This paper presents novel contributions to the field of online optimization, particularly focusing on the adaptation of algorithms from concave optimization to more challenging classes of functions. Key contributions include the introduction of uniform wrappers, a class of meta-algorithms that could be used for algorithmic conversions such as converting algorithms for convex optimization into those for quadratizable optimization. Moreover, we propose a guideline that, given a base algorithm $\mathcal{A}$ for concave optimization and a uniform wrapper $\mathcal{W}$, describes how to convert a proof of the regret bound of $\mathcal{A}$ in the concave setting into a proof of the regret bound of $\mathcal{W}(\mathcal{A})$ for quadratizable setting. Through this framework, the paper demonstrates improved regret guarantees for various classes of DR-submodular functions under zeroth-order feedback. Furthermore, the paper extends zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to existing approaches.
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Optimization · Stochastic & Convex Optimization
We propose and study Sparse Polyak, a variant of Polyak’s adaptive step size, designed to solve high-dimensional statistical estimation problems where the problem dimension is allowed to grow much faster than the sample size. In such settings, the standard Polyak step size performs poorly, requiring an increasing number of iterations to achieve optimal statistical precision-even when, the problem remains well conditioned and/or the achievable precision itself does not degrade with problem size. We trace this limitation to a mismatch in how smoothness is measured: in high dimensions, it is no longer effective to estimate the Lipschitz smoothness constant. Instead, it is more appropriate to estimate the smoothness restricted to specific directions relevant to the problem (restricted Lipschitz smoothnes constant). Sparse Polyak overcomes this issue by modifying the step size to estimate the restricted Lipschitz smoothness constant. We support our approach with both theoretical analysis and numerical experiments, demonstrating its improved performance.
Theory · Learning Theory
Prior work (Klochkov \& Zhivotovskiy, 2021) establishes at most $O\left(\log (n)/n\right)$ excess risk bounds via algorithmic stability for strongly-convex learners with high probability. We show that under the similar common assumptions — Polyak-Lojasiewicz condition, smoothness, and Lipschitz continous for losses — rates of $O\left(\log^2(n)/n^2\right)$ are at most achievable. To our knowledge, our analysis also provides the tightest high-probability bounds for gradient-based generalization gaps in nonconvex settings.
Optimization
In this paper, we establish global non-asymptotic convergence guarantees for the BFGS quasi-Newton method without requiring strong convexity or the Lipschitz continuity of the gradient or Hessian. Instead, we consider the setting where the objective function is strictly convex and strongly self-concordant. For an arbitrary initial point and any arbitrary positive-definite initial Hessian approximation, we prove global linear and superlinear convergence guarantees for BFGS when the step size is determined using a line search scheme satisfying the weak Wolfe conditions. Moreover, all our global guarantees are affine-invariant, with the convergence rates depending solely on the initial error and the strongly self-concordant constant. Our results extend the global non-asymptotic convergence theory of BFGS beyond traditional assumptions and, for the first time, establish affine-invariant convergence guarantees—aligning with the inherent affine invariance of the BFGS method.
Optimization
This work addresses the finite-time analysis of nonsmooth nonconvex stochastic optimization under Riemannian manifold constraints. We adapt the notion of Goldstein stationarity to the Riemannian setting as a performance metric for nonsmooth optimization on manifolds. We then propose a Riemannian Online to NonConvex (RO2NC) algorithm, for which we establish the sample complexity of $O(\epsilon^{-3}\delta^{-1})$ in finding ($\delta,\epsilon$)-stationary points. This result is the first-ever finite-time guarantee for fully nonsmooth, nonconvex optimization on manifolds and matches the optimal complexity in the Euclidean setting. When gradient information is unavailable, we develop a zeroth order version of RO2NC algorithm (ZO-RO2NC), for which we establish the same sample complexity. The numerical results support the theory and demonstrate the practical effectiveness of the algorithms.
