A. T. Lin

Alex Tong Lin, Adrian S. Wong, Robert Martin et al.

We provide a method to identify system parameters of dynamical systems, called ID-ODE -- Inference by Differentiation and Observing Delay Embeddings. In this setting, we are given a dataset of trajectories from a dynamical system with system parameter labels. Our goal is to identify system parameters of new trajectories. The given trajectories may or may not encompass the full state of the system, and we may only observe a one-dimensional time series. In the latter case, we reconstruct the full state by using delay embeddings, and under sufficient conditions, Taken's Embedding Theorem assures us the reconstruction is diffeomorphic to the original. This allows our method to work on time series. Our method works by first learning the velocity operator (as given or reconstructed) with a neural network having both state and system parameters as variable inputs. Then on new trajectories we backpropagate prediction errors to the system parameter inputs giving us a gradient. We then use gradient descent to infer the correct system parameter. We demonstrate the efficacy of our approach on many numerical examples: the Lorenz system, Lorenz96, Lotka-Volterra Predator-Prey, and the Compound Double Pendulum. We also apply our algorithm on a real-world dataset: propulsion of the Hall-effect Thruster (HET).

Stanley Osher, Bao Wang, Penghang Yin et al.

We propose a class of very simple modifications of gradient descent and stochastic gradient descent. We show that when applied to a large variety of machine learning problems, ranging from logistic regression to deep neural nets, the proposed surrogates can dramatically reduce the variance, allow to take a larger step size, and improve the generalization accuracy. The methods only involve multiplying the usual (stochastic) gradient by the inverse of a positive definitive matrix (which can be computed efficiently by FFT) with a low condition number coming from a one-dimensional discrete Laplacian or its high order generalizations. It also preserves the mean and increases the smallest component and decreases the largest component. The theory of Hamilton-Jacobi partial differential equations demonstrates that the implicit version of the new algorithm is almost the same as doing gradient descent on a new function which (i) has the same global minima as the original function and (ii) is ``more convex". Moreover, we show that optimization algorithms with these surrogates converge uniformly in the discrete Sobolev $H_σ^p$ sense and reduce the optimality gap for convex optimization problems. The code is available at: \url{https://github.com/BaoWangMath/LaplacianSmoothing-GradientDescent}

Shay Deutsch, Lionel Yelibi, Alex Tong Lin et al.

Deriving meaningful representations from complex, high-dimensional data in unsupervised settings is crucial across diverse machine learning applications. This paper introduces a framework for multi-scale graph network embedding based on spectral graph wavelets that employs a contrastive learning approach. We theoretically show that in Paley-Wiener spaces on combinatorial graphs, the spectral graph wavelets operator provides greater flexibility and control over smoothness compared to the Laplacian operator, motivating our approach. A key advantage of the proposed embedding is its ability to establish a correspondence between the embedding and input feature spaces, enabling the derivation of feature importance. We validate the effectiveness of our graph embedding framework on multiple public datasets across various downstream tasks, including clustering and unsupervised feature importance.

Alex Tong Lin, Wuchen Li, Stanley Osher et al.

We introduce a new method for training generative adversarial networks by applying the Wasserstein-2 metric proximal on the generators. The approach is based on Wasserstein information geometry. It defines a parametrization invariant natural gradient by pulling back optimal transport structures from probability space to parameter space. We obtain easy-to-implement iterative regularizers for the parameter updates of implicit deep generative models. Our experiments demonstrate that this method improves the speed and stability of training in terms of wall-clock time and Fréchet Inception Distance.

Howard Heaton, Samy Wu Fung, Alex Tong Lin et al.

Inverse problems consist of recovering a signal from a collection of noisy measurements. These are typically cast as optimization problems, with classic approaches using a data fidelity term and an analytic regularizer that stabilizes recovery. Recent Plug-and-Play (PnP) works propose replacing the operator for analytic regularization in optimization methods by a data-driven denoiser. These schemes obtain state of the art results, but at the cost of limited theoretical guarantees. To bridge this gap, we present a new algorithm that takes samples from the manifold of true data as input and outputs an approximation of the projection operator onto this manifold. Under standard assumptions, we prove this algorithm generates a learned operator, called Wasserstein-based projection (WP), that approximates the true projection with high probability. Thus, WPs can be inserted into optimization methods in the same manner as PnP, but now with theoretical guarantees. Provided numerical examples show WPs obtain state of the art results for unsupervised PnP signal recovery.

Alex Tong Lin, Samy Wu Fung, Wuchen Li et al.

We present APAC-Net, an alternating population and agent control neural network for solving stochastic mean field games (MFGs). Our algorithm is geared toward high-dimensional instances of MFGs that are beyond reach with existing solution methods. We achieve this in two steps. First, we take advantage of the underlying variational primal-dual structure that MFGs exhibit and phrase it as a convex-concave saddle point problem. Second, we parameterize the value and density functions by two neural networks, respectively. By phrasing the problem in this manner, solving the MFG can be interpreted as a special case of training a generative adversarial network (GAN). We show the potential of our method on up to 100-dimensional MFG problems.

Alex Tong Lin, Yonatan Dukler, Wuchen Li et al.

We propose regularization strategies for learning discriminative models that are robust to in-class variations of the input data. We use the Wasserstein-2 geometry to capture semantically meaningful neighborhoods in the space of images, and define a corresponding input-dependent additive noise data augmentation model. Expanding and integrating the augmented loss yields an effective Tikhonov-type Wasserstein diffusion smoothness regularizer. This approach allows us to apply high levels of regularization and train functions that have low variability within classes but remain flexible across classes. We provide efficient methods for computing the regularizer at a negligible cost in comparison to training with adversarial data augmentation. Initial experiments demonstrate improvements in generalization performance under adversarial perturbations and also large in-class variations of the input data.

Alex Tong Lin, Mark J. Debord, Katia Estabridis et al.

We consider a multi-agent reinforcement learning problem where each agent seeks to maximize a shared reward while interacting with other agents, and they may or may not be able to communicate. Typically the agents do not have access to other agent policies and thus each agent is situated in a non-stationary and partially-observable environment. In order to obtain multi-agents that act in a decentralized manner, we introduce a novel algorithm under the popular framework of centralized training, but decentralized execution. This training framework first obtains solutions to a multi-agent problem with a single centralized joint-space learner, which is then used to guide imitation learning for independent decentralized multi-agents. This framework has the flexibility to use any reinforcement learning algorithm to obtain the expert as well as any imitation learning algorithm to obtain the decentralized agents. This is in contrast to other multi-agent learning algorithms that, for example, can require more specific structures. We present some theoretical bounds for our method, and we show that one can obtain decentralized solutions to a multi-agent problem through imitation learning.

Bao Wang, Alex T. Lin, Wei Zhu et al.

We improve the robustness of Deep Neural Net (DNN) to adversarial attacks by using an interpolating function as the output activation. This data-dependent activation remarkably improves both the generalization and robustness of DNN. In the CIFAR10 benchmark, we raise the robust accuracy of the adversarially trained ResNet20 from $\sim 46\%$ to $\sim 69\%$ under the state-of-the-art Iterative Fast Gradient Sign Method (IFGSM) based adversarial attack. When we combine this data-dependent activation with total variation minimization on adversarial images and training data augmentation, we achieve an improvement in robust accuracy by 38.9$\%$ for ResNet56 under the strongest IFGSM attack. Furthermore, We provide an intuitive explanation of our defense by analyzing the geometry of the feature space.