Abhishek Halder

Abhishek Halder, Xinbo Geng, P. R. Kumar et al.

Motivated by the growing importance of demand response in modern power system's operations, we propose an architecture and supporting algorithms for privacy preserving thermal inertial load management as a service provided by the load serving entity (LSE). We focus on an LSE managing a population of its customers' air conditioners, and propose a contractual model where the LSE guarantees quality of service to each customer in terms of keeping their indoor temperature trajectories within respective bands around the desired individual comfort temperatures. We show how the LSE can price the contracts differentiated by the flexibility embodied by the width of the specified bands. We address architectural questions of (i) how the LSE can strategize its energy procurement based on price and ambient temperature forecasts, (ii) how an LSE can close the real time control loop at the aggregate level while providing individual comfort guarantees to loads, without ever measuring the states of an air conditioner for privacy reasons. Control algorithms to enable our proposed architecture are given, and their efficacy is demonstrated on real data.

Abhishek Halder, Tryphon T. Georgiou

Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and a systematic way to formulate and solve the same for linear Gaussian systems has appeared in our previous work where the gradient flows were realized via proximal operators with respect to Wasserstein metric arising in optimal mass transport. In this paper, we derive the evolution equations as proximal operators with respect to Fisher-Rao metric arising in information geometry. We develop the linear Gaussian case in detail and show that a template two step optimization procedure proposed earlier by the authors still applies. Our objective is to provide new geometric interpretations of known equations in filtering, and to clarify the implication of different choices of metric.

Abhishek Halder, Tryphon T. Georgiou

The purpose of this work is mostly expository and aims to elucidate the Jordan-Kinderlehrer-Otto (JKO) scheme for uncertainty propagation, and a variant, the Laugesen-Mehta-Meyn-Raginsky (LMMR) scheme for filtering. We point out that these variational schemes can be understood as proximal operators in the space of density functions, realizing gradient flows. These schemes hold the promise of leading to efficient ways for solving the Fokker-Planck equation as well as the equations of non-linear filtering. Our aim in this paper is to develop in detail the underlying ideas in the setting of linear stochastic systems with Gaussian noise and recover known results.

Iman Nodozi, Jared O'Leary, Ali Mesbah et al.

We propose formulating the finite-horizon stochastic optimal control problem for colloidal self-assembly in the space of probability density functions (PDFs) of the underlying state variables (namely, order parameters). The control objective is formulated in terms of steering the state PDFs from a prescribed initial probability measure towards a prescribed terminal probability measure with minimum control effort. For specificity, we use a univariate stochastic state model from the literature. Both the analysis and the computational steps for control synthesis as developed in this paper generalize for multivariate stochastic state dynamics given by generic nonlinear in state and non-affine in control models. We derive the conditions of optimality for the associated optimal control problem. This derivation yields a system of three coupled partial differential equations together with the boundary conditions at the initial and terminal times. The resulting system is a generalized instance of the so-called Schrödinger bridge problem. We then determine the optimal control policy by training a physics-informed deep neural network, where the "physics" are the derived conditions of optimality. The performance of the proposed solution is demonstrated via numerical simulations on a benchmark colloidal self-assembly problem.

Iman Nodozi, Charlie Yan, Mira Khare et al.

We show that the minimum effort control of colloidal self-assembly can be naturally formulated in the order-parameter space as a generalized Schrödinger bridge problem -- a class of fixed-horizon stochastic optimal control problems that originated in the works of Erwin Schrödinger in the early 1930s. In recent years, this class of problems has seen a resurgence of research activities in the control and machine learning communities. Different from the existing literature on the theory and computation for such problems, the controlled drift and diffusion coefficients for colloidal self-assembly are typically nonaffine in control, and are difficult to obtain from physics-based modeling. We deduce the conditions of optimality for such generalized problems, and show that the resulting system of equations is structurally very different from the existing results in a way that standard computational approaches no longer apply. Thus motivated, we propose a data-driven learning and control framework, named `neural Schrödinger bridge', to solve such generalized Schrödinger bridge problems by innovating on recent advances in neural networks. We illustrate the effectiveness of the proposed framework using a numerical case study of colloidal self-assembly. We learn the controlled drift and diffusion coefficients as two neural networks using molecular dynamics simulation data, and then use these two to train a third network with Sinkhorn losses designed for distributional endpoint constraints, specific for this class of control problems.

