The Inference Report

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.

Multi-objective bilevel optimization has wide applications in the AI area such as automated learning and multi-task meta-learning. Although recently some works have been begun to study the multi-objective bilevel optimization, the proposed methods rely on the (strongly) convex lower level problems. In fact, these multi-objective bilevel learning problems are generally nonconvex, and particularly their lower level problems are nonconvex. To fill this gap, we propose a class of Multi-Objective Moreau Envelope based Hessian-free Algorithms (MOMEHA) to solve the multi-objective bilevel learning problems with nonconvex lower level. Specifically, our method uses the Moreau envelope to convert the original problem into a multi-objective single-level optimization with an envelope constraint. In particular, our method retains computational advantages of being single-loop and Hessian-free in the multi-objective setting by incorporating a smooth weighted Tchebycheff scalarization. Furthermore, we propose a momentum-based variant of MOMEHA (i.e., MB-MOMEHA) method to solve the stochastic multi-objective bilevel learning problems. In theory, we provide the convergence properties of our algorithms under both deterministic and stochastic setting. Some experiments on few-shot meta-learning and neural architecture search demonstrate that our methods outperform the existing approaches in Pareto front, validating its effectiveness and robustness.

Laplacian-regularized minimization is fundamental in signal processing and machine learning, but is limited by the dense and ill-conditioned nature of the graph Laplacian pseudoinverse. While the Laplacian itself is sparse, its pseudoinverse is dense and often ill-conditioned, rendering direct computation impractical at scale. Moreover, pseudoinverse learning is more challenging than Laplacian learning. To address this challenge, this paper considers the setting where the graph Laplacian is given and proposes a Difference-of-Convex Regularizer (DCR) graph learning framework that approximates the spectral action of the Laplacian pseudoinverse without direct inversion via regularized Maximum Likelihood Estimation (MLE). By reformulating Laplacian-Regularized Nonnegative Least Squares (LR-NNLS) through a dual representation, DCR decouples pseudoinverse learning from instance-specific inference and enables efficient primal solution reconstruction via a differentiable dual-guided learning scheme. We establish theoretical guarantees on stability and the existence of a unique fixed point for DCR algorithm. Numerical experiments demonstrate improved performance over convex solvers and graph filtering baselines and robust performance across diverse graph topologies.

Distribution steering seeks feedback laws that drive the state law of a dynamical system between prescribed initial and terminal distributions. Optimal transport provides a natural geometric approach, but its implementation generally requires a transport map or coupling in the full state space. Sliced optimal transport avoids this full-dimensional construction through one-dimensional projections. Yet, the resulting projected maps specify only directional displacements and do not by themselves prescribe a realizable feedback law. To this end, we develop a finite-horizon control framework based on sliced optimal transport. At each sampling instant, a projected optimal transport map defines a directional terminal condition, whose minimum-energy realization yields a randomized single-direction controller. Averaging over projection directions gives a deterministic sliced feedback. For the single-integrator dynamics, the averaged feedback makes the sliced Wasserstein distance to the target non-increasing. For Gaussian endpoint laws, it is affine, preserves Gaussianity, and steers the mean and covariance to their prescribed terminal values. We further identify a law-dependent gain that yields linear decay of the sliced Wasserstein distance together with an explicit characterization of the control energy. We also prove that the randomized controller converges to the averaged sliced flow as the sampling period vanishes. Finally, we extend the construction to linear dynamical systems. Reachability-normalized coordinates allow instantaneous realization of the sliced velocity for uniformly fully actuated systems, while local controllability Gramians provide exact finite-step realization for general controllable systems. Numerical examples illustrate the resulting distributional flows.

We study a Restart POMDP (Partially Observable Markov Decision Process) on a general Borel state space, where the controller either lets the hidden state evolve unobserved or restarts the system and observes the new state. Exploiting a sufficient-statistic representation consisting of the last observed state and the elapsed time since restart, we reduce the problem to a fully observed MDP. Under a natural one-step cost deterioration condition, we prove that optimal policies have a threshold structure in the elapsed time for both the discounted and total undiscounted cost criteria. When the state space is partially ordered and the kernel is stochastically monotone, we further show that the optimal threshold is nonincreasing in the state. For the average cost criterion, under additional assumptions of geometric ergodicity and domination of the transient gain, we establish analogous threshold results via the vanishing discount approach, after showing the uniform boundedness of the optimal thresholds and relative value functions.

