In this paper, we study the convergence of Federated Averaging (FedAvg) with partial participation under a general variance condition. In each communication round, the server samples $S$ of the $n$ clients uniformly at random without replacement. The stochastic gradients are assumed to satisfy the so-called ABC condition, which allows their variance to grow with the local objective gaps. For smooth nonconvex objectives, we establish a finite-time stationarity bound in terms of the heterogeneity measure $σ_f^\star:=f^\star-\frac{1}{n}\sum_{i=1}^n f_i^\star$ and the client sampling factor $I_S:=(n-S)/(n-1)$. Here, $f^\star$ and $f_i^\star$ denote the infima of the global average objective and client $i$'s local objective, respectively. The bound exhibits a linear speedup in the number of sampled clients $S$ and recovers the full-participation result when $S=n$. Under diminishing stepsizes, we further prove that the gradient norm converges to zero both in expectation and almost surely. Notably, compared with existing analyses of FedAvg with partial participation, our results require no additional data heterogeneity assumption such as bounded gradient dissimilarity (BGD), and the sampling-induced term in our bound is not amplified by the number of local updates at the leading order. The importance of removing the BGD assumption is illustrated by an example that is smooth and lower bounded but violates BGD, yet is covered by our analysis.
Branch-and-Bound (B&B) and $A^*$ are popular exact search methods originating from Operations Research and Artificial Intelligence, respectively. Although the two methods share fundamental search principles, their different ways of exploring the search space can lead to substantially different computational performance. We investigate these differences using the Resource-Constrained Project Scheduling Problem (RCPSP) as an exemplary problem setting. We first establish the relationships between mixed-integer linear programming, state-action, and timed Petri-net representations. Since B&B and $A^*$ admit numerous algorithmic enhancements, we focus on relatively basic implementations. This allows us to examine the fundamental differences between the methods rather than compare the performance of state-of-the-art solvers. By using the same lower bound and best-first node selection, we isolate the effects of the search mechanisms, particularly the identification of equivalent states reached through different scheduling decision sequences. Experiments on 480 benchmark RCPSP instances show that graph-based $A^*$ substantially outperforms the B&B variants considered. With additional pruning rules, $A^*$ proves optimality for all 480 instances, compared with 445 for best-first B&B under the same time and memory limits. Depth-first B&B uses less memory and can find feasible solutions even when the search does not finish within the time limit. Stronger linear programming relaxations reduce the search space but introduce considerable computational overhead. Our findings highlight the importance of duplicate-state detection and clarify the computational trade-offs between the search approaches. These insights can help researchers select an appropriate method and identify opportunities to incorporate effective mechanisms from one approach into the other.
Rapid AI data-center power fluctuations challenge grid integration. Sustained grid-import smoothing requires allocating limited battery power and energy under uncertain demand. We formulate risk-aware BESS scheduling that coordinates advance reserve commitments and demand-responsive dispatch, accounting for operating costs, grid-ramp penalties, and their conditional value-at-risk. We develop battery scheduling decision-focused learning (BS-DFL) with a constraint-aware neural optimization layer trained directly from this objective without optimal-solution labels. The trained layer is then frozen to guide demand-predictor learning through scheduling costs and ramping risk, enabling solver-free rolling-horizon scheduling. In a 10,000-minute simulation using loads derived from measured AI workloads, BS-DFL reduces mean absolute minute-to-minute grid-import changes by 71.70% and total scheduling cost by 18.01% relative to an MSE-trained predictor coupled with a mixed-integer linear programming solver, with an average decision time of 18.77 ms.
Evaluating a learned AC optimal power flow (AC-OPF) surrogate on one grid serialization cannot show whether its accuracy depends on that encoding. We test this dependence with thirteen metamorphic relations, each specifying when a transformation is valid and how to compare its outputs. Our audit covers 17 model and pipeline configurations: two released trained checkpoints and 15 random-weight architecture or mechanism controls. Every configuration violates at least six relations, even at reporting tolerances ten times tighter or looser. For GridSFM-Open v1.1, source inspection and intermediate-tensor interventions point to an indexing bug and three noncanonical or numerically unstable preprocessing choices; a repaired checkpoint remains untested. In a separate case500 training experiment, augmentation lowers transformed-view error in three of four models and median discrepancy and constraint-violation P95 in all four. These benefits depend on the training protocol: some relations worsen, and every evaluated prediction remains AC infeasible. A GPU/CPU eigensolver mismatch accounts for most of the large raw-Laplacian-baseline gain. We recommend reporting representation consistency alongside accuracy, feasibility, and cost. These results establish neither deployment readiness nor universal invariance.
