We present a tightened convergence analysis of clipped gradient descent on $(L_0, L_1)$-smooth functions, with quantitative constants. Building on the ideas of Koloskova et al (2023), we refactor several case disjunctions to reveal the central role of a control of the bias derived from fundamental properties of $\ell_2$-projection, simplifying proofs. We also extend the domain of validity from $η\leq 1 / (9 β)$ to $η< 1 /β$ where $β= L_0 + c L_1$ for clipping constant $c$, which matches the more traditional analysis of smooth functions. We strengthen the convergence criterion from $\left( \min_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ to $\left( \frac{1}{T} \sum_{t < T} \mathbb{E}[\lVert \nabla f(x_t) \rVert_2] \right)$ with matching speed, and lower the final achievable floor from $\mathcal{O}(\min(σ^2/c, σ))$ to the more precise $6 \min(σ^2 /c, 3 σ)$.
Wolfe's method computes the point of minimum Euclidean norm in the convex hull of finitely many points. The method is parameterized by an insertion rule and it has been an open problem whether Wolfe's original rule guarantees a polynomial number of iterations (De Loera, Haddock, and Rademacher; STOC'18). We resolve this question by proving an exponential lower bound. In fact, we establish a tight connection to the simplex method for linear programming that allows us to transfer many lower bounds for simplex pivot rules to corresponding insertion rules of Wolfe's method. These include not only Wolfe's linopt rule via Dantzig's pivot rule, but also many other classical, randomized and history-based rules. As a corollary, we obtain the first exponential example for the Lawson-Hanson algorithm for nonnegative least squares, as well as PSPACE-hardness of deciding whether the linopt rule ever uses a given point.
Recent years have witnessed a resurgence in using ReLU neural networks (NNs) to represent model predictive control (MPC) policies. However, determining the sufficient network complexity to ensure closed-loop guarantees remains a fundamental open problem. This involves a critical precision-complexity trade-off: undersized networks may fail to capture the MPC policy, while oversized ones may outweigh the benefits of ReLU network approximation. In this work, we project the MPC actions onto a tightened constraint set to satisfy the hard constraints, and establish a state-dependent Lipschitz continuity property for the optimal MPC cost function, which enables sharp convergence analysis of the closed-loop system. For the first time, we derive explicit bounds on ReLU network width and depth for approximating MPC policies with closed-loop guarantees. To reduce network complexity for the convergence to user-specified target sets, we propose a non-uniform error framework with a state-aware scaling function to adaptively adjust both the input and output of the ReLU network. Our contributions provide a foundational step toward certifiable ReLU NN-based MPC.
Jun Zhang, Wei-Jie Yu, Tong Qian, Eui Jeong Kwak, Sang-Woon Jeon, Hua Ding, Qijia Jiang, Hu Jin 等 10 位作者
原文摘要
We study a stylized single-vehicle replenishment-routing problem in which the cities of a national logistics network are both demand points and supply points: a vehicle of limited capacity delivers to every city once, and when its load is insufficient for the next city it must detour to a city already served that can still supply goods. A 100-node benchmark is built from Korean population and geographic data, with the cities divided into 10 levels; the per-visit supply coefficient and the allowed number of stops decrease geometrically with the level. Building on matrix-based ant colony optimization, we keep the states of the whole colony in arrays, couple route construction with a fixed greedy replenishment-decoding policy, and evaluate the candidate moves of a local search in batches. With frozen parameters and 30 runs on seeds never used for tuning, the matrix-based ant colony search with the batch local search (MACO) reaches a mean total distance of 3081.7 km; a genetic algorithm equipped with the same local search at the same frequency reaches 3074.9 km (no significant difference), and a MAX-MIN ant system with the same local search 3142.2 km. Ablation identifies the batch local search as the main contributor to solution quality; a load-aware transition factor gives no significant gain. Within the parameter ranges tested, raising the supply coefficients of the cities below the top level shortens the route more than relaxing the stop limits, which are not binding here.
