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Probabilistic Methods · Monte Carlo and Sampling Methods

Kijung Jeon, Michael Muehlebach, Molei Tao

Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers the cases where the constrained distribution is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods for nonconvex constraint set $\Sigma \subset \mathbb{R}^d$ defined by equality or inequality constraints commonly rely on costly projection steps. Moreover, they cannot handle equality and inequality constraints simultaneously as each method only specialized in one case. In addition, rigorous and quantitative convergence guarantee is often lacking. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and $\Sigma$, OLLA converges exponentially fast in $W_2$ distance to the constrained target density $\rho_\Sigma(x) \propto \exp(-f(x))d\sigma_\Sigma$. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to projection-based constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and reasonable empirical mixing.

Optimization · Large Scale, Parallel and Distributed

Khiem HUYNH, Malcolm Egan, Giovanni Neglia, Jean-Marie Gorce

Federated learning (FL) is now recognized as a key framework for communication-efficient collaborative learning. Most theoretical and empirical studies, however, rely on the assumption that clients have access to pre-collected data sets, with limited investigation into scenarios where clients continuously collect data. In many real-world applications, particularly when data is generated by physical or biological processes, client data streams are often modeled by non-stationary Markov processes. Unlike standard i.i.d. sampling, the performance of FL with Markovian data streams remains poorly understood due to the statistical dependencies between client samples over time. In this paper, we investigate whether FL can still support collaborative learning with Markovian data streams. Specifically, we analyze the performance of Minibatch SGD, Local SGD, and a variant of Local SGD with momentum. We answer affirmatively under standard assumptions and smooth non-convex client objectives: the sample complexity is proportional to the inverse of the number of clients with a communication complexity comparable to the i.i.d. scenario. However, the sample complexity for Markovian data streams remains higher than for i.i.d. sampling. Our analysis is validated via experiments with real pollution monitoring time series data.

Optimization

Kangkang Deng, Jiachen Jin, Jiang Hu, Hongxia Wang

We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. In contrast to conventional Riemannian ADMM methods that require exactly solving a nested subproblem at each iteration, our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order $\mathcal{O}(\epsilon^{-3})$ for finding an $\epsilon$-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality.

Applications · Language, Speech and Dialog

Shicheng Liu, Siyuan Xu, Wenjie Qiu, Hangfan Zhang, Minghui Zhu

A common and effective strategy for humans to improve an unsatisfactory outcome in daily life is to find a cause of this outcome and correct the cause. In this paper, we investigate whether this human improvement strategy can be applied to improving reinforcement learning from human feedback (RLHF) for alignment of language models (LMs). In particular, it is observed in the literature that LMs tuned by RLHF can still output unsatisfactory responses. This paper proposes a method to improve the unsatisfactory responses by correcting their causes. Our method has two parts. The first part proposes a post-hoc explanation method to explain why an unsatisfactory response is generated to a prompt by identifying the training data that lead to this response. We formulate this problem as a constrained combinatorial optimization problem where the objective is to find a set of training data closest to this prompt-response pair in a feature representation space, and the constraint is that the prompt-response pair can be decomposed as a convex combination of this set of training data in the feature space. We propose an efficient iterative data selection algorithm to solve this problem. The second part proposes an unlearning method that improves unsatisfactory responses to some prompts by unlearning the training data that lead to these unsatisfactory responses and, meanwhile, does not significantly degrade satisfactory responses to other prompts. Experimental results demonstrate that our algorithm can improve RLHF.

Optimization · Bayesian & Black-box Optimization

Lam Ngo, Huong Ha, Jeffrey Chan, Hongyu Zhang

Bayesian Optimization (BO) is a powerful tool for optimizing expensive black-box objective functions. While extensive research has been conducted on the single-objective optimization problem, the multi-objective optimization problem remains challenging. In this paper, we propose MOBO-OSD, a multi-objective Bayesian Optimization algorithm designed to generate a diverse set of Pareto optimal solutions by solving multiple constrained optimization problems, referred to as MOBO-OSD subproblems, along orthogonal search directions (OSDs) defined with respect to an approximated convex hull of individual objective minima. By employing a well-distributed set of OSDs, MOBO-OSD ensures broad coverage of the objective space, enhancing both solution diversity and hypervolume performance. To further improve the density of the set of the Pareto optimal candidate solutions without requiring an excessive number of subproblems, we leverage a Pareto Front Estimation technique to generate additional solutions in the neighborhood of existing solutions. Additionally, MOBO-OSD supports batch optimization, enabling parallel function evaluations to accelerate the optimization process when resources are available. Through extensive experiments and analysis on a variety of synthetic and real-world benchmark functions with two to six objectives, we demonstrate that MOBO-OSD consistently outperform the state-of-the-art algorithms.

