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arXiv2026

Convergence rates for the RMSprop optimizer with full control of the hyperparameters

Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW optimizers, where the adaptivity parts in Adam and AdamW basically just coincide with RMSprop. Such adaptive methods involve several hyperparameters including the regularization parameter $ε$ (which ensures that one does not divide by 0 and is often chosen to be very close to zero such as $10^{-8}$ in PyTorch by default) and the second moment decay parameter $β$ (which is often chosen to be very close to $1$ such as 0.99 (RMSprop) and 0.999 (Adam and AdamW) in PyTorch by default). Despite the high relevance of such methods, it remains an open research problem to provide error estimates for such methods with the error constants being not exploding but uniformly bounded with the respect to the hyperparameters, even in the situation of convex stochastic optimization problems. It is the key contribution of this work to essentially solve this problem for RMSprop. Specifically, we bound the expectation of the stopped evaluation of the objective function at the RMSprop process from above by the sum of an initialization term that decays exponentially in the training time, a stochastic approximation remainder of order $γ_n$, and a memory error of order $( 1 - β)^2$ with the error constants being uniformly controlled over all admissible choices of the step sizes, the second moment decay parameter $β$ and the regularization parameter $ε\in[0,1]$ (also covering $ε=0$). Our non-asymptotic error estimates hold not just for all sufficiently large n but hold for every gradient step $n=1,2,3,...$ with all error constants being explicitly specified. The key innovative new feature in the proof of our analysis are suitable inverse moment bounds for the second moment process in RMSprop.

cs.LG↗

arXiv2026

Optimal Sensor and Actuator Selection for Factored Markov Decision Processes: Complexity, Approximability and Algorithms

Factored Markov Decision Processes (fMDPs) are a class of Markov Decision Processes (MDPs) in which the states (and actions) can be factored into a set of state (and action) variables and can be encoded compactly using a factored representation. In this paper, we consider a setting where the state of the fMDP is not directly observable, and the agent relies on a set of potential sensors to gather information. We formulate the problem of selecting a set of sensors for fMDPs (under a limited budget) to maximize the infinite-horizon discounted return provided by the optimal policy. We show the fundamental result that it is NP-hard to approximate this problem to within a factor of $n^{1-c}$ for any $c > 1$, where $n$ is the number of state variables. Our inapproximability results for sensor selection also extend to a general class of Partially Observable MDPs (POMDPs). We also consider the dual problem of budgeted actuator selection (at design-time) to maximize the expected return under the optimal policy, for which we establish similar inapproximability results. Finally, we consider a simple greedy algorithm and empirically show that, despite the lack of formal theoretical guarantees, it performs effectively in practice, achieving on average over $70\%$ of the optimal solution value across a variety of real-world and randomly generated problem instances.

eess.SY↗

arXiv2026

Exact Risk-Complexity Laws for Projective Boundaries in Scenario Optimization and Distribution-Free Certification

Scenario optimization, conformal prediction, and related distribution-free certification methods use finite samples to construct decisions or prediction sets with violation-risk guarantees for fresh observations. In several classical settings, the conditional violation risk follows an exact beta law, whose tail has a beta-binomial representation and whose parameter is a support, calibration, or compression dimension. This paper identifies the deterministic boundary mechanism behind these formulas and derives the corresponding law when the observed boundary size is random. A decision rule is represented by an acceptance set for future observations, together with a boundary map selecting the sample points responsible for that set. The resulting pair is called a {\em proper projective boundary scheme} when held-out samples are accepted precisely if the full-sample boundary is retained, and accepted non-boundary samples can be deleted without changing that boundary. For every such scheme, the conditional law of the violation risk given the observed boundary size is determined by the boundary's cross-sample complexity profile. A stable profile yields the usual beta law, whereas a varying profile produces an exact profile correction. The framework covers scalar order-statistic calibration, support-reconstructive scenario programs, cascaded support-removal certificates, coordinatewise envelopes, and Pareto-frontier calibration with vector scores. It also yields conditional probabilistic certificates and a no-go result explaining why observed complexity alone is insufficient.

