control system developers

Abstract: We present a generic actor-critic method that combines a ridge-regularized pathwise least-squares temporal-difference (LSTD) critic with importance-corrected momentum (ICM) actors. The critic solves a ridge-regularized system per frame by Newton-Schulz inversion and needs no stepsize, no knowledge of the mixing time, and no rank condition. Averaging stochastic policy gradients over frames of $T$ consecutive transitions, the ICM actors incur a frame-dependent gradient bias of order $\frac{1}{T}$. To our knowledge, we establish the first finite-time bounds for ICM actor-critic methods with normalized and unnormalized steps along a single Markovian trajectory. Up to a stationary-to-discounted distribution mismatch, the averaged gradient norm over $K$ updates is bounded by ${\cal O}(K^{-1/3}+\frac{1}{T})$ for normalized steps and by ${\cal O}(K^{-1/4}+\frac{1}{T})$ for unnormalized steps. We specialize these bounds to Muon, normalized stochastic gradient descent (SGD) with momentum, SGD with momentum, and AdaNorm, and compare them in the nuclear norm. With a dimension-free stepsize, Muon confines the factor $\min\{m,n\}^{1/2}$ associated with an $m\times n$ actor parameter matrix to the noise and correction terms. The comparison also characterizes when orthogonalization guarantees a larger noiseless ascent than normalization: exactly when the gradient's nuclear rank exceeds the spectral-to-Frobenius curvature ratio.

Abstract: Linear stochastic approximation (SA) is widely used to estimate fixed points from noisy observations, but most finite-time analyses assume that the underlying operator remains fixed. We study linear SA with time-varying affine operators, where the fixed points change over time and the observation noise may be correlated across rounds. For constant-stepsize updates, we establish a bound on the cumulative expected squared distance between the iterates and the corresponding fixed points. The bound accounts for the initial error, observation noise, and changes in the fixed points over time. We also establish lower bounds for scalar Gaussian models, showing that, under specified conditions, a suitably chosen constant stepsize achieves the smallest possible worst-case cumulative error up to constant factors, even compared with estimators that use future observations. These results further quantify the effect of temporal correlation on estimation accuracy. Since choosing an appropriate stepsize generally requires knowledge of the noise level and how much the fixed points change, we develop an adaptive method that uses an additional bounded, unbiased observation of the fixed point to achieve an almost optimal algorithm without needing the knowledge of the noise level and the fixed point variation over time. Finally, we apply our results to value-function estimation using synchronous tabular temporal-difference learning in changing environments, assuming generative-model access and a common invariant distribution.

Abstract: In this work, we develop a zeroth-order (ZO) method that estimates the gradient with minimal bias and variance. Instead of the random directions often used in the ZO literature, we consider central differences with directions given by a fixed spherical $t$-design, a finite point set that integrates every polynomial of degree at most $t$ exactly. The oracle is deterministic and thus does not induce any variance.We also propose a general finite difference method in the higher-order-smoothness regime and show the bias of both finite difference methods. We then show the convergence of the proposed ZO method via the input-to-state stability concept. Numerical experiments show the superiority of our proposed method over the existing random ZO methods.

Abstract: Stochastic optimization (SO) plays a central role in decision-making under uncertainty. Among SO problems, time-varying stochastic optimization (TV-SO) is particularly important due to its applications in adaptive control and machine learning. Non-parametric approaches have been proposed for time-varying deterministic optimization; however, they have not been developed for their stochastic counterparts. This work addresses that gap by developing a stochastic variational framework based on Malliavin calculus. This framework yields non-parametric optimality conditions for SO problems with stochastic decisions and supports the design of a scalable deep-learning algorithm that is insensitive to the parameterization dimension. This algorithm, called the Stochastic Path Follower (SPF), is applied to two important problems under distribution drift, namely least-squares recovery and logistic regression. Experimental results show that the proposed approach outperforms state-of-the-art learning-based and gradient-based methods in both performance and scalability.

