arXiv subjects

Babak Hassibi

Publications and source records attributed to Babak Hassibi.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Novel Gaussian Min-Max Theorem and its Applications

A celebrated result by Gordon allows one to compare the min-max behavior of two Gaussian processes if certain inequality conditions are met. The consequences of this result include the Gaussian min-max (GMT) and convex Gaussian min-max (CGMT) theorems which have had far-reaching implications in high-dimensional statistics, machine learning, non-smooth optimization, and signal processing. Both theorems rely on a pair of Gaussian processes, first identified by Slepian, that satisfy Gordon's comparison inequalities. In this paper, we identify such a new pair. The resulting theorems extend the classical GMT and CGMT Theorems from the case where the underlying Gaussian matrix in the primary process has iid rows to where it has independent but non-identically-distributed ones. The new CGMT is applied to the problems of multi-source Gaussian regression, as well as to binary classification of general Gaussian mixture models.

cs.LG↗

arXiv2024

One-Bit Quantization and Sparsification for Multiclass Linear Classification with Strong Regularization

We study the use of linear regression for multiclass classification in the over-parametrized regime where some of the training data is mislabeled. In such scenarios it is necessary to add an explicit regularization term, $λf(w)$, for some convex function $f(\cdot)$, to avoid overfitting the mislabeled data. In our analysis, we assume that the data is sampled from a Gaussian Mixture Model with equal class sizes, and that a proportion $c$ of the training labels is corrupted for each class. Under these assumptions, we prove that the best classification performance is achieved when $f(\cdot) = \|\cdot\|^2_2$ and $λ\to \infty$. We then proceed to analyze the classification errors for $f(\cdot) = \|\cdot\|_1$ and $f(\cdot) = \|\cdot\|_\infty$ in the large $λ$ regime and notice that it is often possible to find sparse and one-bit solutions, respectively, that perform almost as well as the one corresponding to $f(\cdot) = \|\cdot\|_2^2$.

cs.LG↗

arXiv2024

Optimal Infinite-Horizon Mixed $\mathit{H}_2/\mathit{H}_\infty$ Control

We study the problem of mixed $\mathit{H}_2/\mathit{H}_\infty$ control in the infinite-horizon setting. We identify the optimal causal controller that minimizes the $\mathit{H}_2$ cost of the closed-loop system subject to an $\mathit{H}_\infty$ constraint. Megretski proved that the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller is non-rational whenever the constraint is active without giving an explicit construction of the controller. In this work, we provide the first exact closed-form solution to the infinite-horizon mixed $\mathit{H}_2/\mathit{H}_\infty$ control in the frequency domain. While the optimal controller is non-rational, our formulation provides a finite-dimensional parameterization of the optimal controller. Leveraging this fact, we introduce an efficient iterative algorithm that finds the optimal causal controller in the frequency domain. We show that this algorithm is convergent when the system is scalar and present numerical evidence for exponential convergence of the proposed algorithm. Finally, we show how to find the best (in $\mathit{H}_\infty$ norm) fixed-order rational approximations of the optimal mixed $\mathit{H}_2/\mathit{H}_\infty$ controller and study its performance.

math.OC↗

arXiv2024

The Distributionally Robust Infinite-Horizon LQR

We explore the infinite-horizon Distributionally Robust (DR) linear-quadratic control. While the probability distribution of disturbances is unknown and potentially correlated over time, it is confined within a Wasserstein-2 ball of a radius $r$ around a known nominal distribution. Our goal is to devise a control policy that minimizes the worst-case expected Linear-Quadratic Regulator (LQR) cost among all probability distributions of disturbances lying in the Wasserstein ambiguity set. We obtain the optimality conditions for the optimal DR controller and show that it is non-rational. Despite lacking a finite-order state-space representation, we introduce a computationally tractable fixed-point iteration algorithm. Our proposed method computes the optimal controller in the frequency domain to any desired fidelity. Moreover, for any given finite order, we use a convex numerical method to compute the best rational approximation (in $H_\infty$-norm) to the optimal non-rational DR controller. This enables efficient time-domain implementation by finite-order state-space controllers and addresses the computational hurdles associated with the finite-horizon approaches to DR-LQR problems, which typically necessitate solving a Semi-Definite Program (SDP) with a dimension scaling with the time horizon. We provide numerical simulations to showcase the effectiveness of our approach.

