Solving optimal classification tree (OCT) problems is computationally challenging because branching, routing, and leaf-assignment decisions must be optimized jointly. We develop an incumbent subtree column generation algorithm (ISCG-OCT) that exploits the recursive structure of classification trees. We first introduce a mixed set-packing and set-partitioning formulation (MSSF). We then construct a restricted master problem whose columns represent feasible subtree patterns. Unlike conventional dual-based column generation, our incumbent-centered pricing evaluates objective improvements relative to the current solution. We establish a recursive decomposition in which fixing a branching decision separates the remaining pricing problem into independent subtree subproblems, enabling exact dynamic programming. We also develop threshold-based variants that trade optimality guarantees for computational efficiency. Computational experiments on 27 benchmark datasets demonstrate competitive training and testing accuracy and substantial reductions in solution time on many instances. When initialized with a CART solution, ISCG-OCT can certify global optimality faster than warm-started mixed-integer programming solvers on selected instances. These results demonstrate how recursive pricing and structured subtree optimization can improve the computational efficiency of exact classification tree methods.
AdamW is ubiquitous in deep learning, yet its behavior remains poorly understood. We analyze its dynamics through the lens of dynamical systems and show that AdamW admits an implicit fixed-point objective: its fixed points coincide with the stationary points of a constrained and regularized optimization problem. However, not all of these fixed points are stable under AdamW's dynamics, and stability depends sensitively on curvature, weight decay, and momentum parameters. Even in one-dimensional settings, equilibria may be unstable, and trajectories can exhibit persistent oscillations rather than convergence to them. We further extend the analysis to higher dimensions, deriving sufficient conditions for local stability of the continuous-time dynamics, and propose a convergent modification that preserves AdamW's fixed points. These results clarify what optimization problem AdamW is associated with, when its convergence can be expected, and how its dynamics could inspire more reliable optimizers.
Single-loop minimax methods are appealing for modern machine learning, but standard stochastic gradient updates can be unstable and oscillatory under tightly coupled primal-dual dynamics, making performance notoriously sensitive to stepsizes. To fundamentally improve stability, we develop an orthogonal update framework tailored to minimax optimization, which applies updates along orthogonalized directions to reduce update anisotropy. However, orthogonalization reshapes the optimization geometry and makes coordinating primal-dual stepsizes even more delicate. To overcome this challenge, we develop AdaSGDA, which equips orthogonal updates with an adaptive stepsize rule driven by accumulated gradient norms. This mechanism automatically balances coupled primal-dual progress under orthogonalization, yielding an effectively parameter-free method for nonconvex-strongly-concave minimax optimization. We prove that AdaSGDA avoids problem-dependent tuning while attaining the state-of-the-art $\mathcal{O}\left(T^{-1/4}\right)$ convergence rate. We further propose AdaSGDA-VR, which incorporates a carefully designed variance reduction scheme and achieves the faster $\mathcal{O}\left(T^{-1/3}\right)$ rate. Extensive experiments demonstrate that our methods deliver competitive performance compared to existing minimax optimization methods.
Risk-sensitive optimal stopping is useful when poor stopping outcomes carry substantial costs, as in option exercise, sequential medical treatment, and safety-critical control. A natural objective is Conditional Value at Risk (CVaR) of the stopped reward. We develop a theoretical framework and algorithm for CVaR deep optimal stopping. The framework and algorithm address four difficulties in applying risk-sensitive reinforcement learning to optimal stopping: state augmentation, score-function pathology, blindness to success, and dependence on auxiliary networks. We isolate three structural properties of optimal stopping: single-point reward, action-independent dynamics, and sum-zero gradient. We exploit these properties to establish Markov optimality without state augmentation and derive a pathwise gradient estimator. Under bounded policy derivatives, the pathwise estimator has bounded second moments near deterministic policies, avoiding the divergence that can occur with score-function estimators. Building on this framework, we propose DIOS (Distribution-Informed Optimal Stopping) to address the remaining two difficulties. A per-time-step soft penalty combined with tail-level annealing mitigates blindness to success. Estimating the reward quantile directly from each batch removes the need for an auxiliary critic. For a fixed tail level, we establish a finite-time stationarity guarantee under regularity assumptions along the training iterates. Across the 24 main Bermudan max-call configurations, DIOS achieves the highest mean CVaR and the shortest recorded training time among the tested CVaR baselines.