Abstract: We develop a comprehensive theoretical analysis of Matrix Shampoo for tracking time-varying optima in nonstationary stochastic optimization. We derive finite-horizon high-probability bounds for average tracking error that disentangle initialization, optimizer drift, momentum-induced drift penalty, stochastic noise, and adaptive preconditioning, with explicit dependence on momentum parameter \(β_1\) and stabilization parameter \(\varepsilon\). These bounds reveal an explicit drift--noise and stability--geometry tradeoffs: noise-dominated regimes favor larger $β_1$, which provides stronger first-moment averaging and greater suppression of gradient noise, whereas drift-dominated regimes favor smaller $β_1$, allowing faster adaptation to moving optimizer. Larger $\varepsilon$ reduces the preconditioner-perturbation term while weakening Shampoo's ability to exploit matrix geometry, thereby reducing adaptivity. We establish a matching minimax lower bound under squared path-variation constraint, proving the information-theoretic optimality of Shampoo over the class of causal optimization procedures. Under a constant stepsize, we establish high-probability bounds for the average projected stationarity gap along general nonstationary nonconvex objectives. We validate our theoretical findings on both synthetic and multiple real-world problems.

Abstract: Existing cumulative regret guarantees for Bayesian optimization (BO) do not necessarily characterize the quality of the algorithm's final output. This gap is critical under noisy evaluations, where the goal is to return a single high-quality recommendation. To address it, we introduce efficient exploration strategies that can be incorporated into any BO procedure by occasionally replacing the acquisition query with an exploration query. To validate this approach, we bound the Gaussian process posterior standard deviation over the search space via the fill distance of the queries, which measures the largest region left unexplored. We then use this result to derive upper bounds on the inference regret, i.e., the gap between the global optimum and the function value at the algorithm's returned point. Empirically, these exploration strategies improve the final recommendations of existing BO algorithms on synthetic benchmarks, hyperparameter optimization, and a control task.

Abstract: To solve a family of large linear systems where the system matrix depends on the family parameter quadratically, we propose a new solution approach. Our method employs the derivatives of solutions with respect to the parameter. It turns out that the accumulated solution derivatives of increasing order, evaluated at a single specially chosen parameter value, provide successively better approximations to solution subspaces for the whole linear system family. We show that the derivatives can be computed by a recurrence, involving, at each step, solving a linear system with the same matrix. Constructing an orthonormal basis for the subspace spanned by the derivatives then leads to an efficient solver where just a single (sparse) LU factorization (or a single preconditioner setup in an iterative scheme) suffices to solve the whole linear system family. We call the new method ODB (orthonormal derivative basis method). As shown in numerical tests, compared to conventional solution methods, the ODB method allows to drastically reduce computational costs.

Abstract: The Lindblad master equation is the standard framework for describing the non-unitary evolution of open quantum systems, where environmental interactions induce dissipation and decoherence. When both the system Hamiltonian and the dissipation rates are partially unknown or explicitly time-dependent, traditional analytical inversion and system-identification techniques become intractable. Recent works have demonstrated that Transformer-based models can infer unknown dissipation rates from observable time series, yet these approaches typically rely on hand-crafted statistical features under idealized and highly restricted conditions. Here we advance the paradigm by introducing a raw time-series Transformer that directly ingests the full trajectories of Pauli expectation values $\langleσ_x(t)\rangle$, $\langleσ_y(t)\rangle$, and $\langleσ_z(t)\rangle$, thereby fully exploiting the self-attention mechanism for temporal modeling. The architecture is further extended to jointly learn unknown Hamiltonian parameters, handle multiple dissipation channels, and operate robustly under realistic measurement noise. Across all tested scenarios the model achieves consistently high reconstruction accuracy while eliminating manual feature engineering. This provides a scalable, robust, and versatile framework for quantum environment sensing in realistic open quantum systems.

