In this work, we study offline convex optimization with smooth objectives, where the classical Nesterov's Accelerated Gradient (NAG) method achieves the optimal accelerated convergence. Extensive research has aimed to understand NAG from various perspectives, and a recent line of work approaches this from the viewpoint of online learning and online-to-batch conversion, emphasizing the role of optimistic online algorithms for acceleration. In this work, we contribute to this perspective by proposing novel optimistic online-to-batch conversions that incorporate optimism theoretically into the analysis, thereby significantly simplifying the online algorithm design while preserving the optimal convergence rates. Specifically, we demonstrate the effectiveness of our conversions through the following results: (i) when combined with simple online gradient descent, our optimistic conversion achieves the optimal accelerated convergence; (ii) our conversion also applies to strongly convex objectives, and by leveraging both optimistic online-to-batch conversion and optimistic online algorithms, we achieve the optimal accelerated convergence rate for strongly convex and smooth objectives, for the first time through the lens of online-to-batch conversion; (iii) our optimistic conversion can achieve universality to smoothness -- applicable to both smooth and non-smooth objectives without requiring knowledge of the smoothness coefficient -- and remains efficient as non-universal methods by using only one gradient query in each iteration. Finally, we highlight the effectiveness of our optimistic online-to-batch conversions by a precise correspondence with NAG.

We propose efficient methods for solving stochastic simple bilevel optimization problems with convex inner levels, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic optimization problem. Existing methods often rely on costly projection or linear optimization oracles over complex sets, limiting their scalability. To overcome this, we propose an iteratively regularized conditional gradient approach that leverages linear optimization oracles exclusively over the base feasible set. Our proposed methods employ a vanishing regularization sequence that progressively emphasizes the inner problem while biasing towards desirable minimal outer objective solutions. In the one-sample stochastic setting and under standard convexity assumptions, we establish non-asymptotic convergence rates of O(t 1/4)for both the outer and inner objectives. In the finite-sum setting with a mini-batch scheme, the corresponding rates become O(t 1/2). When the outer objective is nonconvex, we prove nonasymptotic convergence rates of O(t 1/7) for both the outer and inner objectives in the one-sample stochastic setting, and O(t 1/4) in the finite-sum setting. Experimental results on over-parametrized regression and dictionary learning tasks demonstrate the practical advantages of our approach over existing methods, confirming our theoretical findings.

Existing model-based algorithms for risk measures like the entropic risk measure (ERM) and entropic value-at-risk (EVaR) are effective in small problems, but require full access to transition probabilities. We propose a Q-learning algorithm to compute the optimal stationary policy for total-reward ERM and EVaR objectives with strong convergence and performance guarantees. The algorithm and its optimality are made possible by ERM's dynamic consistency and elicitability. Our numerical results on tabular domains demonstrate quick and reliable convergence of the proposed Q-learning algorithm to the optimal risk-averse value function.

The imperative of user privacy protection and regulatory compliance necessitates sensitive data removal in model training, yet this process often induces distributional shifts that undermine model performance-particularly in out-of-distribution (OOD) scenarios. To address this issue we propose a novel data removal approach that enhances deep predictive models through factor decorrelation and loss perturbation. Our approach introduces: (1) a discriminative-preserving factor decorrelation module employing dynamic adaptive weight adjustment and iterative representation updating to reduce feature redundancy and minimize inter-feature correlations.

We introduce a novel framework for differentially private (DP) statistical estimation via data truncation, addressing a key challenge in DP estimation when the data support is unbounded. Traditional approaches rely on problem-specific sensitivity analysis, limiting their applicability. By leveraging techniques from truncated statistics, we develop computationally efficient DP estimators for exponential family distributions, including Gaussian mean and covariance estimation, achieving near-optimal sample complexity. Previous works on exponential families only consider bounded or one-dimensional families. Our approach mitigates sensitivity through truncation while carefully correcting for the introduced bias using maximum likelihood estimation and DP stochastic gradient descent. Along the way, we establish improved uniform convergence guarantees for the log-likelihood function of exponential families, which may be of independent interest. Our results provide a general blueprint for DP algorithm design via truncated statistics.

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 lowerdimensional 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.

Recent work on first-order optimizers for empirical risk minimization (ERM) has suggested that smoothness of ERM loss functions in the training data, rather than in the optimization parameters, can be leveraged to improve the oracle complexity of gradient descent (GD) methods. In this paper, we propose an inexact gradient method, piecewise polynomial interpolation-based gradient descent (PPI-GD), which approximates the full gradient in each iteration by querying the first-order oracle at equidistant points in the data domain to construct polynomial interpolants of the resulting gradient samples over appropriately sized patches of the data domain. We analyze the oracle complexity of PPI-GD for strongly convex and non-convex loss functions when the data space dimension is bounded by a polylogarithmic function of the number of training samples, and find it to outperform several GD variants in key regimes when the loss function is sufficiently smooth. Furthermore, our analysis extends several techniques from the error analysis of bicubic spline interpolants to the setting of $d$-variate tensor product polynomial interpolants which may be of independent interest in interpolation analysis.

Stochastic gradient optimization is the dominant learning paradigm for a variety of scenarios, from classical supervised learning to modern self-supervised learning. We consider stochastic gradient algorithms for learning problems whose objectives rely on unknown nuisance parameters, and establish non-asymptotic convergence guarantees. Our results show that, while the presence of a nuisance can alter the optimum and upset the optimization trajectory, the classical stochastic gradient algorithm may still converge under appropriate conditions, such as Neyman orthogonality. Moreover, even when Neyman orthogonality is not satisfied, we show that an algorithm variant with approximately orthogonalized updates (with an approximately orthogonalized gradient oracle) may achieve similar convergence rates. Examples from orthogonal statistical learning/double machine learning and causal inference are discussed.

We establish a comprehensive finite-sample and asymptotic theory for stochastic gradient descent (SGD) with constant learning rates. First, we propose a novel linear approximation technique to provide a quenched central limit theorem (CLT) for SGD iterates with refined tail properties, showing that regardless of the chosen initialization, the fluctuations of the algorithm around its target point converge to a multivariate normal distribution. Our conditions are substantially milder than those required in the classical CLTs for SGD, yet offering a stronger convergence result. Furthermore, we derive the first Berry-Esseen bound - the Gaussian approximation error - for the constant learning-rate SGD, which is sharp compared to the decaying learning-rate schemes in the literature. Beyond the moment convergence, we also provide the Nagaev-type inequality for the SGD tail probabilities by adopting the autoregressive approximation techniques, which entails non-asymptotic largedeviation guarantees. These results are verified via numerical simulations, paving the way for theoretically grounded uncertainty quantification, especially with non-asymptotic validity.

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.