Optimization · Stochastic & Convex Optimization

Gokul Rajaraman, Debasish Chatterjee

The problem of optimally covering a given compact subset of $\mathbb{R}^N$ with a preassigned number $n$ of Euclidean metric balls has a long-standing history and it is well-recognized to be computationally hard. This article establishes a numerically viable algorithm for obtaining optimal covers of compact sets via two key contributions. The first is a foundational result establishing Lipschitz continuity of the marginal function of a certain parametric non-convex maximization problem in the optimal covering problem, and it provides the substrate for numerical gradient algorithms to be employed in this context. The second is an adaptation of a stochastically smoothed numerical gradient-based (zeroth-order) algorithm for a non-convex minimization problem, that, equipped with randomized restarts, spurs global convergence to an optimal cover. Several numerical experiments with complicated nonconvex compact sets demonstrate the excellent performance of our techniques.

Optimization

Deeksha Adil, Brian Bullins, Aaron Sidford, Chenyi Zhang

We develop optimization methods which offer new trade-offs between the number of gradient and Hessian computations needed to compute the critical point of a non-convex function. We provide a method that for a twice-differentiable $f\colon \mathbb{R}^d \rightarrow \mathbb{R}$ with $L_2$-Lipschitz Hessian, and input initial point with $\Delta$-bounded sub-optimality and sufficiently small $\epsilon > 0$ outputs an $\epsilon$-critical point, i.e., a point $x$ such that $\|\nabla f(x)\| \leq \epsilon$, using $\tilde{O}(\Delta L_2^{1/4} n_H^{-1/2}\epsilon^{-9/4})$ queries to a gradient oracle and $n_H$ queries to a Hessian oracle. As a consequence, we obtain an improved gradient query complexity of $\tilde{O}(d^{1/3}L_2^{1/2}\Delta\epsilon^{-3/2})$ in the case of bounded dimension and of $\tilde{O}(\Delta^{3/2} L_2^{3/4}\epsilon^{-9/4})$ in the case where we are allowed only a single Hessian query. We obtain these results through a more general algorithm which can handle approximate Hessian computations and recovers known prior state-of-the-art bounds of computing an $\epsilon$-critical point, under the additional assumption that $f$ has an $L_1$-Lipschitz gradient, with $O(\Delta L_2^{1/4}\epsilon^{-7/4})$-gradient queries.

Optimization

Su Hyeong Lee, Sidharth Sharma, Manzil Zaheer, Tian Li

Adaptive optimization is critical in federated learning, where enabling adaptivity on both the server and client sides has proven essential for achieving optimal performance. However, the scalability of such jointly adaptive systems is often hindered by resource limitations in communication and memory. In this paper, we introduce a class of efficient adaptive algorithms, named $FedAda^2$ and its enhanced version $FedAda^2$++, designed specifically for large-scale, cross-device federated environments. $FedAda^2$ optimizes communication efficiency by avoiding the transfer of preconditioners between the server and clients. Additionally, $FedAda^2$++ extends this approach by incorporating memory-efficient adaptive optimizers on the client side, further reducing on-device memory usage. Theoretically, we demonstrate that $FedAda^2$ and $FedAda^2$++ achieve the same convergence rates for general, non-convex objectives as its more resource-intensive counterparts that directly integrate joint adaptivity. Extensive empirical evaluations on image and text datasets demonstrate both the advantages of joint adaptivity and the effectiveness and efficiency of $FedAda^2$/$FedAda^2$++.