Eline Pot, Laurent Lefebvre, Mathieu Segond, Antonio Silveti-Falls
原文摘要
Hyperparameter transfer across model depth is instrumental in training deep models, as it enables learning rates calibrated on shallow networks to be effectively applied to deeper architectures. The recent CompleteP framework establishes a $1/L$ residual scaling rule that demonstrates this for Adam-family optimizers. We study whether similar transfer extends to constrained optimization methods, specifically Frank-Wolfe and projected non-Euclidean normalized steepest descent, where updates are determined by linear minimization oracles. We derive scaling rules for these methods which confirm that the $1/L$ residual scaling applies, although the arguments differ from the Adam-family optimizers. Our analysis supports depth-wise transfer and width-wise transfer of the learning rate for the Scion algorithm while ensuring feature learning and avoiding the kernel-regime. In experiments on NLP transformers (nanoGPT on FineWeb), vision transformers (DeiT on ImageNet), and vision MLPs (MLP-Mixer on CIFAR-10), we observe learning rate transfer across depth and feature learning for appropriately chosen residual scaling.
Computing the $L_\infty$-norm of a time-invariant linear ordinary differential control system is a routine task in robust control, usually delegated to floating-point routines that perform well on most practical systems. Behind the numerical value, however, lies a discrete decision: whether a pole lies on the imaginary axis, and whether a mode of a realization is actually a pole of the transfer matrix. Floating-point arithmetic can only replace this decision with a tolerance. This paper explains, for a control audience, what reliable symbolic-numeric computation offers for this problem: exact arithmetic where a decision has to be made, and certified enclosures where a value has to be computed. We show how the problem is turned into a small polynomial system directly from state-space data, how this system is solved in practice, and what it costs. A benchmark on the {\tt COMPleib} library and on families of systems with known norms shows that two standard numerical routines err in opposite directions on the imaginary-axis decision, and never on the same system, while the certified method decides it correctly on every instance tested and computes the norm at a cost comparable to that of the numerical routines on most systems. We conclude with practical guidelines for users of numerical tools.
Many objectives in machine learning and combinatorial optimization are non-concave. We introduce $γ$-weak $θ$-up-concavity, a first-order condition for nonnegative monotone functions in which a scaling function $θ$ lets the curvature change with the size of the solution. The resulting class strictly contains monotone DR-submodular functions and One-Sided Smooth (OSS) functions. It also contains functions whose marginal gains first increase and then decrease, and functions that vanish faster than any power of the norm at the origin. We construct a surrogate gradient from gradients along the segment from the origin to the current point and prove an upper-linearization inequality with a point-dependent approximation coefficient. Restricting actions to the closed convex hull of the coordinate-wise maximal feasible points makes this coefficient uniformly positive, with a value determined by the curvature parameters and the geometry of the feasible set. Standard reductions to linear optimization then give offline approximation algorithms and online algorithms with regret guarantees. The coefficient matches the optimal one for DR-submodular maximization and improves the best known coefficients for OSS maximization over matroid constraints whenever the OSS parameter is positive.
We study linear quadratic dynamic games where players are uncertain about each other's control policies or goals and consequently seek to be strategically robust. Building on recent work on strategically robust and risk-averse game theory, we first formalize the problem of strategically robust linear quadratic dynamic games. We show that these can be rewritten as simple transformations of linear quadratic games in which each player chooses a controller in a fictitious game in which they are faced with an adversary who is penalized for deviating from the other players' policies. This formulation naturally induces a novel notion of dynamic equilibrium, which we call a strategically robust dynamic equilibrium. We establish existence and uniqueness (in the subgame-perfect sense) of such equilibria and show that the equilibrium policies are Markovian, linear, and can be efficiently computed via coupled backward Riccati equations. Through numerical simulations, including experiments in a network game, we illustrate the benefits of strategic robustness in designing robust and resilient decentralized control schemes. Our experiments also expose a "free-lunch" phenomenon in games in which robustness does not incur a corresponding loss in performance but can yield improvements in players' utilities and social welfare.
