Matrix completion studies the problem of reconstructing a matrix from a (typically random) subset of its entries by exploiting prior structural assumptions such as low rank and incoherence. Roughly speaking, when the underlying n n matrix has low rank and its eigenvectors are sufficiently incoherent, observing Ω(n log n) entries sampled uniformly at random is sufficient for exact recovery via efficient optimization methods [Keshavan et al., 2009, 2010, Candes and Tao, 2010, Candes and Plan, 2010, Recht, 2011, Candes and Recht, 2012, Jain et al., 2013]. This sample complexity is nearly optimal, since specifying a rank-r matrix requires only O(n) degrees of freedom. Tensor completion generalizes this problem to higher-order arrays, aiming to recover a low-rank tensor from a limited set of observed entries, for example, under uniform random sampling. As a natural higher-order analogue of matrix completion, tensor completion has found broad applications in areas such as recommendation systems [Frolov and Oseledets, 2017], signal and image processing [Govindu, 2005, Liu et al., 2012], and data science [Song et al., 2019]. Despite this close analogy, tensor completion behaves fundamentally differently from its matrix counterpart. In contrast to the classical matrix setting, tensor completion exhibits a pronounced trade-off between computational and statistical complexity: while information-theoretic considerations suggest that relatively few samples suffice for recovery, all currently known polynomial-time algorithms require substantially more observations than this optimal limit. Polynomial-time methods A widely used polynomial-time approach to tensor completion is to reduce the problem to matrix completion via matricization.

In this work, we provide a sharp theory of scaling laws for two-layer neural networks trained on a class of hierarchical multi-index targets, in a genuinely representation-limited regime. We derive exact information-theoretic scaling laws for subspace recovery and prediction error, revealing how the hierarchical features of the target are sequentially learned through a cascade of phase transitions. We further show that these optimal rates are achieved by a simple, target-agnostic spectral estimator, which can be interpreted as the small learning-rate limit of gradient descent on the first-layer weights. Once an adapted representation is identified, the readout can be learned statistically optimally, using an efficient procedure. As a consequence, we provide a unified and rigorous explanation of scaling laws, plateau phenomena, and spectral structure in shallow neural networks trained on such hierarchical targets.

We consider the problem of estimating the Fréchet and conditional Fréchet mean from data taking values in separable metric spaces. Unlike Euclidean spaces, where well-established methods are available, there is no practical estimator that works universally for all metric spaces. Therefore, we introduce a computable estimator for the Fréchet mean based on random quantization techniques and establish its universal consistency across any separable metric spaces. Additionally, we propose another estimator for the conditional Fréchet mean, leveraging data-driven partitioning and quantization, and demonstrate its universal consistency when the output space is any Banach space.

Differentially private synthetic data enables the sharing and analysis of sensitive datasets while providing rigorous privacy guarantees for individual contributors. A central challenge is to achieve strong utility guarantees for meaningful downstream analysis. Many existing methods ensure uniform accuracy over broad query classes, such as all Lipschitz functions, but this level of generality often leads to suboptimal rates for statistics of practical interest. Since many common data analysis queries exhibit smoothness beyond what worst-case Lipschitz bounds capture, we ask whether exploiting this additional structure can yield improved utility. We study the problem of generating $(\varepsilon,δ)$-differentially private synthetic data from a dataset of size $n$ supported on the hypercube $[-1,1]^d$, with utility guarantees uniformly for all smooth queries having bounded derivatives up to order $k$. We propose a polynomial-time algorithm that achieves a minimax error rate of $n^{-\min \{1, \frac{k}{d}\}}$, up to a $\log(n)$ factor. This characterization uncovers a phase transition at $k=d$. Our results generalize the Chebyshev moment matching framework of (Musco et al., 2025; Wang et al., 2016) and strictly improve the error rates for $k$-smooth queries established in (Wang et al., 2016). Moreover, we establish the first minimax lower bound for the utility of $(\varepsilon,δ)$-differentially private synthetic data with respect to $k$-smooth queries, extending the Wasserstein lower bound for $\varepsilon$-differential privacy in (Boedihardjo et al., 2024).

We analyze two classical algorithms for solving additively composite convex optimization problems where the objective is the sum of a smooth term and a nonsmooth regularizer: proximal stochastic gradient method for a single regularizer; and the randomized incremental proximal method, which uses the proximal operator of a randomly selected function when the regularizer is given as the sum of many nonsmooth functions. We focus on relaxing the bounded variance assumption that is common, yet stringent, for getting last iterate convergence rates. We prove the $\widetilde{O}(1/\sqrt{T})$ rate of convergence for the last iterate of both algorithms under componentwise convexity and smoothness, which is optimal up to log terms. Our results apply directly to graph-guided regularizers that arise in multi-task and federated learning, where the regularizer decomposes as a sum over edges of a collaboration graph.