Optimization · Stochastic & Convex Optimization
Dataset distillation seeks to condense datasets into smaller but highly representative synthetic samples. While diffusion models now lead all generative benchmarks, current distillation methods avoid them and rely instead on GANs or autoencoders, or, at best, sampling from a fixed diffusion prior. This trend arises because naive backpropagation through the long denoising chain leads to vanishing gradients, which prevents effective synthetic sample optimization. To address this limitation, we introduce Latent Dataset Distillation with Diffusion Models (LD3M), the first method to learn gradient-based distilled latents and class embeddings end-to-end through a pre-trained latent diffusion model. A linearly decaying skip connection, injected from the initial noisy state into every reverse step, preserves the gradient signal across dozens of timesteps without requiring diffusion weight fine-tuning. Across multiple ImageNet subsets at $128\times128$ and $256\times256$, LD3M improves downstream accuracy by up to 4.8 percentage points (1 IPC) and 4.2 points (10 IPC) over the prior state-of-the-art. The code for LD3M is provided at https://github.com/Brian-Moser/prune_and_distill.
Optimization · Stochastic & Convex Optimization
This paper introduces PaZO, a preconditioned accelerated zeroth-order optimization algorithm for fine-tuning large language models (LLMs). First, we theoretically demonstrate the necessity of preconditioning in zeroth-order optimization, proving that zeroth-order stochastic gradient descent (ZO-SGD) alone fails to achieve the ideal convergence rate. Building on this, we propose a Preconditioned Simultaneous Perturbation Stochastic Approximation (PSPSA) and theoretical version of PaZO, and demonstrate that setting the order of preconditioner as $-1/2$ in PSPSA yields the improved convergence rate for PaZO. Moreover, we design a practical version of PaZO that stabilizes training via diagonal Hessian estimate and moving average technique. Extensive experiments on diverse downstream tasks with models like RoBERTa-large and OPT show PaZO’s effectiveness. Compared to other zeroth-order baselines, PaZO achieves better performance across models and tasks.
Optimization
We study quantum algorithms based on quantum (sub)gradient estimation using noisy function evaluation oracles, and demonstrate the first dimension-independent query complexities (up to poly-logarithmic factors) for zeroth-order convex optimization in both smooth and nonsmooth settings. Interestingly, only using noisy function evaluation oracles, we match the first-order query complexities of classical gradient descent, thereby exhibiting exponential separation between quantum and classical zeroth-order optimization. We then generalize these algorithms to work in non-Euclidean settings by using quantum (sub)gradient estimation to instantiate mirror descent and its variants, including dual averaging and mirror prox. By leveraging a connection between semidefinite programming and eigenvalue optimization, we use our quantum mirror descent method to give a new quantum algorithm for solving semidefinite programs, linear programs, and zero-sum games. We identify a parameter regime in which our zero-sum games algorithm is faster than any existing classical or quantum approach.
Optimization
Many high-dimensional optimisation problems exhibit rich geometric structures in their set of minimisers, often forming smooth manifolds due to over-parametrisation or symmetries. When this structure is known, at least locally, it can be exploited through reduction mappings that reparametrise part of the parameter space to lie on the solution manifold. These reductions naturally arise from inner optimisation problems and effectively remove redundant directions, yielding a lower-dimensional objective. In this work, we introduce a general framework to understand how such reductions influence the optimisation landscape. We show that well-designed reduction mappings improve curvature properties of the objective, leading to better-conditioned problems and theoretically faster convergence for gradient-based methods. Our analysis unifies a range of scenarios where structural information at optimality is leveraged to accelerate convergence, offering a principled explanation for the empirical gains observed in such optimisation algorithms.
Applications · Neuroscience, Cognitive Science
Hebbian learning is a key principle underlying learning in biological neural networks. We relate a Hebbian spike-timing-dependent plasticity rule to noisy gradient descent with respect to a non-convex loss function on the probability simplex. Despite the constant injection of noise and the non-convexity of the underlying optimization problem, one can rigorously prove that the considered Hebbian learning dynamic identifies the presynaptic neuron with the highest activity and that the convergence is exponentially fast in the number of iterations. This is non-standard and surprising as typically noisy gradient descent with fixed noise level only converges to a stationary regime where the noise causes the dynamic to fluctuate around a minimiser.
