Inverse reinforcement learning (IRL) typically assumes demonstrations from a single optimal demonstrator, but in many applications data come from multiple imperfect demonstrators with heterogeneous suboptimality levels. We study reward learning in this setting through a feasible-reward-set framework: for each demonstrator, we encode its declared suboptimality level as a linear constraint and intersect the resulting feasible sets across demonstrators. Our theoretical analysis shows that the joint feasible set shrinks monotonically as data are added, and we give an exact characterization of when a new demonstrator strictly tightens it. We further establish two recovery guarantees for the feasible reward set of the ground-truth optimal demonstrator: one bound depends on closeness to the optimal occupancy, while the other requires only sufficient coverage and no near-optimal demonstrator. On the practical side, we introduce strategies to address the inherent reward ambiguity in the obtained reward set and provide an offline algorithm with function approximation for high-dimensional environments. Experiments in tabular grid-world and large language model (LLM) fine-tuning settings are consistent with the theoretical predictions and demonstrate the effectiveness of the proposed framework over baselines.

Large language models often produce hallucinated answers that violate prompt-level constraints. A key diagnostic question is whether these failures reflect missing knowledge, or whether the model has the relevant information but follows the wrong inference path. We study this phenomenon as inference misalignment: a mismatch between the answer supported by the prompt and the answer favored by statistically salient latent associations. We formalize this view with a latent key-task model, in which pretraining-frequency imbalance can cause a shortcut path to dominate the constraint-sensitive path and induce positive inference loss. The framework predicts two failure modes: task-retrieval bias in entity disambiguation and key-selection bias in action choice. We introduce TrapQA, a controlled diagnostic testbed with two components. ScientistQA tests disambiguation among similar scientists with supplementary factual probes, while Real-Life Constrained QA tests everyday constraint following under salient shortcuts. Our results show that hallucination can arise from biased latent inference rather than absent knowledge alone.

Contextual Stochastic Bilevel Optimization (CSBO) extends standard stochastic bilevel optimization (SBO) by incorporating context-dependent lower-level problems. CSBO problems are generally intractable since existing methods require solving a distinct lower-level problem for each sampled context, resulting in prohibitive sample and computational complexity, in addition to relying on impractical conditional sampling oracles. We propose a reduction framework that approximates the lower-level solutions using expressive basis functions, thereby decoupling the lower-level dependence on context and transforming CSBO into a standard SBO problem solvable using only joint samples from the context and noise distribution. First, we show that this reduction preserves hypergradient accuracy and yields an $\epsilon$-stationary solution to CSBO. Then, we relate the sample complexity of the reduced problem to simple metrics of the basis. This establishes sufficient criteria for a basis to yield $\epsilon$-stationary solutions with a near-optimal complexity of $\widetilde{\mathcal{O}}(\epsilon^{-3})$, matching the best-known rate for standard SBO up to logarithmic factors. Moreover, we show that Chebyshev polynomials provide a concrete and efficient choice of basis that satisfies these criteria for a broad class of problems. Empirical results on inverse and hyperparameter optimization demonstrate that our approach outperforms CSBO baselines in convergence, sample efficiency, and memory usage.

Reinforcement learning (RL) has become a key driver of language model reasoning. Among RL algorithms, Group Relative Policy Optimization (GRPO) is the de facto standard, avoiding the need for a critic by using per-prompt baselines and variance normalization. Yet why and when this normalization helps remains unclear. In this work, we provide an explanation through the lens of local curvature of the sequence-level policy gradient: standard deviation normalization implements an adaptive gradient. Theoretically, under mild conditions, GRPO enjoys a strictly improved convergence rate over unnormalized REINFORCE, with gains characterized by the average within-prompt reward standard deviation across prompts and iterations. Empirically, our analysis on GSM8K and MATH benchmarks reveals three distinct training phases governed by the interplay between feature orthogonality and reward variance: (I) an early acceleration phase where high variance and orthogonality favor adaptive scaling; (II) a relatively stable transition phase; and (III) a late-stage regime where the loss of orthogonality limits further gains. Together, these results provide a principled account of when std normalization helps in GRPO, and offer broader insights into the design of critic-free RL algorithms.