Clustering is a fundamental class of data analysis techniques with the most important representatives being centroid-based methods like $k$-means. Such methods are strongly connected to quantization problems, which aim to approximate general probability measures with discrete ones. For example, $k$-means corresponds to quantization with respect to the Wasserstein distance. While Wasserstein quantization clusters points within a fixed space, this paper studies Gromov-Wasserstein (GW) quantization, which additionally aims at clustering the ambient geometry of the space. We show existence of solutions to the GW quantization problem and give a characterization that justifies an analogue to the $k$-means algorithm (Lloyd's algorithm) to approximate them numerically. We further calculate the quantization rate for usual Euclidean geometries that are used in the GW context, and relate it to standard Wasserstein quantization rates. Finally, numerical experiments show that GW quantization opens up many modeling possibilities beyond normal clustering methods (e.g., for geodesic distances of 3D shapes or structured pruning of neural networks) and that the introduced algorithm leads to useful numerical solutions with approximation quality often in line with theoretically optimal rates.

We revisit the Sinkhorn-Knopp (SK) algorithm for the matrix scaling problem. Despite extensive literature on the global convergence of SK and its variants, its local linear convergence behavior remains less understood. We address this gap by providing the first nonasymptotic local analysis of SK that matches the rate obtained from existing asymptotic Jacobian-based arguments. We show that under certain connectivity conditions, SK is a polynomial-time algorithm for doubly stochastic matrix scaling. With the developed tools, we showcase the local suboptimality of SK and provide accelerated variants. Finally, for dense matrices, we improve the complexity of existing first-order matrix scaling algorithms from $O(\tfrac{n^{7/3}}{\varepsilon^{2/3}})$ to $O(\tfrac{n^{9/4}}{\sqrt{\varepsilon}})$.

Bregman proximal stochastic gradient (BPSG) methods bring variance-reduced composite optimization to objectives whose geometry is poorly captured by Euclidean smoothness. Their performance, however, remains sensitive to the step size: raw stochastic curvature estimates can fluctuate sharply, whereas line searches add repeated proximal evaluations. We introduce Ada-BPSG, a line-search-free BPSG method that couples the SAGA gradient table with a stabilized Barzilai--Borwein (BB) candidate. A mediant aggregates incremental secant information so that nearly singular local ratios receive little weight, and an explicit safeguard translates the resulting curvature estimate into the bounded step-size sequence required for convergence. This design yields a direct analytical chain from relative smoothness and component-wise variance control to convergence in finite-dimensional normed spaces. We prove an $O(n/K)$ ergodic rate for convex objectives, a restarted linear rate under relative quadratic growth, and an $O(1/K)$ bound for a Bregman proximal residual in the nonconvex setting. On logistic regression and sparse nonnegative matrix factorization, Ada-BPSG combines low objective values with substantially less sensitivity to the initial step size than standard variance-reduced baselines, while avoiding line search.

Acceleration for deterministic root-finding problems has been extensively studied in recent years; specifically, the anchor-based, or Halpern-type methods achieve optimal convergence rates with respect to the operator norm. However, acceleration via these methods does not directly carry over to stochastic setting due to accumulation of errors, unless one enforces diminishing variance via increasing batch sizes or variance reduction techniques. In this work, we show that another class of acceleration, namely the dual-anchor mechanism, extends to the stochastic setting without such error accumulation, in contrast to anchor-based algorithms. Consequently, we cleanly achieve $O(ε^{-3})$ complexity with iteration-independent batch size, without any variance reduction or double-loop recursive regularization, for stochastic root-finding (resp. fixed-point) problems with cocoercivity (resp. square-nonexpansivity) in expectation. For strongly monotone operators, the same algorithm attains a sharper $\widetilde{O} (ε^{-2})$ complexity, nearly matching the lower bound in terms of $ε$-dependence.