Abhishek Halder

We consider the problem of computing certain parameterized minimum volume outer ellipsoidal (MVOE) approximation of the Minkowski sum of a finite number of ellipsoids. We clarify connections among several parameterizations available in the literature, obtain novel analysis results regarding the conditions of optimality, and based on the same, propose two new algorithms for computing the parameterized MVOE. Numerical results reveal faster runtime for the proposed algorithms than the state-of-the-art semidefinite programming approach of computing the same.

Abhishek Halder

We study the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the $p$-sum of finitely many ellipsoids, for any real $p\geq 1$. The notion of $p$-sum appears in the Brunn-Minkowski-Firey theory in convex analysis, and generalizes several well-known set-valued operations such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the $p$-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying $p\in[1,\infty]$. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for $p=1$, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For $p>1$, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications such as decision making for collision avoidance of autonomous vehicles, where the computational time-scale for reach set calculation needs to be much smaller than the vehicular dynamics time-scale.

Kenneth F. Caluya, Abhishek Halder

We consider the problem of steering the joint state probability density function of a static feedback linearizable control system over finite time horizon. Potential applications include controlling neuronal populations, swarm guidance, and probabilistic motion planning. Our theoretical developments reveal the structure of the minimum energy controller for the same, and can be viewed as a generalization of the Benamou-Brenier theory for dynamic optimal transport. Further analytical results are derived for solving the feasibility problem, i.e., for finding feedback that steers a given joint density function to another in fixed time, subject to the controlled nonlinear dynamics. An algorithm based on the Schrödinger bridge is proposed to approximate a feasible controller; a numerical example is worked out to illustrate the same.

Georgiy A. Bondar, Robert Gifford, Linh Thi Xuan Phan et al.

The solution of the path structured multimarginal Schrödinger bridge problem (MSBP) is the most-likely measure-valued trajectory consistent with a sequence of observed probability measures or distributional snapshots. We leverage recent algorithmic advances in solving such structured MSBPs for learning stochastic hardware resource usage by control software. The solution enables predicting the time-varying distribution of hardware resource availability at a desired time with guaranteed linear convergence. We demonstrate the efficacy of our probabilistic learning approach in a model predictive control software execution case study. The method exhibits rapid convergence to an accurate prediction of hardware resource utilization of the controller. The method can be broadly applied to any software to predict cyber-physical context-dependent performance at arbitrary time.

Alexis M. H. Teter, Yongxin Chen, Abhishek Halder

Schrödinger bridge is a stochastic optimal control problem to steer a given initial state density to another, subject to controlled diffusion and deadline constraints. A popular method to numerically solve the Schrödinger bridge problems, in both classical and in the linear system settings, is via contractive fixed point recursions. These recursions can be seen as dynamic versions of the well-known Sinkhorn iterations, and under mild assumptions, they solve the so-called Schrödinger systems with guaranteed linear convergence. In this work, we study a priori estimates for the contraction coefficients associated with the convergence of respective Schrödinger systems. We provide new geometric and control-theoretic interpretations for the same. Building on these newfound interpretations, we point out the possibility of improved computation for the worst-case contraction coefficients of linear SBPs by preconditioning the endpoint support sets.

Charlie Yan, Iman Nodozi, Abhishek Halder

We consider the finite horizon optimal steering of the joint state probability distribution subject to the angular velocity dynamics governed by the Euler equation. The problem and its solution amounts to controlling the spin of a rigid body via feedback, and is of practical importance, for example, in angular stabilization of a spacecraft with stochastic initial and terminal states. We clarify how this problem is an instance of the optimal mass transport (OMT) problem with bilinear prior drift. We deduce both static and dynamic versions of the Eulerian OMT, and provide analytical and numerical results for the synthesis of the optimal controller.

Alexis Teter, Iman Nodozi, Abhishek Halder

We propose a custom learning algorithm for shallow over-parameterized neural networks, i.e., networks with single hidden layer having infinite width. The infinite width of the hidden layer serves as an abstraction for the over-parameterization. Building on the recent mean field interpretations of learning dynamics in shallow neural networks, we realize mean field learning as a computational algorithm, rather than as an analytical tool. Specifically, we design a Sinkhorn regularized proximal algorithm to approximate the distributional flow for the learning dynamics over weighted point clouds. In this setting, a contractive fixed point recursion computes the time-varying weights, numerically realizing the interacting Wasserstein gradient flow of the parameter distribution supported over the neuronal ensemble. An appealing aspect of the proposed algorithm is that the measure-valued recursions allow meshless computation. We demonstrate the proposed computational framework of interacting weighted particle evolution on binary and multi-class classification. Our algorithm performs gradient descent of the free energy associated with the risk functional.

Shadi Haddad, Abhishek Halder

We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.

Alexis M. H. Teter, Wenqing Wang, Abhishek Halder

Schrödinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schrödinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schrödinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schrödinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials.

Asmaa Eldesoukey, Yongxin Chen, Abhishek Halder

The mean-field Schrödinger bridge (MFSB) problem concerns designing a minimum-effort controller that guides a diffusion process with nonlocal interaction to reach a given distribution from another by a fixed deadline. Unlike the standard Schrödinger bridge, the dynamical constraint for MFSB is the mean-field limit of a population of interacting agents with controls. It serves as a natural model for large-scale multi-agent systems. The MFSB is computationally challenging because the nonlocal interaction makes the problem nonconvex. We propose a generalization of the Hopf-Cole transform for MFSB and, building on it, design a Sinkhorn-type recursive algorithm to solve the associated system of integro-PDEs. Under mild assumptions on the interaction potential, we discuss convergence guarantees for the proposed algorithm. We present numerical examples with repulsive and attractive interactions to illustrate the theoretical contributions.

Hamza Mahmood, Abhishek Halder, Adeel Akhtar

This work studies the Schrödinger bridge problem for the kinematic equation on a compact connected Lie group. The objective is to steer a controlled diffusion between given initial and terminal densities supported over the Lie group while minimizing the control effort. We develop a coordinate-free formulation of this stochastic optimal control problem that respects the underlying geometric structure of the Lie group, thereby avoiding limitations associated with local parameterizations or embeddings in Euclidean spaces. We establish the existence and uniqueness of solution to the corresponding Schrödinger system. Our results are constructive in that they derive a geometric controller that optimally interpolates probability densities supported over the Lie group. To illustrate the results, we provide numerical examples on $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$.

Georgiy A. Bondar, Abigail Eisenklam, Yifan Cai et al.

Modern real-time systems require accurate characterization of task timing behavior to ensure predictable performance, particularly on complex hardware architectures. Existing methods, such as worst-case execution time analysis, often fail to capture the fine-grained timing behaviors of a task under varying resource contexts (e.g., an allocation of cache, memory bandwidth, and CPU frequency), which is necessary to achieve efficient resource utilization. In this paper, we introduce a novel generative profiling approach that synthesizes context-dependent, fine-grained timing profiles for real-time tasks, including those for unmeasured resource allocations. Our approach leverages a nonparametric, conditional multi-marginal Schrödinger Bridge (MSB) formulation to generate accurate execution profiles for unseen resource contexts, with maximum likelihood guarantees. We demonstrate the efficiency and effectiveness of our approach through real-world benchmarks, and showcase its practical utility in a representative case study of adaptive multicore resource allocation for real-time systems.

Tushar Sial, Abhishek Halder

This work uncovers variational principles behind symmetrizing the Bregman divergences induced by generic mirror maps over the cone of positive definite matrices. We show that computing the canonical means for this symmetrization can be posed as minimizing the desired symmetrized divergences over a set of mean functionals defined axiomatically to satisfy certain properties. For the forward symmetrization, we prove that the arithmetic mean over the primal space is canonical for any mirror map over the positive definite cone. For the reverse symmetrization, we show that the canonical mean is the arithmetic mean over the dual space, pulled back to the primal space. Applying this result to three common mirror maps used in practice, we show that the canonical means for reverse symmetrization, in those cases, turn out to be the arithmetic, log-Euclidean and harmonic means. Our results improve understanding of existing symmetrization practices in the literature, and can be seen as a navigational chart to help decide which mean to use when.

Georgiy A. Bondar, Asmaa Eldesoukey, Yongxin Chen et al.

Solutions to the Schrödinger bridge problem and its generalizations yield feedback control policies for optimal density steering over a controlled diffusion. To numerically compute the same, the dynamic Sinkhorn recursion has become a standard approach. The mathematical engine behind this approach is the Hopf-Cole transform that recasts the conditions for optimality into a system of boundary-coupled linear PDEs. Recent works pointed out that for the control-affine Schrödinger bridge problem, this exact linearity via Hopf-Cole transform, and thus the standard Sinkhorn recursion, apply only if the control and noise channels are proportional. When the channels do not match, the Hopf-Cole-transformed PDEs remain nonlinear, and no algorithm is available to solve the same. We advance the state-of-the-art by designing a Sinkhorn recursion with memory that leverages the structure of these nonlinear PDEs, and demonstrate how it solves the control-affine Schrödinger bridge problem with input and noise channel mismatch. We prove the local stability of the proposed algorithm.

Alexis M. H. Teter, Iman Nodozi, Abhishek Halder

The Lambert problem originated in orbital mechanics. It concerns with determining the initial velocity for a boundary value problem involving the dynamical constraint due to gravitational potential with additional time horizon and endpoint position constraints. Its solution has application in transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schrödinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is a generalized SBP, and can be solved numerically using the so-called Schrödinger factors, as we do in this work. Our analysis leads to solving a system of reaction-diffusion PDEs where the gravitational potential appears as the reaction rate.

Alexis M. H. Teter, Wenqing Wang, Sachin Shivakumar et al.

For a controllable linear time-varying (LTV) pair $(\boldsymbol{A}_t,\boldsymbol{B}_t)$ and $\boldsymbol{Q}_{t}$ positive semidefinite, we derive the Markov kernel for the Itô diffusion ${\mathrm{d}}\boldsymbol{x}_{t}=\boldsymbol{A}_{t}\boldsymbol{x}_t {\mathrm{d}} t + \sqrt{2}\boldsymbol{B}_{t}{\mathrm{d}}\boldsymbol{w}_{t}$ with an accompanying killing of probability mass at rate $\frac{1}{2}\boldsymbol{x}^{\top}\boldsymbol{Q}_{t}\boldsymbol{x}$. This Markov kernel is the Green's function for an associated linear reaction-advection-diffusion partial differential equation. Our result generalizes the recently derived kernel for the special case $\left(\boldsymbol{A}_t,\boldsymbol{B}_t\right)=\left(\boldsymbol{0},\boldsymbol{I}\right)$, and depends on the solution of an associated Riccati matrix ODE. A consequence of this result is that the linear quadratic non-Gaussian Schrödinger bridge is exactly solvable. This means that the problem of steering a controlled LTV diffusion from a given non-Gaussian distribution to another over a fixed deadline while minimizing an expected quadratic cost can be solved using dynamic Sinkhorn recursions performed with the derived kernel. Our derivation for the $\left(\boldsymbol{A}_t,\boldsymbol{B}_t,\boldsymbol{Q}_t\right)$-parametrized kernel pursues a new idea that relies on finding a state-time dependent distance-like functional given by the solution of a deterministic optimal control problem. This technique breaks away from existing methods, such as generalizing Hermite polynomials or Weyl calculus, which have seen limited success in the reaction-diffusion context. Our technique uncovers a new connection between Markov kernels, distances, and optimal control. This connection is of interest beyond its immediate application in solving the linear quadratic Schrödinger bridge problem.

Alexis Teter, Abhishek Halder

The purpose of this note is to clarify the importance of the relation $\boldsymbol{gg}^{\top}\propto \boldsymbol{σσ}^{\top}$ in solving control-affine Schrödinger bridge problems via the Hopf-Cole transform, where $\boldsymbol{g},\boldsymbolσ$ are the control and noise coefficients, respectively. We show that the Hopf-Cole transform applied to the conditions of optimality for generic control-affine Schrödinger bridge problems, i.e., without the assumption $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, gives a pair of forward-backward PDEs that are neither linear nor equation-level decoupled. We explain how the resulting PDEs can be interpreted as nonlinear forward-backward advection-diffusion-reaction equations, where the nonlinearity stem from additional drift and reaction terms involving the gradient of the log-likelihood a.k.a. the score. These additional drift and reaction vanish when $\boldsymbol{gg}^{\top}\propto\boldsymbol{σσ}^{\top}$, and the resulting boundary-coupled system of linear PDEs can then be solved by dynamic Sinkhorn recursions. A key takeaway of our work is that the numerical solution of the generic control-affine Schrödinger bridge requires further algorithmic development, possibly generalizing the dynamic Sinkhorn recursion or otherwise.

Iman Nodozi, Djordje Gligorijevic, Abhishek Halder

This work proposes a bid shading strategy for first-price auctions as a measure-valued optimization problem. We consider a standard parametric form for bid shading and formulate the problem as convex optimization over the joint distribution of shading parameters. After each auction, the shading parameter distribution is adapted via a regularized Wasserstein-proximal update with a data-driven energy functional. This energy functional is conditional on the context, i.e., on publisher/user attributes such as domain, ad slot type, device, or location. The proposed algorithm encourages the bid distribution to place more weight on values with higher expected surplus, i.e., where the win probability and the value gap are both large. We show that the resulting measure-valued convex optimization problem admits a closed form solution. A numerical example illustrates the proposed method.

Georgiy A. Bondar, Abhishek Halder

The Multimarginal Schrödinger Bridge (MSB) finds the optimal coupling among a collection of random vectors with known statistics and a known correlation structure. In the MSB formulation, this correlation structure is specified \emph{a priori} as an undirected connected graph with measure-valued vertices. In this work, we formulate and solve the problem of finding the optimal MSB in the sense we seek the optimal coupling over all possible graph structures. We find that computing the optimal MSB amounts to solving the minimum spanning tree problem over measure-valued vertices. We show that the resulting problem can be solved in two steps. The first step constructs a complete graph with edge weight equal to a sum of the optimal value of the corresponding bimarginal SB and the entropies of the endpoints. The second step solves a standard minimum spanning tree problem over that complete weighted graph. Numerical experiments illustrate the proposed solution.

Wenqing Wang, Alexis M. H. Teter, Murat Arcak et al.

Given a controlled diffusion and a connected, bounded, Lipschitz set, when is it possible to guarantee controlled set invariance with probability one? In this work, we answer this question by deriving the necessary and sufficient conditions for the same in terms of gradients of certain log-likelihoods -- a.k.a. score vector fields -- for two cases: given finite time horizon and infinite time horizon. The deduced conditions comprise a score-based test that provably certifies or falsifies the existence of Markovian controllers for given controlled set invariance problem data. Our results are constructive in the sense when the problem data passes the proposed test, we characterize all controllers guaranteeing the desired set invariance. When the problem data fails the proposed test, there does not exist a controller that can accomplish the desired set invariance with probability one. The computation in the proposed tests involve solving certain Dirichlet boundary value problems, and in the finite horizon case, can also account for additional constraint of hitting a target subset at the terminal time. We illustrate the results using several semi-analytical and numerical examples.

Karthik Elamvazhuthi, Abhishek Halder

We revisit the optimal transport problem over angular velocity dynamics given by the controlled Euler equation. The solution of this problem enables stochastic guidance of spin states of a rigid body (e.g., spacecraft) over a hard deadline constraint by transferring a given initial state statistics to a desired terminal state statistics. This is an instance of generalized optimal transport over a nonlinear dynamical system. While prior work has reported existence-uniqueness and numerical solution of this dynamical optimal transport problem, here we present structural results about the equivalent Kantorovich a.k.a. optimal coupling formulation. Specifically, we focus on deriving the ground cost for the associated Kantorovich optimal coupling formulation. The ground cost is equal to the cost of transporting unit amount of mass from a specific realization of the initial or source joint probability measure to a realization of the terminal or target joint probability measure, and determines the Kantorovich formulation. Finding the ground cost leads to solving a structured deterministic nonlinear optimal control problem, which is shown to be amenable to an analysis technique pioneered by Athans et al. We show that such techniques have broader applicability in determining the ground cost (thus Kantorovich formulation) for a class of generalized optimal mass transport problems involving nonlinear dynamics with translated norm-invariant drift.

Sachin Shivakumar, Georgiy A. Bondar, Gabriel Khan et al.

For a given ground cost, approximating the Monge optimal transport map that pushes forward a given probability measure onto another has become a staple in several modern machine learning algorithms. The fourth-order Ma-Trudinger-Wang (MTW) tensor associated with this ground cost function provides a notion of curvature in optimal transport. The non-negativity of this tensor plays a crucial role for establishing continuity for the Monge optimal transport map. It is, however, generally difficult to analytically verify this condition for any given ground cost. To expand the class of cost functions for which MTW non-negativity can be verified, we propose a provably correct computational approach which provides certificates of non-negativity for the MTW tensor using Sum-of-Squares (SOS) programming. We further show that our SOS technique can also be used to compute an inner approximation of the region where MTW non-negativity holds. We apply our proposed SOS programming method to several practical ground cost functions to approximate the regions of regularity of their corresponding optimal transport maps.

Georgiy A. Bondar, Robert Gifford, Linh Thi Xuan Phan et al.

We propose to learn the time-varying stochastic computational resource usage of software as a graph structured Schrödinger bridge problem. In general, learning the computational resource usage from data is challenging because resources such as the number of CPU instructions and the number of last level cache requests are both time-varying and statistically correlated. Our proposed method enables learning the joint time-varying stochasticity in computational resource usage from the measured profile snapshots in a nonparametric manner. The method can be used to predict the most-likely time-varying distribution of computational resource availability at a desired time. We provide detailed algorithms for stochastic learning in both single and multi-core cases, discuss the convergence guarantees, computational complexities, and demonstrate their practical use in two case studies: a single-core nonlinear model predictive controller, and a synthetic multi-core software.

Alexis M. H. Teter, Wenqing Wang, Abhishek Halder

Schrödinger bridge is a diffusion process that steers a given distribution to another in a prescribed time while minimizing the effort to do so. It can be seen as the stochastic dynamical version of the optimal mass transport, and has growing applications in generative diffusion models and stochastic optimal control. {\black{We say a Schrödinger bridge is ``exactly solvable'' if the associated uncontrolled Markov kernel is available in closed form, since then the bridge can be numerically computed using dynamic Sinkhorn recursion for arbitrary endpoint distributions with finite second moments.}} In this work, we propose a regularized variant of the Schrödinger bridge with a quadratic state cost-to-go that incentivizes the optimal sample paths to stay close to a nominal level. Unlike the conventional Schrödinger bridge, the regularization induces a state-dependent rate of killing and creation of probability mass, and its solution requires determining the Markov kernel of a reaction-diffusion partial differential equation. We derive this Markov kernel in closed form, {\black{showing that the regularized Schrödinger bridge is exactly solvable, even for non-Gaussian endpoints. This advances the state-of-the-art because closed form Markov kernel for the regularized Schrödinger bridge is available in existing literature only for Gaussian endpoints}}. Our solution recovers the heat kernel in the vanishing regularization (i.e., diffusion without reaction) limit, thereby recovering the solution of the conventional Schrödinger bridge {\black{as a special case}}. We deduce properties of the new kernel and explain its connections with certain exactly solvable models in quantum mechanics.

Iman Nodozi, Abhishek Halder

We propose a distributed nonparametric algorithm for solving measure-valued optimization problems with additive objectives. Such problems arise in several contexts in stochastic learning and control including Langevin sampling from an unnormalized prior, mean field neural network learning and Wasserstein gradient flows. The proposed algorithm comprises a two-layer alternating direction method of multipliers (ADMM). The outer-layer ADMM generalizes the Euclidean consensus ADMM to the Wasserstein consensus ADMM, and to its entropy-regularized version Sinkhorn consensus ADMM. The inner-layer ADMM turns out to be a specific instance of the standard Euclidean ADMM. The overall algorithm realizes operator splitting for gradient flows in the manifold of probability measures.

Isin M. Balci, Abhishek Halder, Efstathios Bakolas

In this work, we analyze the properties of the solution to the covariance steering problem for discrete time Gaussian linear systems with a squared Wasserstein distance terminal cost. In our previous work, we have shown that by utilizing the state feedback control policy parametrization, this stochastic optimal control problem can be associated with a difference of convex functions program. Here, we revisit the same covariance control problem but this time we focus on the analysis of the problem. Specifically, we establish the existence of solutions to the optimization problem and derive the first and second order conditions for optimality. We provide analytic expressions for the gradient and the Hessian of the performance index by utilizing specialized tools from matrix calculus. Subsequently, we prove that the optimization problem always admits a global minimizer, and finally, we provide a sufficient condition for the performance index to be a strictly convex function (under the latter condition, the problem admits a unique global minimizer). In particular, we show that when the terminal state covariance is upper bounded, with respect to the Löwner partial order, by the covariance matrix of the desired terminal normal distribution, then our problem admits a unique global minimizing state feedback gain. The results of this paper set the stage for the development of specialized control design tools that exploit the structure of the solution to the covariance steering problem with a squared Wasserstein distance terminal cost.