eess.SY↗

arXiv2026

Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency

We study when and how momentum improves large-batch training in the one-pass regime, using power-law kernel regression as a tractable setting. We first characterize risk stability through the critical learning rate, defined as the largest learning rate for stable training, and obtain $η_{\mathrm{SGD}}^{\mathrm{crit}}\eqsim 1$, $η_{\mathrm{Polyak}}^{\mathrm{crit}}\eqsim \min\{1,B(1-ρ)\}$, and $η_{\mathrm{Nesterov}}^{\mathrm{crit}}\eqsim \min\{1,B^β(1-ρ)\}$, where $B$ is the batch size, $ρ$ is the momentum factor, and $β>1$ is the capacity exponent. Within this admissible region, we derive scaling laws for the full risk dynamics, capturing the progression from an early transient, through power-law decay, to a noise floor. We then minimize the final-step risk over the admissible learning rates and momentum factors under a fixed data budget, yielding a three-regime batch-size phase diagram that reveals how the role of momentum changes with batch size. Notably, Polyak enlarges the critical batch size, the largest batch size preserving the best small-batch data-scaling exponent, thereby enabling greater parallelism without sacrificing data efficiency. In contrast, Nesterov achieves better data efficiency in the large-batch regime because its look-ahead mechanism suppresses noise accumulation. Numerical experiments validate the predicted stability boundaries, risk dynamics, and batch-size phase diagram.

stat.ML↗

arXiv2026

Extremum Seeking Control: Three Revolutions and the Road Ahead

The history of extremum seeking is not merely the history of an algorithm; it is the history of an idea. Few ideas in control engineering have demonstrated the remarkable longevity of extremum seeking control (ESC). Invented more than one hundred years ago, ESC has continually reinvented itself while remaining faithful to its original objective: enabling systems to optimize their performance without relying on accurate mathematical models. This perspective article proposes that the evolution of ESC is best understood through three scientific revolutions. The first Engineering Revolution (1922-1999) established the engineering principles of model-free optimization; the second Mathematical Revolution (2000-2010) provided the rigorous mathematical foundations that transformed ESC into a mature discipline of nonlinear control; and the third ongoing Infinite-Dimensional and Cyber-Physical Revolution (2010-present) continues to expand its scope toward delays, partial differential equations, distributed optimization, event-triggered implementations, and increasingly complex cyber-physical systems. Beyond recounting this historical evolution, we offer a personal perspective on why ESC has remained relevant across successive technological eras. We argue that its enduring influence arises because the fundamental engineering challenge has never changed: "how can a dynamical system learn to improve its own performance when the optimum is unknown?" As optimization, learning, and feedback control become increasingly intertwined, ESC appears uniquely positioned to contribute to a new generation of intelligent autonomous systems, pointing toward what may become the field's fourth scientific revolution.

math.OC↗

arXiv2026

Aspiration-based Perturbed Learning Automata in Games with Noisy Utility Measurements. Part A: Stochastic Stability in Non-zero-Sum Games

Reinforcement-based learning has attracted considerable attention both in modeling human behavior as well as in engineering, for designing measurement- or payoff-based optimization schemes. Such learning schemes exhibit several advantages, especially in relation to filtering out noisy observations. However, they may exhibit several limitations when applied in a distributed setup. In multi-player weakly-acyclic games, and when each player applies an independent copy of the learning dynamics, convergence to (usually desirable) pure Nash equilibria cannot be guaranteed. Prior work has only focused on a small class of games, namely potential and coordination games. To address this main limitation, this paper introduces a novel payoff-based learning scheme for distributed optimization, namely aspiration-based perturbed learning automata (APLA). In this class of dynamics, and contrary to standard reinforcement-based learning schemes, each player's probability distribution for selecting actions is reinforced both by repeated selection and an aspiration factor that captures the player's satisfaction level. We provide a stochastic stability analysis of APLA in multi-player positive-utility games under the presence of noisy observations. This is the first part of the paper that characterizes stochastic stability in generic non-zero-sum games by establishing equivalence of the induced infinite-dimensional Markov chain with a finite dimensional one. In the second part, stochastic stability is further specialized to weakly acyclic games.

cs.LG↗

arXiv2026

Joint Network-and-Server Congestion in Multi-Source Traffic Allocation: A Convex Formulation and Price-Based Decentralization (Extended Version)

