# Optimization Path Example: Portfolio Selection via Variance Minimization

## Overview

ROOT is a general functional optimization framework: by supplying a custom objective function, it can be applied to any problem where the goal is to learn an interpretable binary weight function \(w(X) \in \{0, 1\}\) over a set of groups by covariates of interest.

In general optimization mode (`generalizability_path = FALSE`), the user provides a `data.frame` with an optional column `vsq` (a per-unit variance proxy, or the outcome to minimize) and any covariates to split on. ROOT searches over tree-structured weight functions to minimize the supplied objective, then returns a Rashomon set of near-optimal trees and a single summary tree characterizing the final weight assignments. The decision to vote on a single summary tree can be by default majority vote, or the user can specify their own voting functions.

This vignette demonstrates ROOT in general optimization mode using a portfolio selection problem: given a universe of 100 simulated assets characterized by their market beta and annualized volatility, ROOT learns an interpretable rule for which assets to include (\(w = 1\)) in order to minimize portfolio return variance.

## Problem Setup

### Why portfolio selection?

Constructing a minimum-variance portfolio could be framed as an optimization problem. The standard approach could produce weights that are continuous and can be hard to interpret or communicate. ROOT offers a complementary perspective: it returns a binary inclusion rule - a sparse decision tree that describes, in plain language, which types of assets belong in the low-variance portfolio.

### Mapping to ROOT’s framework

The default global objective function ROOT minimizes is as follows: \(L(w, D) = \sqrt{\frac{\sum_i w_i \cdot v_i^2}{\left(\sum_i w_i\right)^2}}\)

where \(v_i^2\) (`vsq`) is a pseudo outcome value, known as the per-unit variance proxy in the optimization example. In the portfolio context, we set `vsq` to the historical return variance of each asset. ROOT then finds the binary weight \(w(X) \in \{0, 1\}\), described as a decision tree over asset features such as beta and volatility that minimizes this quantity. Users can also supply the global objective function they wish to minimize as well.

## Simulating the Data

We simulate 100 assets with two features: annualized volatility and market beta. Returns are generated as a market factor model, and per-asset return variance is computed from 1,000 simulated return observations.

    library(ROOT)
    set.seed(123)

    n_assets <- 100

    # Asset features
    volatility <- runif(n_assets, 0.05, 0.40)  # annualised volatility
    beta       <- runif(n_assets, 0.5,  1.8)   # market beta
    sector     <- sample(c("Tech", "Finance", "Energy", "Health"),
                         n_assets, replace = TRUE)

    # Simulate returns: r_i = beta_i * r_market + epsilon_i
    market      <- rnorm(1000, 0.0005, 0.01)
    returns_mat <- sapply(seq_len(n_assets), function(i)
      beta[i] * market + rnorm(1000, 0, volatility[i] / sqrt(252))
    )

    # Per-asset return variance (the objective proxy ROOT will minimize)
    vsq <- apply(returns_mat, 2, var)

    dat_portfolio <- data.frame(
      vsq    = vsq,
      vol    = volatility,
      beta   = beta,
      sector = as.integer(factor(sector))
    )

    head(dat_portfolio)
    #>            vsq        vol      beta sector
    #> 1 0.0002750578 0.15065213 1.2799856      2
    #> 2 0.0004951413 0.32590680 0.9326706      2
    #> 3 0.0002835768 0.19314192 1.1351969      3
    #> 4 0.0008172707 0.35905609 1.7408160      4
    #> 5 0.0007130039 0.37916355 1.1277731      3
    #> 6 0.0002929372 0.06594477 1.6574553      3

The `vsq` column is recognized by ROOT’s default objective function as the per-unit variance proxy. The columns `vol`, `beta`, and `sector` are the splitting features available to the tree.

### Distribution of asset risk

    plot(
      dat_portfolio$beta, dat_portfolio$vol,
      xlab = "Market beta", ylab = "Annualised volatility",
      pch  = 16, col = "#4E79A7AA",
      main = "Asset universe: volatility vs beta"
    )

We expect ROOT to identify the high-beta, high-volatility corner of this space as the region to exclude.

## Fitting ROOT

    portfolio_fit <- ROOT(
      data        = dat_portfolio,
      num_trees   = 20,
      top_k_trees = TRUE,
      k           = 10,
      seed        = 42
    )

ROOT grows 20 trees and selects the 10 with the lowest objective value as the Rashomon set. Final asset weights \(w_{\text{opt}}\) are determined by majority vote across those 10 trees.

## Inspecting the Results

### Print summary

    print(portfolio_fit)
    #> ROOT object
    #>   Generalizability mode: FALSE
    #>
    #> Summary classifier (f):
    #> n= 100
    #>
    #> node), split, n, loss, yval, (yprob)
    #>       * denotes terminal node
    #>
    #> 1) root 100 4 1 (0.0400000 0.9600000)
    #>   2) beta>=1.658134 12 4 1 (0.3333333 0.6666667)
    #>     4) vol>=0.3285209 4 0 0 (1.0000000 0.0000000) *
    #>     5) vol< 0.3285209 8 0 1 (0.0000000 1.0000000) *
    #>   3) beta< 1.658134 88 0 1 (0.0000000 1.0000000) *