We study the excess risk of constant-stepsize online stochastic gradient descent (SGD) with tail-averaging for high-dimensional linear regression. Building on the instance-dependent spectral framework of Zou et al. (2021), we establish matching initialization-explicit upper and lower bounds on the total excess risk of tail-averaged SGD in terms of the population covariance spectrum $\mathbf{H}$ and initialization $\mathbf{w}_0-\mathbf{w}^*$ whenever a trajectory spectral feedback condition holds. Existing analyses exhibit a fundamental gap in the effective variance: the upper bound of Zou et al. (2021) retains an undamped contribution from the initial error in the head eigenspace $\mathbf{I}_{0:k^\dagger}$, whereas their lower bound omits the variance induced by the head initialization altogether. We show that both the existing upper and lower bounds are loose: the single-hop fast-mode variance contribution from the head initialization is exponentially damped during the burn-in phase, whereas fourth-moment energy scattered from the head initialization into slow spectral modes during the early burn-in steps survives undamped into the averaging window. We prove that our spectral feedback condition is satisfied in two concrete constant-stepsize regimes: (i) for arbitrary covariance spectra $\mathbf{H}$ and any finite horizon, under an explicit horizon-dependent stepsize chosen prior to execution; and (ii) for polynomially decaying spectra $\lambda_i\asymp i^{-a}$ ($a>1$), under a single horizon-independent positive stepsize. Finally, we complement these positive results with explicit one- and two-dimensional separation constructions proving an exponential-versus-polynomial gap in the lower bound of Zou et al. (2021) and showing why fourth-moment assumptions alone cannot determine the asymptotic exponential decay rate without additional structural conditions.
Dantzig--Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying constraint families, and each candidate costs a column generation run. The choice is hard when families strengthen the bound only together, as in classical lot-sizing, network-design, and facility-location formulations. We prove that when an evaluation convexifies at most $s$ of $k$ families, methods that observe only bounds need of order $k^w$ evaluations to find a hidden group of $w$ families in the worst case. A prediction missing $e$ members reduces this to $\Theta(k^e)$ through a staged search that need not know $e$. The orders extend to several groups, prices, and growing evaluation costs. We study two predictors: a lookahead at an optimal LP dual, which vanishes on groups whose families share no variable, and language models reading the model, verified by the staged search in JointDW. Such groups appear in three of ten MIPLIB-NL models whose bound can be improved and in all 59 benchmark instances. Given the coefficients of models with up to 16 families, one language model names a true group first in 249 of 252 answers. On the benchmark instances, decompositions chosen by language models from the text solve faster than those of GCG's detector.
Multiple local updates are widely used in decentralized optimization to enhance communication efficiency, often in conjunction with stochastic gradients when exact gradient computation is costly. However, such algorithms generally converge only to a neighborhood of the optimum, whose size depends on the stepsize and the variance of gradient noise. Variance reduction has been incorporated into ADMM-based methods to achieve exact convergence in expectation, but their stepsizes typically scale inversely with both the square root of the network size and the cube of the number of local updates, leading to slow convergence as the network size or the number of local updates increases. In this paper, we propose a novel primal-dual-based decentralized optimization algorithm that integrates variance reduction with multiple local updates, with a stepsize independent of the network size and inversely proportional to the number of local updates (rather than its cube), thereby enabling significantly larger stepsizes and faster convergence. Moreover, to the best of our knowledge, we establish the first linear convergence guarantee in expectation for decentralized optimization with multiple local updates under stochastic gradients in the strongly convex setting. Beyond the theoretical guarantee, the proposed method exchanges only one variable per round, halving the communication cost compared with gradient-tracking-based methods. Numerical experiments further demonstrate its advantages in convergence speed and accuracy.
Hessian spectra at trained models in deep learning exhibit a persistent pattern: eigenvalues organize into distinct clusters, including a large bulk near zero and a few isolated outliers. This paper shows that a natural account of these spectral phenomena emerges when the original setting is understood as a departure from a nearby, otherwise hidden, highly symmetric reference. Modifications, including changes to the architecture, data distribution, or parameter metric, expose a nearby reference configuration whose Hessian exhibits rich invariances-ones not accounted for by weight symmetries. There, symmetry enables a precise description of the spectra, forcing high-dimensional kernels and eigenvalues of large multiplicity. Returning to the original configuration breaks the Hessian symmetry and thereby produces the observed hierarchy of clusters and outliers. The framework is developed in some generality, with a detailed analysis of three-layer ReLU networks and applications to convolutional, graph, and transformer models, as well as to the NTK. The same mechanism is further shown to yield analogous spectral structures in layerwise Hessians and the Gauss-Newton matrix.
