arXiv subjects

Justin Romberg

Publications and source records attributed to Justin Romberg.

At least 37 records · Page 2Linked to original sources

arXiv2023

Slepian Beamforming: Broadband Beamforming using Streaming Least Squares

In this paper we revisit the classical problem of estimating a signal as it impinges on a multi-sensor array. We focus on the case where the impinging signal's bandwidth is appreciable and is operating in a broadband regime. Estimating broadband signals, often termed broadband (or wideband) beamforming, is traditionally done through filter and summation, true time delay, or a coupling of the two. Our proposed method deviates substantially from these paradigms in that it requires no notion of filtering or true time delay. We use blocks of samples taken directly from the sensor outputs to fit a robust Slepian subspace model using a least squares approach. We then leverage this model to estimate uniformly spaced samples of the impinging signal. Alongside a careful discussion of this model and how to choose its parameters we show how to fit the model to new blocks of samples as they are received, producing a streaming output. We then go on to show how this method naturally extends to adaptive beamforming scenarios, where we leverage signal statistics to attenuate interfering sources. Finally, we discuss how to use our model to estimate from dimensionality reducing measurements. Accompanying these discussions are extensive numerical experiments establishing that our method outperforms existing filter based approaches while being comparable in terms of computational complexity.

eess.SP↗

arXiv2023

Connected Superlevel Set in (Deep) Reinforcement Learning and its Application to Minimax Theorems

The aim of this paper is to improve the understanding of the optimization landscape for policy optimization problems in reinforcement learning. Specifically, we show that the superlevel set of the objective function with respect to the policy parameter is always a connected set both in the tabular setting and under policies represented by a class of neural networks. In addition, we show that the optimization objective as a function of the policy parameter and reward satisfies a stronger "equiconnectedness" property. To our best knowledge, these are novel and previously unknown discoveries. We present an application of the connectedness of these superlevel sets to the derivation of minimax theorems for robust reinforcement learning. We show that any minimax optimization program which is convex on one side and is equiconnected on the other side observes the minimax equality (i.e. has a Nash equilibrium). We find that this exact structure is exhibited by an interesting robust reinforcement learning problem under an adversarial reward attack, and the validity of its minimax equality immediately follows. This is the first time such a result is established in the literature.

cs.LG↗

arXiv2023

Level set methods for gradient-free optimization of metasurface arrays

Global optimization techniques are increasingly preferred over human-driven methods in the design of electromagnetic structures such as metasurfaces, and careful construction and parameterization of the physical structure is critical in ensuring computational efficiency and convergence of the optimization algorithm to a globally optimal solution. While many design variables in physical systems take discrete values, optimization algorithms often benefit from a continuous design space. This work demonstrates the use of level set functions as a continuous basis for designing material distributions for metasurface arrays and introduces an improved parameterization which is termed the periodic level set function. We explore the use of alternate norms in the definition of the level set function and define a new pseudo-inverse technique for upsampling basis coefficients with these norms. The level set method is compared to the fragmented parameterization and shows improved electromagnetic responses for two dissimilar cost functions: a narrowband objective and a broadband objective. Finally, we manufacture an optimized level set metasurface and measure its scattering parameters to demonstrate real-world performance.

physics.app-ph↗

arXiv2023

Approximately low-rank recovery from noisy and local measurements by convex program

Low-rank matrix models have been universally useful for numerous applications, from classical system identification to more modern matrix completion in signal processing and statistics. The nuclear norm has been employed as a convex surrogate of the low-rankness since it induces a low-rank solution to inverse problems. While the nuclear norm for low rankness has an excellent analogy with the $\ell_1$ norm for sparsity through the singular value decomposition, other matrix norms also induce low-rankness. Particularly as one interprets a matrix as a linear operator between Banach spaces, various tensor product norms generalize the role of the nuclear norm. We provide a tensor-norm-constrained estimator for the recovery of approximately low-rank matrices from local measurements corrupted with noise. A tensor-norm regularizer is designed to adapt to the local structure. We derive statistical analysis of the estimator over matrix completion and decentralized sketching by applying Maurey's empirical method to tensor products of Banach spaces. The estimator provides a near-optimal error bound in a minimax sense and admits a polynomial-time algorithm for these applications.