Theory · Everything Else

Takayuki Osogami, Junya Honda, Junpei Komiyama

We study the problem of estimating the mean reward of the best arm in a multi-armed bandit (MAB) setting. Specifically, given a target precision $\varepsilon$ and confidence level $1-\delta$, the goal is to return an $\varepsilon$-accurate estimate of the largest mean reward with probability at least $1-\delta$, while minimizing the number of samples. We first establish an instance-dependent lower bound on the sample complexity, which requires handling the infinitely many possible candidates of the estimated best mean. This lower bound is expressed in a non-convex optimization problem, which becomes the main difficulty of this problem, preventing the direct application of standard techniques such as Track-and-Stop to provably achieve optimality. To overcome this difficulty, we introduce several new algorithmic and analytical techniques and propose an algorithm that achieves the asymptotic lower bound with matching constants in the leading term. Our method combines a confidence ellipsoid-based stopping condition with a two-phase sampling strategy tailored to manage non-convexity proposed algorithm is simple, nearly free of hyperparameters, and achieves the instance-dependent, asymptotically optimal sample complexity. Experimental results support our theoretical guarantees and demonstrate the practical effectiveness of our method.

Optimization

Gonzalo E. Constante, Hao Chen, Can Li

Deep learning models are increasingly deployed in safety-critical tasks where predictions must satisfy hard constraints, such as physical laws, fairness requirements, or safety limits. However, standard architectures lack built-in mechanisms to enforce such constraints, and existing approaches based on regularization or projection are often limited to simple constraints, computationally expensive, or lack feasibility guarantees. This paper proposes a model-agnostic framework for enforcing input-dependent linear equality and inequality constraints on neural network outputs. The architecture combines a task network trained for prediction accuracy with a safe network trained using decision rules from the stochastic and robust optimization literature to ensure feasibility across the entire input space. The final prediction is a convex combination of the two subnetworks, guaranteeing constraint satisfaction during both training and inference without iterative procedures or runtime optimization. We prove that the architecture is a universal approximator of constrained functions and derive computationally tractable formulations based on linear decision rules. Empirical results on benchmark regression tasks show that our method consistently satisfies constraints while maintaining competitive accuracy and low inference latency.

Optimization · Stochastic & Convex Optimization

Yuheng Zhao, Yu-Hu Yan, Kfir Y. Levy, Peng Zhao

Smoothness is known to be crucial for acceleration in offline optimization, and for gradient-variation regret minimization in online learning. Interestingly, these two problems are actually closely connected --- accelerated optimization can be understood through the lens of gradient-variation online learning. In this paper, we investigate online learning with *Hölder* functions, a general class encompassing both smooth and non-smooth (Lipschitz) functions, and explore its implications for offline optimization. For (strongly) convex online functions, we design the corresponding gradient-variation online learning algorithm whose regret smoothly interpolates between the optimal guarantees in smooth and non-smooth regimes. Notably, our algorithms do not require prior knowledge of the Hölder smoothness parameter, exhibiting strong adaptivity over existing methods. Through online-to-batch conversion, this gradient-variation online adaptivity yields an optimal universal method for stochastic convex optimization under Hölder smoothness. However, achieving universality in offline strongly convex optimization is more challenging. We address this by integrating online adaptivity with a detection-based guess-and-check procedure, which, for the first time, yields a universal offline method that achieves accelerated convergence in the smooth regime while maintaining near-optimal convergence in the non-smooth one.

Optimization · Bayesian & Black-box Optimization

WEI XING, Zhenjie Lu, Akeel Shah

Existing Multi-fidelity Bayesian Optimization (MFBO) methods ignore the convergence behavior of the multi-fidelity surrogate as the fidelity increases, leading to inefficient exploration and suboptimal performance. We introduce CAMO (Convergence-Aware Multi-fidelity Optimization), a principled framework based on Linear Fidelity Differential Equations (LFiDEs) that explicitly encodes convergence of fidelity-indexed outputs and employs a closed-form nonstationary kernel. We rigorously prove the existence and pointwise/uniform convergence to the high fidelity surrogate under mild restrictions and provide new convergence results for general FiDEs using smooth, non-smooth and even non-convex Lyapunov functions, establishing a bridge between MFBO and the theory of subgradient flows in non-smooth optimisation theory. Combined with a fidelity-aware acquisition function, CAMO outperforms state-of-the-art MFBO methods on a majority of synthetic and real-world benchmarks, with up to a four-fold improvement in optimisation performance and a dramatic speed-up in convergence. CAMO offers a tractable and theoretically grounded approach to convergence-aware MFBO.