Weighted low rank approximation is a fundamental problem in numerical linear algebra, and it has many applications in machine learning. Given a matrix $M \in \mathbb{R}^{n \times n}$, a non-negative weight matrix $W \in \mathbb{R}_{\geq 0}^{n \times n}$, a parameter $k$, the goal is to output two matrices $X,Y\in \mathbb{R}^{n \times k}$ such that $\| W \circ (M - X Y^\top) \|_F$ is minimized, where $\circ$ denotes the Hadamard product. It naturally generalizes the well-studied low rank matrix completion problem. Such a problem is known to be NP-hard and even hard to approximate assuming the Exponential Time Hypothesis [GG11, RSW16]. Meanwhile, alternating minimization is a good heuristic solution for weighted low rank approximation. In particular, [LLR16] shows that, under mild assumptions, alternating minimization does provide provable guarantees. In this work, we develop an efficient and robust framework for alternating minimization that allows the alternating updates to be computed approximately. For weighted low rank approximation, this improves the cost of each iteration of [LLR16] from $\|W\|_0k^2$ to $\|W\|_0 k$ where $\|W\|_0$ denotes the number of nonzero entries of the weight matrix. At the heart of our framework is a high-accuracy multiple response regression solver together with a robust analysis of alternating minimization. We also replace the random initialization of [LLR16], whose claimed guarantee fails, by subspace iteration from a Gaussian start, so our total running time of $\widetilde O((\|W\|_0 k+nk^3)\log(1/ε))$ includes the initialization. In exchange, our guarantee needs the weight matrix to be within spectral distance $γn$ of the all-ones matrix for a $γ$ that is polynomially small in $n$.
Given a matrix $M\in \mathbb{R}^{m\times n}$, the low rank matrix completion problem asks us to find a rank-$k$ approximation of $M$ as $UV^\top$ for $U\in \mathbb{R}^{m\times k}$ and $V\in \mathbb{R}^{n\times k}$ by only observing a few entries specified by a set of entries $Ω\subseteq [m]\times [n]$. In particular, we examine an approach that is widely used in practice -- the alternating minimization framework. Jain, Netrapalli, and Sanghavi [JNS13] showed that if $M$ has incoherent rows and columns, then alternating minimization provably recovers the matrix $M$ by observing a nearly linear in $n$ number of entries. While the sample complexity has been subsequently improved [GLZ17], alternating minimization steps are required to be computed exactly. This hinders the development of more efficient algorithms and fails to depict the practical implementation of alternating minimization, where the updates are usually performed approximately in favor of efficiency. In this paper, we take a major step towards a more efficient and error-robust alternating minimization framework. To this end, we develop an analytical framework for alternating minimization that can tolerate a moderate amount of errors caused by approximate updates. Moreover, our algorithm runs in time $\widetilde O(|Ω| k)$, which is nearly linear in the time to verify the solution while preserving the sample complexity. This improves upon all prior known alternating minimization approaches which require $\widetilde O(|Ω| k^2)$ time.
Q-learning has been empirically observed to exhibit substantial transient error. This transient remains uncaptured by the leading Gaussian limit because the central-limit normalization removes the smaller-scale displacement generated by the initial nonlinear evolution. On the other hand, worst-case finite-sample guarantees, while accounting for transient effects, can obscure the mechanisms responsible for the error on a particular problem instance. We develop a finer asymptotic analysis that gives an asymptotically exact characterization of this transient for a fixed instance. Specifically, for synchronous Q-learning in finite discounted Markov decision processes, we separate the last-iterate error into an optimal-policy temporal-difference (TD) component and a nonnegative displacement caused by maximization. The centered TD component has Gaussian fluctuations at scale $n^{-1/2}$, whereas the maximization transient is of order $n^{-1}$ and converges almost surely, after rescaling by $n$, to a random limit. This limit represents the propagated effect of maximization errors generated before the greedy policy stabilizes. We combine this characterization with an Edgeworth expansion of the TD component to identify the corrections to its Gaussian approximation, separating the effects of initialization, non-Gaussian TD fluctuations, and maximization. The results reveal how transient errors arise for individual Q-learning instances and how their effects persist below the central-limit scale.
We propose a random feature policy iteration method for reconstructing the obstacle function in a mean-field game (MFG) from observations of its value function, leveraging the method's mesh-free nature and flexibility to handle complex geometries and high-dimensional problems. The method builds on a policy iteration framework for inverse MFGs, which decouples the coupled nonlinear forward-backward PDE system into a sequence of linear PDE and inverse problem solves. The fixed random feature representations preserve this linear structure, allowing the resulting subproblems to be solved by minimizing residuals at sampled collocation points through linear least-squares problems. The mesh-free nature of the method further extends the applicability of the policy iteration framework to inverse MFGs with complex geometry and in high dimensions. Another key advantage is its flexibility in using different spatial approximations for different variables: the obstacle function, value function, and population density can be represented in separate feature spaces, accommodating obstacles that are substantially less regular than the corresponding PDE solutions. Moreover, boundary conditions can be enforced through collocation on irregular domains without constructing boundary-fitted meshes. Numerical experiments involving smooth and discontinuous obstacles, a domain with holes, and higher-dimensional problems demonstrate the accuracy and flexibility of the proposed method.