We study timestep allocation for score-based diffusion sampling, where a learned reverse-time dynamics is discretized on a finite grid. Uniform and hand-crafted schedules are standard choices, but they rely on fixed prescriptions and can therefore be suboptimal. To address this limitation, we propose Adaptive Reparameterized Time (ART), a continuous-time control formulation that learns a time change by treating the speed of the sampling clock as the control, so that a uniform grid on the learned clock induces adaptive timesteps in the original diffusion time. Based on a leading-order Euler error surrogate, ART provides a principled objective for allocating timesteps along the sampling trajectory. To solve this deterministic control problem, we introduce ART-RL, an auxiliary randomized formulation with Gaussian policies that turns schedule learning into a continuous-time reinforcement learning problem. We prove that the randomized ART-RL formulation is equivalent to ART at the optimizer level, in the sense that its optimal Gaussian policy recovers the optimal ART time-warping rate through its mean. We further establish policy evaluation and policy improvement characterizations and derive trajectory-based moment identities that yield implementable actor--critic updates for learning the schedule. Across experiments ranging from controlled low-dimensional settings to image generation, ART-RL can be plugged into existing diffusion samplers by changing only the timestep grid, consistently improving sample quality over strong baseline schedules at matched budgets while leaving the rest of the sampling pipeline unchanged. The learned schedules also exhibit broad generalization, transferring without retraining across sampling budgets, datasets, solvers, pipelines, and representation spaces.
Deep hashing encodes high-dimensional image features into compact binary codes, enabling efficient similarity search in large-scale content-based image retrieval, a problem akin to semantic-preserving lossy compression under extreme bit-rate constraints. A fundamental challenge in deep hashing stems from the discrete nature of quantization in generating the codes. W-type regularizations, such as $z\mapsto\||z|-1\|_1$, are widely adopted and empirically effective at pushing network outputs toward binary values. However, existing methods often directly optimize these regularizations without convergence guarantees. While proximal gradient methods offer a promising solution, the coupling between W-type regularizers and neural network outputs results in composite forms that generally lack closed-form proximal solutions. In this paper, using Fenchel duality, we reformulate the nonconvex W-type regularization term through its convex conjugate, yielding a min-max problem with tractable proximal updates. We show that this min-max problem captures the stationarity of the original deep hashing objective up to the quantization error. Building on this problem, we present a stochastic primal-dual hashing algorithm, referred to as DualHash, with oracle complexity guarantees. We derive two algorithm instances: a momentum-based version with $O(\varepsilon^{-4})$ oracle complexity and an improved $O(\varepsilon^{-3})$ version using variance reduction, both for $\varepsilon$-KKT points of the min-max problem, which yield near-stationarity guarantees for the original objective. Experiments on three standard datasets demonstrate competitive retrieval accuracy and reduced quantization error, validating the practical value of DualHash for image retrieval problems. Our implementation of DualHash is available at https://github.com/RailLee/DualHash.
In this paper, we study differentially private mechanisms for functions whose outputs lie in a Euclidean Jordan algebra. Euclidean Jordan algebras capture many important mathematical structures and form the foundation of linear programming, second-order cone programming, and semidefinite programming. Our main contribution is a generic Gaussian mechanism for such functions, with sensitivity measured in $\ell_2$, $\ell_1$, and $\ell_\infty$ norms. Notably, this framework includes the important case where the function outputs are symmetric matrices, and sensitivity is measured in the Frobenius, nuclear, or spectral norm. We further derive private algorithms for solving symmetric cone programs under various settings, using a combination of the multiplicative weights update method and our generic Gaussian mechanism. As an application, we present differentially private algorithms for semidefinite programming, resolving a major open question posed by [Hsu, Roth, Roughgarden, and Ullman, ICALP 2014].
Variational regularization is a classical technique to solve statistical inference tasks and inverse problems, with modern data-driven approaches parameterizing regularizers via deep neural networks showcasing impressive empirical performance. Recent works along these lines learn task-dependent regularizers. This is done by integrating information about the measurements and ground-truth data in an unsupervised, critic-based loss function, where the regularizer attributes low values to likely data and high values to unlikely data. However, there is little theory about the structure of regularizers learned via this process and how it relates to the two data distributions. To make progress on this challenge, we initiate a study of optimizing critic-based loss functions to learn regularizers over a particular family of regularizers: gauges (or Minkowski functionals) of star-shaped bodies. This family contains regularizers that are commonly employed in practice and shares properties with regularizers parameterized by deep neural networks. We specifically investigate critic-based losses derived from variational representations of statistical distances between probability measures. By leveraging tools from star geometry and dual Brunn-Minkowski theory, we illustrate how these losses can be interpreted as dual mixed volumes that depend on the data distribution. This allows us to derive exact expressions for the optimal regularizer in certain cases. Finally, we identify which neural network architectures give rise to such star body gauges and when do such regularizers have favorable properties for optimization. More broadly, this work highlights how the tools of star geometry can aid in understanding the geometry of unsupervised regularizer learning.
