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We prove a non-asymptotic central limit theorem for vector-valued martingale differences using Stein's method, and use Poisson's equation to extend the result to functions of Markov Chains. We then show that these results can be applied to establish a non-asymptotic central limit theorem for Temporal Difference (TD) learning with averaging.
In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.
We study a PDE-constrained optimization problem for an elliptic equation with the spectral fractional Laplacian and a linear combination of Dirac measures as the forcing term; the controls are the amplitudes of these singular sources. We prove existence and uniqueness of an optimal solution and derive first-order optimality conditions. We then propose a discretization based on finite elements. Since the set of admissible controls is finite dimensional, the control variable itself does not require discretization. We conclude by deriving a priori error bounds
We introduce new notions of controllability and observability for networked linear time-invariant (LTI) systems based on $σ$-semistability of quiver representations. Utilizing King's criterion for $σ$-semistability, we define a network generalization of the Kalman decomposition for networked LTI systems, which systematically decomposes the local and interconnection dynamics while respecting the underlying network structure. Furthermore, we present efficient algorithms for deciding the proposed controllability and observability of a given networked LTI system and for finding the Kalman-type decomposition. We also show efficient algorithms for deciding the $σ$-semistability of representations of acyclic quivers with self-loops if the weight $σ$ has the same sign for all vertices with self-loops. Such quiver representations and weights arise from networked LTI systems.
Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $Σ_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).
Enforcing stochastic state constraints over the full prediction horizon in Model Predictive Control (MPC) can be computationally demanding. Here we study stochastic MPC without terminal ingredients in which chance constraints are enforced only over a shorter constraint horizon. Using stochastic relaxed dynamic programming, we derive an explicit upper bound on the average expected closed-loop cost that depends on both prediction and constraint horizons. For linear quadratic problems with affine chance constraints and bounded uniform disturbances, we provide a deterministic reformulation via coordinate transformation and constraint tightening. Simulations illustrate the trade-off between computational effort and performance.
Solving massive-scale optimization problems requires scalable first-order methods with low per-iteration cost. This tutorial highlights a shift in optimization: using differentiable programming not only to execute algorithms but to learn how to design them. Modern frameworks such as PyTorch, TensorFlow, and JAX enable this paradigm through efficient automatic differentiation. Embedding first-order methods within these systems allows end-to-end training that improves convergence and solution quality. Guided by Fenchel-Rockafellar duality, the tutorial demonstrates how duality-informed iterative schemes such as the alternating direction method of multipliers, and the primal-dual hybrid gradient can be learned and adapted through representative case studies.
We determine the sharp restricted isometry threshold for recovery at global minima of the rank-restricted matrix LASSO. For target rank $r_{\star}$, if the rank-$k$ RIP constant satisfies $δ<δ_{\mathrm{sharp}}(k/r_{\star})$, where $δ_{\mathrm{sharp}}(t)=t/(4-t)$ for $0<t<4/3$ and $δ_{\mathrm{sharp}}(t)=\sqrt{(t-1)/t}$ for $t\ge4/3$, then every global minimizer has Frobenius error $\lesssim\sqrt{r_{\star}}λ$ for all $λ\gtrsim\|\mathcal{A}^{*}(ξ)\|_{\mathrm{op}}$ and at every search rank $r\ge r_{\star}$. The constants depend only on the RIP constant and $t=k/r_{\star}$, and in particular are independent of the search rank. When the rank restriction is inactive, the result specializes to the ordinary convex matrix LASSO. We also obtain the analogous results for sparsity-restricted vector LASSO. Conversely, we show that the threshold $δ<δ_{\mathrm{sharp}}(k/r_{\star})$ cannot be improved, due to the existence of counterexamples whose global minimizers fail to recover the ground truth.
We study the mixing time of weighted Dikin walks for sampling from exponential distributions on polytopes and truncated positive-semidefinite (PSD) cones. Our first result gives a general total-variation mixing bound under strong self-concordance, $\barν$-symmetry, and mixed-trace regularity on the local metric. The key idea is to control the Metropolis--Hastings acceptance probability on a high-probability region rather than at every point. Applying this framework to the Lee--Sidford, Lewis-weight, and John metrics yields an $\widetilde O(d^2)$ mixing bound for sampling from polytopes, while applying it to a hybrid barrier yields an $\widetilde O(d^4)$ mixing bound for sampling from truncated PSD cones. Our second result establishes stronger $χ^2$-divergence guarantees and pointwise acceptance control using a new fourth-order bootstrap condition. For a suitably scaled Lee--Sidford metric, this yields an $\widetilde O(d^2)$ mixing bound in $χ^2$-divergence, improving on the previous $\widetilde O(d^{9/4})$ bound.
An operator--theoretic formulation is developed for the synthesis of parameter-dependent controllers and observers for the Vlasov--Maxwell system. The linearized dynamics are modeled as a non-autonomous evolution system whose generators depend on measurable plasma quantities. Well-posedness of the associated evolution family is established together with uniform growth bounds. Parameter-dependent Lyapunov operators yield operator differential LMIs ensuring uniform exponential stability and observer convergence. An $H_\infty$ extension provides disturbance attenuation conditions consistent with the intrinsic energy structure of the coupled Vlasov--Maxwell equations. Galerkin projections lead to finite-dimensional LMIs consistent with the operator inequalities, enabling reliable numerical synthesis while preserving the analytical structure of the original model. Numerical results on a reduced Vlasov--Maxwell benchmark confirm the predicted convergence properties.