math.ST↗

arXiv2022

Iterative Broadband Source Localization

In this paper we consider the problem of localizing a set of broadband sources from a finite window of measurements. In the case of narrowband sources this can be reduced to the problem of spectral line estimation, where our goal is simply to estimate the active frequencies from a weighted mixture of pure sinusoids. There exists a plethora of modern and classical methods that effectively solve this problem. However, for a wide variety of applications the underlying sources are not narrowband and can have an appreciable amount of bandwidth. In this work, we extend classical greedy algorithms for sparse recovery (e.g., orthogonal matching pursuit) to localize broadband sources. We leverage models for samples of broadband signals based on a union of Slepian subspaces, which are more aptly suited for dealing with spectral leakage and dynamic range disparities. We show that by using these models, our adapted algorithms can successfully localize broadband sources under a variety of adverse operating scenarios. Furthermore, we show that our algorithms outperform complementary methods that use more standard Fourier models. We also show that we can perform estimation from compressed measurements with little loss in fidelity as long as the number of measurements are on the order of the signal's implicit degrees of freedom. We conclude with an in-depth application of these ideas to the problem of localization in multi-sensor arrays.

eess.SP↗

arXiv2022

Loop Unrolled Shallow Equilibrium Regularizer (LUSER) -- A Memory-Efficient Inverse Problem Solver

In inverse problems we aim to reconstruct some underlying signal of interest from potentially corrupted and often ill-posed measurements. Classical optimization-based techniques proceed by optimizing a data consistency metric together with a regularizer. Current state-of-the-art machine learning approaches draw inspiration from such techniques by unrolling the iterative updates for an optimization-based solver and then learning a regularizer from data. This loop unrolling (LU) method has shown tremendous success, but often requires a deep model for the best performance leading to high memory costs during training. Thus, to address the balance between computation cost and network expressiveness, we propose an LU algorithm with shallow equilibrium regularizers (LUSER). These implicit models are as expressive as deeper convolutional networks, but far more memory efficient during training. The proposed method is evaluated on image deblurring, computed tomography (CT), as well as single-coil Magnetic Resonance Imaging (MRI) tasks and shows similar, or even better, performance while requiring up to 8 times less computational resources during training when compared against a more typical LU architecture with feedforward convolutional regularizers.

eess.IV↗

arXiv2022

Regularized Gradient Descent Ascent for Two-Player Zero-Sum Markov Games

We study the problem of finding the Nash equilibrium in a two-player zero-sum Markov game. Due to its formulation as a minimax optimization program, a natural approach to solve the problem is to perform gradient descent/ascent with respect to each player in an alternating fashion. However, due to the non-convexity/non-concavity of the underlying objective function, theoretical understandings of this method are limited. In our paper, we consider solving an entropy-regularized variant of the Markov game. The regularization introduces structure into the optimization landscape that make the solutions more identifiable and allow the problem to be solved more efficiently. Our main contribution is to show that under proper choices of the regularization parameter, the gradient descent ascent algorithm converges to the Nash equilibrium of the original unregularized problem. We explicitly characterize the finite-time performance of the last iterate of our algorithm, which vastly improves over the existing convergence bound of the gradient descent ascent algorithm without regularization. Finally, we complement the analysis with numerical simulations that illustrate the accelerated convergence of the algorithm.

math.OC↗

arXiv2022

Optimal convex lifted sparse phase retrieval and PCA with an atomic matrix norm regularizer

We present novel analysis and algorithms for solving sparse phase retrieval and sparse principal component analysis (PCA) with convex lifted matrix formulations. The key innovation is a new mixed atomic matrix norm that, when used as regularization, promotes low-rank matrices with sparse factors. We show that convex programs with this atomic norm as a regularizer provide near-optimal sample complexity and error rate guarantees for sparse phase retrieval and sparse PCA. While we do not know how to solve the convex programs exactly with an efficient algorithm, for the phase retrieval case we carefully analyze the program and its dual and thereby derive a practical heuristic algorithm. We show empirically that this practical algorithm performs similarly to existing state-of-the-art algorithms.

math.ST↗

arXiv2022

A Dual Accelerated Method for Online Stochastic Distributed Averaging: From Consensus to Decentralized Policy Evaluation

Motivated by decentralized sensing and policy evaluation problems, we consider a particular type of distributed stochastic optimization problem over a network, called the online stochastic distributed averaging problem. We design a dual-based method for this distributed consensus problem with Polyak--Ruppert averaging and analyze its behavior. We show that the proposed algorithm attains an accelerated deterministic error depending optimally on the condition number of the network, and also that it has an order-optimal stochastic error. This improves on the guarantees of state-of-the-art distributed stochastic optimization algorithms when specialized to this setting, and yields -- among other things -- corollaries for decentralized policy evaluation. Our proofs rely on explicitly studying the evolution of several relevant linear systems, and may be of independent interest. Numerical experiments are provided, which validate our theoretical results and demonstrate that our approach outperforms existing methods in finite-sample scenarios on several natural network topologies.

