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Provable benefits of score matching
Chirag Pabbaraju · Dhruv Rohatgi · Anish Prasad Sevekari · Holden Lee · Ankur Moitra · Andrej Risteski

Thu Dec 14 03:00 PM -- 05:00 PM (PST) @ Great Hall & Hall B1+B2 #1727

Score matching is an alternative to maximum likelihood (ML) for estimating a probability distribution parametrized up to a constant of proportionality. By fitting the ''score'' of the distribution, it sidesteps the need to compute this constant of proportionality (which is often intractable).While score matching and variants thereof are popular in practice, precise theoretical understanding of the benefits and tradeoffs with maximum likelihood---both computational and statistical---are not well understood. In this work, we give the first example of a natural exponential family of distributions such that the score matching loss is computationally efficient to optimize, and has a comparable statistical efficiency to ML, while the ML loss is intractable to optimize using a gradient-based method. The family consists of exponentials of polynomials of fixed degree, and our result can be viewed as a continuous analogue of recent developments in the discrete setting. Precisely, we show: (1) Designing a zeroth-order or first-order oracle for optimizing the maximum likelihood loss is NP-hard. (2) Maximum likelihood has a statistical efficiency polynomial in the ambient dimension and the radius of the parameters of the family. (3) Minimizing the score matching loss is both computationally and statistically efficient, with complexity polynomial in the ambient dimension.

Author Information

Chirag Pabbaraju (Stanford University)
Dhruv Rohatgi (Massachusetts Institute of Technology)
Anish Prasad Sevekari (Carnegie Mellon University)
Holden Lee (Johns Hopkins University)
Ankur Moitra (MIT)
Andrej Risteski (CMU)

Assistant Professor in the ML department at CMU. Prior to that I was a Wiener Fellow at MIT, and prior to that finished my PhD at Princeton University.

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