Papers

This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance with respect to the current Nash equilibrium (NE). To address this issue, we introduce a zero-sum game regret function with a forgetting factor, which assigns exponentially decaying weights to past saddle gaps and emphasizes recent performance. This metric directly links regret minimization to the tracking of time-varying NEs. Within this framework, we investigate three online algorithms under different computational and information settings. For first-order feedback, we analyze projected gradient descent-ascent and design a projection-free online Frank-Wolfe method to reduce the computational cost of projections. For zeroth-order feedback, we develop a deterministic finite-difference method that only uses function-value queries. For all three algorithms, we establish forgetting-factor regret bounds that explicitly characterize the effects of NE variation, payoff variation, and gradient-estimation error. We further provide sufficient conditions under which the proposed regret converges to zero, thereby certifying asymptotic tracking of time-varying NEs. The numerical example validates the theoretical results and illustrates the tracking advantage of the proposed regret metric.

This paper introduces the weak augmented Zarankiewicz number $z_{wA}(m,n)$ and the weak limited augmented Zarankiewicz number $z_{wL}(m,n)$, which are combinatorial extensions of the classical Zarankiewicz number obtained by relaxing the original admissibility conditions for augmented bipartite graphs. We show that the resulting weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields the inequality chain \[ \mathrm{BSR}(m,n) \geq z_{wA}(m,n) \geq z_{wL}(m,n) \geq z_L(m,n) \geq z(m,n). \] We provide three complementary constructions demonstrating the power of the weak framework. First, a $5\times 3$ construction using degenerate 2-edges yields $z_{wL}(5,3)\ge 10>9=z_L(5,3)$, giving $\mathrm{BSR}(5,3)\ge 10$. Second, a $15\times 6$ construction on the incidence graph of $K_6$ with 14 nondegenerate 2-edges gives $z_{wL}(15,6)\ge 44>43$, improving the previously known bound. Third, a critical $6\times 3$ construction with complementary 2-cycles gives $z_{wL}(6,3)\ge 12>11=z_L(6,3)$, yielding $\mathrm{BSR}(6,3)\ge 12$ and demonstrating that complementary 2-cycles are safe.

Model Predictive Path Integral (MPPI) control is directly implementable on nonlinear systems because its online update requires only forward rollouts of the dynamics, not gradients, linearizations, or convex optimization. However, this algorithmic flexibility does not by itself provide a closed-loop stability certificate. This paper establishes such a certificate through a stability-inheritance argument. We assume that there exists a deterministic nonlinear MPC policy whose disturbance-free closed loop is certified by a Control Lyapunov Function terminal cost and a contraction metric, and we show that finite-sample MPPI inherits the nominal contraction when its sampling-based update approximates this reference policy with sufficient accuracy. The approximation error decomposes into a finite-temperature bias floor and a Monte Carlo term that vanishes at the inverse square-root rate in the sample count. Under an explicit small-gain condition, the resulting MPPI closed loop satisfies a finite-horizon, high-probability localized mean practical stability bound with residual floors due to MPPI approximation error, Gaussian process noise, and bad sampling events. The paper also gives an ISS-type restatement and a finite-horizon design procedure for choosing the localization set, temperature, and sample count.

Reinforcement learning (RL) is increasingly grounded in tools from probability, optimization, and operator theory. This survey organizes the mathematical structures that underpin the design and analysis of modern algorithms in RL. We begin from Markov decision processes (MDPs) and the Bellman operators, emphasizing contraction mappings, monotonicity, and fixed-point theory that yield convergence guarantees and rates for value and policy iteration, and temporal-difference schemes. We then develop the optimization perspective: stochastic approximation and martingale methods, convex duality and the role of regularization linking mirror/proximal methods. Function approximation is treated through linear and non-linear settings, covering stabilization, error decomposition, and sample-complexity via concentration inequalities for dependent data and mixing processes. We further cover off-policy evaluation/learning, constrained RL and constrained MDPs (CMDPs). Throughout we unify algorithmic templates under common operator and variational lenses, highlighting both finite-sample bounds and asymptotic results. Our presentation is intended to provide a unified mathematical entry point for researchers in probability, optimization, and statistics interested in reinforcement learning.