math.OC↗

arXiv2022

Streaming Reconstruction from Non-uniform Samples

We present an online algorithm for reconstructing a signal from a set of non-uniform samples. By representing the signal using compactly supported basis functions, we show how estimating the expansion coefficients using least-squares can be implemented in a streaming manner: as batches of samples over subsequent time intervals are presented, the algorithm forms an initial estimate of the signal over the sampling interval then updates its estimates over previous intervals. We give conditions under which this reconstruction procedure is stable and show that the least-squares estimates in each interval converge exponentially, meaning that the updates can be performed with finite memory with almost no loss in accuracy. We also discuss how our framework extends to more general types of measurements including time-varying convolution with a compactly supported kernel.

eess.SP↗

arXiv2022

Finite-Time Convergence Rates of Decentralized Stochastic Approximation with Applications in Multi-Agent and Multi-Task Learning

We study a decentralized variant of stochastic approximation, a data-driven approach for finding the root of an operator under noisy measurements. A network of agents, each with its own operator and data observations, cooperatively find the fixed point of the aggregate operator over a decentralized communication graph. Our main contribution is to provide a finite-time analysis of this decentralized stochastic approximation method when the data observed at each agent are sampled from a Markov process; this lack of independence makes the iterates biased and (potentially) unbounded. Under fairly standard assumptions, we show that the convergence rate of the proposed method is essentially the same as if the samples were independent, differing only by a log factor that accounts for the mixing time of the Markov processes. The key idea in our analysis is to introduce a novel Razumikhin-Lyapunov function, motivated by the one used in analyzing the stability of delayed ordinary differential equations. We also discuss applications of the proposed method on a number of interesting learning problems in multi-agent systems.

cs.LG↗

arXiv2022

Broadband Beamforming via Linear Embedding

In modern applications multi-sensor arrays are subject to an ever-present demand to accommodate signals with higher bandwidths. Standard methods for broadband beamforming, namely digital beamforming and true-time delay, are difficult and expensive to implement at scale. In this work, we explore an alternative method of broadband beamforming that uses a set of linear measurements and a robust low-dimensional signal subspace model. The linear measurements, taken directly from the sensors, serve as a method for dimensionality reduction and serve to limit the array readout. From these embedded samples, we show how the original samples can be recovered to within a provably small residual error using a Slepian subspace model. Previous work in multi-sensor array subspace models have largely analyzed performance from a qualitative or asymptotic perspective. In contrast, we give quantitative estimates of how well different dimensionality reduction strategies preserve the array gain. We also show how spatial and temporal correlations can be used to relax the standard Nyquist sampling criterion, how recovery can be achieved through fast algorithms, and how "hardware friendly" linear measurements can be designed.

eess.SP↗

arXiv2022

Streaming Solutions for Time-Varying Optimization Problems

This paper studies streaming optimization problems that have objectives of the form $ \sum_{t=1}^Tf(\mathbf{x}_{t-1},\mathbf{x}_t)$. In particular, we are interested in how the solution $\hat{\mathbf{x} }_{t|T}$ for the $t$th frame of variables changes as $T$ increases. While incrementing $T$ and adding a new functional and a new set of variables does in general change the solution everywhere, we give conditions under which $\hat{\mathbf{x} }_{t|T}$ converges to a limit point $\mathbf{x}^*_t$ at a linear rate as $T\rightarrow\infty$. As a consequence, we are able to derive theoretical guarantees for algorithms with limited memory, showing that limiting the solution updates to only a small number of frames in the past sacrifices almost nothing in accuracy. We also present a new efficient Newton online algorithm (NOA), inspired by these results, that updates the solution with fixed complexity of $ \mathcal{O}( {3Bn^3})$, independent of $T$, where $B$ corresponds to how far in the past the variables are updated, and $n$ is the size of a single block-vector. Two streaming optimization examples, online reconstruction from non-uniform samples and non-homogeneous Poisson intensity estimation, support the theoretical results and show how the algorithm can be used in practice.

math.OC↗

arXiv2022

Finite Sample Analysis of Two-Time-Scale Natural Actor-Critic Algorithm

Actor-critic style two-time-scale algorithms are one of the most popular methods in reinforcement learning, and have seen great empirical success. However, their performance is not completely understood theoretically. In this paper, we characterize the \emph{global} convergence of an online natural actor-critic algorithm in the tabular setting using a single trajectory of samples. Our analysis applies to very general settings, as we only assume ergodicity of the underlying Markov decision process. In order to ensure enough exploration, we employ an $ε$-greedy sampling of the trajectory. For a fixed and small enough exploration parameter $ε$, we show that the two-time-scale natural actor-critic algorithm has a rate of convergence of $\tilde{\mathcal{O}}(1/T^{1/4})$, where $T$ is the number of samples, and this leads to a sample complexity of $\Tilde{\mathcal{O}}(1/δ^{8})$ samples to find a policy that is within an error of $δ$ from the \emph{global optimum}. Moreover, by carefully decreasing the exploration parameter $ε$ as the iterations proceed, we present an improved sample complexity of $\Tilde{\mathcal{O}}(1/δ^{6})$ for convergence to the global optimum.