Optimization · Optimization and Learning under Uncertainty
Semi-Infinite Programming (SIP) has emerged as a powerful framework for modeling problems with infinite constraints, however, its theoretical development in the context of nonconvex and large-scale optimization remains limited. In this paper, we investigate a class of nonconvex min-max optimization problems with nonconvex infinite constraints, motivated by applications such as adversarial robustness and safety-constrained learning. We propose a novel inexact dynamic barrier primal-dual algorithm and establish its convergence properties. Specifically, under the assumption that the squared infeasibility residual function satisfies the Lojasiewicz inequality with exponent $\theta \in (0,1)$, we prove that the proposed method achieves $\mathcal{O}(\epsilon^{-3})$, $\mathcal{O}(\epsilon^{-6\theta})$, and $\mathcal{O}(\epsilon^{-3\theta/(1-\theta)})$ iteration complexities to achieve an $\epsilon$-approximate stationarity, infeasibility, and complementarity slackness, respectively. Numerical experiments on robust multitask learning with task priority further illustrate the practical effectiveness of the algorithm.
Optimization · Stochastic & Convex Optimization
As the economic and environmental costs of training and deploying large vision or language models increase dramatically, analog in-memory computing (AIMC) emerges as a promising energy-efficient solution. However, the training perspective, especially its training dynamic, is underexplored. In AIMC hardware, the trainable weights are represented by the conductance of resistive elements and updated using consecutive electrical pulses. While the conductance changes by a constant in response to each pulse, in reality, the change is scaled by asymmetric and non-linear response functions, leading to a non-ideal training dynamic. This paper provides a theoretical foundation for gradient-based training on AIMC hardware with non-ideal response functions. We demonstrate that asymmetric response functions negatively impact Analog SGD by imposing an implicit penalty on the objective. To overcome the issue, we propose residual learning algorithm, which provably converges exactly to a critical point by solving a bilevel optimization problem. We show that the proposed method can be extended to deal with other hardware imperfections like limited response granularity. As far as we know, it is the first paper to investigate the impact of a class of generic non-ideal response functions. The conclusion is supported by simulations validating our theoretical insights.
Theory · Learning Theory
Consider the task of locating an unknown target point using approximate distance queries: in each round, a reconstructor selects a reference point and receives a noisy version of its distance to the target. This problem arises naturally in various contexts—from localization in GPS and sensor networks to privacy-aware data access—making it relevant from the perspective of both the reconstructor (seeking accurate recovery) and the responder (aiming to limit information disclosure, e.g., for privacy or security reasons). We study this reconstruction game through a learning-theoretic lens, focusing on the rate and limits of the best possible reconstruction error. Our first result provides a tight geometric characterization of the optimal error in terms of the Chebyshev radius, a classical concept from geometry. This characterization applies to all compact metric spaces (in fact, to all totally bounded spaces) and yields explicit formulas for natural subsets of the Euclidean metric. Our second result addresses the asymptotic behavior of reconstruction, distinguishing between pseudo-finite spaces, where the optimal error is attained after finitely many queries, and spaces where the approximation curve exhibits a nontrivial decay. We characterize pseudo-finiteness for convex subsets of Euclidean spaces.
Optimization · Stochastic & Convex Optimization
We study stochastic gradient descent (SGD) with gradient clipping on convex functions under a generalized smoothness assumption called $(L_0,L_1)$-smoothness. Using gradient clipping, we establish a high probability convergence rate that matches the SGD rate in the $L$ smooth case up to polylogarithmic factors and additive terms. We also propose a variation of adaptive SGD with gradient clipping, which achieves the same guarantee. We perform empirical experiments to examine our theory and algorithmic choices.