Stochastic first-order methods use sampled gradients to reduce per-iteration costs, but estimation errors can slow convergence and, in some settings, cause divergence. To improve gradient accuracy while retaining acceleration, we develop scalar and diagonal versions of VR-Adam-SHANG for smooth stochastic convex optimization. The methods combine STORM-type recursive estimation, Adam-style preconditioning, and SHANG's accelerated interpolation and correction, using at most two stochastic gradient evaluations per iteration without full-gradient refreshes. Under unbiased sampling, bounded variance, and mean-square smoothness of gradient noise, polynomial steps with prescribed initialization yield $O(1/K)$ mean-square gradient-estimation error and expected objective-gap bounds combining $O(1/K^2)$ initial-energy decay with an $O(1/\sqrt K)$ stochastic term after $K$ iterations. These guarantees require neither monotonicity of the preconditioner nor a uniform positive lower bound on its eigenvalues. The analysis also establishes polynomial-step convergence of projected nonrecursive Adam-SHANG-s, filling a gap in its original theory. Experiments compare the methods at matched sample-gradient budgets. In every tested setting, at least one of the two versions achieves the lowest mean terminal objective gap among all compared methods.
LLMs are powerful but incur substantial inference cost, which can be exploited by malicious users to induce overly long outputs, increasing latency and operational expense. Prior resource consumption attacks either coerce repetition or suppress the end-of-sequence token by lowering its logit value. However, with autoregressive, sampling-based generation, the search space grows exponentially with length and small deviations or unstable trajectories can derail optimization, limiting attack effectiveness and stability. To address this issue, we introduce Chatter Attack, a novel class of resource consumption attacks that optimize adversarial prompts to induce predefined short target phrases, thereby avoiding search-space explosion while still eliciting long responses. Specifically, Chatter Attack triggers self-doubt using short doubt-inducing phrases as the target, and sustains this state throughout generation via inference attention loss and global entropy loss. To extend to black-box settings, we further propose a novel Bayesian Prompt Optimization (BPO) method, which models the objective function globally by constructing a discrete token kernel, efficiently exploring the entire search space using prior information. Extensive experiments across multiple models and datasets show that our method outperforms baselines, with maximum response length increases to 31.5$\times$, and revealing practical resource-depletion risks for real-world LLM inference services.
Elements arrive one at a time, and after each arrival an algorithm holds a set of at most $k$ arrived elements. Its goal is a held set whose value under a monotone submodular function is close to the best value that $k$ arrived elements can achieve. The recourse of an update is the number of elements inserted into or removed from the held set, and we require it to be bounded independently of $k$ and of the number $n$ of arrivals. With evaluations only on sets of arrived elements and polynomially many of them, the best ratio a randomized algorithm can achieve is $2-\sqrt{2}\approx0.5858$. We show that seeing future arrivals raises the best ratio for polynomially many evaluations to $1-1/e\approx0.6321$, for deterministic and for randomized algorithms; no algorithm with polynomially many evaluations exceeds $1-1/e$, even when all elements are available at once. For every $\varepsilon>0$, a deterministic algorithm that sees the next $\lceil\varepsilon k/3\rceil$ arrivals keeps a set of value at least $1-1/e-\varepsilon$ times the optimum at every time, changes $O(\varepsilon^{-3})$ elements per arrival and makes $(n+1)^{O(\varepsilon^{-2})}$ evaluations. With the whole arrival sequence known in advance, a deterministic algorithm attains the same ratio with $O(\varepsilon^{-2})$ changes per arrival. A randomized algorithm with the same short lookahead makes $O(n^2k)$ evaluations, changes $O(\varepsilon^{-2})$ elements per arrival, and attains the same ratio in expectation at each fixed time. The main tool is a greedy algorithm with an explicit guarantee for each of its partial solutions combined with each initial segment of the known future arrivals. A second tool moves between two such solutions one element at a time while keeping the value high, and the deterministic algorithm times each move using upper bounds on the optimum from failed searches.
