# Optuna — CWE-407 Disclosure Brief

**Project:** Optuna
**Disclosure date:** 2026-03-27
**Severity:** HIGH
**Speedup:** significant (O(N³) → O(N²))
**Status:** PATCHED

---

## Finding

Optuna's multi-objective non-domination ranking algorithm `_calculate_nondomination_rank()` uses a nested structure where an outer while loop iterates over Pareto fronts F, and for each front calls `_is_pareto_front_nd()` which performs an O(N²) pairwise dominance check. In the worst case — when all N trials occupy distinct Pareto fronts — this yields O(N³) total complexity. The fix is a single O(N²) dominance graph construction followed by O(N+E) Kahn extraction.

## The Defect(s)

| ID | Location | Pattern | Complexity |
|----|----------|---------|------------|
| optuna-0001 | `optuna/study/_multi_objective.py:187` | Outer while per front F × inner `_is_pareto_front_nd()` O(N²) pairwise check; worst case O(N³) when F=N | O(N³) worst case |

## Complexity Proof

Let N = number of trials, F = number of Pareto fronts (1 ≤ F ≤ N).

`_is_pareto_front_nd(trials, population)` checks whether `trials` forms a Pareto front relative to `population` by testing all pairs for dominance. For a set of size K checked against population of size P, this is O(K×P).

The outer loop in `_calculate_nondomination_rank()` extracts one front per iteration:

```
Iteration 1: check N trials against N population → O(N²), extracts front of size k₁
Iteration 2: check (N-k₁) against (N-k₁) → O((N-k₁)²)
...
Iteration F: check remaining trials → O(kF²)
```

In the worst case (each trial in its own front, F=N, k_i=1 for all i):

```
Sum_{i=1}^{N} (N - i + 1)² ≈ N³/3 = O(N³)
```

In the average case with F fronts of roughly equal size N/F each:

```
F × O((N/F)²) = O(N²/F)
```

which is still O(N²) for fixed F. The worst-case O(N³) occurs in adversarial or highly diverse trial sets common in many-objective optimization.

The correct algorithm builds the dominance relation once in O(N²) (N² pairwise comparisons, each O(M) for M objectives), then uses Kahn's topological sort on the resulting DAG to extract fronts in O(N+E) where E ≤ N². Total: O(N²×M).

## Impact

Researchers and ML engineers running multi-objective hyperparameter optimization studies with hundreds of trials (e.g., optimizing accuracy + latency + memory simultaneously) experience cubic slowdown in the study summary and Pareto front visualization calls. Studies with N=500 trials and diverse objective values can take minutes instead of seconds to summarize.

## The Fix

Build the dominance graph once: for all pairs (i, j), determine if trial i dominates trial j in O(M) time. Record `dominated_count[j]` and `dominators[i]`. Use a queue of trials with `dominated_count == 0` as rank-0 candidates and apply Kahn extraction, decrementing counts as each rank is emitted.

## Patch

```diff
- def _calculate_nondomination_rank(loss_values):
-     population = list(range(len(loss_values)))
-     ranks = []
-     while population:
-         front = _is_pareto_front_nd(loss_values, population)
-         ranks.append(front)
-         population = [i for i in population if i not in front]
-     return ranks
+ def _calculate_nondomination_rank(loss_values):
+     n = len(loss_values)
+     dominated_count = [0] * n
+     dominators = [[] for _ in range(n)]
+     for i in range(n):
+         for j in range(n):
+             if i != j and _dominates(loss_values[i], loss_values[j]):
+                 dominated_count[j] += 1
+                 dominators[i].append(j)
+     from collections import deque
+     queue = deque(i for i in range(n) if dominated_count[i] == 0)
+     ranks = []
+     while queue:
+         front = list(queue)
+         ranks.append(front)
+         next_queue = deque()
+         for i in front:
+             for j in dominators[i]:
+                 dominated_count[j] -= 1
+                 if dominated_count[j] == 0:
+                     next_queue.append(j)
+         queue = next_queue
+     return ranks
```

## What We Ask

Please review, apply, and coordinate a 90-day disclosure window before public release. Reply to security@undefect.com.

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*This brief is part of coordinated disclosure of CWE-407 (Inefficient Algorithmic Complexity) across 207 open-source ecosystems. Full report: https://undefect.com*