Abstract: We identify and study a structural mechanism for Markovian nonconvex ADMM in reinforcement learning. Using finite discounted MDPs as a canonical proving ground, we show that the discounted Bellman resolvent $(I-γP_π)^{-1}$ can provide the multiplier stability that classical nonconvex ADMM analyses often obtain from a designated smooth block. Starting from this mechanism, we establish convergence under controlled Markov sampling and then under stochastic observations using an empirical Bellman surrogate that jointly represents the random residual and its Jacobian. Markov mixing, initialization drift, observation noise, and decaying bias enter as one operator perturbation, avoiding unbiased product and double sampling requirements. When the perturbations are square summable, the true KKT residual converges almost surely to zero. Under a finite conditional fourth moment condition, a companion iterate satisfies $ \mathbb{E}[\widetilde G_{K+1}] \le A/T+(B/T)\sum_{k<T}m_k^{-1}, $ which becomes $O(T^{-1}+T/N)$ for total Markov sample budget $N$, giving $O(ε^{-1})$ iteration complexity and $O(ε^{-2})$ sample complexity for squared KKT accuracy $ε$. Beyond stationarity, discounted occupancy coverage yields $J^\star-J(π)=O(\sqrt G)$ for direct tabular policies, so covered exact KKT points are globally optimal, while a statewise quadratic Bellman-improvement condition sharpens the relation to $O(G)$. Finally, nonlinear policy, projected Bellman, and explicit occupancy formulations exhibit the same chain of operator invertibility, dual representation, and multiplier stability. This supports discounted operator invertibility as a reusable structural principle for primal-dual reinforcement learning.

Abstract: Differentiable optimization brings the structural guarantees of mathematical optimization to network pipelines, allowing them to be trained end-to-end. However, its application remains challenging for nonconvex constrained problems, as existing differentiable solvers often suffer from limited modeling expressiveness due to their reliance on specialized problem structures, while also incurring substantial computation time and memory overhead in both the forward and backward passes. To address these challenges, we propose lapanda, a matrix-free differentiable solver for nonconvex optimization with general constraints. It reformulates the problem to a sequence of augmented Lagrangian subproblems, each handled by a first-order inner solver through a proximal averaged quasi-Newton algorithm with adaptive linesearch, thus enabling efficient forward optimization. We establish local well-posedness of the solution map and convergence of the outer iterations, and further derive a sensitivity alignment between the original problem and the final subproblem in the backward pass, demonstrating that the subproblem sensitivity, which can be computed efficiently in a matrix-free manner, provides a principled approximation to the exact optimizer sensitivity. We evaluate lapanda on nonconvex constrained Rosenbrock benchmarks, imitation learning with several representative constrained optimal control problems, and embedded robotic obstacle-avoidance tasks. Compared with state-of-the-art differentiable solvers, lapanda delivers substantial reductions in computation time and memory footprint while maintaining reliable constraint satisfaction and learning performance.

Abstract: Derivative-informed training improves neural operators by directly supervising their input-output sensitivities, which is crucial when neural operators are used as differentiable surrogates for inverse problems, PDE-constrained optimization, design, and control. However, existing methods rely on offline-generated derivative labels, making data generation slow, storage-intensive, and difficult to adapt across datasets, resolutions, or perturbation bases. We propose sketched tangent consistency loss (sTCL), an on-the-fly derivative-informed training objective that, for PDEs with a known and differentiable residual, enforces sensitivity consistency directly from the governing equation without offline tangent labels or neural-operator architecture changes. sTCL uses randomly sketched input perturbations to provide a lightweight derivative-level physics constraint during training. However, raw forward-sensitivity residual penalties can fail for stiff, ill-conditioned, indefinite, or coupled saddle-point tangent operators. To address this, we introduce lightweight operator-aware loss-conditioning mechanisms selected by a simple tangent-operator decision rule. Across Helmholtz, nonlinear diffusion-reaction, Burgers, Allen-Cahn, and Navier-Stokes, with the same neural-operator backbone for all methods, the PDE-specific sTCL losses achieve solution and Jacobian accuracy comparable to offline derivative-informed training (DIFNO) while eliminating the offline derivative-data generation and storage pipeline. These results show that on-the-fly derivative-informed training need not merely amortize offline tangent-solve cost into training; with appropriate sketching and loss design, sTCL provides an effective drop-in path to derivative-informed neural operators. Code is available at https://github.com/yang9579/Derivative-informed-traning-on-the-fly.