We consider the problem of finding stationary points for stochastic convex optimization problems. Rather than surrogates to stationarity, such as a proximity-to-stationarity guarantee or small gradient of the Moreau envelope, we ask for a stronger notion: that the subdifferential of the objective actually contains a small element. This criterion is non-trivial, because subdifferentials of convex functions fail to converge uniformly, even in arbitrarily small neighborhoods of the optimum. Our convergence guarantees rely on dimension theory to decompose the graph of the subdifferential of a convex function, showing how stochastic sampling preserves "pieces" of these graphs, and allowing effective application of proximal-point-like methods.

This paper addresses the problem of non-adaptive global robust disturbance rejection for two distinct classes of nonlinear systems. The first class, denoted by C1, consists of nonlinear systems in strict-feedback form, with linear and Hurwitz zero-dynamics (whose states are unavailable for feedback), and enhanced - within this work - by forcing, unmatched, additive disturbances. Nonlinear tools are herein employed to demonstrate that the proposed control architecture - based on the high-gain paradigm - achieves closed-loop input-to-state stability with respect to the forcing disturbances, along with global asymptotic convergence towards an attractor which can be rendered as small as desired. Then, owing to the established input-to-state stability property, a uniformly bounded control action is additionally embedded within the control architecture. The additional unit, designed following the sliding-mode paradigm, is aimed at improving the disturbance rejection task, by potentially lowering the required high-gain control expenditure. The second class of systems, denoted by C2, is constituted by minimum-phase, uncertain, nonlinear systems with relative degree greater than one, featuring possibly unbounded, with possibly unbounded derivatives, output-dependent nonlinearities, with matched additive forcing disturbances. To solve the problem of output-feedback, non-adaptive, global robust disturbance rejection for systems within C2, first, an open-loop observer is employed in lieu of a classic dynamic extension adopted in earlier works, as the latter is no longer implementable due to the presence of unknown forcing disturbances. Subsequently, the results derived for C1 are then adapted to C2, to yield an output-feedback dynamic controller providing closed-loop global uniform boundedness, along with asymptotic regulation towards an attractor which can be rendered as small as desired.

A cheap swarm of unreliable agents can be steered to a correct consensus by a few strong, expensive "oracle" correctors. We ask how much one must spend, and where to place the oracles. We model the swarm as a consensus on a graph in which each oracle pins one node toward the truth at a cost-coupled, concave strength, and measure quality by the coherence H(R)=tr M(R)^{-1}. Our first result is that H stays submodular (each added oracle helps less than the last) even when the oracles differ in strength, so a cost-benefit greedy comes within 1-1/e of the best placement at any budget. Inverting the budget gives the budget-correctness frontier B*(eps), the least spend that guarantees an eps-correct consensus: closed-form on the complete graph, and a minimal oracle count k* when oracles cost the same. Whether a budget then buys a few strong oracles or many medium onese curvature of the cost-quality law: diminishing returns favour spreadsharply increasion. Measured onthe Qwen3 ladder (0.6-32B), the law is concave for math verificatio convex foremergent code tracing, so the verdict is genuinely task-dependent.https://github.com/YehudaItkin/budgeted-oracle-placemen

Sepsis is a leading cause of mortality, yet optimal treatment policies remain contested. Existing reinforcement learning (RL) approaches learn fixed strategies for sepsis treatment, limiting adaptability to changing clinical objectives during inference. We propose EHRMPC, a framework that decouples learning patient dynamics from optimizing treatment by training a patient digital twin in the form of a generative electronic health record (EHR) model. The digital twin predicts clinical trajectories under interventions and enables model predictive control (MPC) to optimize treatments via inference-time planning over simulations. We evaluate EHR-MPC on a multicenter ICU sepsis cohort spanning 8 hospitals in the Mass General Brigham health system using both off-policy importance sampling and on-policy simulation-based evaluation. Relative to RL baselines, EHR-MPC achieves comparable off-policy performance and improved simulation performance. Unlike RL, this work frames sepsis treatment optimization as inference-time control over learned patient dynamics, establishing a general framework for decision making with generative clinical models.

