Distributed Start Framework

Large Multi-Dimensional Space

Generally speaking the optimization methods used in Colossus and similar software are based on using derivatives to improve the log-likelihood or some measure of deviance. These methods search for an optimum, but it is not guaranteed to be a global optimum. The final solution may be dependent on the initial regression conditions. For a model with a small number of covariates or a well understood relationship between covariate and event, the user may be able to start the regression near the global optimum. If the model is composed of hundreds of covariates, this becomes less likely.

The solution an experienced user might come to is to try multiple starting points and see if they all converge to the same solution. Colossus tries to make this easier by automating the process. The user provides Colossus with an initial starting point to try and parameters controlling how many random points to try and how to generate them. Currently Colossus only supports uniformly generated points with user provided minimum and maximum values. The log-linear terms generally have a different range of acceptable values than the linear terms, so different minimum and maximum values can be given for log-linear terms than the other options.

There is one special case of large multi-dimensional spaces that has been given a separate function. This would be general multi-term models. This case assumes that the model can be split into multiple independent terms that can be solved separately. Colossus automates splitting the model into a simplified form, searching for a solution, substituting the final solution for the simplified model into the full model, and then searching for a solution to the full model near the solution to the simplified model.

Infeasible Parameter Spaces

The risks calculated for Cox Proportional Hazards and Poisson model regressions are generally assumed to be strictly positive values. The use of log-likelihoods as a scoring metric would not be possible without this assumption. However it is possible that, during a regression, the risk may be calculated with a set of parameters that would give a negative probability of an event. To illustrate consider the following Poisson model.

\[ \begin{aligned} \lambda(\alpha,z, \beta, x) = (1+\alpha*z \times \exp{(\beta \times x)})\\ E(\alpha,z, \beta, x, t) = \lambda(\alpha,z, \beta, x) * t \end{aligned} \]

The number of events predicted for an interval is proportional to the risk and number of person-years. The exponential term is always strictly positive, but if \(\alpha\) is negative the risk and number of events can also be negative. Suppose we have several ranges of parameters that give negative event rates such that we get the following plot of score by parameter values:

x <- c(
  -2.0, -1.667, -1.333, -1.0, -0.667, -0.333, 0.0, -2.0, -1.667, -1.333, -1.0, -0.667,
  -0.333, 0.0, -2.0, -1.667, -1.333, -1.0, -0.667, -0.333, 0.0, -2.0, -1.667, -1.333,
  -1.0, -0.667, -0.333, 0.0, -2.333, -2.0, -1.667, -1.333, -0.667, -0.333, 0.0, -3.0,
  -2.667, -2.333, -2.0, -1.667, -0.333, 0.0, -3.0, -2.667, -2.333, -2.0, 0.0, -3.0,
  -2.667, -2.333, -2.0, -1.667, -0.333, 0.0, -3.0, -2.667, -2.333, -2.0, -1.667,
  -1.333, -0.667, -0.333, 0.0, -3.0, -2.667, -2.333, -2.0, -1.667, -1.333, -1.0,
  -0.667, -0.333, 0.0
)
y <- c(
  -3.0, -3.0, -3.0, -3.0, -3.0, -3.0, -3.0, -2.667, -2.667, -2.667, -2.667, -2.667,
  -2.667, -2.667, -2.333, -2.333, -2.333, -2.333, -2.333, -2.333, -2.333, -2.0, -2.0,
  -2.0, -2.0, -2.0, -2.0, -2.0, -1.667, -1.667, -1.667, -1.667, -1.667, -1.667, -1.667,
  -1.333, -1.333, -1.333, -1.333, -1.333, -1.333, -1.333, -1.0, -1.0, -1.0, -1.0,
  -1.0, -0.667, -0.667, -0.667, -0.667, -0.667, -0.667, -0.667, -0.333, -0.333,
  -0.333, -0.333, -0.333, -0.333, -0.333, -0.333, -0.333, 0.0, 0.0, 0.0, 0.0, 0.0,
  0.0, 0.0, 0.0, 0.0, 0.0
)
c <- c(
  3.0, 3.85, 4.46, 4.896, 5.209, 5.433, 5.594, 2.278, 3.333, 4.089, 4.631, 5.019, 5.297,
  5.496, 1.455, 2.744, 3.667, 4.328, 4.802, 5.142, 5.385, 0.563, 2.105, 3.209, 4.0, 4.567,
  4.973, 5.264, 3.674, 2.754, 1.47, 2.754, 4.333, 4.806, 5.144, 5.315, 5.045, 4.667, 4.139,
  3.403, 4.667, 5.045, 5.632, 5.487, 5.283, 5.0, 5.0, 5.824, 5.755, 5.658, 5.522, 5.333,
  4.702, 5.07, 5.937, 5.912, 5.877, 5.829, 5.761, 5.667, 5.351, 5.094, 5.351, 6.0, 6.0,
  6.0, 6.0, 6.0, 6.0, 6.0, 6.0, 6.0, 6.0
)
dft <- data.table("x" = x, "y" = y, "c" = c)
g <- ggplot() +
  geom_point(data = dft, aes(
    x = .data$x, y = .data$y,
    color = .data$c
  ), size = 4) +
  scale_fill_continuous(guide = guide_colourbar(title = "-2*Log-Likelihood")) +
  xlab("Linear Parameter") +
  ylab("Log-Linear Parameter")
g + scale_colour_viridis_c()

-2*Log-Likelihood With Infeasible Points Removed

In this plot, the ranges of missing data show points which are infeasible. The goal of regression is reduce the Log-Likelihood close to 0, so the solution would be near the point \((-2,-2)\). If the regression started near the origin it may end up in either of the infeasible ranges. The user might catch this issue before-hand and start the regression carefully to avoid the infeasible ranges. Colossus can automate this process by sampling across a range of possible values and automatically remove infeasible points.