math.OC↗

arXiv2024

Distributionally Robust Kalman Filtering over Finite and Infinite Horizon

This paper investigates the distributionally robust filtering of signals generated by state-space models driven by exogenous disturbances with noisy observations in finite and infinite horizon scenarios. The exact joint probability distribution of the disturbances and noise is unknown but assumed to reside within a Wasserstein-2 ambiguity ball centered around a given nominal distribution. We aim to derive a causal estimator that minimizes the worst-case mean squared estimation error among all possible distributions within this ambiguity set. We remove the iid restriction in prior works by permitting arbitrarily time-correlated disturbances and noises. In the finite horizon setting, we reduce this problem to a semi-definite program (SDP), with computational complexity scaling with the time horizon. For infinite horizon settings, we characterize the optimal estimator using Karush-Kuhn-Tucker (KKT) conditions. Although the optimal estimator lacks a rational form, i.e., a finite-dimensional state-space realization, it can be fully described by a finite-dimensional parameter. {Leveraging this parametrization, we propose efficient algorithms that compute the optimal estimator with arbitrary fidelity in the frequency domain.} Moreover, given any finite degree, we provide an efficient convex optimization algorithm that finds the finite-dimensional state-space estimator that best approximates the optimal non-rational filter in ${\cal H}_\infty$ norm. This facilitates the practical implementation of the infinite horizon filter without having to grapple with the ill-scaled SDP from finite time. Finally, numerical simulations demonstrate the effectiveness of our approach in practical scenarios.

math.OC↗

arXiv2024

Infinite-Horizon Distributionally Robust Regret-Optimal Control

We study the infinite-horizon distributionally robust (DR) control of linear systems with quadratic costs, where disturbances have unknown, possibly time-correlated distribution within a Wasserstein-2 ambiguity set. We aim to minimize the worst-case expected regret-the excess cost of a causal policy compared to a non-causal one with access to future disturbance. Though the optimal policy lacks a finite-order state-space realization (i.e., it is non-rational), it can be characterized by a finite-dimensional parameter. Leveraging this, we develop an efficient frequency-domain algorithm to compute this optimal control policy and present a convex optimization method to construct a near-optimal state-space controller that approximates the optimal non-rational controller in the $\mathit{H}_\infty$-norm. This approach avoids solving a computationally expensive semi-definite program (SDP) that scales with the time horizon in the finite-horizon setting.

math.OC↗

arXiv2024

A Distributionally Robust Approach to Shannon Limits using the Wasserstein Distance

We consider the rate-distortion function for lossy source compression, as well as the channel capacity for error correction, through the lens of distributional robustness. We assume that the distribution of the source or of the additive channel noise is unknown and lies within a Wasserstein-2 ambiguity set of a given radius centered around a specified nominal distribution, and we look for the worst-case asymptotically optimal coding rate over such an ambiguity set. Varying the radius of the ambiguity set allows us to interpolate between the worst-case and stochastic scenarios using probabilistic tools. Our problem setting fits into the paradigm of compound source / channel models introduced by Sakrison and Blackwell, respectively. This paper shows that if the nominal distribution is Gaussian, then so is the worst-case source / noise distribution, and the compound rate-distortion / channel capacity functions admit convex formulations with Linear Matrix Inequality (LMI) constraints. These formulations yield simple closed-form expressions in the scalar case, offering insights into the behavior of Shannon limits with the changing radius of the Wasserstein-2 ambiguity set.