Stochastic Gradient Descent (SGD) is a workhorse algorithm for continuous optimization. It is a folk belief among deep learning practitioners that learning rate decay in SGD substantially improves performance. However, it has remained unclear whether learning rate decay offers an asymptotic advantage for global nonconvex optimization. We answer this question in the affirmative by demonstrating a natural class of nonconvex functions for which, we prove that SGD with learning rate decay requires exponentially fewer gradient queries for global optimization than SGD with any fixed learning rate in the small-step size regime. To the best of our knowledge, this is the first exponential advantage that has been demonstrated for learning rate decay. The class of functions we consider includes many popular benchmark nonconvex functions. Our results provide a robust explanatory theory for the empirically observed benefits of learning rate decay for global optimization. We show that annealing performs the non-logconcave sampling task associated to the optimization problem in polynomial time, via a novel application of weak Poincar\'{e} inequalities.
Exact score-based causal structure learning supports explicit structural constraints and optimisation certificates, but its branch-and-cut search can be computationally demanding. We augment exact causal structure learning methods with an upper bound on the maximum clique size of the learned undirected skeleton, enforced at integer candidates through lazy cycle and clique constraints. On static linear-Gaussian Erd\H{o}s-R\'enyi and scale-free DAGs, we evaluate callback order, alternative clique-cut policies, incumbent repair, fractional separation, and Big-M bounds. We report causal-recovery metrics jointly with optimality gaps, primal and dual bounds, node throughput, MIPSOL events, and callback cost. Two variants receive particular attention: B3 selects the most violated minimal (k+1)-clique inequality at each callback, whereas C3 repairs an infeasible integer candidate by deleting violating edges and refitting affected regression columns using QR-based least squares. The results do not indicate that clique constraints uniformly accelerate optimisation. However, their benefit depends on graph density, sample size, and cut-management policy, while large collections of correlated constraints can reduce node throughput. Tighter valid Big-M bounds reduced the fixed-budget optimality gap in most tested settings, without necessarily changing SHD or F1 at a fixed time limit. These findings characterize the computational trade-offs of clique-constrained exact causal learning, and identify selective separation and incumbent repair as promising directions.
Alternating minimization is a standard tool for optimization problems with two coupled variable blocks. Its cost is dominated by the inner subproblem solvers, which rarely have a closed-form solution. In this paper, we propose to learn an inexact alternating scheme in which neural networks approximate the subproblem solution at each iteration. For convex problems with quadratic couplings between the variable blocks that satisfy a quadratic growth condition, we derive a worst-case bound on the optimality gap that exposes the per-iteration error that each network is trained to minimize in a greedy fashion. Combining the bound with a PAC-Bayes argument, we obtain a fixed-horizon, high-probability bound on the expected optimality gap over the distribution of problem instances. We apply the learned scheme to robust matrix completion and low-dose CT reconstruction with dictionary regularization where we achieve a 5x-13x speedup over the inexact first-order baseline at matched accuracy. A brief end-to-end fine-tuning stage extends this up to 30x and outperforms other learned schemes. The code is available at https://github.com/paulhausner/learned-alternating-minimization.
Over the past ten years Tensor Network Kernel Machines have enabled efficient nonlinear system identification through the use of low-rank tensor networks. The goal of this article is to provide the reader with a basic understanding of this model class in terms of its two defining properties. Furthermore, both the optimization and regularization of such models are discussed, together with recent developments and starting points for further exploration.
The heavy-ball method with a constant step size and a constant momentum parameter accelerates gradient descent on quadratics, but no choice of the two parameters does so on the class of smooth strongly convex functions. We determine how much faster than gradient descent it can be made on this class. Writing the best worst-case rate over all tunings as $1-c(\kappa)/\kappa$, where $\kappa$ is the condition number, we prove that $c(\kappa)$ tends to $8$ in every dimension of two or more. The lower bound on the rate holds for every tuning, because a tuning that is faster than $1-8/\kappa$ on quadratics follows a slowly decaying spiral on some function of two variables. The matching upper bound comes from an explicit tuning, with nearly the momentum of Polyak's tuning and a step size about $\sqrt{\kappa}/8$ times smaller, and from a Lyapunov function whose positivity is verified in exact arithmetic. Optimally tuned heavy-ball is therefore asymptotically four times faster than gradient descent on this class.