This paper studies an important rate allocation problem that arises in many networked and distributed systems: steady-state traffic rate allocation from multiple sources to multiple service nodes when both (i) the access-path delay on each source-node route is rate-dependent (capacity-constrained) and convex, and (ii) each service node (also capacity-constrained) experiences a load-dependent queueing delay driven by aggregate load from all sources. We show that the resulting flow-weighted end-to-end delay minimization is a convex program, yielding a global system-optimal solution characterized by KKT conditions that equalize total marginal costs (a path marginal access term plus a node congestion price) across all utilized routes. This condition admits a Wardrop-type interpretation: for each source, all utilized options equalize total marginal cost, while any option with strictly larger total marginal cost receives no flow. Building on this structure, we develop a lightweight distributed pricing-based algorithm in which each service node locally computes and broadcasts a scalar congestion price from its observed aggregate load, while each source updates its traffic split by solving a small separable convex allocation problem under the advertised prices. Numerical illustrations demonstrate convergence of the distributed iteration to the centralized optimum and highlight the trade-offs induced by jointly modeling access and service congestion.

cs.DC↗

arXiv2026

Distribution-Agnostic Robust Trajectory Optimization via Chance-Constrained Reinforcement Learning

This paper presents a distribution-agnostic robust trajectory-optimization framework based on chance-constrained reinforcement learning. The uncertainty is represented here through initial conditions and process noise, with the only requirement being that it can be sampled. A deterministic nominal trajectory is first computed offline, and reinforcement learning is then used only to robustify that baseline through a structured affine closed-loop correction law comprising a feedforward control adjustment and time-varying feedback gains. Probabilistic feasibility is enforced empirically through rollout-based upper-tail quantiles, while terminal dispersion is regulated through covariance-feasibility penalties. The framework is assessed on two materially different trajectory design problems. The flagship case study is a three-dimensional multi-impulse Earth-Mars transfer, where the learned policy is benchmarked against a recent robust trajectory-optimization reference under Gaussian uncertainty and then evaluated under bounded uniform uncertainty and under process disturbances not seen during training. The second case study is a stochastic atmospheric pinpoint rocket landing problem, used to assess portability to a short-horizon continuous-thrust setting with drag, mass depletion, and glide-slope constraints. The results show that the proposed framework can remain competitive in upper-tail fuel cost while preserving probabilistic feasibility, and that the same robustification scaffold can be carried across heterogeneous spacecraft trajectory planning problems without redesign of its core stochastic-control structure.

math.OC↗

arXiv2026

Infrequent Resolving Algorithm for Online Linear Programming

Online linear programming (OLP) has gained significant attention from both researchers and practitioners due to its extensive applications such as online auctions, network revenue management, order fulfillment and advertising. Existing OLP algorithms fall into two categories: LP-based algorithms and LP-free algorithms. The former typically guarantees better performance but requires solving a large number of LPs, which could be computationally expensive. In contrast, LP-free algorithms only require first-order computations but induce a worse performance. In this work, we bridge the gap between these two extremes by proposing a well-performing algorithm that solves LPs at a few selected time points and conducts first-order computations at other time points. Specifically, for the case where the inputs are drawn from an unknown finite-support distribution, the proposed algorithm achieves a constant regret (even for the hard "degenerate" case) while solving LPs only $O(\log\log T)$ times over the time horizon $T$. Moreover, when we are allowed to solve LPs only $M$ times, we design the corresponding schedule such that the proposed algorithm can guarantee a nearly $O\left(T^{(1/2)^{M-1}}\right)$ regret. Our work highlights the value of resolving both at the beginning and the end of the selling horizon, and provides a novel framework to prove the performance guarantee of the proposed policy under different infrequent resolving schedules. Numerical experiments are conducted to demonstrate the efficiency of the proposed algorithms.

