General solution to unsupervised learning problems: a meta-algorithm-based framework

Researchers from Microsoft Research and Princeton University on November 13 proposed a general framework for designing efficient algorithms for unsupervised learning problems, such as a mixture of Gaussian distributions and subspace clustering.

#The framework proposed by the researchers uses a meta-algorithm to solve the noise problem, which uses the calculation method of the lower bound learning calculation formula. This framework is designed based on the recent work of Garg, Kayal, and Saha (FOCS'20), who proposed this framework for learning arithmetic formulas without any noise. A key element of the meta-algorithm is an efficient algorithm for solving a new problem called "robust vector space decomposition"

Research has shown that when some matrix has a sufficiently large minimum non-zero Meta-algorithms work well when dealing with singular values. "We conjecture that this condition holds for smoothed instances of our problem, and therefore our framework will yield efficient algorithms for these problems in smooth settings."

The study is titled "In Learning Arithmetic Formulas in the Presence of Noise: A General Framework and Applications to Unsupervised Learning, published on the arXiv preprint platform on November 13

Unsupervised learning involves discovering hidden patterns and structures in data without using any labels or direct human supervision.

Here, researchers consider data that has a good mathematical structure or is generated from a mathematically well-defined distribution. An example of the former is that data points can be grouped into meaningful clusters based on certain similarity patterns, and the goal is to find the underlying clusters. An example of the latter is mixture modeling, which assumes that data are generated by a mixture of succinctly described probability distributions (e.g. Gaussian distributions), and the goal is to learn the parameters of these distributions from samples.

A common framework for solving many unsupervised learning problems is the moment method, which exploits the statistical moments of the data to infer the underlying structure or underlying parameters of the model. For many unsupervised learning problem scenarios, where the underlying data has some nice mathematical structure, the moments of the data are well-defined functions of the parameters. Heuristic arguments suggest that the opposite should generally hold, i.e. the parameters of a structure/distribution are usually uniquely determined by some low-order moments of the data. In this general direction, the main challenge is to design algorithms to (approximately) recover the underlying parameters from (empirical) moments.

We also want the algorithm to be efficient, tolerant to noise (i.e., work well even if the moments are only known approximately, rather than exactly), and even tolerant of anomalies (i.e., Works well even if a few data points don't fit the underlying structure/distribution). But even the simplest problems in the field tend to be NP-hard, and remain so even in the absence of noise and outliers.

So one cannot actually expect an algorithm with provable worst-case guarantees. But one can hope that the algorithm is guaranteed to work generally well, i.e. for random problem instances, or more ideally for instances chosen in a smooth way. Therefore, many different algorithms have been designed for each such problem in unsupervised learning, with varying levels of efficiency, noise tolerance, outlier tolerance, and provable guarantees.

In this work, the researchers present a meta-algorithm applicable to many such unsupervised learning problems. The starting point of this study is the observation that many such problems boil down to the task of learning appropriate subclasses of arithmetic formulas.

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