Conventional preference learning methods often prioritize opinions held more widely when aggregating preferences from multiple evaluators. This may result in policies that are biased in favor of some types of opinions or groups and susceptible to strategic manipulation. To address this issue, we develop a novel preference learning framework capable of aligning aggregate opinions and policies proportionally with the true population distribution of evaluator preferences. Grounded in social choice theory, our approach infers the feasible set of evaluator population distributions directly from pairwise comparison data. Using these estimates, the algorithm constructs a policy that satisfies foundational axioms from social choice theory, namely monotonicity and Pareto efficiency, as well as our newly-introduced axioms of population-proportional alignment and population-bounded manipulability. Moreover, we propose a soft-max relaxation method that smoothly trade-offs population-proportional alignment with the selection of the Condorcet winner (which beats all other options in pairwise comparisons). Finally, we validate the effectiveness and scalability of our approach through experiments on both tabular recommendation tasks and large language model alignment.

Diffusion models are increasingly used not only for sampling from learned data distributions, but also for generating samples that optimize task-specific objectives. A common approach is to guide the reverse diffusion process using gradients of an external objective. However, when the data distribution is supported on a structured feasible set, such as a manifold or a constraint set, gradient guidance can move samples away from the learned data geometry. In this paper, we study a simple projected-gradient-guided diffusion update based on the observation that the Stein denoising operator can act as an approximate projection onto the data geometry. The proposed update incorporates the objective gradient inside the denoising step, yielding an inference-time method that uses only a pretrained denoiser and gradient evaluations. We analyze this update as an inexact projected-gradient method for constrained optimization over learned feasible geometries. Our theory covers three settings: linear manifolds, compact convex feasible sets, and compact Riemannian submanifolds. In all these settings, we prove descent and finite-time convergence guarantees. Numerical experiments support the theoretical interpretation and illustrate how the proposed update balances objective descent with preservation of the learned geometry.

We study bilevel optimization problems where the lower-level problems are strongly convex and have coupled linear constraints. To overcome the potential non-smoothness of the hyper-objective and the computational challenges associated with the Hessian matrix, we utilize penalty and augmented Lagrangian methods to reformulate the original problem as a single-level one. Especially, we establish a strong theoretical connection between the reformulated function and the original hyper-objective by characterizing the closeness of their values and derivatives. Based on this reformulation, we propose a single-loop, first-order algorithm for linearly constrained bilevel optimization (SFLCB). We provide rigorous analyses of its non-asymptotic convergence rates, showing an improvement over prior double-loop algorithms -- form $O(\epsilon^{-3}\log(\epsilon^{-1}))$ to $O(\epsilon^{-3})$. The experiments corroborate our theoretical findings and demonstrate the practical efficiency of the proposed SFLCB algorithm. Simulation code is provided at https://github.com/ShenGroup/SFLCB.

Momentum accelerated stochastic gradient descent (SGDM) has gained significant popularity in several signal processing and machine learning tasks. Despite its widespread success, the step size of SGDM remains a critical hyperparameter affecting its performance and often requires manual tuning. Recently, some works have introduced the Polyak step size to SGDM and provided corresponding convergence analysis. However, the convergence guarantee of existing Polyak step sizes for SGDM are limited to convex objectives and lack theoretical support for more widely applicable non-convex problems. To bridge this gap, we design a novel Polyak adaptive step size for SGDM. The proposed algorithm, termed SGDM-APS, incorporates a moving average form tailored for the momentum mechanism in SGDM. We establish the convergence guarantees of SGDM-APS for both convex and non-convex objectives, providing theoretical analysis of its effectiveness. To the best of our knowledge, SGDM-APS is the first Polyak step size for SGDM with general convergence guarantee. Our analysis can also be extended to constant step size SGDM, enriching the theoretical comprehension of the classic SGDM algorithm. Through extensive experiments on diverse benchmarks, we demonstrate that SGDM-APS achieves competitive convergence rates and generalization performance compared to several popular optimization algorithms.

