We introduce general tools for designing efficient private estimation algorithms, in the high-dimensional settings, whose statistical guarantees almost match those of the best known non-private algorithms. To illustrate our techniques, we consider two problems: recovery of stochastic block models and learning mixtures of spherical Gaussians. For the former, we present the first efficient (ε,δ)-differentially private algorithms for both weak recovery and exact recovery. Previously known algorithms achieving comparable guarantees required quasi-polynomial time. We complement these results with an information-theoretic lower bound that highlights how the guarantees of our algorithms are almost tight. For the latter, we design an (ε,δ)-differentially private algorithm that recovers the centers of the k-mixture when the minimum separation is at least O(k1/t t). For all choices of t, this algorithm requires sample complexity n kO(1)dO(t) and time complexity (nd)O(t). Prior work required either an additional additive Ω( logn) term in the minimum separation or an explicit upper bound on the Euclidean norm of the centers.

Inference in Markov random fields subject to consistency structure is a fundamental problem that arises in many real-life applications. In order to enforce consistency, classical approaches utilize consistency potentials or encode constraints over feasible instances. Unfortunately this comes at the price of a tremendous computational burden. In this paper we suggest to tackle consistency by incorporating constraints on beliefs. This permits derivation of a closed-form message-passing algorithm which we refer to as the Constraints Based Convex Belief Propagation (CBCBP). Experiments show that CBCBP outperforms the conventional consistency potential based approach, while being at least an order of magnitude faster.

Neural Information Processing Systems http://nips.cc/

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. In recent prior work Sinha and Vaze [2024], algorithms with simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ or (CCV of $O(1)$ in specific cases Vaze and Sinha [2025], e.g. when $d=1$) have been proposed. It is widely believed that CCV is $Ω(\sqrt{T})$ for all algorithms that ensure that regret is $O(\sqrt{T})$ with the worst case input for any $d\ge 2$. In this paper, we refute this and show that the algorithm of Vaze and Sinha [2025] simultaneously achieves regret of $O(\sqrt{T})$ regret and CCV of $O(T^{1/3})$ when $d=2$.

We study Constrained Online Convex Optimization with Memory (COCO-M), where both the loss and the constraints depend on a finite window of past decisions made by the learner. This setting extends the previously studied unconstrained online optimization with memory framework and captures practical problems such as the control of constrained dynamical systems and scheduling with reconfiguration budgets. For this problem, we propose the first algorithms that achieve sublinear regret and sublinear cumulative constraint violation under time-varying constraints, both with and without predictions of future loss and constraint functions. Without predictions, we introduce an adaptive penalty approach that guarantees sublinear regret and constraint violation. When short-horizon and potentially unreliable predictions are available, we reinterpret the problem as online learning with delayed feedback and design an optimistic algorithm whose performance improves as prediction accuracy improves, while remaining robust when predictions are inaccurate. Our results bridge the gap between classical constrained online convex optimization and memory-dependent settings, and provide a versatile learning toolbox with diverse applications.