Social Aspects · Privacy-preserving Statistics and Machine Learning

Petros Prastakos, Kayhan Behdin, Rahul Mazumder

Sparse variable selection improves interpretability and generalization in high-dimensional learning by selecting a small subset of informative features. Recent advances in Mixed Integer Programming (MIP) have enabled solving large-scale non-private sparse regression—known as Best Subset Selection (BSS)—with millions of variables in minutes. However, extending these algorithmic advances to the setting of Differential Privacy (DP) has remained largely unexplored. In this paper, we introduce two new differentially private estimators for sparse variable selection, levering modern MIP techniques. Our framework is general and applies broadly to problems like sparse regression or classification, and we provide theoretical support recovery guarantees in the case of BSS. Inspired by the exponential mechanism, we develop structured sampling procedures that efficiently explore the non-convex objective landscape, avoiding the exhaustive combinatorial search in the exponential mechanism. We complement our theoretical findings with extensive numerical experiments, using both least squares and hinge loss for our objective function, and demonstrate that our methods achieve state-of-the-art empirical support recovery, outperforming competing algorithms in settings with up to $p=10^4$.

Deep Learning · Generative Models and Autoencoders

Nina Vesseron, Louis Bethune, Marco Cuturi

The canonical approach in generative modeling is to split model fitting into two blocks: define first how to sample noise (e.g. Gaussian) and choose next what to do with it (e.g. using a single map or flows). We explore in this work an alternative route that ties sampling and mapping. We find inspiration in moment measures, a result that states that for any measure $\rho$, there exists a unique convex potential $u$ such that $\rho=\nabla u \sharp e^{-u}$. While this does seem to tie effectively sampling (from log-concave distribution $e^{-u}$) and action (pushing particles through $\nabla u$), we observe on simple examples (e.g., Gaussians or 1D distributions) that this choice is ill-suited for practical tasks. We study an alternative factorization, where $\rho$ is factorized as $\nabla w^* \sharp e^{-w}$, where $w^\ast$ is the convex conjugate of a convex potential $w$. We call this approach conjugate moment measures, and show far more intuitive results on these examples. Because $\nabla w^*$ is the Monge map between the log-concave distribution $e^{-w}$ and $\rho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $\rho$, and parameterize $w$ as an input-convex neural network. We also address the common sampling scenario in which the density of $\rho$ is known only up to a normalizing constant, and propose an algorithm to learn $w$ in this setting.

Theory · Everything Else

Georgy Noarov, Soham Mallick, Tao Wang, Sunay Joshi, Yan Sun, Yangxinyu Xie, Mengxin Yu, Edgar Dobriban

Top-$k$ decoding is a widely used method for sampling from LLMs: at each token, only the largest $k$ next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-$k$ and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-$k$ decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-$k$ decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We introduce *Bregman decoders* obtained by minimizing a separable Bregman divergence (for both the *primal* and *dual* cases) with a sparsity-inducing $\ell_0$-regularization; in particular, these decoders are *adaptive* in the sense that the sparsity parameter $k$ is chosen depending on the underlying token distribution. Despite the combinatorial nature of the sparse Bregman objective, we show how to optimize it efficiently for a large class of divergences. We prove that (i) the optimal decoding strategies are greedy, and further that (ii) the objective is discretely convex in $k$, such that the optimal $k$ can be identified in logarithmic time. We note that standard top-$k$ decoding arises as a special case for the KL divergence, and construct new decoding strategies with substantially different behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).

Deep Learning · Theory

Jingfeng Wu, Pierre Marion, Peter Bartlett

We study *gradient descent* (GD) with a constant stepsize for $\ell_2$-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in $\widetilde{\mathcal{O}}(\kappa)$ steps with $\kappa$ being the condition number. Surprisingly, we show that this can be *accelerated* to $\widetilde{\mathcal{O}}(\sqrt{\kappa})$ by simply using a large stepsize---for which the objective evolves *nonmonotonically*. The acceleration brought by large stepsizes extends to minimizing the population risk for separable distributions, improving on the best-known upper bounds on the number of steps to reach a near-optimum. Finally, we characterize the largest stepsize for the local convergence of GD, which also determines the global convergence in special scenarios. Our results extend the analysis of Wu et al. (2024) from convex settings with minimizers at infinity to strongly convex cases with finite minimizers.