### Detailed summary

    summary(portfolio_fit)
    #> ROOT object
    #>   Generalizability mode: FALSE
    #>
    #> Summary classifier (f):
    #> n= 100
    #>
    #> node), split, n, loss, yval, (yprob)
    #>       * denotes terminal node
    #>
    #> 1) root 100 4 1 (0.0400000 0.9600000)
    #>   2) beta>=1.658134 12 4 1 (0.3333333 0.6666667)
    #>     4) vol>=0.3285209 4 0 0 (1.0000000 0.0000000) *
    #>     5) vol< 0.3285209 8 0 1 (0.0000000 1.0000000) *
    #>   3) beta< 1.658134 88 0 1 (0.0000000 1.0000000) *
    #>
    #> Global objective function:
    #>   User-supplied: No (default objective used)
    #>
    #> Diagnostics:
    #>   Number of trees grown: 20
    #>   Rashomon set size: 10
    #>   % observations with w_opt == 1: 96.0%

The Diagnostics section confirms that 20 trees were grown, 10 were selected into the Rashomon set, and 96% of assets received \(w_{\text{opt}} = 1\) (included in the portfolio). Only the 4 most risk-concentrated assets are screened out.

## Visualizing the Characterized Tree

    plot(portfolio_fit)

The tree encodes a simple, actionable rule:

- `beta < 1.7`: include the asset unconditionally (88 assets).
- `beta >= 1.7` and `vol < 0.33`: include the asset despite high market sensitivity, because some risk may be controlled (8 assets).
- `beta >= 1.7` and `vol >= 0.33`: exclude the asset — high market sensitivity combined with high volatility drives up portfolio variance (4 assets, \(w = 0\)).

## Examining the Weights

The final weight assignments are stored in `portfolio_fit$D_rash$w_opt`. We can compare the included and excluded assets directly.

    dat_portfolio$w_opt <- portfolio_fit$D_rash$w_opt

    # Summary statistics by inclusion decision
    included <- dat_portfolio[dat_portfolio$w_opt == 1, ]
    excluded <- dat_portfolio[dat_portfolio$w_opt == 0, ]

    cat("Included assets (w = 1):", nrow(included), "\n")
    #> Included assets (w = 1): 96
    cat("  Mean beta:       ", round(mean(included$beta), 3), "\n")
    #>   Mean beta:        1.146
    cat("  Mean volatility: ", round(mean(included$vol),  3), "\n\n")
    #>   Mean volatility:  0.219

    cat("Excluded assets (w = 0):", nrow(excluded), "\n")
    #> Excluded assets (w = 0): 4
    cat("  Mean beta:       ", round(mean(excluded$beta), 3), "\n")
    #>   Mean beta:        1.701
    cat("  Mean volatility: ", round(mean(excluded$vol),  3), "\n")
    #>   Mean volatility:  0.368

The excluded assets have substantially higher beta and volatility than the included ones, confirming that ROOT correctly targets the risk-concentrated corner of the asset universe.

### Visualizing the inclusion decision

    plot(
      dat_portfolio$beta, dat_portfolio$vol,
      xlab = "Market beta", ylab = "Annualised volatility",
      pch  = ifelse(dat_portfolio$w_opt == 1, 16, 4),
      col  = ifelse(dat_portfolio$w_opt == 1, "#4E79A7", "#E15759"),
      main = "Portfolio inclusion decisions"
    )
    legend(
      "topleft",
      legend = c("w = 1 (included)", "w = 0 (excluded)"),
      pch    = c(16, 4),
      col    = c("#4E79A7", "#E15759"),
      bty    = "n"
    )

## Using a Custom Objective Function

ROOT’s general optimization mode is not limited to the default variance objective. You can supply any function of the form `function(D) -> numeric` where `D` is the working data frame with a column `w` containing the current weight assignments.

For example, suppose we want to minimize the interquartile range of portfolio returns rather than variance. We can define a custom objective:

    iqr_objective <- function(D) {
      w <- D$w
      if (sum(w) == 0) return(Inf)
      # Weighted IQR: compute quantiles using the included assets only
      included_vsq <- D$vsq[w == 1]
      diff(quantile(included_vsq, probs = c(0.25, 0.75)))
    }

    portfolio_fit_iqr <- ROOT(
      data                = dat_portfolio,
      global_objective_fn = iqr_objective,
      num_trees           = 20,
      top_k_trees         = TRUE,
      k                   = 10,
      seed                = 112
    )

The custom objective illustrates ROOT’s flexibility: any scalar-valued function of the weighted dataset can be used as the optimization target, and the resulting characterized tree would still be interpretable.

## Key Parameters

Parameter | Role | Default
--- | --- | ---
`num_trees` | Number of trees to grow in the forest | `10`
`top_k_trees` | If `TRUE`, select the top `k` trees by objective value | `FALSE`
`k` | Rashomon set size when `top_k_trees = TRUE` | `10`
`cutoff` | Rashomon threshold when `top_k_trees = FALSE`; `"baseline"` uses objective at \(w \equiv 1\) | `"baseline"`
`vote_threshold` | Fraction of Rashomon-set trees that must vote \(w = 1\) for inclusion | `2/3`
`global_objective_fn` | Custom objective `function(D) -> numeric`; if `NULL`, uses default variance objective | `NULL`
`seed` | Random seed for reproducibility | `NULL`
`feature_est` | Feature importance method for split selection (`"Ridge"`, `"GBM"`, or custom) | `"Ridge"`
`leaf_proba` | Controls tree depth by increasing the probability of stopping at a leaf | `0.25`

## Reference

Parikh, H., Ross, R. K., Stuart, E., & Rudolph, K. E. (2025). Who Are We Missing?: A Principled Approach to Characterizing the Underrepresented Population. Journal of the American Statistical Association, 120(551), 1414–1423. https://doi.org/10.1080/01621459.2025.2495319