We study the query complexity of sampling from high-dimensional Gaussian distributions using gradient information. In the standard oracle model, exact gradients expose only matrix-vector products with the precision matrix, leading to polynomial approximation barriers and a characteristic \(\sqrtκ\) dependence on the condition number. We show that this barrier disappears when the sampler is allowed to query \emph{smoothed scores}, namely gradients of the logarithms of the Gaussian-convolved densities. For a Gaussian target with precision matrix \(Λ\), a smoothed-score query at noise level \(τ\) gives access to the resolvent \((Λ+τ^{-1}I)^{-1}\). Combining geometrically spaced noise levels with sinc-quadrature rational approximation, we obtain a sampler with $q=O\!\left(\bigl(\logκ+\log(e\sqrt d/δ_{\rm TV})\bigr)\log(e\sqrt d/δ_{\rm TV})\right)$ smoothed-score queries for total variation error \(δ_{\rm TV}\), improving the condition-number dependence from \(\sqrtκ\) to logarithmic. We also study finite-bit gradient oracles. Using coordinatewise quantization of the transformed smoothed-score answers and a final dithering step, we obtain a sampling scheme whose total communicated gradient information is polylogarithmic in \(κ\); in particular, for fixed dimension and accuracy, the bit complexity is \(O(\log^2κ)\). To complement these upper bounds, we introduce a channel-synthesis, or reverse-Shannon, converse technique for sampling lower bounds. This converts total-variation simulation guarantees into communication requirements and yields an \(Ω(\logκ)\) lower bound on the required gradient information. Together, these results identify smoothed scores as a provably more informative oracle for sampling and give nearly matching upper and lower bounds for its finite-bit complexity.

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

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

We consider the problem of Imitation Learning (IL) by actively querying noisy expert for feedback. While imitation learning has been empirically successful, much of prior work assumes access to noiseless expert feedback which is not practical in many applications. In fact, when one only has access to noisy expert feedback, algorithms that rely on purely offline data (non-interactive IL) can be shown to need a prohibitively large number of samples to be successful. In contrast, in this work, we provide an interactive algorithm for IL that uses selective sampling to actively query the noisy expert for feedback. Our contributions are twofold: First, we provide a new selective sampling algorithm that works with general function classes and multiple actions, and obtains the best-known bounds for the regret and the number of queries.

To prove Lemma 2 we start by proving a few inequalities. Since Ais an ( 1, 2,Q)-solver, using Definition 4 and Taylor's expansion, we get for any i [n] and j [k], In this section we present and prove a few auxiliary results which will be used in the proofs our main results. We start with the following standard concentration inequalities. R2, (32) if n clog(1/δ)2, where c > 0 is some absolute constant. The following locality lemma states that the fuzzy k-means function is strictly increasing. Lemma 5. Let (X,P?) be a clustering instance, where P? refers to the optimal solution for the fuzzy k-mean problem (namely, minimizes the objective in (2)). Output: bµj 1: Initialize S φ. 2: for s= 1,2,...,mdo 3: Sample iuniformly at random from [n] and update S S {i}. Next, we analyze the performance of Algorithm 6, which estimates the center of a given cluster using a set of randomly sampled elements. Note that this algorithm is used as a sub-routine in Algorithm 1. Lemma 6 (Estimate of mean using uniform sampling). Let (X,P) be a consistent center-based clustering instance, and let δ (0,1).

The fuzzy or soft k-means objective is a popular generalization of the well-known kmeans problem, extending the clustering capability of the k-means to datasets that are uncertain, vague and otherwise hard to cluster. In this paper, we propose a semisupervised active clustering framework, where the learner is allowed to interact with an oracle (domain expert), asking for the similarity between a certain set of chosen items. We study the query and computational complexities of clustering in this framework. We prove that having a few of such similarity queries enables one to get a polynomial-time approximation algorithm to an otherwise conjecturally NP-hard problem. In particular, we provide algorithms for fuzzy clustering in this setting that ask O(poly(k)logn)similarity queries and run with polynomialtime-complexity, where nis the number of items. The fuzzy k-means objective is nonconvex, with k-means as a special case, and is equivalent to some other generic nonconvex problem such as non-negative matrix factorization. The ubiquitous Lloyd-type algorithms (or alternating-minimization algorithms) can get stuck at a local minima. Our results show that by making few similarity queries, the problem becomes easier to solve. Finally, we test our algorithms over real-world datasets, showing their effectiveness in real-world applications.