We discuss a method for estimating the convergence speed of Anderson Acceleration (AA) applied to the Alternating Direction Method of Multipliers (ADMM), for the case where ADMM by itself converges linearly. It has been observed empirically that AA may speed up asymptotic ADMM convergence by a large amount for various applications in data science and machine learning, but there are no known convergence bounds for AA with finite window size that would allow quantification of this improvement in asymptotic convergence speed. We discuss an approach to quantify the improved linear asymptotic convergence factor that is based on computing the optimalasymptotic convergence factor for a stationary version of AA applied to ADMM . Numerical tests show that this allows us to estimate the asymptotic linear convergence factor of AA with finite window size applied to ADMM. The numerical results also explain how and by how much nonlinear acceleration methods like AA can accelerate ADMM, in terms of the spectral properties of the Jacobians of ADMM and stationary AA evaluated at the fixed point .
We explain how Anderson Acceleration (AA) speeds up the Alternating Direction Method of Multipliers (ADMM), for the case where ADMM by itself converges linearly. We do so by considering the spectral properties of the Jacobians of ADMM and a stationary version of AA evaluated at the fixed point, where the coefficients of the stationary version are computed such that its asymptotic linear convergence factor is optimal . Numerical tests show that this allows us to quantify the improved linear asymptotic convergence speed of AA-ADMM as compared to the convergence factor of ADMM used by itself. This way of estimating AA-ADMM convergence speed is useful because there are no known convergence bounds for AA with finite window size that would allow quantification of this improvement in asymptotic convergence speed .
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It has been observed empirically that AA may speed up asymptotic ADMM convergence by a large amount for various applications in data science and machine learning, but there are no known convergence bounds for AA with finite window size that would allow quantification of this improvement in asymptotic convergence speed. We discuss an approach to quantify the improved linear asymptotic convergence factor that is based on computing the optimalasymptotic convergence factor for a stationary version of AA applied to ADMM
We do so by considering the spectral properties of the Jacobians of ADMM and a stationary version of AA evaluated at the fixed point, where the coefficients of the stationary version are computed such that its asymptotic linear convergence factor is optimal
R
estimate the asymptotic linear convergence factor of
quantify the improved linear asymptotic convergence speed of AA-ADMM as compared to the convergence factor of ADMM used by itself. This way of estimating AA-ADMM convergence speed is useful because there are no known convergence bounds for
R
applied to ADMM. The numerical results also explain how and by how much nonlinear acceleration methods like AA can accelerate ADMM, in terms of the spectral properties of the Jacobians of ADMM and stationary AA evaluated at the fixed point
that would allow quantification of this improvement in asymptotic convergence speed