We propose a Jacobi-like relative value iteration (RVI) algorithm and a Gauss-Seidel-like implementation for the ergodic risk-sensitive control (ERSC) problem of a controlled discrete time Markov chain (DTMC) on a finite state space. Under the assumption that the DTMC is irreducible and recurrent under every stationary Markov policy, we prove that the iterates of the proposed RVI algorithms converge at a geometric rate. The main challenge stems from the multiplicative structure of the ERSC cost criterion and the associated Bellman-like operators, which prevents us from adapting the analogous global contraction and bi-Lipschitz continuity properties that underlie the proof of convergence in the average cost setting. We overcome this by establishing local contraction properties for the risk-sensitive Bellman-like operators and a local bi-Lipschitz continuity property for their fixed points, and use these properties to show the iterates converge geometrically. We conclude by implementing our proposed RVI algorithms on two examples: service effort control for a single-server queue of finite capacity, and maximizing the exit rate from a finite domain (on a graph).

Quick-commerce dark stores dispatch e-grocery orders within 15 to 30 minutes, so operators such as Getir, Glovo, and GoPuff must commit drivers before orders arrive. Demand follows a latent regime that persists across hours, while unfulfilled orders spill forward as a compounding backlog. This decision binds whether drivers are employed on fixed shifts or drawn from a gig platform whose incentives are set ahead of the hour, yet existing models do not learn the regime as orders arrive. We formulate the single-store problem as a partially observable Markov decision process in which the firm infers the regime from realized orders. We show that optimal staffing rises with backlog and prove the single-store problem is indexable, a property open for multi-action partially observed problems in general. We then extend the framework to a driver pool shared across stores through a Lagrangian relaxation that decouples the network into per-store subproblems. The result is a two-level allocation policy that prices the value of tracking demand in real time and ranks stores by a provably valid priority index. On 2021 to 2022 data from the European firm SuperGlovo, which staffs full-time drivers to comply with Spain's Rider's Law, belief updating adds 10.9%, about \$188K across 27 stores, over the same program with the belief held fixed. The gain concentrates in the stores whose regimes are most persistent, observable before deployment. Online allocation reduces to a table lookup and a ranked list with a cutoff price.

Power System State Estimation (PSSE) is commonly formulated as a nonconvex weighted least-squares (WLS) problem, making global optimality difficult both to attain and to certify. Recent work has shown that, when voltage magnitudes are known to sufficient accuracy, the remaining angle estimation subproblem can be reduced to phase synchronization and solved effectively using spectral initialization and spectral certification. This paper explains why these spectral methods succeed. We prove that their behavior is governed by the measurement error, normalized against the usual observability margin from the classical literature. Below a fixed threshold, spectral initialization recovers the true voltage angles to first-order accuracy, and the WLS estimator is unique up to a global phase. Moreover, a zero-duality-gap spectral certificate verifies recovery of this unique WLS estimate. In the noiseless observable regime, spectral initialization exactly recovers the true angles and certification is exact without local refinement. Numerical experiments on standard benchmark systems support the theory and show that the spectral methods remain effective beyond the conservative regime covered by the guarantees.

