We present a general framework for designing efficient algorithms for unsupervised learning problems, such as mixtures of Gaussians and subspace clustering. Our framework is based on a meta algorithm that learns arithmetic circuits in the presence of noise, using lower bounds. This builds upon the recent work of Garg, Kayal and Saha (FOCS 20), who designed such a framework for learning arithmetic circuits without any noise. A key ingredient of our meta algorithm is an efficient algorithm for a novel problem called Robust Vector Space Decomposition. We show that our meta algorithm works well when certain matrices have sufficiently large smallest non-zero singular values. We conjecture that this condition holds for smoothed instances of our problems, and thus our framework would yield efficient algorithms for these problems in the smoothed setting.

Since solving State-of-the-art algorithms for Approximate Nearest Neighbor Search the problem exactly requires an expensive exhaustive scan of the (ANNS) such as DiskANN, FAISS-IVF, and HNSW build data dependent database - which would be impractical for real-world indices that indices that offer substantially better accuracy and search span billions of objects - practical interactive search systems use efficiency over data-agnostic indices by overfitting to the index Approximate Nearest Neighbor Search (ANNS) algorithms with data distribution. When the query data is drawn from a different highly sub-linear query complexity [10, 18, 24, 30] to answer such distribution - e.g., when index represents image embeddings and queries. The quality of such ANN indices is often measured by query represents textual embeddings - such algorithms lose much k-recall@k which is the overlap between the top-results of the of this performance advantage. On a variety of datasets, for a fixed index search with the ground truth -nearest neighbors (-NNs) in recall target, latency is worse by an order of magnitude or more for the corpus for the query, averaged over a representative query set. Out-Of-Distribution (OOD) queries as compared to In-Distribution State-of-the-art algorithms for ANNS, such as graph-based indices (ID) queries. The question we address in this work is whether ANNS [16, 24, 30] which use data-dependent index construction, algorithms can be made efficient for OOD queries if the index construction achieve better query efficiency over prior data-agnostic methods is given access to a small sample set of these queries. We like LSH [6, 18] (see Section A.1 for more details). Such efficiency answer positively by presenting OOD-DiskANN, which uses a sparing enables these indices to serve queries with > 90% recall with a sample (1% of index set size) of OOD queries, and provides up to latency of a few milliseconds, required in interactive web scenarios.

We study the tradeoff between the statistical error and communication cost of distributed statistical estimation problems in high dimensions. In the distributed sparse Gaussian mean estimation problem, each of the $m$ machines receives $n$ data points from a $d$-dimensional Gaussian distribution with unknown mean $\theta$ which is promised to be $k$-sparse. The machines communicate by message passing and aim to estimate the mean $\theta$. We provide a tight (up to logarithmic factors) tradeoff between the estimation error and the number of bits communicated between the machines. This directly leads to a lower bound for the distributed \textit{sparse linear regression} problem: to achieve the statistical minimax error, the total communication is at least $\Omega(\min\{n,d\}m)$, where $n$ is the number of observations that each machine receives and $d$ is the ambient dimension. These lower results improve upon [Sha14,SD'14] by allowing multi-round iterative communication model. We also give the first optimal simultaneous protocol in the dense case for mean estimation. As our main technique, we prove a \textit{distributed data processing inequality}, as a generalization of usual data processing inequalities, which might be of independent interest and useful for other problems.

We explore the connection between dimensionality and communication cost in distributed learning problems. Specifically we study the problem of estimating the mean $\vectheta$ of an unknown $d$ dimensional gaussian distribution in the distributed setting. In this problem, the samples from the unknown distribution are distributed among $m$ different machines. The goal is to estimate the mean $\vectheta$ at the optimal minimax rate while communicating as few bits as possible. We show that in this setting, the communication cost scales linearly in the number of dimensions i.e. one needs to deal with different dimensions individually. Applying this result to previous lower bounds for one dimension in the interactive setting \cite{ZDJW13} and to our improved bounds for the simultaneous setting, we prove new lower bounds of $\Omega(md/\log(m))$ and $\Omega(md)$ for the bits of communication needed to achieve the minimax squared loss, in the interactive and simultaneous settings respectively. To complement, we also demonstrate an interactive protocol achieving the minimax squared loss with $O(md)$ bits of communication, which improves upon the simple simultaneous protocol by a logarithmic factor. Given the strong lower bounds in the general setting, we initiate the study of the distributed parameter estimation problems with structured parameters. Specifically, when the parameter is promised to be $s$-sparse, we show a simple thresholding based protocol that achieves the same squared loss while saving a $d/s$ factor of communication. We conjecture that the tradeoff between communication and squared loss demonstrated by this protocol is essentially optimal up to logarithmic factor.