Lenstra showed that mixed-integer linear programming can be solved in polynomial time when the number of integer variables is fixed and the number of continuous variables is arbitrary. Mixed-integer quadratic programming (MIQP) can be solved in polynomial time when the total number of variables is fixed. However, since MIQP is NP-hard already without integer variables, the analog of Lenstra's result fails unless P = NP. Hence, polynomial-time algorithms for MIQP with an arbitrary number of continuous variables must exploit additional structure. We study MIQPs with two-stage structure, which underlies Benders decomposition and two-stage stochastic programming. A few linking variables, including all integer variables, are coupled with many blocks of continuous variables that interact only through them, both in the constraints and in the objective. We give a polynomial-time algorithm for MIQP with an arbitrary quadratic objective when the number of linking variables and the block size are fixed, while the number of blocks, and hence the number of variables, is arbitrary. The case with no blocks is MIQP in fixed dimension. None of our assumptions can be dropped. Unless P = NP, the number of linking variables and the block size must both be fixed, the blocks cannot interact or contain integer variables, and large blocks cannot be allowed even if their objective is linear. Unless FPT = W[1], the exponent of the running time must depend on both the number of linking variables and the block size.
In this work, we study last-iterate convergence bounds for stochastic gradient descent (SGD) on convex and $β$-smooth loss functions and improve the bound from $\cO((\log T)^3/T)$ to $\cO((\log T)^3/(T\log\log T))$ in the interpolation regime. The improvement comes from applying a horizon-dependent multiscale estimate of gradients to the comparison of trajectories. The bound applies to both sampling with replacement and sampling without replacement within one pass.
Muhammad Khan, Xiaodong Cui, Brian Kingsbury, George Saon
原文摘要
BEST-RQ, based on random projection and quantization, has been a well-established self-supervised learning approach to learn speech representations from large-scale unlabeled audio due to its high performance and efficiency. However, in BEST-RQ representations learned by random quantization can be sensitive to the choice of quantization targets. To address this issue, we propose a distributionally robust extension of BEST-RQ that seeks representations robust across multiple quantization configurations. Specifically, we define groups by codebooks and formulate the training as a group Distributionally Robust Optimization (DRO) problem. Rather than optimizing solely for the average loss across groups, our approach adaptively reweights groups to emphasize those performing poorly. Additionally, we regularize DRO with a Kullback-Leibler divergence to prevent excessive distribution deviation. We evaluate our approach on the Multilingual LibriSpeech dataset and show that the distributionally robust BEST-RQ achieves superior performance over the conventional uniform average across multiple codebooks.
Firooz Shahriari-Mehr, Javad Aliakbari, Alexandre Graell i Amat, Ashkan Panahi
原文摘要
The joint use of graph structure and node-specific information is central to transductive node classification, yet a fundamental theoretical question remains: when does graph structure actually help? We provide the first deterministic, non-asymptotic, distribution-free answer to this question. We build on nuclear-norm graph clustering and generalized convex clustering. Our formulation of the latter incorporates node features, partial labels, and general convex loss functions, making it suitable for transductive node classification. We analyze each problem separately to characterize when graph structure alone, or node information alone, is sufficient for perfect recovery. We then formulate a unified convex optimization problem through a shared atomic norm representation and prove that combining the two sources expands the range of conditions under which perfect recovery is guaranteed beyond what either source alone can certify. This shows when node information can improve clustering and graph structure can improve classification.