cs.DS↗

arXiv2026

Turnpike properties in nonlinear system identification

We analyze the problem of learning general discrete-time nonlinear state-space models using the simulation error minimization (SEM) method. In this setting, model parameters are typically learned by minimizing the mismatch between simulated and measured outputs over a training dataset, or shorter subsequences extracted from it. Specifically, we study the cumulative output turnpike property of the underlying SEM optimization problem, which requires optimal output sequences emanating from a fixed initial state to approach and remain close to an optimal output sequence of the corresponding SEM problem with free initial state. In the presence of non-unique optimal output sequences---as may arise, for instance, in fully black-box system identification using neural networks---the property is formulated with respect to the closest such sequence. Turnpike behavior is generally desirable in practice, as it provides a theoretical justification for employing computationally more tractable SEM formulations with fixed initial states while ensuring that their optimal output sequences remain close to unconstrained optimal ones. Under a mild reachability condition, we establish equivalence between the cumulative turnpike property, coercivity of the value function, and a tailored notion of strict dissipativity. We additionally introduce a cardinality turnpike property and show that it is strictly weaker than the cumulative notion. Finally, we establish sufficient conditions for turnpike behavior based on incremental output stability, convexity of the stage cost, and a suitable optimality condition, and illustrate the theory by means of a numerical example.

eess.SY↗

arXiv2026

No-Regret Bayesian Optimization with Finite-Library Input-Warped Kernels

Gaussian-process Bayesian optimization (GP-BO) excels at black-box optimization of costly functions, e.g., hyperparameter optimization (HPO) and multi-agent system (MAS) design. Convergence-rate guarantees exist for select methods, notably GP upper confidence bound (GP-UCB), but require a fixed kernel. Critically, the kernel encodes how input proximity affects objective value similarity. When raw coordinates poorly match this geometry - as with log-scaled hyperparameters or localized peaks - input warping can greatly improve sample efficiency, yet known GP-UCB proofs require a fixed kernel. We propose Finite-Library Input-Warped Bayesian Optimization (FLIWBO), which selects warps from a finite library of smooth input maps by any history-dependent rule. It adapts the input geometry to accelerate learning while retaining high-probability convergence guarantees under mild hypotheses, with an explicit $\sqrt(N_\varepsilon)$ library-size cost. Controlled diagnostics show that finite-library warping repairs planted geometry mismatches and identify FLIWBO failure cases. Across four repeated benchmarks - warped synthetic objectives, a confidence-fence trap, and Fashion-MNIST HPO - FLIWBO-UCB beats raw-coordinate GP-UCB under misspecified geometry, escapes traps that defeat even oracle-warp expected improvement, and recovers much of the gain from manual log scaling, while leading the tested methods that admit a matching regret guarantee. A 20-dimensional MAS design study further shows feasibility under costly noisy evaluations. Code for experiments is available: https://github.com/edvin-ketabati/bogp-paper-experiments.

cs.LG↗

arXiv2026

Learning-Assisted Congestion-Aware Route Scheduling for Semiconductor Fab Material Control Systems

Automated material handling systems in semiconductor fabs are operated by a material control system (MCS) that must schedule a relay route for every transport command online, before execution. This is a data-driven scheduling problem in which route cost is dominated in the upper tail by queueing at heterogeneous, partially observable relay equipment, so route selection requires estimating both delivery time and congestion risk at the decision moment. This paper proposes a transport-network-aware dynamic congestion representation (TN-DCR). Built on a static directed transport graph induced by historically observed relay segments, TN-DCR combines structural route priors, multi-window network-wide congestion context, route-level bottleneck exposure, and an inductive graph-aware route embedding, all constructed under a prediction-time-safety invariant that admits only information observed strictly before the prediction moment. The representation feeds separate queue- and transfer-time regressors and an ordinal multi-label classifier producing calibrated multi-threshold exceedance scores, with an empirical-Bayes stock-key residual correction reducing systematic queue-time underprediction. The predictions serve as costs in a risk-constrained route-scheduling rule that minimizes predicted delivery time subject to a bound on extreme-congestion probability, embedding the learned predictors within a lightweight operations-research decision model. In a controlled closed-loop evaluation, mean delivery time falls by 16.4\% and internal resource waiting time by 22.6\% while throughput remains essentially unchanged.

cs.AI↗

arXiv2026

Modeling of Network Constraints in Large-scale Capacity Expansion Optimization of Power Grids

