The best replacement for least squares is linear regression.

The best alternatives to least squares are rated as following:
★★★★★  linear regression
★★★★★  gradient descent
★★★★★  ordinary least squares
★★★★  least absolute deviation
★★★  total least squares
★★★  maximum likelihood estimator
★★  maximum likelihood
★  kalman filter
  compass rule

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least squares
linear regression
gradient descent