Optimization · Stochastic & Convex Optimization

Jingxin Zhan, Yuchen Xin, Chenjie Sun, Zhihua Zhang

We consider a common case of the combinatorial semi-bandit problem, the $m$-set semi-bandit, where the learner exactly selects $m$ arms from the total $d$ arms. In the adversarial setting, the best regret bound, known to be $\mathcal{O}(\sqrt{nmd})$ for time horizon $n$, is achieved by the well-known Follow-the-Regularized-Leader (FTRL) policy. However, this requires to explicitly compute the arm-selection probabilities via optimizing problems at each time step and sample according to them. This problem can be avoided by the Follow-the-Perturbed-Leader (FTPL) policy, which simply pulls the $m$ arms that rank among the $m$ smallest (estimated) loss with random perturbation. In this paper, we show that FTPL with a Fr\'echet perturbation also enjoys the near optimal regret bound $\mathcal{O}(\sqrt{nm}(\sqrt{d\log(d)}+m^{5/6}))$ in the adversarial setting and approaches best-of-both-world regret bounds, i.e., achieves a logarithmic regret for the stochastic setting. Moreover, our lower bounds show that the extra factors are unavoidable with our approach; any improvement would require a fundamentally different and more challenging method.

Theory · Learning Theory

Paul Liautaud, Pierre Gaillard, Olivier Wintenberger

We study online adversarial regression with convex losses against a rich class of continuous yet highly irregular competitor functions,% prediction rules, modeled by Besov spaces $B_{pq}^s$ with general parameters $1 \leq p,q \leq \infty$ and smoothness $s > \tfrac{d}{p}$. We introduce an adaptive wavelet-based algorithm that performs sequential prediction without prior knowledge of $(s,p,q)$, and establish minimax-optimal regret bounds against any comparator in $B_{pq}^s$. We further design a locally adaptive extension capable of sequentially adapting to spatially inhomogeneous smoothness. This adaptive mechanism adjusts the resolution of the predictions over both time and space, yielding refined regret bounds in terms of local regularity. Consequently, in heterogeneous environments, our adaptive guarantees can significantly surpass those obtained by standard global methods.

Applications · Robotics

Yiyuan Pan, Yunzhe XU, Zhe Liu, Hesheng Wang

Visual navigation is a fundamental problem in embodied AI, yet practical deployments demand long-horizon planning capabilities to address multi-objective tasks. A major bottleneck is data scarcity: policies learned from limited data often overfit and fail to generalize OOD. Existing neural network-based agents typically increase architectural complexity that paradoxically become counterproductive in the small-sample regime. This paper introduce NeuRO, a integrated learning-to-optimize framework that tightly couples perception networks with downstream task-level robust optimization. Specifically, NeuRO addresses core difficulties in this integration: (i) it transforms noisy visual predictions under data scarcity into convex uncertainty sets using Partially Input Convex Neural Networks (PICNNs) with conformal calibration, which directly parameterize the optimization constraints; and (ii) it reformulates planning under partial observability as a robust optimization problem, enabling uncertainty-aware policies that transfer across environments. Extensive experiments on both unordered and sequential multi-object navigation tasks demonstrate that NeuRO establishes SoTA performance, particularly in generalization to unseen environments. Our work thus presents a significant advancement for developing robust, generalizable autonomous agents.

Theory · Domain Adaptation and Transfer Learning

Christoph Dann, Yishay Mansour, Teodor Vanislavov Marinov, Mehryar Mohri

The rapid proliferation of domain-specialized machine learning models presents a challenge: while individual models excel in specific domains, their performance varies significantly across diverse applications. This makes selecting the optimal model when faced with an unknown mixture of tasks, especially with limited or no data to estimate the mixture, a difficult problem. We address this challenge by formulating it as a multiple-source domain adaptation (MSA) problem. We introduce a novel, scalable algorithm that effectively routes each input to the best-suited model from a pool of available models. Our approach provides a strong performance guarantee: remarkably, for any mixture domain, the accuracy achieved by the best source model is maintained. This guarantee is established through a theoretical bound on the regret for new domains, expressed as a convex combination of the best regrets in the source domains, plus a concentration term that diminishes as the amount of source data increases. While our primary contributions are theoretical and algorithmic, we also present empirical results demonstrating the effectiveness of our approach.