We study a linear contextual optimization problem where a decision maker has access to historical data and contextual features to learn a cost prediction model aimed at minimizing decision error. We adopt the predict-then-optimize framework for this analysis. Given that perfect model alignment with reality is often unrealistic in practice, we focus on scenarios where the chosen hypothesis set is misspecified. In this context, it remains unclear whether current contextual optimization approaches can effectively address such model misspecification. In this paper, we present a novel integrated learning and optimization approach designed to tackle model misspecification in contextual optimization. This approach offers theoretical generalizability, tractability, and optimality guarantees, along with strong practical performance. Our method involves minimizing a tractable surrogate loss that aligns with the performance value from cost vector predictions, regardless of whether the model misspecified or not, and can be optimized in reasonable time. To our knowledge, no previous work has provided an approach with such guarantees in the context of model misspecification.

Contextual optimization problems are prevalent in decision-making applications where historical data and contextual features are used to learn predictive models that inform optimal actions. However, practical applications often suffer from model misspecification due to incomplete knowledge of the underlying data-generating process, leading to suboptimal decisions. Existing approaches primarily address the well-specified case, leaving a critical gap in handling misspecified models. In this paper, we propose a novel Integrated Learning and Optimization (ILO) framework that explicitly accounts for model misspecification by introducing a tractable surrogate loss function with strong theoretical guarantees on generalizability, tractability, and optimality. Our surrogate loss aligns with the true decision performance objective, ensuring robustness to misspecification without imposing restrictive assumptions. The proposed approach effectively mitigates the challenges of non-convexity and non-smoothness in the target loss function, leading to efficient optimization procedures. We provide rigorous theoretical analysis and experimental validation, demonstrating superior performance compared to state-of-the-art methods. Our work offers a principled solution to the practically relevant challenge of model misspecification in contextual optimization.

With increasing size of large language models (LLMs), full-parameter fine-tuning imposes substantial memory demands. To alleviate this, we propose a novel memory-efficient training paradigm called Momentum Low-rank compression (MLorc). The key idea of MLorc is to compress and reconstruct the momentum of matrix parameters during training to reduce memory consumption. Compared to LoRA, MLorc avoids enforcing a fixed-rank constraint on weight update matrices and thus enables full-parameter learning. Compared to GaLore, MLorc directly compress the momentum rather than gradients, thereby better preserving the training dynamics of full-parameter fine-tuning. We provide a theoretical guarantee for its convergence under mild assumptions. Empirically, MLorc consistently outperforms other memory-efficient training methods, matches or even exceeds the performance of full fine-tuning at small ranks (e.g., r=4), and generalizes well across different optimizers – all while not compromising time or memory efficiency.

Acquiring accurate channel state information (CSI) at low overhead is crucial for millimeter wave MIMO communications but is challenging in dynamic environments. In this work, we exploit the emerging integrated sensing and communication (ISAC) beamforming technique for concurrent CSI sensing and data transmission. Despite its low overhead, the corresponding ISAC transmit beamforming design faces a complex trade-off between CSI sensing accuracy and communication interference management. To address this, we formulate the beamforming design as an optimization problem minimizing the maximum Cram & eacute;r-Rao bound (CRB) of CSI sensing errors subject to the users' worst-case communication rates under CSI errors. To efficiently solve the problem, we step-by-step propose three algorithms. The first algorithm is based on the semidefinite relaxation and successive convex optimization techniques, which can serve as a benchmark algorithm but suffers high computational complexity. To efficiently handle the worst-case objective and rate constraints, we propose a complexity-reduced algorithm based on the primal-dual optimization method and first-order min-max algorithm. Furthermore, we dismiss SDR and employ the block coordinate descent method combined with cheap gradient descent steps to achieve a low-complexity algorithm. Extensive simulations show the proposed ISAC beamforming design and low-complexity algorithms can provide robust communication performance and significantly outperform existing schemes.