Abstract: Neural ODEs learn dynamics from trajectory losses, but their adjoint gradients lack the regressor-times-parameter-error structure on which Lyapunov analyses of online adaptation rest, so training on streaming data comes without stability guarantees. We show that this structure is in fact present: the adjoint gradient decomposes exactly into a positive semi-definite trajectory operator acting on the parameter error plus a nonlinear perturbation with explicit, horizon-dependent bounds. A quadratic Lyapunov function then certifies online Neural ODE training over sliding windows under computable gain and horizon conditions, and the same certificate extends to stored data: its drift branch recovers concurrent learning, and its trajectory branch yields NODE-CL, a stored-segment Gauss-Newton method built on batched forward sensitivities that needs no state-derivative estimates. On four DeepMind Control Suite domains, NODE-CL attains the lowest median prediction error on three under velocity measurement noise, where observer-based concurrent learning degrades by up to 8x; with clean measurements it is best on the pendulum and within a factor of 1.6 of the best stored-data baseline on the cartpole and reacher.

Abstract: Modern neural predictors can model highly nonlinear maps, but many scientific and engineering tasks require reasoning in the opposite direction: given a performance or safety level, the goal is to characterize the preimage, that is, the complete set of inputs which meet the desired target level and optimize over that set. For expressive neural predictors, however, such preimages typically have no explicit representation and are expensive to recover or optimize over. This creates a fundamental three-way challenge between expressive forward prediction, accurate preimage approximation, and tractable optimization over the preimage for downstream tasks. We introduce TRIO (tractable representations for preimage learning and inverse optimization), a framework for learning representations that make these objectives compatible by construction. Our key contribution is a preimage factorization: the forward model remains expressive through nonlinear radial transformations (including neural networks), while, under inversion, each transformation reduces to a single scalar radius, which yields simple geometric level sets. This yields an explicit geometric representation that is reusable for downstream optimization over the preimage, and, for linear objectives, we show that this admits a closed-form global solution. We finally prove a universal approximation theorem which shows that TRIO can approximate any continuous forward map and its entire family of potentially disconnected, nonconvex preimages arbitrarily well. Hence, TRIO combines expressive forward modeling, exact preimage recovery, and tractable global downstream optimization over preimages by design.

Abstract: Many real-world optimization problems rely on expensive simulations or experiments, making the efficient use of available data essential. Multi-fidelity optimization of high-dimensional black-box functions subject to black-box constraints is increasingly relevant as the cost of objective evaluations continues to rise in applications such as machine learning, engineering, and control. To our knowledge, no existing method simultaneously addresses high-dimensionality, black-box constraints, an arbitrary number of fidelity levels, and non-nested sampling. In this work, we extend the Scalable Constrained Bayesian Optimization method to the multi-fidelity setting, resulting in the MF-SCBO method. The proposed approach is evaluated on standard benchmark functions as well as challenging problems. The experimental results demonstrate that MF-SCBO generally achieves better convergence than both the single-fidelity SCBO and the other multi-fidelity method considered in this high-dimensional and constrained settings.

Abstract: Optimization problems (OPs) are key to solving many challenging research problems in robotics. However, reproducibility still remains a major issue. In this paper, we present Amplify, a lightweight nonlinear programming library aimed at reproducible results of robotic-related trajectory optimization problems. The minimalistic requirements for the 537-line library (80 characters per line) are an Internet connection, familiarity with the AMPL modeling language, and a text editor. Our primary contribution is the formulation of a library where trajectory optimization algorithms are represented directly within the optimization model. Specifically, we implement the algorithms used to compute the dynamics, trajectories, and reference motions as constraints of the OP in a declarative programming paradigm. We outline how our formulation of objectives, decisions variables, and constraints can be implemented in other transcription libraries that want to be lightweight and reproducible. We also compare the Amplify framework with 3 other libraries across examples of benchmark optimization problems across several fields, including bipedal locomotion and grasp planning.