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

We prove a sharp lower bound for smooth nonconvex stochastic optimization with uniformly bounded gradient noise. In the \(K=1\) fresh-sample model, every randomized adaptive algorithm requires $$Ω\left( \frac{ΔL}{ε^2} + \frac{ΔLσ^2}{ε^4} \right)$$ queries to find a point with expected gradient norm at most \(ε\). This matches the standard upper bound and, to the best of our knowledge, resolves the question raised by [Arjevani et al. 2023] of whether almost-surely bounded oracle error permits a better rate than bounded variance. The proof was independently generated with GPT-5.6 Sol in Codex's Ultra mode during a two-hour session. The human author supplied the prompt and was responsible only forchecking the proof and revising and polishing the manuscript.

Embedding trained neural networks as surrogates within optimisation problems is an established practice in operations research. The prevailing approach uses feedforward neural networks (FNNs) with ReLU activations, whose piecewise-linear structure admits an exact but computationally intensive mixed-integer programming (MIP) reformulation as the networks grow. We advocate input convex neural networks (ICNNs) as structurally superior surrogates when the underlying response is approximately convex or concave. The convex architecture offers two computational advantages. First, the ICNN-MIP formulation tends to yield a tighter linear programming (LP) relaxation than its FNN-MIP counterpart, with no integrality gap in favourable instances. Second, ICNNs uniquely admit an LP-based reformulation via epigraph representations of ReLU activations, though this embedding is not always exact. When it is not, we exploit the properties of ICNNs to construct the strongest continuous relaxation over box domains, namely, the convex hull of the ICNN's graph, bounded below by the epigraph and above by the concave envelope; this construction is tractable under input convexity but hard for general ReLU networks. On this basis, we develop a branch-and-bound algorithm that builds this relaxation at each node, branches directly on input variables rather than intermediate variables as in MIP reformulations, and terminates at the root node whenever the epigraph embedding is valid. Case studies on humanitarian food aid, oil well routing, and wine blending show that ICNN surrogates match FNN accuracy and deliver gains in solve time and scalability, supporting ICNN as the default surrogate when the underlying function is convex, concave, or well-approximated as such.

We prove that mirror descent converges to a KKT point for the nonconvex problem without excluding boundary limits. The result holds under verifiable conditions that jointly couple the objective, the Legendre kernel, and the feasible geometry. The key ingredient to establish the convergence is a metric-flattening reparameterization \(S\) that admits a definable boundary extension. Applying the KL argument to the reparameterized objective yields convergence of \(S(x_k)\). Continuity of \(S^{-1}\) then recovers convergence to the KKT point of the original sequence. We further apply our general framework to some concrete examples: Shannon entropy, Fermi--Dirac entropy, and power kernels. Future work may consider more general constraint geometries and genuinely nonseparable kernels, and extend mirror descent to broader Bregman-type methods, e.g. Bregman proximal point algorithms and Bregman ADMM, and their inexact variants.

Production planning in the manufacturing industry often relies on the use of optimization models, but defining an appropriate objective function can be a challenge. In practice, planners must balance competing goals, manage uncertainty, and account for qualitative business preferences that are difficult to quantify. As a result, many optimization models fail to match expert behavior, limiting trust and adoption. In this work, we propose a data-driven inverse optimization framework to infer the objective function implicitly captured in expert planners' decisions. We formulate the production planning problem as a mixed-integer linear program, where the unknown objective function is represented as a weighted sum of hypothesized cost terms. A suboptimality-loss-based inverse optimization method is then applied to learn the objective weights from historical production plans. The proposed approach is applied to a real industrial case provided by Dow, where the inferred weights reveal that avoiding inventory shortages and maintaining consistent cycle lengths dominate the planners' decision-making. Time- and product-dependent extensions further improve predictive accuracy and uncover evolving priorities. Expert interviews confirm the practical validity of these insights. Overall, this study shows that inverse optimization can transform tacit human expertise into interpretable models, enabling more accurate and trusted decision-support tools for complex industrial systems.

Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space. We study entropy-regularized discounted linear-quadratic (LQ) control. A Bellman verification argument shows that the unrestricted problem has a linear-Gaussian optimal policy, and the discounted-occupancy-weighted statewise Wasserstein gradient is tangent to this policy class. WPG therefore reduces exactly to a finite-dimensional ODE for the feedback gain and action covariance. We prove that this ODE is globally well posed and converges exponentially from every admissible initialization. For each fixed LQ problem, the exponent has a positive limit as the entropy temperature tends to zero and contains no perturbative factor of the form $\exp(-c/τ)$, while retaining the usual dependence on the conditioning of the control problem.

In this paper, we consider a class of multiblock nonconvex nonsmooth optimization problems, which covers many applications such as the analysis of pre-earthquake anomalies and machine learning. To solve this class of problems, we propose the inertial block proximal linearized method with two-phase adaptive momentum (IBPL$^+$-TP). Compared to the current methods, our method possesses three main advantages: (1) it introduces a two-phase adaptive momentum strategy to effectively update the extrapolation parameters, (2) it allows using two different extrapolation points to accelerate the convergence, (3) it allows the extrapolation parameters of these two extrapolation points to be independent of and unconstrained by all other parameters. While maintaining the above advantages, we prove that our method ensures the monotonic convergence of the objective function of this class of problems, and we also prove that the sequence generated by our method globally converges to a critical point, as well as establish the convergence rate of our method. To demonstrate the effectiveness of our method, we apply it to solve two nonconvex and nonsmooth machine learning problems, namely sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. The numerical experimental results on solving these problems show that our method outperforms several state-of-the-art methods.

We study stochastic extragradient (SEG) methods for solving monotone variational inequality problems (VIPs) over a feasible set. Although extragradient is a foundational algorithm for VIPs and its deterministic convergence theory is well developed, its stochastic counterpart remains less understood. Most existing analyses focus on independent-sample SEG (I-SEG) and assume either that the domain is compact or that the variance of the stochastic operator is uniformly bounded. The behavior of same-sample SEG (S-SEG), a natural variant with materially different properties, has received far less attention. In this work, we address these gaps in the literature. We first show that S-SEG is sensitive to samplewise Lipschitz parameters: mean Lipschitzness and bounded variance alone do not ensure convergence, even on a compact set. Then, for possibly unbounded domains, we establish a high-probability restricted-gap convergence for each SEG variant under a relaxed set of assumptions, and show that certain fundamental improvements to these results are impossible in general. Finally, we show that a known asymmetric double step-size selection that guarantees almost sure last-iterate convergence for I-SEG can fail for S-SEG: there exists a stochastic monotone VIP for which S-SEG diverges almost surely even under the modified step-sizes.

We study Muon, a recently proposed matrix-aware optimization method, in the context of the Stiefel manifold. This manifold consists of matrices with orthonormal columns and is ubiquitous in machine learning and scientific computing. Existing extensions of Muon to this manifold rely on heuristic, approximate, or iterative updates with varying computational efficiency. We show that the corresponding Stiefel Muon update admits an exact closed-form solution and use this result to develop Skewon, a practical algorithm for orthogonality-constrained optimization with an efficient implementation. We further establish first-order convergence guarantees for Skewon in the smooth non-convex setting.

We give a negative solution to MAIS-O60. We first construct an example in which an initially active ReLU neuron becomes completely inactive in finite time and thereafter remains frozen at a limit whose Fourier energy is equally distributed among all nonzero real frequency classes. The counterexample holds on an open set of initial conditions and therefore occurs with positive probability under Gaussian initialization. An appendix prepared by GPT-5.6 Sol strengthens the counterexample by showing that the same failure can occur for every Clarke trajectory from an open set of initial conditions, under the convention $\mathrm{ReLU}'(0)=0$, for smooth dead-zone approximations of ReLU, and for fixed-step full-batch gradient descent. Thus, single-frequency alignment is not a general consequence of training a single neuron on modular addition.