We propose smoothed primal-dual algorithms for solving stochastic nonconvex optimization problems with linear inequality constraints. Our algorithms are single-loop and only require a single (or two) samples of stochastic gradients at each iteration. A defining feature of our algorithm is that it is based on an inexact gradient descent framework for the Moreau envelope, where the gradient of the Moreau envelope is estimated using one step of a stochastic primal-dual (linearized) augmented Lagrangian algorithm. To handle inequality constraints and stochasticity, we combine the recently established global error bounds in constrained optimization with a Moreau envelope-based analysis of stochastic proximal algorithms. We establish the optimal (in their respective cases) O(epsilon(-4)) and O(epsilon(-3)) sample complexity guarantees for our algorithms and provide extensions to stochastic linear constraints. Unlike existing methods, iterations of our algorithms are free of subproblems, large batch sizes or increasing penalty parameters in their iterations and they use dual variable updates to ensure feasibility.

The majority of parameters in neural networks are naturally represented as matrices. However, most commonly used optimizers treat these matrix parameters as flattened vectors during optimization, potentially overlooking their inherent structural properties. Recently, an optimizer called Muon has been proposed, specifically designed to optimize matrix-structured parameters. Extensive empirical evidence shows that Muon can significantly outperform traditional optimizers when training neural networks. Nonetheless, the theoretical understanding of Muon's convergence behavior and the reasons behind its superior performance remain limited. In this work, we present a comprehensive convergence rate analysis of Muon and its comparison with Gradient Descent (GD). We characterize the conditions under which Muon can outperform GD. Our theoretical results reveal that Muon can benefit from the low-rank structure of Hessian matrices, a phenomenon widely observed in practical neural network training. Our experimental results support and corroborate the theoretical findings.

Inverse Reinforcement Learning (IRL) and Reinforcement Learning from Human Feedback (RLHF) are pivotal methodologies in reward learning, which involve inferring and shaping the underlying reward function of sequential decision-making problems based on observed human demonstrations and feedback. Most prior work in reward learning has relied on prior knowledge or assumptions about decision or preference models, potentially leading to robustness issues. In response, this paper introduces a novel linear programming (LP) framework tailored for offline reward learning. Utilizing pre-collected trajectories without online exploration, this framework estimates a feasible reward set from the primal-dual optimality conditions of a suitably designed LP, and offers an optimality guarantee with provable sample efficiency. Our LP framework also enables aligning the reward functions with human feedback, such as pairwise trajectory comparison data, while maintaining computational tractability and sample efficiency. We demonstrate that our framework potentially achieves better performance compared to the conventional maximum likelihood estimation (MLE) approach through analytical examples and numerical experiments.

This paper introduces a smoothed proximal Lagrangian method for minimizing a nonconvex smooth function over a convex domain with additional explicit convex nonlinear constraints. Two key features are 1) the proposed method is single-looped, and 2) an first-order iteration complexity of 𝒪(ϵ^-2) is established under mild regularity assumptions. The first feature suggests the practical efficiency of the proposed method, while the second feature highlights its theoretical superiority. Numerical experiments on various problem scales demonstrate the advantages of the proposed method in terms of speed and solution quality.

