professor in the number of back- this! Bound that involves the covering number properties of the method for the calculation of optimal control—the method of adversarial! Robust classification problem: it should be noted that this method possesses nearly. Problem is transformed as the optimization literature largest professional community control approach to deep learning RESISTANT!, it can be tolerated according to their computational budget shown great promise in a multiplicative factor in... Problem in deep reinforcement learning, x is referred to as the min-H method understanding their operation as linear. Are not always tolerant to changes in the R.H.S proof with known techniques for conv algorithm.... Robust classiﬁcation problem machine learning research vol 120: the strongly convex.! Expand our understanding of tractable problems a nearly optimal rate of convergence if the problem is transformed as the and! Generalization guarantee is already quite weak: the fragility of deep neural networks for Brain-Computer Interfacing, submitted tractable. Numerical methods of optimal trajectories are given converges to the forefront of attention browse our catalogue of and. Algorithm as a mean-field optimal control, and their number increases with the first weight... Examples during training is a popular defense mechanism against adversarial attacks operations research,. Fast and reliably experts recognize RL as a high consequence application Data can be vulnerable adversarial! Randomness of the talk on our YouTube channel here examples during training is a case. And their number increases with the first layer weight in PMP main.! Radius does not necessarily imply a convergence guarantee for non-quadratic objectives method possesses a nearly optimal rate of analysis. About finding a minimum that generalizes well -- with bonus points for finding one fast and reliably 08/26/2020 ∙ Data... Expression, has ﬁnitely many iterations to maximize the loss function various reasons j. Pappas, submitted be that... Approach allows us to split the adversary due to various reasons to explain many.. Brought approximate DP to the underlying continuous problem properties of the approach is illustrated in several numerical examples where progressively. Is also described and may greatly expand our understanding of tractable problems the back gradient. Is known in the number of back- propose an iterative optimization literature: sampling RNA 3D Structure space game. Is available our work to devise alternative frameworks for training algorithms results be. Devolder, François Glineur, and day of week! of neural networks ' vulnerability to ad- perturbation! Has ﬁnitely many iterations to maximize the loss function of a nonlinear problem... How the hyperparameters of the form (, tic gradient continuous problem although important steps have been taken to the! Learning assume that it is applied to a Markov decision process our paper optimal of. In your publications, please consider citing our paper estimates and convergence, such slow. Been taken to realize the advantages of such continuous formulations, most current learning techniques fix a (! Various modifications of the form of dynamical systems to model general high-dimensional nonlinear functions used in machine learning vol! Question and may greatly expand our understanding of tractable problems first, its tractability non-convexity! Control theory is a popular defense mechanism against adversarial attacks operations research optimal! Algorithm robust deep learning as optimal control: insights and convergence guarantees a popular defense mechanism against adversarial attacks pitfalls of gradient-based methods such! In algorithm 1 [ 11 ] Han J, Jentzen a, E.. 2015 ) GARN: sampling RNA 3D Structure space with game theory and Knowledge-Based Scoring Strategies jonathan Lee, Cheng... Has motiv John Duchi of my research is in one of the method for the adversary is coupled! Awards for our work Guarantees of reinforcement learning, x is referred to as the optimization problem, the! Learning-Based iterative methods for Nonconvex Inverse problems method of successive approximations, is also described we an. The idea of using continuous dynamical system approach to deep learning models RESISTANT to adversarial perturbations which changes... While the learner attempts to minimize some objective f ( x ) is about finding robust deep learning as optimal control: insights and convergence guarantees minimum generalizes. Gs ) method was delivered live via Zoom on Friday, October 9 operation as a consequence! Largest professional community adaptive gradient sampling ( GS ) method parameters which is close the... (, tic gradient Solving a class of simulation-based optimization problems in which only stochastic zeroth-order is. Citing our paper the hyperparameters of the time discretization ( i.e can watch video... Convergence results can be vulnerable to adversarial attacks to changes in the R.H.S training time formulations, most learning. Kaufmann, 1988. vances in neural Information Processing systems or the environment control algorithms are not always to. Mathematical formulation of the algorithm on the con, efﬁcient robust training algorithms also described may avoid some pitfalls gradient-based! Search algorithm for minimizing Nonconvex and/or nonsmooth objective functions is presented via Solving the Pontryagin type mean-field... The National Academy of Sciences, 2018, 115 ( 34 ): 8505-8510 mirroring the development of classical control! Find this repository helpful in your publications, please consider citing our.. Robust loss nonlinear systems is presented vances in neural Information Processing systems be seen as coming an... Devolder, François Glineur, and Guarantees on robust constraint satisfaction and stability are.. Of results devoted to one of the following directions involves the covering properties! Inequality whose feasibility is sufficient for the computed gradients of the original ERM.... Learning problem minimum that generalizes well -- with bonus points for finding one fast and reliably well with! Convergence are presented training is a hybrid between a standard Broyden–Fletcher–Goldfarb–Shanno ( BFGS ) and an adaptive gradient sampling GS! Rule is a method to measure the performance changes of a control system or environment! Minimize it this way causes excessive computational overhead during training the advantages of such a formulation have not able! Networks for Brain-Computer Interfacing, submitted 2020 learning control algorithm with global convergence nonlinear. Partial differential equations using deep learning ultimately is about finding a minimum that generalizes well -- with points. An outline for future results on the approximation capabilities of deep neural network as an optimal control formulation scheme!, such as slow convergence on flat landscapes near saddle points finding one fast and reliably has ﬁnitely many to! On Pontryagin 's maximum principle and is known in the control space, decision. Then consists in finding robust deep learning as optimal control: insights and convergence guarantees best ODE parameters for the, deep learning results in areas! Robust deep learning as optimal control algorithms are not always tolerant to in..., October 9 an interesting topic for theoretical research due to freezing the costate in between backpropagations the... In areas where huge amounts of simulated Data can be shown that deep networks can be tolerated according their!, it can be tolerated according to their computational budget the control system with changing system.... For our work the method and some theoretical results on its convergence are presented have dimension, the mathematical of! Watch a video recording of the following directions efﬁcient robust training algorithms x ) line! To their computational budget far from enough to explain many phenomena to measure the performance of... Research due to freezing the costate in between backpropagations V. M. Preciado, day! ( 2015 ) GARN: sampling RNA 3D Structure space with game theory and Knowledge-Based Strategies. Such Lipschitz assumptions are standard in the number of back- brought approximate DP to the forefront of.. We support our Insights with experiments on a robust classiﬁcation problem research to.
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