This paper proposes Variational Inference-based Bayesian Estimation with Sobol screening (VIBES), a two-stage scalable framework for Bayesian uncertainty quantification (UQ). The proposed approach combines Sobol global sensitivity analysis (GSA) for screening and dimensionality reduction, followed by variational inference (VI) for UQ of kinetic, design/operational, and economic parameters. In the first stage, Sobol GSA is performed to identify dominant variables and parameters governing uncertainty in process outputs. In the second stage, Bayesian inference is performed only on the reduced dimensional space using VI, thus reducing computational burden and enhancing scalability. The framework is demonstrated on a process for bioadhesive production through base-catalyzed depolymerization of kraft lignin and subsequent crosslinking with isolated soy protein. A Python-Aspen interface is developed for automated simulation and parameter estimation, enabling Bayesian calibration through stochastic gradient-based optimization and automatic-differentiation. The methodology is generic and readily generalizable to other biomass conversion pathways. The results show that application of VIBES consistently reduces predictive uncertainty bounds across all model outputs by more than 80%, even when only the reduced-space input variables and parameters are optimized during Bayesian estimation. The framework can be potentially applied for scalable, uncertainty-aware decision-making in high-dimensional, complex chemical process systems.

This paper studies linear-quadratic mean field games on compact Riemannian manifolds with a hybrid interaction topology. The network structure is a superposition of a deterministic graph for local geometric connectivity and a stochastic directed graph for non-local interactions. The global graph is constructed via random sampling based on a continuous kernel $K$. The out-degree of each node scales as $\Theta(\log N)$ or as $\Theta(N)$ to represent a sparse or dense network, respectively. In the infinite-population limit, the continuum system is governed by a coupled system of forward-backward partial differential equations, where the dynamics of the expected state incorporate the integral operator corresponding to the non-local sampling. The existence of a Nash equilibrium is established for this limit system. Furthermore, the approximation error is analyzed using operator concentration inequalities and analytic semigroup theory. Non-asymptotic high-probability error bounds between the finite-population empirical state and the continuum limit are derived. The convergence rates differ depending on the two topological regimes. Under the dense regime, the tracking error exhibits a polynomial decay rate dependent on the manifold dimension and Sobolev regularity, while under the sparse regime, the error decays at a rate of $\mathcal{O}((\log N)^{-1/2})$.

Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

We extend the linear programming-based algorithm of De Rosa et al~\cite{derKhaWan24} for K-means clustering to two important clustering paradigms: fair K-means clustering and spectral clustering. For fair K-means clustering, we show that widely used notions of group fairness can be incorporated into the partition-matrix formulation of K-means clustering through a linear number of linear inequalities. For spectral clustering, we consider a linear programming relaxation of the minimum ratio-cut problem that fits naturally within the same framework. We complement these formulations with problem-specific initialization and rounding procedures and evaluate the resulting algorithms on a large collection of real-world data sets. Denoting by $n$ the number of data points, our computational results demonstrate that the proposed approach solves $90\%$ of benchmark instances with $n \leq 3000$ to within $1\%$ optimality in at most three hours. This in turn demonstrates the remarkable strength of the proposed LP relaxations in both applications. Moreover, for more than $56\%$ of the instances, the proposed algorithm finds better solutions than those produced by popular fair Lloyd-type and spectral clustering heuristics.

Portfolio optimization under uncertainty is inherently a multi-objective decision problem involving complex interactions among return, risk, market dynamics, and practical investment constraints. Existing reliability based portfolio optimization approaches primarily rely on static optimization frameworks and often fail to capture sequential decision making, tail risk, and market frictions such as transaction costs. To address these limitations, we propose a deep reinforcement learning framework for multi-objective reliability based portfolio optimization (MORP-DRL). The proposed framework jointly optimizes expected return and downside risk using three complementary risk measures: variance, Conditional Value-at-Risk (CVaR), and Entropic Value-at-Risk (EVaR). To model uncertainty and heavy-tailed market behavior, asset returns are represented using GARCH(1,1), Extreme Value Theory, and a t-copula dependence structure, while realistic scenarios are generated through quasi-Monte Carlo simulation. A Proximal Policy Optimization (PPO) based strategy is developed under practical constraints including transaction costs and portfolio bounds, and is benchmarked against NSGA-II. Experiments on ten global equity indices across pre-COVID, COVID, and post-COVID market regimes demonstrate that MORP-DRL achieves competitive risk-return performance, reduced downside risk during periods of market stress, and scalability to high-dimensional portfolio settings.