Stochastic gradient descent (SGD) optimization methods are the standard instruments for the training of deep neural networks (DNNs). In many relevant artificial intelligence (AI) systems - such as popular large language models (LLMs)-not the standard SGD scheme is used as the optimization method but instead suitable accelerated variants of SGD are employed. One of the most popular methods of such accelerated SGD variants is the momentum orthogonalized by Newton-Schulz (MUON) optimizer proposed by Jordan et al. in 2024. The MUON optimizer exploits the special matrix structure of the weight parameters in the training of the DNNs and, in its original form, employs five Newton-Schultz (NS) matrix steps in each MUON iteration. In this work we propose and study a generalized variant of the MUON optimizer involving an arbitrary number of generalized NS steps with polynomials of possibly arbitrary high degree. The considered optimizer covers MUON with the original NS polynomial as well as MUON combined with the recently proposed Polar Express method as special cases. For a simple class of stochastic optimization problems (SOPs) we show for almost every mini-batch size that MUON fails to converge to the solution of the SOP as the number of gradient steps converges to infinity. We also establish an error analysis for MUON with the generalized NS steps that provides convergence rates in terms of the number of gradient steps and in terms of the size of the mini-batch. We illustrate our general error analysis for MUON in the case of several concrete examples including quadratic stochastic optimization problems (SOPs) as well as $\ell_2$ regularized logistic regression for binary classification.

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov--Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

In this paper, we consider stochastic optimal control problems with infinite-horizon joint chance constraints. By means of an appropriate state augmentation, we reformulate the original problem as a constrained Markov decision process, in which both the cost and the constraint function exhibit an additive structure. We then prove that this formulation enjoys strong duality, thereby enabling us to reformulate the problem as an equivalent unconstrained one in the Lagrange dual framework. We propose a dual-ascent algorithm to solve the resulting problem and show that it converges to a deterministic Markov policy defined over the augmented state space that is both optimal and feasible. To accommodate continuous state-input spaces, we propose a dedicated learning algorithm to approximate the value function in an offline training setting, thereby significantly reducing the computational complexity of the online control phase. We then test our approach on a numerical example and demonstrate its effectiveness compared to online predictive control methods in terms of performance and computational complexity.

Quadratic Gradient (QG) is a Newton-type optimization framework that bridges first-order gradient descent and second-order optimization by incorporating curvature information into gradient updates. Simplified Quadratic Gradient (SQG) reduces the complexity of QG construction while preserving its optimization capability, whereas Quasi-Quadratic Gradient (QQG) extends the quadratic gradient principle to quasi-Newton methods such as BFGS. In this paper, we propose **Generalized Quadratic Gradient (GQG)**, a unified framework that extends the quadratic gradient principle to a broader class of Newton-type optimization algorithms. By abstracting the common structure of existing quadratic gradient methods, we show that the fundamental requirement of quadratic gradient construction is not limited to specific Hessian approximations, such as constant Hessian matrices, diagonal Hessian approximations, or BFGS-based Hessian surrogates. Instead, it can be generalized to any positive-definite curvature matrix satisfying the stationary condition of a local quadratic model. Based on this perspective, we investigate the construction of generalized quadratic gradients using various positive-definite Hessian surrogates beyond BFGS, providing a broader foundation for developing curvature-aware optimization algorithms.

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush--Kuhn--Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $Δ_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $α_k\asymp k^{-β}$ with $β\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.

We investigate Optimization under Input Uncertainty (OIU), in which the input to the objective function, rather than the objective function itself, is subject to uncertainty. OIU appears in manufacturing processes with production tolerance, control of physical systems with actuation noise, Mixture of Experts, and Reinforcement Learning (RL). Most of the existing approaches solve OIU by using the value of the objective function but discard the information of the realized input, even though the realized input is observable in various applications. The question here is whether the discarded information of the realized input is useful to accelerate the optimization process. We affirmatively answer this question for Evolutionary Strategy (ES) by theoretically showing that the information of the realized input can reduce the variance of the gradient estimator via Rao-Blackwellization. Using the Rao-Blackwellized gradient estimator, we propose Phenotype-Accelerated Evolutionary Strategy (PAES), which is a refinement of ES for OIU. Numerical experiments show that PAES converges faster than the usual ES from simple continuous optimization problems to RL benchmarks.