cs.IT↗

arXiv2023

Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon

We investigate the Distributionally Robust Regret-Optimal (DR-RO) control of discrete-time linear dynamical systems with quadratic cost over an infinite horizon. Regret is the difference in cost obtained by a causal controller and a clairvoyant controller with access to future disturbances. We focus on the infinite-horizon framework, which results in stability guarantees. In this DR setting, the probability distribution of the disturbances resides within a Wasserstein-2 ambiguity set centered at a specified nominal distribution. Our objective is to identify a control policy that minimizes the worst-case expected regret over an infinite horizon, considering all potential disturbance distributions within the ambiguity set. In contrast to prior works, which assume time-independent disturbances, we relax this constraint to allow for time-correlated disturbances, thus actual distributional robustness. While we show that the resulting optimal controller is non-rational and lacks a finite-dimensional state-space realization, we demonstrate that it can still be uniquely characterized by a finite dimensional parameter. Exploiting this fact, we introduce an efficient numerical method to compute the controller in the frequency domain using fixed-point iterations. This method circumvents the computational bottleneck associated with the finite-horizon problem, where the semi-definite programming (SDP) solution dimension scales with the time horizon. Numerical experiments demonstrate the effectiveness and performance of our framework.

eess.SY↗

arXiv2023

Regret-Optimal Control under Partial Observability

This paper studies online solutions for regret-optimal control in partially observable systems over an infinite-horizon. Regret-optimal control aims to minimize the difference in LQR cost between causal and non-causal controllers while considering the worst-case regret across all $\ell_2$-norm-bounded disturbance and measurement sequences. Building on ideas from Sabag et al., 2023, on the the full-information setting, our work extends the framework to the scenario of partial observability (measurement-feedback). We derive an explicit state-space solution when the non-causal solution is the one that minimizes the $\mathcal H_2$ criterion, and demonstrate its practical utility on several practical examples. These results underscore the framework's significant relevance and applicability in real-world systems.

eess.SY↗

arXiv2023

Regularized Linear Regression for Binary Classification

Regularized linear regression is a promising approach for binary classification problems in which the training set has noisy labels since the regularization term can help to avoid interpolating the mislabeled data points. In this paper we provide a systematic study of the effects of the regularization strength on the performance of linear classifiers that are trained to solve binary classification problems by minimizing a regularized least-squares objective. We consider the over-parametrized regime and assume that the classes are generated from a Gaussian Mixture Model (GMM) where a fraction $c<\frac{1}{2}$ of the training data is mislabeled. Under these assumptions, we rigorously analyze the classification errors resulting from the application of ridge, $\ell_1$, and $\ell_\infty$ regression. In particular, we demonstrate that ridge regression invariably improves the classification error. We prove that $\ell_1$ regularization induces sparsity and observe that in many cases one can sparsify the solution by up to two orders of magnitude without any considerable loss of performance, even though the GMM has no underlying sparsity structure. For $\ell_\infty$ regularization we show that, for large enough regularization strength, the optimal weights concentrate around two values of opposite sign. We observe that in many cases the corresponding "compression" of each weight to a single bit leads to very little loss in performance. These latter observations can have significant practical ramifications.

cs.LG↗

arXiv2023

Coded Kalman Filtering Over Gaussian Channels with Feedback

This paper investigates the problem of zero-delay joint source-channel coding of a vector Gauss-Markov source over a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel with feedback. In contrast to the classical problem of causal estimation using noisy observations, we examine a system where the source can be encoded before transmission. An encoder, equipped with feedback of past channel outputs, observes the source state and encodes the information in a causal manner as inputs to the channel while adhering to a power constraint. The objective of the code is to estimate the source state with minimum mean square error at the infinite horizon. This work shows a fundamental theorem for two scenarios: for the transmission of an unstable vector Gauss-Markov source over either a multiple-input single-output (MISO) or a single-input multiple-output (SIMO) AWGN channel, finite estimation error is achievable if and only if the sum of logs of the unstable eigenvalues of the state gain matrix is less than the Shannon channel capacity. We prove these results by showing an optimal linear innovations encoder that can be applied to sources and channels of any dimension and analyzing it together with the corresponding Kalman filter decoder.