Mixed-integer linear programming (MILP) instances used for solver development are hard to obtain when models come from private or application-specific pipelines. A generator must keep the structure that solvers and learned policies rely on. Existing general generators usually choose their generation unit from a formulation template, summary statistics, local graph edits, or blocks found after recombination. These units do not explicitly record how a local part of the MILP is coupled to the rest of the instance. We propose GraphBU, a graph-native generator whose basic unit is a local subproblem plus its interface. The method promotes coupling nodes into master constraints or boundary variables and uses the resulting block units for compatibility-checked replacement. The analysis focuses on the properties needed by this construction: promotion separates interfaces, replacement can preserve feasibility under an interface-slack condition, and the graph construction is invariant to row-column permutations. On MILP instances generation, this unit keeps graph statistics close to the source family, preserves feasibility on most datasets, and improves downstream Predict-and-Search training. Genrated by GraphBU, The average graph-statistical similarity was approximately 0.934, the average feasibility was approximately 96.7%, and the average increase in the main index of downstream PS was approximately 8.0%.

It has been known since the seminal work of Guillemin and Sternberg that Lie subalgebras of finite codimension of an arbitrary real or complex Lie algebra can be realized as subalgebras of formal vector fields over formal power series. In this note, we characterize the Lie subalgebras which admit a convergent realization in the sense of locally analytic vector fields. We give generalizations of these properties for the problem of output realization. We give reformulations and applications of these algebraic results in the context of control theory. In particular, we recover and clarify previous results on the realization of Chen–Fliess series for control-affine systems, the equivalence of control systems, and the existence of embedded or canonical systems.

Adaptive control learns the plant online; neural-operator control learns the control gains offline. We bring the two together for a class of nonlinear hyperbolic PDEs whose dynamics are governed by an unknown Volterra series of arbitrarily many kernels. An observer-based passive identifier learns a truncation of this series online. The infinite-dimensional map that synthesizes the backstepping kernels from the parameter estimates – a cascade of PDEs on simplex domains of increasing dimension, prohibitive to solve in real time – is approximated once, offline, by a neural operator. The closed loop then carries two learning processes in series: online learning of the plant feeds an offline-learned PDE solver, whose output is the online control gains. We prove closed-loop stability and asymptotic regulation of the plant state, observer state, and input, on a basin that recovers the exact-kernel basin as the neural-operator accuracy improves. With a single Lyapunov function we absorb at once the perturbations – all vanishing – of truncating an infinite Volterra series, of identifying the plant online, and of approximating the gains.

This paper investigates feature learning within the framework of the deep Ritz method for solving the stationary Schrödinger equation with Neumann boundary conditions. We first analyze the convergence of Riemannian gradient descent in an agnostic setting, where the hypothesis function is restricted to a single-index model while the PDE solution is arbitrary. We prove that gradient descent reaches an approximate global minimum: after T = O(log(1/\epsilon)) iterations, the loss is within \epsilon of a constant multiple of the optimal loss. We then examine the loss landscape when the source term of the PDE itself follows a single-index model, considering hypothesis functions given by either a single-index model or a two-neuron multi-index model. In the single-index case, we show that the minimum Ritz energy is attained at the feature vector aligned with that of the source term. In the two-neuron case, we study the landscape of regularized Ritz losses and characterize how a second feature emerges, given that the first feature is aligned with the source, as the regularization parameter varies. Finally, numerical experiments are presented to validate the feature emergence theory in the two-neuron setting.