Effective reinforcement learning (RL) for complex stochastic systems requires leveraging historical data collected in previous iterations to accelerate policy optimization. Classical experience replay treats all past observations uniformly and fails to account for their varying contributions to learning. To overcome this limitation, we propose Variance Reduction Experience Replay (VRER), a principled framework that selectively reuses informative samples to reduce variance in policy gradient estimation. VRER is algorithm-agnostic and integrates seamlessly with existing policy optimization methods, forming the basis of our sample-efficient off-policy algorithm, Policy Gradient with VRER (PG-VRER). Motivated by the lack of rigorous theoretical analysis of experience replay, we develop a novel framework that explicitly captures dependencies introduced by Markovian dynamics and behavior-policy interactions. Using this framework, we establish finite-time convergence guarantees for PG-VRER and reveal a fundamental bias-variance trade-off: reusing older experience increases bias but simultaneously reduces gradient variance. Extensive empirical experiments demonstrate that VRER consistently accelerates policy learning and improves performance over state-of-the-art policy optimization algorithms.

Muon updates matrix parameters via the matrix sign of the gradient and has shown strong empirical gains, yet its dynamics and scaling behavior remain unclear in theory. We study Muon in a linear associative memory model with softmax retrieval and a hierarchical frequency spectrum over query-answer pairs, with and without label noise. In this setting, we show that Gradient Descent (GD) learns frequency components at highly imbalanced rates, leading to slow convergence bottlenecked by low-frequency components. In contrast, the Muon optimizer mitigates this imbalance, leading to faster and more uniform progress. Specifically, in the noiseless case, Muon achieves an exponential speedup over GD; in the noisy case with a power-decay frequency spectrum, we derive Muon's optimization scaling law and demonstrate its superior scaling efficiency over GD. Furthermore, we show that Muon can be interpreted as an implicit matrix preconditioner arising from adaptive task alignment and block-symmetric gradient structure. In contrast, the preconditioner with coordinate-wise sign operator could match Muon under oracle access to unknown task representations, which is infeasible for SignGD in practice. Experiments on synthetic long-tail classification and LLaMA-style pre-training corroborate the theory.

Membership Inference Attacks (MIAs) are widely used to quantify training data memorization and assess privacy risks. Standard evaluation requires repeated retraining, which is computationally costly for large models. One-run methods (single training with randomized data inclusion) and zero-run methods (post hoc evaluation) are often used instead, though their statistical validity remains unclear. To address this gap, we frame MIA evaluation as a causal inference problem, defining memorization as the causal effect of including a data point in the training set. This novel formulation reveals and formalizes key sources of bias in existing protocols: one-run methods suffer from interference between jointly included points, while zero-run evaluations popular for LLMs are confounded by non-random membership assignment. We derive causal analogues of standard MIA metrics and propose practical estimators for multi-run, one-run, and zero-run regimes with non-asymptotic consistency guarantees. Experiments on real-world data show that our approach enables reliable memorization measurement even when retraining is impractical and under distribution shift, providing a principled foundation for privacy evaluation in modern AI systems.

The stochastic block model (SBM) is a fundamental model for studying graph clustering or community detection in networks. It has received great attention in the last decade and the balanced case, i.e., assuming all clusters have large size, has been well studied. However, our understanding of SBM with unbalanced communities (arguably, more relevant in practice) is still limited. In this paper, we provide a simple SVD-based algorithm for recovering the communities in the SBM with communities of varying sizes.We improve upon a result of Ailon, Chen and Xu [ICML 2013; JMLR 2015] by removing the assumption that there is a large interval such that the sizes of clusters do not fall in, and also remove the dependency of the size of the recoverable clusters on the number of underlying clusters. We further complement our theoretical improvements with experimental comparisons.Under the planted clique conjecture, the size of the clusters that can be recovered by our algorithm is nearly optimal (up to poly-logarithmic factors) when the probability parameters are constant. As a byproduct, we obtain an efficient clustering algorithm with sublinear query complexity in a faulty oracle model, which is capable of detecting all clusters larger than $\tilde{\Omega}({\sqrt{n}})$, even in the presence of $\Omega(n)$ small clusters in the graph. In contrast, previous efficient algorithms that use a sublinear number of queries are incapable of recovering any large clusters if there are more than $\tilde{\Omega}(n^{2/5})$ small clusters.

Two of the most prominent algorithms for solving unconstrained smooth games are the classical stochastic gradient descent-ascent (SGDA) and the recently introduced stochastic consensus optimization (SCO) [Mescheder et al., 2017]. SGDA is known to converge to a stationary point for specific classes of games, but current convergence analyses require a bounded variance assumption. SCO is used successfully for solving large-scale adversarial problems, but its convergence guarantees are limited to its deterministic variant. In this work, we introduce the expected co-coercivity condition, explain its benefits, and provide the first last-iterate convergence guarantees of SGDA and SCO under this condition for solving a class of stochastic variational inequality problems that are potentially non-monotone. We prove linear convergence of both methods to a neighborhood of the solution when they use constant step-size, and we propose insightful stepsize-switching rules to guarantee convergence to the exact solution. In addition, our convergence guarantees hold under the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching.