Tao Xu 徐韬

This paper develops a decision-aware learning framework for constrained linear quadratic regulator (LQR) design. It addresses the challenge of tuning unknown context-dependent cost weighting matrices by directly optimizing downstream control performance. When the prediction accuracy is poor, traditional two-stage methods, separating weight prediction and control optimization, are difficult to meet the demands of downstream high-precision tasks. Thus, we introduce the Smart Predict-then-Optimize (SPO) framework to the constrained LQR problem, which optimizes downstream control quality via direct decision regret minimization. First, we generalize the theory of the SPO framework to fit the LQR control problem. Second, we derive a convex surrogate for SPO loss, which retains interpretability via explicit weight prediction. Third, we propose an end-to-end training pipeline based on gradient-based optimization. Finally, we validate our approach via simulations, showing that our method reduces the cost ratio by approximately 34% compared to the Mean Squared Error (MSE) baseline, achieving significantly better downstream control performance.

Distributed nonconvex optimization with certification constraints is a crucial demand in multi-agent networks, especially sparse ones. However, a globally suboptimal consensus value may invalidate stability or safety requirements. To address this issue, we develop a Chebyshev-Accelerated Surrogate Optimization (CASO) pipeline for distributed optimization of univariate objectives. CASO consists of four stages: distributed power iteration, local surrogate construction, accelerated coefficient consensus and algebraic global minimization of the final surrogate. With a distributedly verifiable stopping rule, CASO provides an $\epsilon$-global optimality guarantee, and reduces communication costs across sparse networks. Simulations verify the theoretical rigor and methodology effectiveness of CASO.

Deriving analytic solutions for optimal mixed strategies in zero-sum linear-quadratic differential games (ZSLQDGs) remains an open problem. In this paper, we analytically synthesize near-optimal mixed strategies for ZSLQDGs and establish rigorous performance certifications. We construct a surrogate pure-strategy stochastic differential game (SDG) by matching the first two moments of the mixed strategies, achieving an $O(\bar{\pi}^{2})$ weak approximation of state distributions and expected costs with respect to the maximum commitment delay $\bar{\pi}$. By analytically resolving the surrogate SDG, we derive closed-form optimal control laws for the matched moments, revealing that the surrogate game is governed by a Generalized Riccati Differential Equation (GRDE) that dictates a dynamic energy allocation law for variance injection. Building on these solutions, we propose a robust dual-routing architecture to execute the near-optimal mixed strategies, and certify that both the global value approximation error and the strategy suboptimality gaps are bounded by $O(\bar{\pi}^{1/2})$. Numerical experiments on a double-integrator pursuit–evasion game illustrate the induced physical behaviors and validate the theoretical bounds.

Synthesizing near-optimal mixed strategies for zero-sum differential games (ZSDGs) has been a longstanding challenge. Existing research mainly focuses on characterizing the theoretical value function, while the practical design of executable mixed strategies remains open. We propose a novel weak approximation framework: map the original mixed-strategy game into a surrogate stochastic differential game (SDG) under pure strategies, ensuring state distributions and cost expectations closely match the original. Based on the solution of this auxiliary SDG, the game value can be approximated and near-optimal mixed strategies synthesized. To operationalize the framework, we develop a constructive control-space discretization algorithm for general ZSDGs, parameterizing the infinite-dimensional measure optimization into probability simplices solved by local linear programs. We rigorously prove the global weak approximation error is $O(\bar{\pi})$ in the maximum commitment delay $\bar{\pi}$, and derive explicit analytical upper bounds for strategy suboptimality gaps. Numerical examples validate the theoretical results.

Probabilistic prediction of stochastic dynamical systems (SDSs) aims to accurately predict the conditional probability distributions of future states. However, accurate probabilistic predictions hinge on accurate distributional information from a nominal model, which is rarely available in practice. We propose a novel functional-maximin-based distributionally robust probabilistic prediction (DRPP) framework, in which one can design probabilistic predictors with worst-case performance guarantees over a pre-defined ambiguity set of SDSs. Although DRPP requires optimizing over the space of probability measures — generally intractable — we develop a methodology that equivalently transforms the maximin problem from function spaces to Euclidean spaces. Two suboptimal solutions are derived: Noise-DRPP (relaxing constraints on the ambiguity set) and Eig-DRPP (relaxing constraints on the predictor), together with their optimality gaps against the global optimum. Numerical simulations compare the performance of the predictors under different SDSs.