Branch-and-cut interleaves branching with cutting-plane generation. How the two operations share the work of proving a bound is a basic theoretical question. We study an abstract model in which a tree certifies a target bound $Z$. Each branch node improves the bound by $\ell$ on one child and by $r$ on the other, where $0<\ell\le r$. The $i$th cut along a root-to-node path improves it by $c_i\ge0$, with cumulative improvement $C_k=\sum_{i=1}^k c_i$. Asymmetric branching enters through the rate $\lambda^{\star}>0$ defined by $e^{-\lambda^{\star}\ell}+e^{-\lambda^{\star}r}=1$. We establish uniform two-sided bounds of order $e^{\lambda^{\star}Z}$ on the minimal leaf count of pure branching trees. We then identify $\log k$ as the sharp threshold scale for the power of cutting. For cut schedules with extended limit $\gamma=\lim_{k\to\infty}C_k/\log k\in[0,\infty]$, minimal-size trees obey a trichotomy. If $\gamma=\infty$, cuts prove asymptotically all of the target. If $0\le\gamma<\infty$, the limiting fraction of the bound proved by cuts is $\gamma\lambda^{\star}/(1+\gamma\lambda^{\star})$. If $\gamma=0$, branch-and-cut has the same exponential size rate as pure branch-and-bound. This resolves open questions raised by Kazachkov, Le Bodic, and Sankaranarayanan on minimal-size trees under harmonically-worsening cuts, and generalizes their results to asymmetric branching and to all cut schedules in the model with this logarithmic limit. Finally, we show that branch-and-cut attains polynomial size in terms of $Z$ if and only if polynomially many cuts reduce the residual bound to $O(\log Z)$.

In non-convex markets, a competitive equilibrium may fail to exist. This turns out to be an important issue in real-world non-convex auction markets, such as electricity markets, as it complicates pricing and requires the auctioneer to resort to out-of-market discriminatory side payments to sustain an equilibrium. We investigate whether the introduction of convex financial trading induces a smoothing effect, mitigating the issues arising from non-convexities. We develop a two-stage non-convex market model (a forward market followed by a spot market) in which convex financial traders participate in the forward market. Our model predicts that financial trading reduces the magnitude of side payments required to support the cleared allocation. To test the prediction of our model, we examine the introduction of a transaction fee on financial traders in 2020 by PJM, the US's largest electricity market. We show that the substantial decline in financial trading volume caused by this policy coincided with a significant increase in side payments, in line with our theoretical predictions.

Let $f:\mathbb{R}^d \rightarrow \mathbb{R}$. We consider gradient descent $x_{n+1} = x_n - \tau_n \nabla f(x_n)$, where the stepsize $\tau_n = \tau \cdot e^{rn}$ is exponentially growing (with $\tau > 0$ and $0 < r \ll 1$). This diverges for almost all initial values. We show that restarting the algorithm whenever $\|x_{n+1} - x_n\| \geq e^r\|x_n - x_{n-1}\|$ has good properties: it works very well in practice; we determine the limiting convergence rate in the case of convergence to a non-degenerate local minimum: it improves on classic gradient descent even though computational cost is comparable. The precise choice of $0 < r \ll 1$ does not matter much and the method is virtually independent of an initial stepsize $\tau$ that is too small: while the convergence rate for gradient descent decays linearly as $\tau \rightarrow 0$, it decays as $1/\log(1/\tau)$ in this modified version; numerical examples illustrate the results.

In this paper, we study the optimal control problems for stochastic Volterra integral equations driven by G-Brownian motion under Volatility Ambiguity. With the help of G-stochastic analysis techniques and the weak convergence methods, we obtain the variation of the cost functional and the variational inequality. Under the convexity assumptions, we establish the stochastic maximum principle, which serves as both a necessary and sufficient condition for optimal control.

