Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees Jacob H. Seidman , Mahyar Fazlyab , Victor M. Preciado , George J. Pappas 08 Jun 2020 L4DC 2020 Readers: Everyone You can watch a video recording of the talk on our YouTube channel here. 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton) , 480-487. method with an inexact gradient oracle. Robust deep learning as optimal control: Insights and convergence guarantees JH Seidman, M Fazlyab, VM Preciado, GJ Pappas arXiv preprint arXiv:2005.00616 , 2020 Abstract. prove our main theorem we will apply the bound from the previous result to the follo, Combining the previous three results allows us to prov, The first term on the right side is typical for conver, accumulate over the algorithm. 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.. This paper introduces the mathematical formulation of the population risk minimization problem in deep learning as a mean-field optimal control problem. In particular, our framework enables us to obtain a new and simple proof of the O(1/k) convergence rate of the algorithm when the objective function is not strongly convex. Our result further suggest that for a fixed number of backprop-, agations, increasing the number of adversary updates past a certain point can have a ne. H�,�� $�\AD �$���Xr %���U##�Ɓ&x������[� � � endstream endobj startxref 0 %%EOF 434 0 obj <>stream Finally, the proposed algorithms are applied to the periodic linear quadratic optimal control of the well-known lossy Mathieu equation, which shows the … 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). denote the corresponding stochastic gradient of the robust loss. 11:00AM – 11:30AM – Warren Dixon, University of Florida (wdixon@ufl.edu) Title: Multiple Timescale Deep Learning Abstract: A Deep Neural Network (DNN) adaptive control architecture is presented for general uncertain nonlinear (2015) Convergence of heterogeneous distributed learning in stochastic routing games. Optimization is a critical component in deep learning. Early attempts at explaining this phenomenon focused on nonlinearity and overfitting. Most of my research is in one of the following directions. Learning the system dynamics is often the basis of associated control or policy decision problems in tasks varying from linear-quadratic control to deep reinforcement learning. 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. The inner method finds an adversarial perturbation by performing gradient, are able to bound the oracle error and appeal to known inexact oracle con, The outer method then makes a parameter update to the network based on the perturbation found, an exact gradient update. In this work we formalize two new criteria of robustness to action uncertainty. In this paper, we provide the first conv, ing algorithm by combining techniques from robust optimal control and inexact oracle methods, its stability and convergence. Policy Search/Deep Learning (DL) Control Limitations: • Large number of computations at each time step • No standard training procedures for control tasks • Few analysis tools for deep neural network architectures • Non-convex form of optimization provides few guarantees • Lack of research in optimization with regards to robustness To solve it, previous works directly run gradient descent on the "adversarial loss", i.e. Recent literature has provided finite-sample statistical analysis for simple least-squares regression models applied to this problem. We support our insights with experiments on a robust classification, Deep neural networks have repeatedly demonstrated their capacity to achie, mance on benchmark machine learning problems, can be significantly affected by small input perturbations that can drastically change the network’, Therefore, an important line of work has emerged to train deep neural networks to be robust to, Among the most empirically successful methods is an optimization-based approach, where ad-, versarial training is formulated as a min-max non-conv, tions, typically using an iterative method such as Projected Gradient Descent (PGD), backpropagation through the network. This approach has the advantage that rigorous error estimates and convergence results can be established. Geometric Insights into the Convergence of Nonlinear TD Learning. approximate stationary point of a nonlinear programming problem. problem under consideration, and their number increases with the accuracy of the time discretization. Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees. 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. A variety of AI algorithms have shown great promise in a large nu… argument we construct provides an outline for future results on the con, efficient robust training algorithms. First-order methods with inexact oracle: the strongly convex case. Preprint. TOWARDS DEEP LEARNING MODELS RESISTANT TO ADVERSARIAL ATTACKS Performance of the approach … * Robust deep learning, Adversarial attacks * Optimal transport, GANs, geometry learning * Reinforcement learning * Optimal control and prediction * Applications-- Prediction and control of Covid-19 pandemic-- Data driven and Science informed discovery -- Solving high dimensional partial differential equations 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 … Policy iteration is guaranteed to converge and at convergence, the current policy 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. J. H. Seidman, M. Fazlyab, V. M. Preciado, and George J. Pappas, Submitted. However, optimal control algorithms are not always tolerant to changes in the control system or the environment. 