To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. We propose an $\epsilon$-logarithm score that generalizes the logarithm score by considering a neighborhood of radius $\epsilon$. We characterize the probabilistic predictability of an SDS by optimizing the expected score over the space of probability measures, and show how predictability is quantitatively determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. For any predictor, we provide approximations of the expected score with error $O(\epsilon)$. We also analyze the asymptotic behaviors of the score on individual trajectories, proving convergence to the expected score for i.i.d. process noises with speed $O(T^{-1/2})$ in probability. Numerical examples elaborate the results.

Equivariant graph neural networks (Equiv-GNNs) have demonstrated effectiveness in modeling dynamics of multi-object systems by explicitly encoding symmetries. Among them, scalarization-based methods are widely adopted for their computational efficiency, particularly in comparison to high-steerable models. However, most existing scalarization-based approaches rely on empirical design of invariant functions, lacking rigorous theoretical guarantees. Moreover, these methods typically only consider directional information from object positions, neglecting that from higher-order differential components. To address these limitations, we propose a general and efficient SE(3)-equivariant graph framework with Complete Differential Invariants and Frames (CDIF). Specifically, we show how to construct a set of differential invariants to universally express any invariant functions through network layers. Additionally, we illustrate the complete recovery of directional information from the aforementioned invariants via frames that integrate both positional and differential components. Extensive experiments across diverse domains, including molecular dynamics, formation control, motion capture and particle simulation, validate that our method is simple, scalable, and outperforming state-of-the-art baselines.

The predict-then-optimize (PTO) framework addresses practical stochastic decision-making: first predict unknown parameters of an optimization model, then solve the problem using the predictions. Elmachtoub and Grigas introduced the Smart Predict-then-Optimize (SPO) loss — which gauges decision error from predicted parameters — and a convex surrogate, SPO+, incorporating the optimization model's structure. Consistency of these losses is guaranteed under i.i.d. training data, but real data are often dependent (e.g., power-load fluctuations over time), which can degrade performance. We present an autoregressive SPO method that directly targets the optimization problem at the decision stage, where consistency conditions no longer hold. We analyze generalization bounds of the SPO loss in the autoregressive model, extend uniform calibration results, and empirically demonstrate the effectiveness of SPO+ versus absolute and least-squares losses when cost vectors are determined by stationary dynamical systems.

This paper utilizes the weak approximation method to analyze differential games that involve mixed strategies. Mixed strategies can produce unique strategic behaviors that traditional models and tools in pure-strategy games cannot directly capture. Based on stochastic processes with independent increments, we define the mixed strategy without assuming knowledge of the opponents' strategy and system state. This general mixed strategy poses challenges in evaluating game payoff and game value; to overcome them, we use the weak approximation method to employ a stochastic differential game to characterize the dynamics of the mixed-strategy game. We demonstrate that the game's payoff function can be precisely approximated with an error of the same scale as the step size, and estimate the upper and lower value functions of the weak-approximated game to analyze the existence of game value. Numerical examples illustrate our findings.

In this paper, we study the problem of inferring the state-feedback cooperative control of continuous-time NDSs from noisy state observations. To practice, we first propose a causality-based estimator to obtain the discretized system matrix, adaptive to both stable and explosive state evolution cases. Then, we derive the observation period guarantees and leverage the matrix logarithm to accurately reconstruct the continuous closed-loop matrix from the discrete one, circumventing the insufficiency of conventional sampling-recovery methods in this situation (e.g., Shannon sampling theorem). Finally, we exploit the element-wise coupling relation between the local feedback gain and the unknown interaction topology, and construct a duel-level least squares method to obtain the feedback matrix. Simulations are conducted to verify the proposed inference method.

To characterize the predictability of discrete-time continuous-state stochastic dynamical systems (SDS) and use predictability to explain the probability of making accurate predictions, we propose a new performance metric — the predictability exponent. This metric quantifies the asymptotic exponential decaying rate of the probability that prediction errors never exceed a fixed $\epsilon$. Its novelties lie in an asymptotic probabilistic perspective: it minimizes the exponential decaying rate of prediction probability, holds an approximate expression deeply related to differential entropy within an error of $O(\epsilon)$, and possesses a fast converging speed from finite decaying rate of $O(e^{-K})$ in probability. Numerical examples illustrate the efficiency of the obtained results.