We study a human-AI service system in which tasks arrive sequentially and are processed through a two-stage architecture: an automated chatbot followed, when necessary, by a human agent. We consider $T$ sequentially arriving tasks, each belonging to one of $K$ heterogeneous types. For each task the decision maker chooses how many resources to allocate to the chatbot, whose type-dependent success probabilities are initially unknown. Tasks not resolved by the chatbot enter type-dependent human-service queues, where they are processed by a human agent with unknown service rates. This model captures a central tradeoff in hybrid service systems: relying more on automation reduces human congestion but increases chatbot costs, while insufficient automation may overload the human agent. We propose the UCB-DPP policy, which combines Upper Confidence Bounds with Drift-Plus-Penalty control to learn the unknown parameters of the system while making queue-aware decisions. We prove that UCB-DPP achieves regret $\widetilde{\mathcal{O}}(K\sqrt{T})$ and guarantees mean-rate stability of the human-service queues. Simulations on synthetic instances show that the proposed policy outperforms natural baselines.

Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S_t^{-1/2} normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-epsilon crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.

Empirical risk minimization is a standard and effective paradigm for learning predictive models by minimizing average loss. In high-stakes decision-making, however, an average-loss criterion may underrepresent rare but severe losses. Spectral risk measures (SRMs) provide a principled framework by incorporating weighted order statistics of losses, but the induced nonsmoothness and nonseparability from sorting make the resulting optimization problems challenging. We propose a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems. Exploiting a dual reformulation and properties of the Moreau envelope, we reduce the subproblems to structured dual-variable formulations, significantly simplifying computation. We provide explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation. Numerical results on synthetic and real-data instances show that the proposed method attains lower running times than the tested ADMM baseline while producing comparable stationarity residuals and sparse solutions.

The Picard iteration converges to the unique fixed point of a $q$-contractive operator at a linear rate $q^N$, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.

We use symmetric gauge theory to develop a general class of cutting-plane algorithms for semidefinite programming. We formulate a separation problem based on spectral normalizations induced by gauges and derive a closed-form separation oracle. This oracle yields an implementable cut-generation procedure that, by varying the gauge, recovers standard cut families and generates new ones with tunable spectral structure. We embed the oracle within Kelley's method and characterize convergence as a function of the chosen gauge and initial conic relaxation. Numerical experiments on small and large instances of box-constrained quadratic programming and sparse principal component analysis illustrate the versatility and performance of the proposed framework.

Memory-efficient optimizers such as GaLore train large language models by projecting gradients onto a rank-r subspace recomputed every T steps, assuming this subspace is a slowly drifting object that can be tracked. We show that beyond a small reproducible core, there is no such object. Two estimates of the top-r subspace computed at the same step from disjoint minibatches disagree as much as estimates computed T steps apart (0.73 vs 0.74 of the maximal chordal distance sqrt(2r), at Pythia-160M with r=128): the apparent rotation at each refresh is dominated by estimator noise. This holds across four model families in three architecture classes from 70M to 6.9B parameters, strengthening with scale, and more weakly in a vision transformer. Only ~39 of 128 directions are reproducible across minibatches, and averaging cannot recover the rest: under N-fold averaging the gradient's spectral tail shrinks as N^(-1/4) rather than the N^(-1/2) of pure noise, so no averaging budget makes the subspace well defined. What helps instead follows from treating each refresh as a change of coordinates for Adam's state. Carrying the second moment blindly is provably about (r-k*)/2 worse than the best rotation-blind estimator, while the first moment transports exactly through the rotation, the optimal linear map under isotropic gradients and the rule LDAdam uses. At 1B over 40k steps (3 seeds), full LDAdam reaches 18.7 perplexity at beta2=0.999, beating untransported GaLore after its best beta2 fix (19.3); shortening the second-moment memory to beta2=0.99 helps the refreshing optimizers, though for canonical GaLore the effect is small and a full-rank control reverses it. One measurable fact, subspace non-identifiability, clarifies why GaLore works, which patches work, and what to check before trusting a low-rank assumption: the reproducible rank k*.