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. However, since in general the adv, to the true optimal point in finitely many iterations, and is an inexact method itself, the update for. Research on artificial intelligence (AI) has advanced significantly in recent years. However, the mathematical aspects of such a formulation have not been systematically explored. View Nir Levine’s profile on LinkedIn, the world's largest professional community. For adversarial example defense, our experiment shows that YOPO can achieve comparable defense accuracy using around 1/5 GPU time of the original projected gradient descent training. 11/26/2020 ∙ 24 PGD steps also guarantee the convergence to optimal solution under the convex settings. Bayesian Deep Learning workshop, NIPS, 2018 [Paper] [BibTex] Evaluating Uncertainty Quantification in End-to-End Autonomous Driving Control Self-driving has benefited from significant performance improvements with the rise of deep learning, with millions of miles having been driven with no human intervention. A new iterative learning control algorithm with global convergence for nonlinear systems is presented. 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. functions of low complexity. I, and to high profile developments in deep reinforcement learning, which have brought approximate DP to the forefront of attention. This explanation is supported by new quantitative results while giving the first explanation of the most intriguing fact about them: their generalization across architectures and training sets. Robust Learning Model Predictive Control for Periodically Correlated Building Control Jicheng Shi†, Yingzhao Lian†, and Colin N. Jones Abstract—Accounting for more than 40% of global energy consumption, residential and commercial buildings will be key players in any future green energy systems. Robust Optimization. of gradient oracle errors for the adversary due to freezing the costate in between backpropagations. Artificial Intelligence and Machine Learning Innovation Engineer. Li Q, Hao S. An optimal control approach to deep learning and applications to discrete-weight neural networks[J]. [C89] Robust Deep Learning as Optimal Control: Insights and Convergence Guarantees J.H. 01/26/2019 ∙ by Chen Tessler, et al. algorithm, namely the randomized stochastic gradient (RSG) method, for solving Risheng Liu, Shichao Cheng, Yi He, Xin Fan, Zhouchen Lin, Zhongxuan Luo IEEE TPAMI (CCF-A) A Theoretically Guaranteed Deep Optimization Framework for Robust Compressive Sensing MRI. 1optimal control is a special case of our DR optimal control formulation. Controlling a 2D Robotic Arm with Deep Reinforcement Learning an article which shows how to build your own robotic arm best friend by diving into deep reinforcement learning Spinning Up a Pong AI With Deep Reinforcement Learning an article which shows you to code a vanilla policy gradient model that plays the beloved early 1970s classic video game Pong in a step-by-step manner 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. The performance of these algorithms depends on the choice of step size parameters, for which the optimal values are known in some specific cases, and otherwise are set heuristically. and Rob Fergus. replacing the input data with the corresponding adversaries. 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. Although theoretical foundations have been developed on the optimization side through mean-field optimal control theory, the function approximation properties of such models remain largely unexplored, especially when the dynamical systems are controlled by, Douglas-Rachford splitting, and its equivalent dual formulation ADMM, are widely used iterative methods in composite optimization problems arising in control and machine learning applications. First-order methods with inexact oracle: the Figure on the left shows how accuracy changes with m = 5 fixed and varying n, figure on the right is the same for m = 10 and varying n. In both figures we see that performance degrades quickly in n after a certain point, as predicted by Theorem 5. We think optimization for neural networks is an interesting topic for theoretical research due to various reasons. %PDF-1.5 %���� The learning task then consists in finding the best ODE parameters for the, Deep learning achieves state-of-the-art results in many areas. Throughout we will reserve, Due to their compositional structure, feed-forward deep neural networks can be viewed as dynam-, dynamics and use the interpretation to suggest new training algorithms. 05/24/2019 ∙ by Viktor Reshniak, et al. where the expectation is taken over the randomness of the mini-batch sampling. 4. Robust control theory is a method to measure the performance changes of a control system with changing system parameters. Sheds light on how the hyperparameters of the method of convergence if the problem is.. Parameters can also be seen as coming from an inexact gradient oracle errors for the computed gradients of models... 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