Capacity expansion modeling plays a critical role in optimizing the deployment of new generation, storage, and transmission, typically at national and regional levels. To support long-term planning, these models consider a large set of energy technologies and policies, along with decades of weather and demand data. Realistic capacity expansion models thus become high-dimensional optimization problems, with hundreds of millions of variables and constraints, which are challenging to solve. A common strategy to address this complexity is to omit non-linear, non-convex AC optimal power flow (ACOPF) constraints and instead use linearized power balance equations or transport formulations. While these simplifications improve tractability, they limit our understanding of how power flow and the physical properties of power networks impact investment decisions across generation, storage, and transmission infrastructure. This paper addresses this gap by extending the GenX capacity expansion model to incorporate fixed point theorem-based network constraints. These embed ACOPF-based considerations while maintaining the tractability of the planning model, nearly preserving the dimensionality of the transport formulation and incurring only modest runtime increases. This approach is much cheaper than embedding ACOPF directly, making it appropriate for large-scale capacity planning problems. We compare our approach to the original transport-based GenX model as well as a non-linear, non-convex version that incorporates the full ACOPF constraints, for a case study of the ISO New England grid.

math.OC↗

arXiv2026

The Multiple Timescales of Gradient Descent on the Edge of Stability: A Perturbative Derivation of the Central Flow

The central flow of Cohen et al. (2025) is an empirically accurate continuous-time model of gradient descent at the edge of stability in deep learning, However, its derivation is heuristic. We propose a perturbative regime in which the central flow is the limit of gradient descent: we assume that the loss decomposes as $f = g + \varepsilon h$; in the limit $\varepsilon \to 0$, the dynamics of gradient descent with learning rate $η$ converge to the gradient flow of $h$ constrained to the minimizers of $g$ of sharpness at most $2/η$. Our approach is formal rather than rigorous; it treats gradient descent as a singularly perturbed dynamical system in $\varepsilon$. Three timescales emerge: a fast timescale of oscillations along the sharpest direction, an intermediate timescale of the self-stabilization mechanism, and a slow timescale of the dynamics along the minimizers of $g$-the central flow. Using the method of multiple scales, a classical formal method from singular perturbation theory, we derive the expansion of the dynamics in $\varepsilon$: the central flow emerges as the leading-order term in the expansion, while the self-stabilization mechanism appears in the next-order term. We study this mechanism beyond previous analyses: with a single eigenvalue at the edge of stability, we compute the slow drift of the energy of the fluctuations; with several eigenvalues at the edge of stability, we derive the self-stabilization system and explain why fluctuations persist.

cs.LG↗

arXiv2026

Centered Permutation Prefixes for SGD with Random Reshuffling: Sharp Rates, Hölder Geometry, and Composite Proximal Extensions

We study stochastic gradient descent with random reshuffling for finite sums \[ F(x)=\frac1n\sum_{i=1}^n f_i(x). \] For fresh reshuffling with a constant component stepsize, if each $f_i$ has an $L$-Lipschitz gradient and the average $F$ is $μ$-strongly convex with a Lipschitz-continuous Hessian, we prove the last-epoch rate \[ \mathbb E[F(y_K)-F(x_\star)] =\widetilde O\!\left(T^{-2}+n^2T^{-3}\right), \qquad T=nK, \] matching the known quadratic lower bound in its $(n,K)$-dependence. The components may be nonconvex, and no componentwise Hessian continuity or separate bounded-iterate assumption is required. More generally, a $ν$-Hölder-continuous average Hessian adds only $\widetilde O(n^{1+ν}T^{-2-2ν})$, so every $ν\ge 1/2$ preserves the quadratic rate. Under convex components, a decreasing-stepsize result removes the large-epoch requirement and recovers the same two-term scale once $nK$ exceeds the condition-number scale. We also analyze epoch-wise ProxRR for $\mathcal P=F+ψ$. Writing $x^\dagger$ for the composite minimizer and $β_\star=\|\nabla F(x^\dagger)\|$, we prove \[ \mathbb E\|y_K-x^\dagger\|^2 =\widetilde O\!\left( \frac{β_\star^2}{K^2} +T^{-2}+n^2T^{-3} +n^{1+ν}T^{-2-2ν} \right). \] For $ν\ge 1/2$, we show that the $β_\star^2/K^2$ splitting term is unavoidable and obtain a matching lower bound up to logarithms in the stated constant-stepsize regime.

