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Register a new userOptimizing Neural Networks via Koopman Operator Theory
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Koopman operator theory, a powerful framework for discovering the underlying dynamics of nonlinear dynamical systems, was recently shown to be intimately connected with neural network training. In this work, we take the first steps in making use of this connection. As Koopman operator theory is a linear theory, a successful implementation of it in evolving network weights and biases offers the promise of accelerated training, especially in the context of deep networks, where optimization is inherently a non-convex problem. We show that Koopman operator theoretic methods allow for accurate predictions of weights and biases of feedforward, fully connected deep networks over a non-trivial range of training time. During this window, we find that our approach is >10x faster than various gradient descent based methods (e.g. Adam, Adadelta, Adagrad), in line with our complexity analysis. We end by highlighting open questions in this exciting intersection between dynamical systems and neural network theory. We highlight additional methods by which our results could be expanded to broader classes of networks and larger training intervals, which shall be the focus of future work.
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A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last century, the Koopman operator has provided a mathematical framework for the study of dynamical systems, which facilitates conjugacy arguments and can provide efficient reduced descriptions. More recently, numerical approximations of the operator have enabled the analysis of a large number of deterministic and stochastic dynamical systems in a completely data-driven, essentially equation-free pipeline. Discrete or continuous time numerical algorithms (integrators, nonlinear equation solvers, optimization algorithms) are themselves dynamical systems. In this paper, we use this insight to leverage the Koopman operator framework in the data-driven study of such algorithms and discuss benefits for analysis and acceleration of numerical computation. For algorithms acting on high-dimensional spaces by quickly contracting them towards low-dimensional manifolds, we demonstrate how basis functions adapted to the data help to construct efficient reduced representations of the operator. Our illustrative examples include the gradient descent and Nesterov optimization algorithms, as well as the Newton-Raphson algorithm.
Optimization for deep networks is currently a very active area of research. As neural networks become deeper, the ability in manually optimizing the network becomes harder. Mini-batch normalization, identification of effective respective fields, momentum updates, introduction of residual blocks, learning rate adoption, etc. have been proposed to speed up the rate of convergent in manual training process while keeping the higher accuracy level. However, the problem of finding optimal topological structure for a given problem is becoming a challenging task need to be addressed immediately. Few researchers have attempted to optimize the network structure using evolutionary computing approaches. Among them, few have successfully evolved networks with reinforcement learning and long-short-term memory. A very few has applied evolutionary programming into deep convolution neural networks. These attempts are mainly evolved the network structure and then subsequently optimized the hyper-parameters of the network. However, a mechanism to evolve the deep network structure under the techniques currently being practiced in manual process is still absent. Incorporation of such techniques into chromosomes level of evolutionary computing, certainly can take us to better topological deep structures. The paper concludes by identifying the gap between evolutionary based deep neural networks and deep neural networks. Further, it proposes some insights for optimizing deep neural networks using evolutionary computing techniques.
This paper is concerned with the utilization of deterministically modeled chemical reaction networks for the implementation of (feed-forward) neural networks. We develop a general mathematical framework and prove that the ordinary differential equations (ODEs) associated with certain reaction network implementations of neural networks have desirable properties including (i) existence of unique positive fixed points that are smooth in the parameters of the model (necessary for gradient descent), and (ii) fast convergence to the fixed point regardless of initial condition (necessary for efficient implementation). We do so by first making a connection between neural networks and fixed points for systems of ODEs, and then by constructing reaction networks with the correct associated set of ODEs. We demonstrate the theory by constructing a reaction network that implements a neural network with a smoothed ReLU activation function, though we also demonstrate how to generalize the construction to allow for other activation functions (each with the desirable properties listed previously). As there are multiple types of networks utilized in this paper, we also give a careful introduction to both reaction networks and neural networks, in order to disambiguate the overlapping vocabulary in the two settings and to clearly highlight the role of each networks properties.
The Koopman operator allows for handling nonlinear systems through a (globally) linear representation. In general, the operator is infinite-dimensional - necessitating finite approximations - for which there is no overarching framework. Although there are principled ways of learning such finite approximations, they are in many instances overlooked in favor of, often ill-posed and unstructured methods. Also, Koopman operator theory has long-standing connections to known system-theoretic and dynamical system notions that are not universally recognized. Given the former and latter realities, this work aims to bridge the gap between various concepts regarding both theory and tractable realizations. Firstly, we review data-driven representations (both unstructured and structured) for Koopman operator dynamical models, categorizing various existing methodologies and highlighting their differences. Furthermore, we provide concise insight into the paradigms relation to system-theoretic notions and analyze the prospect of using the paradigm for modeling control systems. Additionally, we outline the current challenges and comment on future perspectives.
A Deep Neural Network is applied to classify physical signatures obtained from physical sensor measurements of running gasoline and diesel-powered vehicles and other devices. The classification provides information on the target identities as to vehicle type and even vehicle model. The physical measurements include acoustic, acceleration (vibration), geophonic, and magnetic.
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