Optimization · Large Scale, Parallel and Distributed
We investigate the problem of finding second-order stationary points (SOSP) in differentially private (DP) stochastic non-convex optimization. Existing methods suffer from two key limitations: \textbf{(i)} inaccurate convergence error rate due to overlooking gradient variance in the saddle point escape analysis, and \textbf{(ii)} dependence on auxiliary private model selection procedures for identifying DP-SOSP, which can significantly impair utility, particularly in distributed settings. To address these issues, we propose a generic perturbed stochastic gradient descent (PSGD) framework built upon Gaussian noise injection and general gradient oracles. A core innovation of our framework is using model drift distance to determine whether PSGD escapes saddle points, ensuring convergence to approximate local minima without relying on second-order information or additional DP-SOSP identification. By leveraging the adaptive DP-SPIDER estimator as a specific gradient oracle, we develop a new DP algorithm that rectifies the convergence error rates reported in prior work. We further extend this algorithm to distributed learning with arbitrarily heterogeneous data, providing the first formal guarantees for finding DP-SOSP in such settings. Our analysis also highlights the detrimental impacts of private selection procedures in distributed learning under high-dimensional models, underscoring the practical benefits of our design. Numerical experiments on real-world datasets validate the efficacy of our approach.
Theory · Domain Adaptation and Transfer Learning
Curriculum learning has emerged as an effective strategy to enhance the training efficiency and generalization of machine learning models. However, its theoretical underpinnings remain relatively underexplored. In this work, we develop a theoretical framework for curriculum learning based on biased regularized empirical risk minimization (RERM), identifying conditions under which curriculum learning provably improves generalization. We introduce a sufficient condition that characterizes a "good" curriculum and analyze a multi-task curriculum framework, where solving a sequence of convex tasks can facilitate better generalization. We also demonstrate how these theoretical insights translate to practical benefits when using stochastic gradient descent (SGD) as an optimization method. Beyond convex settings, we explore the utility of curriculum learning for non-convex tasks. Empirical evaluations on synthetic datasets and MNIST validate our theoretical findings and highlight the practical efficacy of curriculum-based training.
Theory · Learning Theory
We study the online calibration of multi-dimensional forecasts over an arbitrary convex set $\mathcal{P} \subset \mathbb{R}^d$ relative to an arbitrary norm $\Vert\cdot\Vert$. We connect this with the problem of external regret minimization for online linear optimization, showing that if it is possible to guarantee $O(\sqrt{\rho T})$ worst-case regret after $T$ rounds when actions are drawn from $\mathcal{P}$ and losses are drawn from the dual $\Vert \cdot \Vert_*$ unit norm ball, then it is also possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\rho /\epsilon^2))$ rounds. When $\mathcal{P}$ is the $d$-dimensional simplex and $\Vert \cdot \Vert$ is the $\ell_1$-norm, the existence of $O(\sqrt{T\log d})$ algorithms for learning with experts implies that it is possible to obtain $\epsilon$-calibrated forecasts after $T = \exp(O(\log{d}/\epsilon^2)) = d^{O(1/\epsilon^2)}$ rounds, recovering a recent result of Peng 2025. Interestingly, our algorithm obtains this guarantee without requiring access to any online linear optimization subroutine or knowledge of the optimal rate $\rho$ -- in fact, our algorithm is identical for every setting of $\mathcal{P}$ and $\Vert \cdot \Vert$. Instead, we show that the optimal regularizer for the above OLO problem can be used to upper bound the above calibration error by a swap regret, which we then minimize by running the recent TreeSwap algorithm with Follow-The-Leader as a subroutine. The resulting algorithm is highly efficient and plays a distribution over simple averages of past observations in each round. Finally, we prove that any online calibration algorithm that guarantees $\epsilon T$ $\ell_1$-calibration error over the $d$-dimensional simplex requires $T \geq \exp(\mathrm{poly}(1/\epsilon))$ (assuming $d \geq \mathrm{poly}(1/\epsilon)$). This strengthens the corresponding $d^{\Omega(\log{1/\epsilon})}$ lower bound of Peng 2025, and shows that an exponential dependence on $1/\epsilon$ is necessary.