Mohammad Zarei, Amirhossein MohammadpourAzari, Moslem Habibi
原文摘要
Reinforcement-learning portfolio policies can optimize allocations from prices, covariances, and prior positions, yet they lack a calibrated, replayable mechanism for certifying that a proposed allocation respects a bounded risk envelope before execution. Regulators increasingly expect automated portfolio systems to provide exactly this kind of pre-trade audit trail, interpretable risk limit, and human-oversight mechanism. Gated Conformal Semantic Risk Control addresses this gap by separating return-seeking allocation from text-conditioned downside-risk verification. An instruction-tuned language model maps timestamped disclosures into schema-validated semantic risk states over macro, liquidity, earnings, and sector channels. These states induce correlated ellipsoidal factor-shock uncertainty sets whose conformal radii are calibrated by rolling finite-sample prediction over localized semantic nonconformity buckets. An asymmetric second-order cone gate then checks each raw PPO proposal against a certified worst-case semantic loss budget. The gate stays inactive for compliant proposals and intervenes only when that budget is exceeded. In walk-forward out-of-sample tests on nine liquid U.S. Select Sector SPDR ETFs, Raw PPO earns a 12.7% annual return while incurring a 4.05% semantic violation rate. Gated-CSRC removes all observed out-of-sample semantic limit breaches while retaining an 11.7% annual return. This 100-basis-point auditable safety premium is smaller than the yield loss imposed by unconditional hard projection, and exact McNemar testing confirms the compliance gain at $p < 10^{-15}$. Loughran-McDonald lexicon controls and time-shuffled text controls fail to reproduce this certificate.
Oğuz Kaan Yüksel, Rodrigo Alvarez Lucendo, Nicolas Flammarion
原文摘要
Transformers integrate information from multiple past positions through sparse attention patterns that copy from specific locations. We study how these patterns form during training in a high-order Markov-chain setting, where the next token depends on several groups of past positions with differing statistical importance. Learning proceeds in discrete stages, each marked by the emergence of a sparse attention pattern for the next most important group. The dynamics are initially competitive, with all attention heads focusing on the dominant pattern, but later become cooperative, with different heads specializing in distinct patterns. For a tractable surrogate, the dynamics reduce to those of a tensor factorization problem, enabling us to prove stagewise convergence. Empirically, the number of stages completed depends on training set size, suggesting a form of implicit complexity control through early stopping. Overall, we give a theoretical account of how structured attention patterns and head specialization emerge without an explicit curriculum.
Branch-and-Bound (B&B) and $A^*$ are popular exact search methods originating from Operations Research and Artificial Intelligence, respectively. Although the two methods share fundamental search principles, their different ways of exploring the search space can lead to substantially different computational performance. We investigate these differences using the Resource-Constrained Project Scheduling Problem (RCPSP) as an exemplary problem setting. We first establish the relationships between mixed-integer linear programming, state-action, and timed Petri-net representations. Since B&B and $A^*$ admit numerous algorithmic enhancements, we focus on relatively basic implementations. This allows us to examine the fundamental differences between the methods rather than compare the performance of state-of-the-art solvers. By using the same lower bound and best-first node selection, we isolate the effects of the search mechanisms, particularly the identification of equivalent states reached through different scheduling decision sequences. Experiments on 480 benchmark RCPSP instances show that graph-based $A^*$ substantially outperforms the B&B variants considered. With additional pruning rules, $A^*$ proves optimality for all 480 instances, compared with 445 for best-first B&B under the same time and memory limits. Depth-first B&B uses less memory and can find feasible solutions even when the search does not finish within the time limit. Stronger linear programming relaxations reduce the search space but introduce considerable computational overhead. Our findings highlight the importance of duplicate-state detection and clarify the computational trade-offs between the search approaches. These insights can help researchers select an appropriate method and identify opportunities to incorporate effective mechanisms from one approach into the other.
The nonnegative rank of a nonnegative matrix $X$ is the smallest number of nonnegative rank-one factors that sum to $X$. Since computing the nonnegative rank is NP-hard, it is common to circumvent this issue by computing lower and upper bounds. In this paper, we propose non-convex formulations and practical implementations for four important lower bounds for the nonnegative rank, namely the fooling set bound (FSB), the rectangle covering bound (RCB), the hyperplane separation bound (HSB), and the self-scaled bound (SSB). In particular, to the best of our knowledge, our algorithm for computing the SSB is the first available in the literature. We compute the SSB for several matrices for which its value had not previously been reported. In some cases, the SSB provides the best known upper bound, while the other three bounds do not. Moreover, on canonical benchmarks, we show that our non-convex approaches are often competitive with existing methods. The paper also provides a consolidated reference for the current state of several classical lower bounds on a large number of benchmark matrices.