Learning-based safety certification offers a promising route for verifying complex dynamical systems when explicit models are unavailable, but existing data-driven methods often scale poorly: they treat the system as a black box, rely on dense discretizations or Lipschitz bounds, and consequently suffer from sample complexity that grows exponentially with dimension. This paper shows that a common structural property of dynamical systems--monotonicity--can be used to break this curse of dimensionality. We develop a data-driven framework for robust safety verification and safe controller synthesis for unknown monotone systems using only simulator queries. Our main theoretical result proves that, for monotone systems with upper-closed unsafe sets, safety is completely characterized by the existence of monotone inductive barrier certificates. This structure reduces certificate verification over continuous state spaces to localized boundary checks over finitely many cells. To learn such certificates, we introduce Max-Monotone Networks, a monotone neural architecture with universal approximation guarantees for monotone functions. We then propose a training and refinement algorithm that, upon successful termination, returns formally valid neural barrier certificates and, under mild partition assumptions, requires only $O(n)$ simulator queries. Across high-dimensional benchmarks, including oscillator networks with up to $13{,}659$ states and traffic networks with $1{,}000$ states, our method certifies safety in regimes where prior approaches either do not converge or exceed memory limits. These results demonstrate that exploiting order structure can make neural safety certification both scalable and formally sound.
Real-world deployment of federated learning (FL) requires resilience against adversarial clients and data privacy. Byzantine fault captures the worst-case behavior of adversarial clients, while differential privacy (DP) provides statistical guarantees against privacy loss. In particular, user-level DP guards a client's entire contribution and provides more realistic protection against information leakage. Despite intensive efforts, the minimax-optimal convergence bound of Byzantine-resilient FL with user-level DP remains open. We establish an information-theoretic lower bound for any Byzantine-resilient empirical risk minimization problem under user-level DP and heterogeneous clients. We propose ByzULDP, a central DP algorithm featuring a two-level momentum across the client and server. Client-side momentum mitigates stochastic gradient variance during robust aggregation. Unlike the fault-free case, naively combining non-linear Byzantine-resilient aggregation with central DP noise can be insufficient to guarantee convergence to a bounded neighborhood of a stationary point. Our key contribution is a server-side momentum that leverages a property specific to user-level DP, where privacy sensitivity depends on how client updates are constructed, to empower convergence. ByzULDP achieves the order-optimal convergence bound for strongly-convex objectives, with vanishing error terms decaying at the optimal $O(1/T)$ and $O(1/\sqrt{T})$ rates for strongly-convex and non-convex objectives, respectively. We corroborate our analysis with numerical experiments on real-world datasets across diverse privacy budgets.
We study the problem of controlling instability in the Vlasov--Poisson plasma systems, a fundamental challenge for nuclear fusion control. Motivated by the partial observability of plasma states in practice, we investigate imitation learning algorithms that learn from a fully-observable expert controller, but operate under macroscopic measurements. We theoretically establish a separation between online and offline imitation learning --- even a small behavior cloning error leads to error compounding that exponentially amplifies instability over time. By way of contrast, the online imitation learning loss can polynomially control the stability of rollout trajectories. We then propose a DAgger-style algorithm that learns the stabilizing controller under partial observability. Simulation results on a 1D Vlasov--Poisson system demonstrate that our algorithm can effectively stabilize plasma instabilities over longer time horizons than behavior cloning.
Zeroth-order (ZO) optimization estimates descent directions from function values alone, which removes the activation storage required by backpropagation and makes it attractive for memory-efficient fine-tuning of large language models (LLMs). Existing low-rank ZO methods either parameterize the additive update through Euclidean factors with a non-unique representation, or restrict each finite-difference direction to be low-rank. In this work, we propose to constrain each additive update block to a fixed-rank matrix manifold while keeping the pretrained weights frozen, so the learned update remains rank-constrained throughout fine-tuning. On this geometry, we build a Riemannian ZO gradient estimator that projects a random matrix onto the tangent space, normalizes it to correct the scale mismatch induced by projection, and further propose Zeroth-order Riemannian Gradient Descent (ZO-RGD) and its smoothed variant, ZO-RGD-S. We show that the ambient finite difference matches its retracted counterpart, the normalized estimator recovers the Riemannian gradient in expectation, and ZO-RGD converges under trajectory-regularity conditions. Experiments on LLM fine-tuning tasks show that the proposed algorithms are competitive with strong forward-only baselines across model families and tasks, with gains in several settings.