Theory · Learning Theory
Bandeira et al. (2022) introduced the Franz-Parisi (FP) criterion for characterizing the computational hard phases in statistical detection problems. The FP criterion, based on an annealed version of the celebrated Franz-Parisi potential from statistical physics, was shown to be equivalent to low-degree polynomial (LDP) lower bounds for Gaussian additive models, thereby connecting two distinct approaches to understanding the computational hardness in statistical inference. In this paper, we propose a refined FP criterion that aims to better capture the geometric ``overlap" structure of statistical models. Our main result establishes that this optimized FP criterion is equivalent to Statistical Query (SQ) lower bounds---another foundational framework in computational complexity of statistical inference. Crucially, this equivalence holds under a mild, verifiable assumption satisfied by a broad class of statistical models, including Gaussian additive models, planted sparse models, as well as non-Gaussian component analysis (NGCA), single-index (SI) models, and convex truncation detection settings. For instance, in the case of convex truncation tasks, the assumption is equivalent with the Gaussian correlation inequality (Royen, 2014) from convex geometry. In addition to the above, our equivalence not only unifies and simplifies the derivation of several known SQ lower bounds—such as for the NGCA model (Diakonikolas et al., 2017) and the SI model (Damian et al., 2024)—but also yields new SQ lower bounds of independent interest, including for the computational gaps in mixed sparse linear regression (Arpino et al., 2023) and convex truncation (De et al., 2023).
Theory · Learning Theory
In dueling bandits, an agent explores and exploits choices (i.e., arms) by learning from their stochastic feedback in the form of relative preferences. Prior related studies focused on unbiased feedback. In practice, however, the feedback provided by evaluators can be biased. For example, human users are likely to provide biased evaluation towards large language models due to their heterogeneous background. In this work, we aim to minimize the regret in dueling bandits considering evaluators’ biased feedback. We begin with a benchmark case where evaluators’ bias information is known. Solving the known-bias case is nontrivial, because the bias cannot be easily decoupled from the feedback. We overcome this challenge and propose an unbiased arm performance estimator and a bias-sensitive dueling bandits algorithm. We manage to analyze the regret, dealing with the complex form of the estimator, and show that the feedback either matching or opposing the ground-truth reduces the regret. Then, we study the case where evaluators’ bias information is unknown. The associated estimator can hardly be solved in closed-form due to the non-convexity of the estimator solving problem. We address this challenge and propose an extended bias-sensitive algorithm by incorporating block coordinate descent. This algorithm is proven to achieve the same order of regret (as in the known bias case) with a bounded error. Experiments show that when compared with baselines, our algorithms reduces the regret by up to 86.9%.
Social Aspects · Privacy-preserving Statistics and Machine Learning
The hidden state threat model of differential privacy (DP) assumes that the adversary has access only to the final trained machine learning (ML) model, without seeing intermediate states during training. However, the current privacy analyses under this model are restricted to convex optimization problems, reducing their applicability to multi-layer neural networks, which are essential in modern deep learning applications. Notably, the most successful applications of the hidden state privacy analyses in classification tasks have only been for logistic regression models. We demonstrate that it is possible to privately train convex problems with privacy-utility trade-offs comparable to those of 2-layer ReLU networks trained with DP stochastic gradient descent (DP-SGD). This is achieved through a stochastic approximation of a dual formulation of the ReLU minimization problem, resulting in a strongly convex problem. This enables the use of existing hidden state privacy analyses and provides accurate privacy bounds also for the noisy cyclic mini-batch gradient descent (NoisyCGD) method with fixed disjoint mini-batches. Empirical results on benchmark classification tasks demonstrate that NoisyCGD can achieve privacy-utility trade-offs on par with DP-SGD applied to 2-layer ReLU networks.