Worst-case complexity bounds often fail to reflect the practical performance of optimization algorithms, while average-case complexity heavily relies on the distribution of the input. In this work, we develop a smoothed-analysis framework for classical continuous optimization algorithms by studying their behaviour on random perturbations of arbitrary functions. Our main contribution is to apply methods suited for non-degenerate functions to perturbations of degenerate functions, thereby recovering the faster best-known rates for the non-degenerate class. We derive smoothed complexity guarantees for several standard methods, including (Accelerated) Gradient Descent, Conjugate Gradient, and Newton-type methods, under various settings including nonconvex ones, and show that these bounds can substantially improve upon their worst-case guarantees, bringing them closer to the practical behaviour of the respective methods.
In this paper, we study the convergence of Federated Averaging (FedAvg) with partial participation under a general variance condition. In each communication round, the server samples $S$ of the $n$ clients uniformly at random without replacement. The stochastic gradients are assumed to satisfy the so-called ABC condition, which allows their variance to grow with the local objective gaps. For smooth nonconvex objectives, we establish a finite-time stationarity bound in terms of the heterogeneity measure $σ_f^\star:=f^\star-\frac{1}{n}\sum_{i=1}^n f_i^\star$ and the client sampling factor $I_S:=(n-S)/(n-1)$. Here, $f^\star$ and $f_i^\star$ denote the infima of the global average objective and client $i$'s local objective, respectively. The bound exhibits a linear speedup in the number of sampled clients $S$ and recovers the full-participation result when $S=n$. Under diminishing stepsizes, we further prove that the gradient norm converges to zero both in expectation and almost surely. Notably, compared with existing analyses of FedAvg with partial participation, our results require no additional data heterogeneity assumption such as bounded gradient dissimilarity (BGD), and the sampling-induced term in our bound is not amplified by the number of local updates at the leading order. The importance of removing the BGD assumption is illustrated by an example that is smooth and lower bounded but violates BGD, yet is covered by our analysis.
We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propose Adaptive Reparameterized Time (ART), a continuous-time control formulation that learns a time change by treating the speed of the sampling clock as the control, so that a uniform grid on the learned clock induces adaptive timesteps in the original diffusion time. Based on a leading-order Euler error surrogate, ART provides a principled objective for allocating timesteps along the sampling trajectory. To solve this deterministic control problem, we introduce ART-RL, an auxiliary randomized formulation with Gaussian policies that turns schedule learning into a continuous-time reinforcement learning problem. We prove that the randomized ART-RL formulation is equivalent to ART at the optimizer level, in the sense that its optimal Gaussian policy recovers the optimal ART time-warping rate through its mean. We further establish policy evaluation and policy improvement characterizations and derive trajectory-based moment identities that yield implementable actor--critic updates for learning the schedule. Across experiments ranging from controlled low-dimensional settings to image generation, ART-RL can be plugged into existing diffusion samplers by changing only the timestep grid, consistently improving sample quality over strong baseline schedules at matched budgets while leaving the rest of the sampling pipeline unchanged. The learned schedules also exhibit broad generalization, transferring without retraining across sampling budgets, datasets, solvers, pipelines, and representation spaces.
In this paper we show that mixed-integer linear programming is solvable in strongly polynomial time if the constraint matrix is totally unimodular with at most two nonzero entries in every row. This is done by reducing the problem to an instance of minimization of an $L$-convex function.
Robotic additive manufacturing requires joint paths that reconcile robot feasibility with process-specific geometry. We develop an offline optimization-based path planner that combines joint-path regularity and self-collision penalties with process costs and constraints, so that a new process enters through its own reference, residuals, and tolerances. Full-path and boundary-coupled chunked optimization handle references of different lengths. Iterative Weight Tuning (IWT) determines the cost weights from the gaps between Key Performance Indicators (KPIs) and their targets. In carbon-fiber winding and robotic 3D printing, the planned paths meet the geometric tolerances and all KPI targets, and both paths are executed on a UR10e, without extrusion for printing, with joint-level tracking recorded during execution.