We present a novel GPU-accelerated global solver for constrained nonlinear programming (NLP) inspired by the efficient linear bound propagation framework in neural network (NN) verification. Existing work on NN verification, a problem that can be cast as feasibility detection for a nonlinear objective, demonstrated orders-of-magnitude speedups with the linear bound propagation-based framework when compared to traditional solvers. Extending this paradigm to global nonlinear programming requires addressing many challenges: a global constrained NLP solver must effectively exploit all, possibly unstructured, nonlinear constraints to improve the objective, whereas NN verification typically handles simple input box constraints over a well-structured NN and does not directly optimize an objective function. We address these challenges by fully exploiting the linear bounds produced by bound propagation: these linear bounds can be viewed as relaxations of nonlinear constraints and used to provide lower bounds, check infeasibility, and guide the search for a primal solution. In addition, we propose tighter linear relaxations for bilinear, quotient, and other commonly used nonlinear functions in NLP problems, and a projected augmented Lagrangian method tightly coupled to our dual solving procedure to produce primal solutions. Our method solves 421 of 505 bounded and continuous NLP problems in the GAMS and MINLPLib benchmarks under a 180-second limit, more than each of the six global solvers we compare with, including Gurobi and BARON (commercial), and SCIP (open-source). Sixteen of these problems are certified by none of the six global solvers; all of them contain exponential, logarithmic, or trigonometric terms, which are often challenging for traditional approaches. Project webpage is at https://verified-intelligence.github.io/GEAR.
Zeroth-order methods provide a way to optimize stochastic nonconvex objectives using only noisy function evaluations. However, the smoothing inherent in their gradient estimators effectively modifies the objective being optimized. In particular, Gaussian smoothing with smoothing parameter $\eta>0$ induces, to leading order, an implicit Hessian-trace regularization, which motivates the study of the curvature-regularized objective of the form $f(x)+(\eta^2/2) tr(\nabla^2 f(x))$. We develop a non-asymptotic theory in the nonconvex setting for a class of algorithms called stochastic zeroth-order Langevin dynamics (ZOLD), which use only noisy function evaluations and do not have access to first- or higher-order derivative information, and derive excess-risk guarantees for the curvature-regularized objective. We first control the finite-time distribution of the ZOLD iterates in Wasserstein distance with respect to the Gibbs measure associated with the Gaussian-smoothed objective. An explicit coupling between the smoothing and inverse temperature parameters then quantifies the proximity of this measure to the Gibbs measure associated with the corresponding Hessian-trace-regularized objective. These distributional estimates in Wasserstein distance lead to excess-risk bounds for the curvature-regularized objective. Our analysis accommodates assumptions under which the stochastic function evaluations are allowed to have non-uniform dependence on the data stream. In addition, the constants in the non-asymptotic bounds do not rely on log-Sobolev or Poincar\'e inequalities as in prior zeroth-order Langevin analyses. To the best of our knowledge, this is the first non-asymptotic analysis of a zeroth-order Langevin method that simultaneously establishes excess-risk guarantees for a curvature-regularized objective and Wasserstein guarantees for the associated Gibbs measure in the nonconvex setting.