cs.LG↗

arXiv2021

Decentralized Feature-Distributed Optimization for Generalized Linear Models

We consider the "all-for-one" decentralized learning problem for generalized linear models. The features of each sample are partitioned among several collaborating agents in a connected network, but only one agent observes the response variables. To solve the regularized empirical risk minimization in this distributed setting, we apply the Chambolle--Pock primal--dual algorithm to an equivalent saddle-point formulation of the problem. The primal and dual iterations are either in closed-form or reduce to coordinate-wise minimization of scalar convex functions. We establish convergence rates for the empirical risk minimization under two different assumptions on the loss function (Lipschitz and square root Lipschitz), and show how they depend on the characteristics of the design matrix and the Laplacian of the network.

math.OC↗

arXiv2021

A Decentralized Policy Gradient Approach to Multi-task Reinforcement Learning

We develop a mathematical framework for solving multi-task reinforcement learning (MTRL) problems based on a type of policy gradient method. The goal in MTRL is to learn a common policy that operates effectively in different environments; these environments have similar (or overlapping) state spaces, but have different rewards and dynamics. We highlight two fundamental challenges in MTRL that are not present in its single task counterpart, and illustrate them with simple examples. We then develop a decentralized entropy-regularized policy gradient method for solving the MTRL problem, and study its finite-time convergence rate. We demonstrate the effectiveness of the proposed method using a series of numerical experiments. These experiments range from small-scale "GridWorld" problems that readily demonstrate the trade-offs involved in multi-task learning to large-scale problems, where common policies are learned to navigate an airborne drone in multiple (simulated) environments.

cs.LG↗

arXiv2021

Thomson's Multitaper Method Revisited

Thomson's multitaper method estimates the power spectrum of a signal from $N$ equally spaced samples by averaging $K$ tapered periodograms. Discrete prolate spheroidal sequences (DPSS) are used as tapers since they provide excellent protection against spectral leakage. Thomson's multitaper method is widely used in applications, but most of the existing theory is qualitative or asymptotic. Furthermore, many practitioners use a DPSS bandwidth $W$ and number of tapers that are smaller than what the theory suggests is optimal because the computational requirements increase with the number of tapers. We revisit Thomson's multitaper method from a linear algebra perspective involving subspace projections. This provides additional insight and helps us establish nonasymptotic bounds on some statistical properties of the multitaper spectral estimate, which are similar to existing asymptotic results. We show using $K=2NW-O(\log(NW))$ tapers instead of the traditional $2NW-O(1)$ tapers better protects against spectral leakage, especially when the power spectrum has a high dynamic range. Our perspective also allows us to derive an $ε$-approximation to the multitaper spectral estimate which can be evaluated on a grid of frequencies using $O(\log(NW)\log\tfrac{1}ε)$ FFTs instead of $K=O(NW)$ FFTs. This is useful in problems where many samples are taken, and thus, using many tapers is desirable.

eess.SP↗

arXiv2020

STAN: Spatio-Temporal Attention Network for Pandemic Prediction Using Real World Evidence

Objective: The COVID-19 pandemic has created many challenges that need immediate attention. Various epidemiological and deep learning models have been developed to predict the COVID-19 outbreak, but all have limitations that affect the accuracy and robustness of the predictions. Our method aims at addressing these limitations and making earlier and more accurate pandemic outbreak predictions by (1) using patients' EHR data from different counties and states that encode local disease status and medical resource utilization condition; (2) considering demographic similarity and geographical proximity between locations; and (3) integrating pandemic transmission dynamics into deep learning models. Materials and Methods: We proposed a spatio-temporal attention network (STAN) for pandemic prediction. It uses an attention-based graph convolutional network to capture geographical and temporal trends and predict the number of cases for a fixed number of days into the future. We also designed a physical law-based loss term for enhancing long-term prediction. STAN was tested using both massive real-world patient data and open source COVID-19 statistics provided by Johns Hopkins university across all U.S. counties. Results: STAN outperforms epidemiological modeling methods such as SIR and SEIR and deep learning models on both long-term and short-term predictions, achieving up to 87% lower mean squared error compared to the best baseline prediction model. Conclusions: By using information from real-world patient data and geographical data, STAN can better capture the disease status and medical resource utilization information and thus provides more accurate pandemic modeling. With pandemic transmission law based regularization, STAN also achieves good long-term prediction performance.

cs.SI↗