In online convex optimization (OCO), a single loss function sequence is revealed over a time horizon of $T$, and an online algorithm has to choose its action at time $t$, before the loss function at time $t$ is revealed. The goal of the online algorithm is to incur minimal penalty (called $\textit{regret}$ compared to a static optimal action made by an optimal offline algorithm knowing all functions of the sequence in advance. In this paper, we broaden the horizon of OCO, and consider multi-objective OCO, where there are $K$ distinct loss function sequences, and an algorithm has to choose its action at time $t$, before the $K$ loss functions at time $t$ are revealed. To capture the tradeoff between tracking the $K$ different sequences, we consider the $\textit{min-max}$ regret, where the benchmark (optimal offline algorithm) takes a static action across all time slots that minimizes the maximum of the total loss (summed across time slots) incurred by each of the $K$ sequences. An online algorithm is allowed to change its action across time slots, and its {\it min-max} regret is defined as the difference between its $\textit{min-max}$ cost and that of the benchmark. The $\textit{min-max}$ regret is a stringent performance measure and an algorithm with small regret needs to `track' all loss function sequences closely at all times. We consider this $\textit{min-max}$ regret in the i.i.d. input setting where all loss functions are i.i.d. generated from an unknown distribution. For the i.i.d. model we propose a simple algorithm that combines the well-known $\textit{Hedge}$ and online gradient descent (OGD) and show via a remarkably simple proof that its expected $\textit{min-max}$ regret is $O(\sqrt{T \log K})$.

Many real-world machine learning tasks require outputs that satisfy hard constraints, such as physical conservation laws, structured dependencies in graphs, or column-level relationships in tabular data. Existing approaches rely either on domain-specific architectures and losses or on strong assumptions on the constraint space, restricting their applicability to linear or convex constraints. We propose a general-purpose framework for constraint-aware refinement that leverages denoising diffusion implicit models (DDIMs). Starting from a coarse prediction, our method iteratively refines it through a deterministic diffusion trajectory guided by a learned prior and augmented by constraint gradient corrections. The approach accommodates a wide class of non-convex and nonlinear equality constraints and can be applied post hoc to any base model. We demonstrate the method in two representative domains: constrained adversarial attack generation on tabular data with column-level dependencies and in AC power flow prediction under Kirchhoff's laws. Across both settings, our diffusion-guided refinement improves both constraint satisfaction and performance while remaining lightweight and model-agnostic.

Quality-Diversity algorithms have transformed optimization by prioritizing the discovery of diverse, high-performing solutions over a single optimal result. However, traditional Quality-Diversity methods, such as MAP-Elites, rely heavily on predefined behavior descriptors and complete prior knowledge of the task to define the behavior space grid, limiting their flexibility and applicability. In this work, we introduce Vector Quantized-Elites (VQ-Elites), a novel Quality-Diversity algorithm that autonomously constructs a structured behavior space grid using unsupervised learning, eliminating the need for prior task-specific knowledge. At the core of VQ-Elites is the integration of Vector Quantized Variational Autoencoders, which enables the dynamic learning of behavior descriptors and the generation of a structured, rather than unstructured, behavior space grid -- a significant advancement over existing unsupervised Quality-Diversity approaches. This design establishes VQ-Elites as a flexible, robust, and task-agnostic optimization framework. To further enhance the performance of unsupervised Quality-Diversity algorithms, we introduce behavior space bounding and cooperation mechanisms, which significantly improve convergence and performance, as well as the Effective Diversity Ratio and Coverage Diversity Score, two novel metrics that quantify the actual diversity in the unsupervised setting. We validate VQ-Elites on robotic arm pose-reaching, mobile robot space-covering, and MiniGrid exploration tasks. The results demonstrate its ability to efficiently generate diverse, high-quality solutions, emphasizing its adaptability, scalability, robustness to hyperparameters, and potential to extend Quality-Diversity optimization to complex, previously inaccessible domains.

We propose ECPv2, a scalable and theoretically grounded algorithm for global optimization of Lipschitz-continuous functions with unknown Lipschitz constants. Building on the Every Call is Precious (ECP) framework, which ensures that each accepted function evaluation is potentially informative, ECPv2 addresses key limitations of ECP, including high computational cost and overly conservative early behavior. ECPv2 introduces three innovations: (i) an adaptive lower bound to avoid vacuous acceptance regions, (ii) a Worst-m memory mechanism that restricts comparisons to a fixed-size subset of past evaluations, and (iii) a fixed random projection to accelerate distance computations in high dimensions. We theoretically show that ECPv2 retains ECP's no-regret guarantees with optimal finite-time bounds and expands the acceptance region with high probability. We further empirically validate these findings through extensive experiments and ablation studies. Using principled hyperparameter settings, we evaluate ECPv2 across a wide range of high-dimensional, non-convex optimization problems. Across benchmarks, ECPv2 consistently matches or outperforms state-of-the-art optimizers, while significantly reducing wall-clock time.

We investigate how to optimally design local differential privacy (LDP) mechanisms that reduce data unfairness and thereby improve fairness in downstream classification. We first derive a closed-form optimal mechanism for binary sensitive attributes and then develop a tractable optimization framework that yields the corresponding optimal mechanism for multi-valued attributes. As a theoretical contribution, we establish that for discrimination-accuracy optimal classifiers, reducing data unfairness necessarily leads to lower classification unfairness, thus providing a direct link between privacy-aware pre-processing and classification fairness. Empirically, we demonstrate that our approach consistently outperforms existing LDP mechanisms in reducing data unfairness across diverse datasets and fairness metrics, while maintaining accuracy close to that of non-private models. Moreover, compared with leading pre-processing and post-processing fairness methods, our mechanism achieves a more favorable accuracy-fairness trade-off while simultaneously preserving the privacy of sensitive attributes. Taken together, these results highlight LDP as a principled and effective pre-processing fairness intervention technique.

Often the collection of elements to be summarized is generated continuously, and it is important to maintain at real time a summary of the part of the collection generated so far.

We propose stochastic optimization algorithms that can find local minima faster than existing algorithms for nonconvex optimization problems, by exploiting the third-order smoothness to escape non-degenerate saddle points more efficiently.

We propose a novel randomized first order optimization method---SEGA (SkEtched GrAdient method)---which progressively throughout its iterations builds a variance-reduced estimate of the gradient from random linear measurements (sketches) of the gradient provided at each iteration by an oracle. In each iteration, SEGA updates the current estimate of the gradient through a sketch-and-project operation using the information provided by the latest sketch, and this is subsequently used to compute an unbiased estimate of the true gradient through a random relaxation procedure. This unbiased estimate is then used to perform a gradient step. Unlike standard subspace descent methods, such as coordinate descent, SEGA can be used for optimization problems with a non-separable proximal term. We provide a general convergence analysis and prove linear convergence for strongly convex objectives. In the special case of coordinate sketches, SEGA can be enhanced with various techniques such as importance sampling, minibatching and acceleration, and its rate is up to a small constant factor identical to the best-known rate of coordinate descent.