math.OC↗

arXiv2026

Scaled Null-Adjusted Persistence: A Multiscale Bridge between Modularity and Persistence

Community detection methods must balance two competing objectives: identifying small, cohesive groups while avoiding excessive fragmentation. Modularity, the most widely adopted optimization criterion, typically merges small communities in large networks due to its resolution limit. In contrast, a persistence-based criterion promotes more granular partitions. We introduce Scaled Null-Adjusted Persistence (Scaled-NAP), a parametric family of quality functions that incorporates both these criteria. The definition exploits the exact identity between a cluster's modularity contribution and its Null-Adjusted Persistence (NAP) multiplied by its relative volume. Raising this volume factor to a parameter $α\in[0,1]$ yields NAP at $α=0$ and modularity at $α=1$, while intermediate values control the scale of the detected partition. We derive conditions under which merging two communities improves the objective function and characterize the emergence of scale dependence, including resolution-limit behaviour on Caveman graphs. We develop the Milano algorithm, a multilevel Louvain-style heuristic for optimizing Scaled-NAP on large networks. Experiments on weighted and unweighted Lancichinetti-Fortunato-Radicchi benchmarks show that Scaled-NAP achieves the highest or tied-highest recovery wherever the ground truth structure is detectable, with its advantage increasing under community-size heterogeneity. Tests on three real networks with up to 1.1 million nodes confirm its capability to identify fine-grained ground-truth communities. The Milano algorithm also turned out to be the fastest method evaluated on large networks. These results show that Scaled-NAP provides an effective and scalable bridge between modularity-based and persistence-based community detection methodologies.

cs.SI↗

arXiv2026

B$^3$-PWL: GPU-Batched Branch-and-Bound for Piecewise-Linear Optimization with SOS2 Constraints

Piecewise-linear (PWL) optimization problems arise in many mixed-integer programming (MIP) optimization applications, including portfolio optimization, workforce scheduling, and resource allocation. But solving them to global optimality remains computationally expensive because branch-and-bound repeatedly solves LP relaxation subproblems. Existing solvers are largely CPU-centric, leaving the scalability of modern GPUs underutilized. Few prior GPU-accelerated branch-and-bound either targets neural network which is not suitable for general PWL optimization, or accelerates only auxiliary subroutines such as strong branching heuristics within CPU-centric MIP solvers. To bridge this gap, we propose B$^3$-PWL, a GPU-centric batched branch-and-bound framework for piecewise-linear optimization with Special Ordered Set of type 2 (SOS2) constraints. Our method solves batches of LP relaxation subproblems concurrently on the GPU using a first-order primal-dual solver, enabled by a specialized batched block-tiled sparse matrix kernel. To complement bound computation, we further introduce a unified feasibility search module that combines an SOS2 repair primal heuristic with a batched feasibility pump to rapidly obtain feasible incumbents and improve pruning efficiency. On a benchmark of 43 PWL-MIP instances, B$^3$-PWL achieves a 9.25x geometric-mean speedup over NVIDIA cuOpt while reaching high-quality feasible incumbents on every tested instance. On a public valve-point unit-commitment benchmark, it further outperforms NVIDIA cuOpt and the open-source CPU solvers SCIP and HiGHS, demonstrating the potential of first-order LP methods as the central engine of GPU-accelerated branch-and-bound.

math.OC↗

arXiv2026

Conformal Risk-Averse Decision Making with Optimized Certainty Equivalent Risk Control

We study risk-averse decision making, in which an agent selects actions while being uncertain about the true system state. The risk is measured via optimized certainty equivalent (OCE) metrics, which generalize popular criteria such as mean-variance risk and conditional value-at-risk (CVaR). We characterize the optimal policy under known distributions, and show that it reduces to a prediction set-based solution for the CVaR. This provides an operational interpretation of conformal prediction-type prediction sets. For unknown distributions, we develop a data-driven calibration strategy, based on a synthetic model for the likelihood and held-out calibration data, yielding high-probability control of the OCE risk. The approach is evaluated on two wireless beamforming settings.

stat.ML↗