cs.IT↗

arXiv2023

Wasserstein Distributionally Robust Regret-Optimal Control under Partial Observability

This paper presents a framework for Wasserstein distributionally robust (DR) regret-optimal (RO) control in the context of partially observable systems. DR-RO control considers the regret in LQR cost between a causal and non-causal controller and aims to minimize the worst-case regret over all disturbances whose probability distribution is within a certain Wasserstein-2 ball of a nominal distribution. Our work builds upon the full-information DR-RO problem that was introduced and solved in Yan et al., 2023, and extends it to handle partial observability and measurement-feedback (MF). We solve the finite horizon partially observable DR-RO and show that it reduces to a tractable semi-definite program whose size is proportional to the time horizon. Through simulations, the effectiveness and performance of the framework are demonstrated, showcasing its practical relevance to real-world control systems. The proposed approach enables robust control decisions, enhances system performance in uncertain and partially observable environments, and provides resilience against measurement noise and model discrepancies.

math.OC↗

arXiv2023

Regret-Optimal LQR Control

We consider the infinite-horizon LQR control problem. Motivated by competitive analysis in online learning, as a criterion for controller design we introduce the dynamic regret, defined as the difference between the LQR cost of a causal controller (that has only access to past disturbances) and the LQR cost of the \emph{unique} clairvoyant one (that has also access to future disturbances) that is known to dominate all other controllers. The regret itself is a function of the disturbances, and we propose to find a causal controller that minimizes the worst-case regret over all bounded energy disturbances. The resulting controller has the interpretation of guaranteeing the smallest regret compared to the best non-causal controller that can see the future. We derive explicit formulas for the optimal regret and for the regret-optimal controller for the state-space setting. These explicit solutions are obtained by showing that the regret-optimal control problem can be reduced to a Nehari extension problem that can be solved explicitly. The regret-optimal controller is shown to be linear and can be expressed as the sum of the classical $H_2$ state-feedback law and an $n$-th order controller ($n$ is the state dimension), and its construction simply requires a solution to the standard LQR Riccati equation and two Lyapunov equations. Simulations over a range of plants demonstrate that the regret-optimal controller interpolates nicely between the $H_2$ and the $H_\infty$ optimal controllers, and generally has $H_2$ and $H_\infty$ costs that are simultaneously close to their optimal values. The regret-optimal controller thus presents itself as a viable option for control systems design.

math.OC↗

arXiv2023

The Generalization Error of Stochastic Mirror Descent on Over-Parametrized Linear Models

Despite being highly over-parametrized, and having the ability to fully interpolate the training data, deep networks are known to generalize well to unseen data. It is now understood that part of the reason for this is that the training algorithms used have certain implicit regularization properties that ensure interpolating solutions with "good" properties are found. This is best understood in linear over-parametrized models where it has been shown that the celebrated stochastic gradient descent (SGD) algorithm finds an interpolating solution that is closest in Euclidean distance to the initial weight vector. Different regularizers, replacing Euclidean distance with Bregman divergence, can be obtained if we replace SGD with stochastic mirror descent (SMD). Empirical observations have shown that in the deep network setting, SMD achieves a generalization performance that is different from that of SGD (and which depends on the choice of SMD's potential function. In an attempt to begin to understand this behavior, we obtain the generalization error of SMD for over-parametrized linear models for a binary classification problem where the two classes are drawn from a Gaussian mixture model. We present simulation results that validate the theory and, in particular, introduce two data models, one for which SMD with an $\ell_2$ regularizer (i.e., SGD) outperforms SMD with an $\ell_1$ regularizer, and one for which the reverse happens.

