Bradley-Terry-Luce (BTL) model estimation is a well-established strategy to rank a collection of items given a dataset of pairwise comparisons. Although the theoretical performance of BTL estimation methods, such as spectral and maximum likelihood estimation, is well studied in the regime of uniformly sampled graphs, generalizing such results to a wider class of random graphs has proved challenging. In this work, we investigate the entry-wise error of spectral algorithms against a semi-random adversary that can arbitrarily boost the sampling probabilities of certain edges. We find that the performance of the unweighted spectral method is heavily dependent on the spectral properties of the generated graph. Furthermore, we show that asymptotic performance approaching that of uniformly sampled graphs can be recovered by appropriately reweighting the observed edges to counteract the adversary and restore the spectral gap. Finally, we provide numerical simulations that support our theoretical findings.

Dataset distillation, a training-aware data compression technique, has recently attracted increasing attention as an effective tool for mitigating costs of optimization and data storage. However, progress remains largely empirical. Mechanisms underlying the extraction of task-relevant information from the training process and the efficient encoding of such information into synthetic data points remain elusive. In this paper, we theoretically analyze practical algorithms of dataset distillation applied to the gradient-based training of two-layer neural networks with width $L$. By focusing on a non-linear task structure called multi-index model, we prove that the low-dimensional structure of the problem is efficiently encoded into the resulting distilled data. This dataset reproduces a model with high generalization ability for a required memory complexity of $\tildeΘ$$(r^2d+L)$, where $d$ and $r$ are the input and intrinsic dimensions of the task. To the best of our knowledge, this is one of the first theoretical works that include a specific task structure, leverage its intrinsic dimensionality to quantify the compression rate and study dataset distillation implemented solely via gradient-based algorithms.

Active seriation aims at recovering an unknown ordering of $n$ items by adaptively querying pairwise similarities. The observations are noisy measurements of entries of an underlying $n$ x $n$ permuted Robinson matrix, whose permutation encodes the latent ordering. The framework allows the algorithm to start with partial information on the latent ordering, including seriation from scratch as a special case. We propose an active seriation algorithm that provably recovers the latent ordering with high probability. Under a uniform separation condition on the similarity matrix, optimal performance guarantees are established, both in terms of the probability of error and the number of observations required for successful recovery.

For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings andtensor decompositions, providing anearly-consistent noise rate estimator.

Ournew bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enoughn).

Wealso provide provable error bounds fordifferent norms forreconstructing noisy observations. Our empirical validation demonstrates that we obtain better reconstructions when the latent dimensionislarge.

Many potential applications of reinforcement learning (RL) have intractably-large state spaces.

Such first order expansion implies that the risk ofˆβ is asymptotically the same as the risk ofη which leads to a precise characterization of the MSE ofˆβ; this characterization takes aparticularly simple form for isotropic design. Such first order expansion also leads to inference results based onˆβ. We provide sufficient conditions for theexistence ofsuch first order expansion forthree regularizers: theLasso inits constrainedform,thelassoinitspenalizedform,andtheGroup-Lasso.Theresults apply to general loss functions under some conditions and those conditions are satisfied for the squared loss in linear regression and for the logistic loss in the logisticmodel.

A graph G is a collection of nodes and links connecting pairs of nodes.