The paper introduces a successive backstepping observer design for strictly feedforward parabolic PDE-ODE systems, in which the coupling structure determines the order of error stabilization and the corresponding transformations. First, a transformation based on a virtual measurement stabilizes the ODE observer error subsystem, which is most distal from the measurement, while decoupling it from the PDE error state. Second, a Volterra integral transformation is employed to stabilize the PDE error subsystem and to map the overall error dynamics into a cascade of exponentially stable ODE and PDE subsystems. The design is shown to be dual to a recently proposed multi-step state feedback design for parabolic PDE-ODE systems in strict feedback form, thus explaining the structure of the presented observer design.
Separable nonnegative matrix factorization (SNMF) has been widely used for low-rank representation and clustering of nonnegative data, owing to its ability to produce part-based and interpretable decompositions. In particular, SNMF is closely related to graph clustering and community detection. To enhance sparsity and identifiability of the learned factors, we propose an $\ell_1^p/\ell_2$-regularized SNMF model based on a powered ratio-of-norms regularizer. The resulting formulation is nonconvex and nonsmooth, which poses significant challenges for optimization. To address this, we develop efficient algorithms based on the difference-of-convex function algorithm (DCA) and the alternating direction method of multipliers (ADMM). The proposed methods decompose the original problem into tractable subproblems, leveraging closed-form proximal operators associated with the powered norm terms. We establish descent and limiting criticality properties for the DCA scheme and convergence under standard assumptions for the ADMM scheme. Extensive numerical experiments on synthetic datasets and hand gesture classification tasks demonstrate that the proposed approach achieves competitive or improved performance in anchor identification and classification accuracy compared with existing SNMF methods, while maintaining competitive computational efficiency.
In this paper, constrained parameter update laws for adaptive control are developed using barrier constraints. An interpretation of the parameter update law from a constrained optimization problem, in which a regularized Barrier saddle function is formulated to incorporate parameter constraints using inverse and logarithmic barrier functions from interior-point methods. The resulting constrained update law is integrated with an adaptive trajectory tracking controller, enabling online learning of the unknown system model parameters. Forward invariance of the parameter estimate is established and Lyapunov stability of the closed-loop system with the constrained parameter update law is derived. The effectiveness of the proposed constrained adaptive control law is demonstrated through simulations, which validate its ability to maintain parameter estimates within prescribed bounds while ensuring convergence to the true parameter values and achieving steady state tracking performance.
Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding ground states of these Hamiltonians. We prove that for such Hamiltonians, all local minima are global minima. Therefore, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.
We propose a numerical algorithm for computing feasible and approximately optimal solutions of the matching for teams problem. Specifically, we introduce the notion of approximate matching equilibrium as a feasible approximation of a matching equilibrium with relaxed rationality, and we show that a true equilibrium is recovered in the limit of a sequence of approximate matching equilibria with sub-optimality approaching 0. In our approximation scheme, we parametrize the so-called transfer functions, and we show that tackling the resulting parametric primal and dual optimization problems yields two approximate matching equilibria as well as provable and computable lower and upper bounds for the optimal social welfare. Under a flexible Euclidean setting, we show that the approximation error of our scheme can be controlled to be arbitrarily close to 0, we derive an explicit computational complexity bound, and we develop an algorithm for computing approximate matching equilibria that is efficient for large-scale problems involving a large number of agent populations. We study three problems in our numerical experiments: a retail business problem, the Wasserstein barycenter problem, and a large-scale problem involving up to 1000 agent populations. We show that the proposed algorithm can produce nearly optimal approximate matching equilibria to provide quantitative managerial insights for policymakers, and that the computed sub-optimality estimates are much less conservative than theoretical estimates.
We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.
This paper studies the identification of linearly parameterized nonlinear (LPN) systems. Although LPN systems share the same linear parameterization structure as linear systems, they are more challenging to identify. In particular, previous work has shown, through a counterexample based on a piecewise-affine system, that non-active exploration is generally insufficient for LPN system identification. In this paper, we consider LPN systems with real-analytic feature functions. We show that non-active exploration is sufficient for the identification of this class of systems by establishing non-asymptotic convergence rates of least-squares estimation and set-membership estimation. In addition, we provide counterexamples to show that non-active exploration may not be sufficient for system identification for non-real-analytic systems, even if those systems are infinitely differentiable. We present numerical experiments to further support and validate our theoretical results.
The uncontrolled equations of motion in the Nechvile frame for the restricted three-body problem have been well-known since at least the 1960s. It would seem that adding an external force to these equations is quite trivial: simply add an external force per mass term to the acceleration equations. Here we show that the last statement is not true. In fact, we show that the additive generic external force must be multiplied by the inverse of $(1 + e \cosθ)^3$ where $e$ is the relative eccentricity of the primaries and $θ$ is the true anomaly of the rotating frame located at the barycenter. Furthermore, when this result is combined with the mass flow rate equation, it generates several surprising results due to the mismatch between the resulting quadratic term and the cubic term in the equations of motion. This leads to a corresponding modification of the rocket equation itself. A Birkhoff-theoretic solution to an illustrative cislunar space mission problem shows propellent savings of 80% with the use of the correct cost functional. The popular quadratic cost utilizes more than $2X$ the minimum propellant consumption.