cs.LG↗

arXiv2023

Precise Asymptotic Analysis of Deep Random Feature Models

We provide exact asymptotic expressions for the performance of regression by an $L-$layer deep random feature (RF) model, where the input is mapped through multiple random embedding and non-linear activation functions. For this purpose, we establish two key steps: First, we prove a novel universality result for RF models and deterministic data, by which we demonstrate that a deep random feature model is equivalent to a deep linear Gaussian model that matches it in the first and second moments, at each layer. Second, we make use of the convex Gaussian Min-Max theorem multiple times to obtain the exact behavior of deep RF models. We further characterize the variation of the eigendistribution in different layers of the equivalent Gaussian model, demonstrating that depth has a tangible effect on model performance despite the fact that only the last layer of the model is being trained.

stat.ML↗

arXiv2023

Feedback Capacity of MIMO Gaussian Channels

Finding a computable expression for the feedback capacity of channels with colored Gaussian, additive noise is a long standing open problem. In this paper, we solve this problem in the scenario where the channel has multiple inputs and multiple outputs (MIMO) and the noise process is generated as the output of a time-invariant state-space model. Our main result is a computable expression for the feedback capacity in terms of a finite-dimensional convex optimization. The solution to the feedback capacity problem is obtained by formulating the finite-block counterpart of the capacity problem as a \emph{sequential convex optimization problem} which leads in turn to a single-letter upper bound. This converse derivation integrates tools and ideas from information theory, control, filtering and convex optimization. A tight lower bound is realized by optimizing over a family of time-invariant policies thus showing that time-invariant inputs are optimal even when the noise process may not be stationary. The optimal time-invariant policy is used to construct a capacity-achieving and simple coding scheme for scalar channels, and its analysis reveals an interesting relation between a smoothing problem and the feedback capacity expression.

cs.IT↗

arXiv2022

Stochastic Mirror Descent in Average Ensemble Models

The stochastic mirror descent (SMD) algorithm is a general class of training algorithms, which includes the celebrated stochastic gradient descent (SGD), as a special case. It utilizes a mirror potential to influence the implicit bias of the training algorithm. In this paper we explore the performance of the SMD iterates on mean-field ensemble models. Our results generalize earlier ones obtained for SGD on such models. The evolution of the distribution of parameters is mapped to a continuous time process in the space of probability distributions. Our main result gives a nonlinear partial differential equation to which the continuous time process converges in the asymptotic regime of large networks. The impact of the mirror potential appears through a multiplicative term that is equal to the inverse of its Hessian and which can be interpreted as defining a gradient flow over an appropriately defined Riemannian manifold. We provide numerical simulations which allow us to study and characterize the effect of the mirror potential on the performance of networks trained with SMD for some binary classification problems.

cs.LG↗

arXiv2022

Measurement-Feedback Control with Optimal Data-Dependent Regret

Inspired by online learning, data-dependent regret has recently been proposed as a criterion for controller design. In the regret-optimal control paradigm, causal controllers are designed to minimize regret against a hypothetical optimal noncausal controller, which selects the globally cost-minimizing sequence of control actions given noncausal access to the disturbance sequence. We extend regret-optimal control to the more challenging measurement-feedback setting, where the online controller must compete against the optimal noncausal controller without directly observing the state or the driving disturbance. We show that no measurement-feedback controller can have bounded competitive ratio or regret which is bounded by the pathlength of the measurement disturbance. We do derive, however, a controller whose regret has optimal dependence on the joint energy of the driving and measurement disturbances, and another controller whose regret has optimal dependence on the pathlength of the driving disturbance and the energy of the measurement disturbance. The key technique we introduce is a reduction from regret-optimal measurement-feedback control to $H_{\infty}$-optimal measurement-feedback control in a synthetic system. We present numerical simulations which illustrate the efficacy of our proposed control algorithms.

math.OC↗