Performance of empirical risk minimization in linear aggregation

Guillaume Lecué; Shahar Mendelson

doi:10.3150/15-BEJ701

Performance of empirical risk minimization in linear aggregation

Guillaume Lecué, Shahar Mendelson

Research output: Contribution to journal › Article › peer-review

23 Citations (Scopus)

Abstract

We study conditions under which, given a dictionary F = {f₁, . . . , f_M} and an i.i.d. sample (X_i,Y_i )N i=1, the empirical minimizer in span(F ) relative to the squared loss, satisfies that with high probability R(∼f ERM) ≤ inf f span(F )R(f )+ rN(M),where R() is the squared risk and rN(M) is of the order of M/N. Among other results, we prove that a uniform small-ball estimate for functions in span(F) is enough to achieve that goal when the noise is independent of the design.

Original language	English
Pages (from-to)	1520-1534
Number of pages	15
Journal	Bernoulli
Volume	22
Issue number	3
DOIs	https://doi.org/10.3150/15-BEJ701
Publication status	Published - Aug 2016
Externally published	Yes

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title = "Performance of empirical risk minimization in linear aggregation",

abstract = "We study conditions under which, given a dictionary F = {f1, . . . , fM} and an i.i.d. sample (Xi,Yi )N i=1, the empirical minimizer in span(F ) relative to the squared loss, satisfies that with high probability R(∼f ERM) ≤ inf f span(F )R(f )+ rN(M),where R() is the squared risk and rN(M) is of the order of M/N. Among other results, we prove that a uniform small-ball estimate for functions in span(F) is enough to achieve that goal when the noise is independent of the design.",

keywords = "Aggregation theory, Empirical processes, Empirical risk minimization, Learning theory",

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year = "2016",

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AU - Lecué, Guillaume

AU - Mendelson, Shahar

PY - 2016/8

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N2 - We study conditions under which, given a dictionary F = {f1, . . . , fM} and an i.i.d. sample (Xi,Yi )N i=1, the empirical minimizer in span(F ) relative to the squared loss, satisfies that with high probability R(∼f ERM) ≤ inf f span(F )R(f )+ rN(M),where R() is the squared risk and rN(M) is of the order of M/N. Among other results, we prove that a uniform small-ball estimate for functions in span(F) is enough to achieve that goal when the noise is independent of the design.

AB - We study conditions under which, given a dictionary F = {f1, . . . , fM} and an i.i.d. sample (Xi,Yi )N i=1, the empirical minimizer in span(F ) relative to the squared loss, satisfies that with high probability R(∼f ERM) ≤ inf f span(F )R(f )+ rN(M),where R() is the squared risk and rN(M) is of the order of M/N. Among other results, we prove that a uniform small-ball estimate for functions in span(F) is enough to achieve that goal when the noise is independent of the design.

KW - Aggregation theory

KW - Empirical processes

KW - Empirical risk minimization

KW - Learning theory

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Performance of empirical risk minimization in linear aggregation

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