This paper studies the distributed online control problem over a network of linear time-invariant (LTI) systems in the presence of adversarial disturbances and time-varying convex costs. The network cost is characterized by the summation of local cost functions, where each local function is sequentially revealed only to the corresponding agent. The goal of each agent is to generate a control sequence, using only local observations and neighbor communication, that competes with the best {\it centralized} linear policy in hindsight. We extend the recently proposed Online Spectral Control framework from the centralized setting to the distributed setting. In particular, each agent applies a spectral controller obtained by convolving past disturbances with the leading eigenvectors of a Hankel matrix, while the controller parameters are updated through a distributed online gradient descent step over the local surrogate costs. We formulate this problem this problem as a {\it regret} minimization problem based on the spectral parameterization, and under standard assumptions, we establish a sublinear regret bound of $O(\frac{\sqrt{T}\text{poly}(\log T)}{γ^3})$, where $T$ is the time horizon and $γ$ denotes the stability margin. The resulting bound also captures the dependence on the network size and connectivity.

SignMuon compresses the Muon update to one bit per parameter by taking its elementwise sign, providing the most direct way to run a matrix-aware optimizer under an extremely low communication budget. It outperforms SignSGD in practice, yet it can ascend even on a linear function. Signing the gradient before the Linear Minimization Oracle (LMO), rather than after, does not repair this: we construct a small explicit instance on which sign-before (MuonUSign) and sign-on-both-sides (MuonSign) ascend as well, so no placement of the sign around the oracle descends in general. Error feedback, the standard remedy for a biased compressor, does not rescue SignMuon: when applied to Muon's output, error feedback can fail for every smoothness constant, step size, and momentum. Applied to the gradient, error feedback does work, and EF21-MuonUSign and EF21-MuonSign attain the standard $\mathcal{O}(T^{-1/2})$ rate for the squared gradient norm on smooth nonconvex problems, the latter at one bit in each direction. Experiments then reverse the ordering: across centralized CIFAR-10, federated CIFAR-10, and the nanoGPT speedrun, the strongest compressed method is consistently sign-after-the-LMO, precisely the placement we prove divergent, with the provably convergent variants trailing it. Compressing after the LMO, a heuristic, matters more at these scales than the guarantee does.

We study optimization under performative prediction, where deploying a model affects the future data distribution. For this setting, several gradient-based approaches have been proposed. However, they typically assume specific data distributions or loss functions, which limit their practical applicability. To overcome these limitations, we propose a gradient-based optimization method with convergence guarantees under substantially weaker assumptions. Our method explicitly estimates the induced distribution shift through finite differences. It enables higher-dimensional optimization across broader classes of loss functions and data distributions. We also propose a practical variant that reduces the number of samples required. Numerical experiments demonstrate that our proposed algorithms converge faster and more consistently than existing ones.

This study presents a theoretical analysis of partitional clustering on networks, analyzing both hard and soft assignment schemes with different objective functions. Cluster centers are not restricted to vertices but can also be located along the edges. We examine four key models: P-Median (PMP) and Sum of Squares Clustering (SSC) under hard assignment, and Probabilistic Distance Clustering (PDC) and Fuzzy C-Means (FCM) under soft assignment. Through mathematical analysis, we uncover structural properties that differentiate these models, such as the significance of assignment bottleneck points and the role of vertex-restricted solutions in determining optimal cluster centers. Our findings reveal that, while SSC and FCM can yield optimal centers along edges, PMP and PDC inherently favor vertex placement, leading to insights into clustering behavior on networks. These insights offer new directions for designing efficient algorithms and have implications ranging from facility location and network design to clustering on the embedding graphs that power similarity search in modern retrieval systems.