In adversarial machine learning, neural networks suffer from a significant issue known as robust overfitting, where the robust test accuracy decreases over epochs (Rice et al., 2020). Recent research conducted by Xing et al., 2021;Xiao et al., 2022 has focused on studying the uniform stability of adversarial training. Their investigations revealed that SGD-based adversarial training fails to exhibit uniform stability, and the derived stability bounds align with the observed phenomenon of robust overfitting in experiments. This finding motivates us to develop uniformly stable algorithms specifically tailored for adversarial training. To this aim, we introduce Moreau envelope-$\mathcal{A}$ (ME-$\mathcal{A}$), a variant of the Moreau Envelope-type algorithm. We employ a Moreau envelope function to reframe the original problem as a min-min problem, separating the non-strong convexity and non-smoothness of the adversarial loss. Then, this approach alternates between solving the inner and outer minimization problems to achieve uniform stability without incurring additional computational overhead. In practical scenarios, we demonstrate the efficacy of ME-$\mathcal{A}$ in mitigating the issue of robust overfitting. Beyond its application in adversarial training, this represents a fundamental result in uniform stability analysis, as ME-$\mathcal{A}$ is the first algorithm to exhibit uniform stability for weakly-convex, non-smooth problems.

Decentralized computation has received considerable research interest lately, due to its wide applications in information processing systems. However, one key requirement to establish convergence for almost all decentralized algorithms, for convex and non-convex problems alike, is that the loss function has Lipschitz-continuous gradient (LipGrad). This is a strong assumption, which does not hold for many practical problems, such as matrix/tensor factorization, neural network training, etc. On the contrary, in the centralized setting, one can utilize techniques such as the Bregman proximal gradient (BPG) method to deal with the lack of LipGrad. This work fills the gap between centralized and decentralized cases by developing a novel smoothed decentralized BPG algorithm to deal with a class of nonconvex decentralized problem, where the local problems do not have LipGrad objective functions. By leveraging the recent notion of relative smoothness and primal-dual error bounds, we show that the proposed algorithm achieves a certain ε-stationary solution by using $\mathcal{O}\left( {{\varepsilon ^{ - 2}}} \right)$ iterations, matching the rate of the centralized Bregman proximal gradient method. To our knowledge, this is the first decentralized algorithm that matches the centralized convergence rate bounds under the class of considered problems. Our numerical results on the decentralized quadratic regression example demonstrate the effectiveness of proposed algorithm.

In recent years, federated minimax optimization has attracted growing interest due to its extensive applications in various machine learning tasks. While Smoothed Alternative Gradient Descent Ascent (Smoothed-AGDA) has proved its success in centralized nonconvex minimax optimization, how and whether smoothing technique could be helpful in federated setting remains unexplored. In this paper, we propose a new algorithm termed Federated Stochastic Smoothed Gradient Descent Ascent (FESS-GDA), which utilizes the smoothing technique for federated minimax optimization. We prove that FESS-GDA can be uniformly used to solve several classes of federated minimax problems and prove new or better analytical convergence results for these settings. We showcase the practical efficiency of FESS-GDA in practical federated learning tasks of training generative adversarial networks (GANs) and fair classification.

In recent years, deep network pruning has attracted significant attention in order to enable the rapid deployment of AI into small devices with computation and memory constraints. Pruning is often achieved by dropping redundant weights, neurons, or layers of a deep network while attempting to retain a comparable test performance. Many deep pruning algorithms have been proposed with impressive empirical success. However, existing approaches lack a quantifiable measure to estimate the compressibility of a sub-network during each pruning iteration and thus may under-prune or over-prune the model. In this work, we propose PQ Index (PQI) to measure the potential compressibility of deep neural networks and use this to develop a Sparsity-informed Adaptive Pruning (SAP) algorithm. Our extensive experiments corroborate the hypothesis that for a generic pruning procedure, PQI decreases first when a large model is being effectively regularized and then increases when its compressibility reaches a limit that appears to correspond to the beginning of underfitting. Subsequently, PQI decreases again when the model collapse and significant deterioration in the performance of the model start to occur. Additionally, our experiments demonstrate that the proposed adaptive pruning algorithm with proper choice of hyper-parameters is superior to the iterative pruning algorithms such as the lottery ticket-based pruning methods, in terms of both compression efficiency and robustness.