Bilevel optimization is a fundamental framework for machine learning problems, where an upper-level (UL) objective depends on the solution of a nested lower-level (LL) problem. Existing bilevel algorithms typically assume that the LL objective is available for evaluation, differentiation, or unrolling. However, this assumption can fail when the LL response is produced by a black-box solver, simulator, adversary, or learned predictor. In this paper, we study LL agnostic bilevel optimization, where the UL learner has access only to an approximate LL solution $\bar{\mathbf{y}}(\mathbf{x})$ and cannot query the LL objective, its gradients, Hessian/Jacobian information, or the procedure that generates the LL response. We propose LeGo-BiO, a coordinate-wise pseudo-hypergradient method that estimates the missing Jacobian $\nabla _{\mathbf{x}}\bar{\mathbf{y}}(\mathbf{x})$ using finite differences of $\bar{\mathbf{y}}(\cdot)$. For each UL coordinate, LeGo-BiO constructs a decision vector subject to 1-sparse updates, thereby isolating the corresponding Jacobian column needed in the UL gradient computation. This avoids the directional-projection limitation of full-parameter response differences and, unlike derivative-free/zeroth-order approximations of the entire UL gradient, preserves the available first-order gradient information with respect to the UL variables for more accurate UL gradient evaluation. We establish convergence guarantees for LeGo-BiO in both deterministic and stochastic settings under an LL error-bound/Polyak-Lojasiewicz-type condition, without requiring LL strong convexity. Our bounds yield an $O(T^{-1})$ rate in the deterministic setting and an $O(T^{-1/2})$ rate in the stochastic setting when the LL approximation error decreases geometrically. Experiments demonstrate competitive performance of LeGo-BiO despite being agnostic to the LL objective.
In recent years, singular-spectrum-based optimizers have attracted increasing attention due to their ability to preserve the matrix structure of neural network gradients, thereby enabling geometry-aware optimization. However, most singular-spectrum-based (SSB) approaches remain centralized, limiting their use in distributed learning infrastructures that exploit massive parallelism. Distributed SSB optimization is still at an early stage, with open questions about convergence guarantees and communication efficiency. To fill this gap, we propose the Distributed Momentum-based Spectral Estimator (DisMUSE), an efficient distributed SSB optimization framework for both data-center and federated settings. Using Muon-style exact orthogonalized momentum as the spectral update, DisMUSE integrates a recursive variance-reduced momentum estimator and achieves an $O((KT)^{-1/3})$ convergence rate, where $K$ and $T$ denote the number of nodes and local iterations per node, respectively. This rate matches the best-known rates for variance-reduced nonconvex distributed optimization. Building on this variance-reduced framework, we study two extensions. First, our DisMUSE method with finite Newton-Schulz (NS) iterations preserves the same convergence order under a uniform normalized support-spectrum condition, up to a multiplicative factor. Second, we develop a communication-efficient variant based on low-rank (LR) momentum transmission and quantify the error it introduces to the stationarity bound, characterizing the trade-off between accuracy and communication cost.
We introduce P-IMLA, a preconditioned implicit-midpoint Langevin sampler for high-dimensional log-concave targets with non-smooth regularization. The obstruction is that the large steps enabled by matrix preconditioning are precisely where preconditioned Euler-Maruyama inflates stationary variance. Implicit midpoint removes this surplus via a Cayley-transform identity, making P-IMLA Gaussian-exact at any step size. Beyond Gaussians, whitening transfers the sharp IMLA contraction rate to $\kappa_{\mathrm{eff}}$, and backward-error analysis gives drift-only rather than Euler-type diffusion bias under explicit smoothness assumptions. An $M$-norm Moreau envelope yields a proximal non-smooth variant with a three-term error budget. Controlled Gaussian and TV-regularized imaging experiments confirm the predicted separation: the unpreconditioned midpoint scheme retains long transients as $\kappa_f$ grows, whereas P-IMLA remains $\kappa_{\mathrm{eff}}$-governed and near variance-exact.
We study online newsvendor learning with bounded demand, censored observations, and convex nonlinear shortage costs. Because stockouts can conceal the shortage magnitudes needed by gradient-based algorithms, we use randomized ordering to construct unbiased subgradient estimates. We characterize admissible randomization distributions within this framework and develop an online inventory policy that combines the resulting subgradient estimates with gradient-based updates. Our regret analysis quantifies the cost of randomization via the estimation variance and an additional loss term. We characterize the randomization distributions that minimize this objective. For piecewise-linear shortage costs, the optimal distribution is computed through a linear-time recursion; for costs with bounded curvature density, it is characterized by a scalar ordinary differential equation. The policy achieves expected regret of $O(T^{2/3})$ against the best fixed order in hindsight over $T$ periods. Under independent demand and suitable curvature of expected losses, the bound improves to $O(\sqrt{T})$. Matching minimax lower bounds establish rate optimality over the corresponding classes with a fixed marginal-cost bound. Both rates extend to nonperishable inventory under uniformly positive conditional expected demand. Numerical experiments on synthetic and retail data show improvements over alternative policies using censored observations.