The Adam optimizer is widely used to train neural networks with highly nonlinear and nonconvex objective functions. Remarkably, Adam often drives the empirical training loss nearly to zero. This phenomenon has been partially explained by neural tangent kernel (NTK) analyses of gradient descent (GD) and stochastic gradient descent (SGD). However, analyzing Adam presents additional challenges. Its first-moment momentum couples stochastic gradients across iterations, while its adaptive second-moment preconditioner depends on the same stochastic gradient used in the update. In this paper, we establish linear convergence guarantees for Adam in training sufficiently wide two-layer neural networks with either full-batch or minibatch gradients. For full-batch Adam, we construct a Lyapunov function to control the effect of historical momentum. For minibatch Adam, our analysis builds on an exact virtual-iterate representation, a conditional analysis of the dependence between stochastic gradients and adaptive preconditioners, and uniform concentration bounds for the second-moment estimates. We prove geometric decay of the empirical loss for full-batch Adam and geometric decay in restricted expectation on a high-probability event for minibatch Adam. Beyond regression, we extend our convergence analysis to physics-informed neural networks (PINNs) for solving the Poisson equation, where differential operators and separate sampling of interior and boundary points introduce additional challenges. We establish analogous global convergence guarantees for both deterministic and stochastic Adam under explicit conditions on network width and Adam's parameters, characterizing the effects of minibatch size on convergence. These results help bridge the gap between Adam's empirical success and its theoretical understanding, providing sharper convergence guarantees for PINNs than existing analyses.
We study how to combine a human binary call (a referee's offside or onside decision) with an AI numerical measurement (VAR's estimate of an attacker's position relative to the offside line). We derive an explicit maximum-likelihood rule that determines when the referee's call should stand and when a conflicting VAR measurement should overturn it. Our model treats the true offside margin as fixed and unknown and assumes independent additive Gaussian errors in the human's perception and the AI measurement. When both errors have zero means and known standard deviations $\sigma_H$ and $\sigma_{AI}$, respectively, the reversal threshold is $\sqrt{2/\pi}\,\sigma^2_{AI}/\sigma_H$. An onside call is overturned precisely when VAR's measured offside margin exceeds this threshold---not merely because VAR places the attacker beyond the offside line. We also derive the rule when either source has a systematic bias. We prove that the maximum-likelihood estimate of the true margin exists and is unique, while the final binary decision requires only a threshold comparison, without computing that estimate numerically. We illustrate the rule through numerical examples and show how to estimate the error parameters from calibration data. The same analysis applies beyond soccer to binary classification problems in which a human call and an AI measurement satisfy the stated model. This paper is intended for educational purposes: we aim to illustrate how undergraduate students and advanced high school students familiar with basic probability and calculus can investigate research questions in AI and its applications.
We study Minty set inclusion on a compact convex set $\mathcal{X} \subseteq \mathbb{R}^d$ with a set-valued operator $A : \mathcal{X} \rightrightarrows \mathbb{R}^d$. Under the Minty promise, the goal is to compute $x \in \mathcal{X}$ and $u \in A(x)$ such that $\sup_{y \in \mathcal{X}} \langle u, x - y \rangle \le \varepsilon$. Our approach is geometric and we work in the fixed-scale constrained-resolvent oracle model, where a query at anchor $a$ is assumed to return a resolvent point $x^+$ together with the Yosida vector $g_{\eta}(a) = (a - x^+)/\eta$. We prove that each oracle response yields a sharp dichotomy: either the returned point provides an $\varepsilon$-SVI certificate, or the Yosida vector defines a strict separating normal for the Minty set. Thus, a single black-box resolvent query supplies either a certified candidate solution or a valid geometric cut. Combining this separation principle with the ellipsoid method, we obtain RESOLVENT-ELLIPSOID, an algorithm that computes an $\varepsilon$-SVI in $\mathcal{O}\!\left(\operatorname{poly}(d, \log(1/\varepsilon))\right)$ black-box oracle calls to the fixed-scale constrained resolvent. For monotone inclusions, our algorithm gives, to our knowledge, the first $\mathcal{O}\!\left(\mathrm{poly}(d, \log(1/\varepsilon))\right)$ black-box guarantee in the fixed-scale resolvent model.