We consider the problem of estimating the power spectral density of a Vector-valued Autoregressive Moving-Average (VARMA) process. A well-established approach to accomplishing this task consists of formulating a moment problem that selects the spectral density maximizing an entropy-like index while matching certain moments of the underlying process. However, existing methods assume that the moving-average (MA) component is either fixed or scalar, thereby reducing the class of ARMA models over which the optimal spectral density is sought. In this paper, we propose a new moment problem, based on the notion of transportation entropy, for learning the spectral density of a VARMA process without fixing the MA component or restricting it to be scalar.
Overactuated multirotors can produce the same wrench with many rotor-speed combinations, some of which leave the vehicle unable to change its wrench quickly, e.g., to reject a gust. Readiness barriers keep this ability, the readiness, above a floor, but only in continuous time; with sampled commands and varying speed-controller delays, the available robustness bound covers less than one millisecond. We remove this limit by exploiting the structure of the problem: each rotor responds monotonically and in closed form to its own torque, so the speeds reachable under uncertain switching times lie in a cheaply computed box, and the lowest readiness over that box sits at a corner known in advance. This yields a Lipschitz-free test that a command keeps the readiness above its floor until the next sample, plus a fallback torque that keeps the vehicle on a certified plan. In hexarotor simulations with 2--40~ms delays and $\pm2$~ms jitter, the filter never violated the floor. At $10\pm2$~ms, it matched a delay-compensating predictor within the design envelope and cost 2--30\% more tracking error beyond it, where the predictor alone violated the floor in 2 to 16 of 16 missions. A tuned robust predictor was also safe with synchronous switching, but not with per-rotor jitter.
Minh Hoang Trinh, Chuong Van Nguyen, Quoc Van Tran, Tuynh Van Pham
原文摘要
In this paper, we consider the problem of regulating and maintaining a target formation characterized by a set of bidirectional bearing constraints under disturbances. The agents in the formation are modeled by single integrators with deterministic bounded continuous disturbances of which both actual function and upper bound are unavailable for the control design. Due to the time-varying disturbances, the target formation is time-varying. We propose displacement/bearing-only based adaptive sliding mode control laws to uniformly asymptotically stabilize the time-varying target formation and reject the matched disturbances, and provide sufficient conditions to prevent possible collisions. A time-dependent adaptive design method is provided to convert the displacement-based control law to a bearing-only one. In addition, to alleviate chattering phenomena from sliding mode control, smooth adaptive control laws are introduced to guarantee uniform global boundedness of the desired formation. Finally, simulation results are given to support the analysis.
Jan Quan, Konstantinos Oikonomidis, Alexander Bodard, Panagiotis Patrinos
原文摘要
This paper develops a dual-space preconditioning framework for variational inequalities and root-finding problems beyond classical cocoercivity and Lipschitz continuity assumptions. To this end, we introduce both a novel cocoercivity-type condition and a relaxed Lipschitz continuity condition, and derive convergence of deterministic and stochastic methods under these generalized conditions. For isotropic preconditioners, the resulting algorithms can be interpreted as variable-stepsize variants of classical methods, extending recent analyses based on $(L_0, L_1)$-type growth conditions while allowing simpler proofs and larger stepsize ranges. We also show that the framework naturally handles variational inequalities through weak Minty-type arguments. Finally, we study the associated continuous-time dynamics and derive a variational and optimal control interpretation of the preconditioned flow. Overall, the results show that dual-space preconditioning provides a flexible mechanism for designing and analyzing operator methods beyond the standard Euclidean Lipschitz setting.
Hessian spectra at trained models in deep learning exhibit a persistent pattern: eigenvalues organize into distinct clusters, including a large bulk near zero and a few isolated outliers. This paper shows that a natural account of these spectral phenomena emerges when the original setting is understood as a departure from a nearby, otherwise hidden, highly symmetric reference. Modifications, including changes to the architecture, data distribution, or parameter metric, expose a nearby reference configuration whose Hessian exhibits rich invariances-ones not accounted for by weight symmetries. There, symmetry enables a precise description of the spectra, forcing high-dimensional kernels and eigenvalues of large multiplicity. Returning to the original configuration breaks the Hessian symmetry and thereby produces the observed hierarchy of clusters and outliers. The framework is developed in some generality, with a detailed analysis of three-layer ReLU networks and applications to convolutional, graph, and transformer models, as well as to the NTK. The same mechanism is further shown to yield analogous spectral structures in layerwise Hessians and the Gauss-Newton matrix.