Abstract: Muon updates matrix-valued neural-network parameters by orthogonalizing a gradient-based momentum matrix. Its reliance on derivatives limits its use when gradients are unavailable or unreliable. We develop a derivative-free framework that constructs Muon-style updates from structured finite differences. Four variants are considered: full entrywise recovery, random low-rank surrogates, basis-aligned rank-one probing, and direct structured search. Exhaustive basis-aligned probing is equivalent, up to positive scaling before ideal polar orthogonalization, to coordinate finite differences. Matrix-regression experiments show that random rank-one probing can reduce the number of function evaluations substantially, at the cost of less accurate updates. Controlled noisy-gradient experiments on regression and a neural network illustrate when accurate function values can compensate for an unreliable gradient oracle. A small CartPole study further examines orthogonal rank-one probes under a fixed episode budget. These results support structured probing as a practical option for selected black-box problems; they do not establish a general convergence guarantee or an advantage over accurate, inexpensive gradients.

Abstract: Reservoir computing (RC) trains only a linear readout over a fixed recurrent layer, making it fast and data-efficient for online prediction. However, a static reservoir degrades under system drift, readout-only adaptation is then insufficient, and unconstrained reservoir adaptation can destroy the echo-state and incremental stability properties that make RC reliable. This paper proposes LoRA-RC, which adapts the recurrent matrix through a low-rank correction driven by streaming prediction errors. The base reservoir and adaptation bases are fixed offline; a small core matrix is adapted online, projected onto a spectral-norm ball, and low-pass filtered at each step. The projection guarantees that every applied recurrent matrix remains within a certified contraction set, and an incremental input-to-state stability bound is established for the reservoir along each online adaptation path, with path-independent rate and gain. On a Lorenz system with an abrupt parameter drift, LoRA-RC cuts post-drift prediction error by 56% versus a fixed RC and 51% versus readout-only adaptation; ablations over 20 seeds show that removing the projection inflates this error by more than a factor of 40.

Abstract: Open-loop multilegged locomotion over rough terrain has been interpreted as matter transport over a noisy channel: leg-ground interactions are discrete basic active contacts, terrain deletes or perturbs those contacts, and spatial redundancy concentrates the resulting thrust and arrival time. That construction is repetition-like because every module carries the same scalar locomotion task. It consequently provides neither a positive task rate nor a decoder that changes with the surviving contact set. Here we formulate locomotion instead as a quantized finite-frame expansion with erasures. A d-dimensional body-level command is mapped into N>d heterogeneous local contact commands. Rough terrain erases or corrupts frame coefficients, while a contact-gated compliant morphology physically realizes the weighted active-subframe decoder. For a linear-Gaussian model, mechanical equilibrium is exactly the posterior mean, tangent stiffness is posterior precision, and mechanical compliance is posterior covariance. Equal-norm Parseval frames are shown to be minimax optimal against one missing contact, two-contact robustness is governed by frame coherence, and a harmonic frame gives a directly realizable gait family. For independently surviving contacts of probability q, random Gaussian gait frames admit exact reconstruction at every analog dimension rate R<q, with a binomial reliability exponent, whereas recovery of arbitrary commands is impossible for R>q. Residual contact noise yields an asymptotic per-mode amplification 1/(q-R) and a vanishing mechanical stiffness margin at the threshold. An information-locomotion inequality and an exact incremental-redundancy rule direct the next gait component toward the softest task-relevant unresolved mode. The resulting analog frame-coding theorem establishes a finite relative redundancy and converse as part of a fundamental limit theory of legged locomotion.