Learned barrier certificates provide a principled mechanism for establishing safety of stochastic systems, but their validity may deteriorate during deployment when the underlying dynamics differ from those used for training. We propose an anytime-valid runtime monitor that audits a learned barrier certificate using observed closed-loop trajectories. The monitor is intended to detect loss of statistical support for the deployed barrier condition; it does not by itself guarantee detection before unsafe entry or safety of a subsequent intervention. Specifically, the conditional barrier-drift condition is formulated as an episode-wise sequential hypothesis test, and observed barrier increments are transformed into a nonnegative test supermartingale, yielding finite-sample false-alarm control under arbitrary stopping times. Inspired by finite-horizon safety verification, we further introduce a deterministic temporal shaping of the monitoring boundary, making the monitor more conservative early in a prescribed deployment horizon while preserving anytime-validity. The monitor is designed to detect statistical evidence that the safety certificate is no longer consistent with the observed dynamics, rather than to predict unsafe entry. To assess the monitor empirically, we evaluate empirical alarm behavior under valid and invalid certificates. Code and data are available at https://github.com/FormalCert-AI/runtime-barrier-monitor.
Outliers are essential for evaluating and improving the robustness of machine learning systems, especially when future distributions may differ significantly from historical training data. In high-stakes applications, robustness often depends on rare cases that finite datasets fail to capture, making simple resampling or perturbation insufficient for stress scenario generation. Existing outlier synthesis methods typically rely on sparse neighborhoods, low support latent regions, or classifier boundary crossings, which can be heuristic, unstable, and tied to specific modalities or architectures. We therefore propose Sinkhorn Boundary Outlier Generation (SBOG), a structured framework for latent-space outlier generation that couples Sinkhorn optimal transport geometry with distributionally robust boundary modeling. The resulting Sinkhorn-induced support cost guides the sampler toward weakly supported boundary regions, while semantic constraints prevent uncontrolled drift from the intended context, yielding controlled deviations from the in-distribution reference measure rather than arbitrary sparse-region samples. Experiments on time series anomaly generation and image outlier synthesis show that our framework produces informative, semantically controlled outliers and improves downstream robustness evaluation across modalities, providing a foundation for stress scenario generation beyond empirical support.
We characterize, up to logarithmic factors, the optimal higher-order oracle complexity of finding approximate solutions to composite monotone inclusion problems, formed by the sum of a smooth single-valued monotone operator and a maximal monotone set-valued operator. We introduce the Anchored Extra-Proximal (AEP) framework, which combines an anchored extrapolation step with an inexact anchored proximal update satisfying a relative-error condition. This framework recovers the composite Fast Extragradient method in the first-order setting and yields natural higher-order extensions through Taylor approximations of the smooth operator at the extrapolated point. For every $p\geq 2$, assuming that the $(p-1)$th derivative of the single-valued operator is $L_p$-Lipschitz continuous and the initial distance to a solution is at most $R$, we combine AEP with a bisection line search to obtain a $p$th-order method that attains a tangent residual of at most $\varepsilon$ in $\widetilde{O}((L_p R^{p}/\varepsilon)^{\frac{2}{3p-1}})$ oracle calls. We complement this result with a worst-case lower bound of $\Omega((L_p R^p/\varepsilon)^{\frac{2}{3p-1}})$ for every deterministic algorithm in the $p$th-order oracle model, without imposing any prescribed update structure. For the subclass of smooth nonexpansive fixed-point problems, we further establish a lower bound of $\Omega(\min\{R/\varepsilon, {(L_p R^{p}/\varepsilon)}^{\frac{2}{3p-1}}\})$. By choosing between AEP and Halpern iteration, we match this lower bound up to logarithmic factors, thereby also settling the deterministic oracle complexity of smooth nonexpansive fixed-point problems.