We propose a decentralized market model in which agents can negotiate bilateral contracts. This builds on a similar, but centralized, model of trading networks introduced by Hatfield et al. in 2013. Prior work has established that fully-substitutable preferences guarantee the existence of competitive equilibria which can be centrally computed. Our motivation comes from the fact that prices in markets such as over-the-counter markets and used car markets arise from decentralized negotiation among agents, which has left open an important question as to whether equilibrium prices can emerge from agent-to-agent bilateral negotiations. We design a best response dynamic intended to capture such negotiations between market participants. We assume fully substitutable preferences for market participants. In this setting, we provide proofs of convergence for sparse markets (covering many real world markets of interest), and experimental results for more general cases, demonstrating that prices indeed reach equilibrium, quickly, via bilateral negotiations. Our best response dynamic, and its convergence behavior, forms an important first step in understanding how decentralized markets reach, and retain, equilibrium.

Large Language Models (LLMs) embed sensitive, human-generated data, prompting the need for unlearning methods. Although certified unlearning offers strong privacy guarantees, its restrictive assumptions make it unsuitable for LLMs, giving rise to various heuristic approaches typically assessed through empirical evaluations. These standard evaluations randomly select data for removal, apply unlearning techniques, and use membership inference attacks (MIAs) to compare unlearned models against models retrained without the removed data. However, to ensure robust privacy protections for every data point, it is essential to account for scenarios in which certain data subsets face elevated risks. Prior research suggests that outliers, particularly including data tied to minority groups, often exhibit higher memorization propensity which indicates they may be more difficult to unlearn. Building on these insights, we introduce a complementary, minority-aware evaluation framework to highlight blind spots in existing frameworks. We substantiate our findings with carefully designed experiments, using canaries with personally identifiable information (PII) to represent these minority subsets and demonstrate that they suffer at least 20\% higher privacy leakage across various unlearning methods, MIAs, datasets, and LLM scales. Our proposed minority-aware evaluation framework marks an essential step toward more equitable and comprehensive assessments of LLM unlearning efficacy.

Adversarially robust optimization (ARO) has emerged as the *de facto* standard for training models that hedge against adversarial attacks in the test stage. While these models are robust against adversarial attacks, they tend to suffer severely from overfitting. To address this issue, some successful methods replace the empirical distribution in the training stage with alternatives including *(i)* a worst-case distribution residing in an ambiguity set, resulting in a distributionally robust (DR) counterpart of ARO; *(ii)* a mixture of the empirical distribution with a distribution induced by an auxiliary (*e.g.*, synthetic, external, out-of-domain) dataset. Inspired by the former, we study the Wasserstein DR counterpart of ARO for logistic regression and show it admits a tractable convex optimization reformulation. Adopting the latter setting, we revise the DR approach by intersecting its ambiguity set with another ambiguity set built using the auxiliary dataset, which offers a significant improvement whenever the Wasserstein distance between the data generating and auxiliary distributions can be estimated. We study the underlying optimization problem, develop efficient solution algorithms, and demonstrate that the proposed method outperforms benchmark approaches on standard datasets.

Creation of a synthetic dataset that faithfully represents the data distribution and simultaneously preserves privacy is a major research challenge. Many space partitioning based approaches have emerged in recent years for answering statistical queries in a differentially private manner. However, for synthetic data generation problem, recent research has been mainly focused on deep generative models. In contrast, we exploit space partitioning techniques together with noise perturbation and thus achieve intuitive and transparent algorithms. We propose both data independent and data dependent algorithms for $\epsilon$-differentially private synthetic data generation whose kernel density resembles that of the real dataset. Additionally, we provide theoretical results on the utility-privacy trade-offs and show how our data dependent approach overcomes the curse of dimensionality and leads to a scalable algorithm. We show empirical utility improvements over the prior work, and discuss performance of our algorithm on a downstream classification task on a real dataset.