Walid Krichene, Kenneth F. Caluya, Abhishek Halder

Recent results have shown that for two-layer fully connected neural networks, gradient flow converges to a global optimum in the infinite width limit, by making a connection between the mean field dynamics and the Wasserstein gradient flow. These results were derived for first-order gradient flow, and a natural question is whether second-order dynamics, i.e., dynamics with momentum, exhibit a similar guarantee. We show that the answer is positive for the heavy ball method. In this case, the resulting integro-PDE is a nonlinear kinetic Fokker Planck equation, and unlike the first-order case, it has no apparent connection with the Wasserstein gradient flow. Instead, we study the variations of a Lyapunov functional along the solution trajectories to characterize the stationary points and to prove convergence. While our results are asymptotic in the mean field limit, numerical simulations indicate that global convergence may already occur for reasonably small networks.

Kenneth F. Caluya, Abhishek Halder

How to steer a given joint state probability density function to another over finite horizon subject to a controlled stochastic dynamics with hard state (sample path) constraints? In applications, state constraints may encode safety requirements such as obstacle avoidance. In this paper, we perform the feedback synthesis for minimum control effort density steering (a.k.a. Schrödinger bridge) problem subject to state constraints. We extend the theory of Schrödinger bridges to account the reflecting boundary conditions for the sample paths, and provide a computational framework building on our previous work on proximal recursions, to solve the same.

Abhishek Halder, Kenneth F. Caluya, Bertrand Travacca et al.

We provide gradient flow interpretations for the continuous-time continuous-state Hopfield neural network (HNN). The ordinary and stochastic differential equations associated with the HNN were introduced in the literature as analog optimizers, and were reported to exhibit good performance in numerical experiments. In this work, we point out that the deterministic HNN can be transcribed into Amari's natural gradient descent, and thereby uncover the explicit relation between the underlying Riemannian metric and the activation functions. By exploiting an equivalence between the natural gradient descent and the mirror descent, we show how the choice of activation function governs the geometry of the HNN dynamics. For the stochastic HNN, we show that the so-called "diffusion machine", while not a gradient flow itself, induces a gradient flow when lifted in the space of probability measures. We characterize this infinite dimensional flow as the gradient descent of certain free energy with respect to a Wasserstein metric that depends on the geodesic distance on the ground manifold. Furthermore, we demonstrate how this gradient flow interpretation can be used for fast computation via recently developed proximal algorithms.

Kenneth F. Caluya, Abhishek Halder

We develop a new computational framework to solve the partial differential equations (PDEs) governing the flow of the joint probability density functions (PDFs) in continuous-time stochastic nonlinear systems. The need for computing the transient joint PDFs subject to prior dynamics arises in uncertainty propagation, nonlinear filtering and stochastic control. Our methodology breaks away from the traditional approach of spatial discretization or function approximation -- both of which, in general, suffer from the "curse-of-dimensionality". In the proposed framework, we discretize time but not the state space. We solve infinite dimensional proximal recursions in the manifold of joint PDFs, which in the small time-step limit, is theoretically equivalent to solving the underlying transport PDEs. The resulting computation has the geometric interpretation of gradient flow of certain free energy functional with respect to the Wasserstein metric arising from the theory of optimal mass transport. We show that dualization along with an entropic regularization, leads to a cone-preserving fixed point recursion that is proved to be contractive in Thompson metric. A block co-ordinate iteration scheme is proposed to solve the resulting nonlinear recursions with guaranteed convergence. This approach enables remarkably fast computation for non-parametric transient joint PDF propagation. Numerical examples and various extensions are provided to illustrate the scope and efficacy of the proposed approach.

Abhishek Halder

We provide a variational interpretation of the DeGroot-Friedkin map in opinion dynamics. Specifically, we show that the nonlinear dynamics for the DeGroot-Friedkin map can be viewed as mirror descent on the standard simplex with the associated Bregman divergence being equal to the generalized Kullback-Leibler divergence, i.e., an entropic mirror descent. Our results reveal that the DeGroot-Friedkin map elicits an individual's social power to be close to her social influence while minimizing the so called "extropy" -- the entropy of the complimentary opinion.

Zahra Askarzadeh, Rui Fu, Abhishek Halder et al.

We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as {\em nonlinear Markov chains}. In this paper we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the $\ell_1$-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear, respectively--both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.