Extremal Preserve/Delete Objective

• The extremal preserve/delete objective is a formal framework that optimizes the balance between retaining informative components and eliminating obstructive elements.
• It integrates mathematical formulations with explicit trade-off hyperparameters across combinatorics, robust set selection, model unlearning, and visual attribution to guarantee performance.
• Algorithmic strategies—including RL-based evolution, greedy methods, and differentiable contour optimization—deliver practical, efficient solutions in diverse applications.

An extremal preserve/delete objective refers to the formalization and optimization of tasks where one must, under resource or adversarial constraints, maximally preserve desired elements or features while deleting, deactivating, or ignoring obstructive ones. This paradigm occurs across combinatorial optimization, learning-to-unlearn in models, robust set function maximization, and interpretable machine vision, with a core structural motif: explicit trade-off or control between retaining informative components and preventing loss due to deletion, obsolescence, or adversarial action. The following sections detail foundational instances, mathematical frameworks, key algorithms, and theoretical and empirical results in prominent domains.

1. Formal Paradigms and Definitions

The extremal preserve/delete formulation is instantiated in multiple settings under the commonality of selective retention and elimination:

• Combinatorics (Zero–One Matrix Patterns): Given two $0$–$1$ matrices $A$ and $M$, $M$ is said to contain $A$ if $A$ can be produced from $M$ by deleting rows, deleting columns, and changing some $1$s to $0$s. The extremal number $\ex(n,A)$ is the maximum number of $1$-entries in an $n\times n$ matrix $M$ that does not contain $A$ (Janzer et al., 7 Mar 2024).
• Robust Set Selection (Adversarial Deletion): For a monotone set function $g:2^V\rightarrow \mathbb{R}_+$, selecting $S$ of size $k$ under the possibility of $\tau$ adversarial deletions leads to the utility

$$f(S) := \min_{D \subset S, |D| = \tau} g(S \setminus D)$$

with the extremal objective being maximization of $f(S)$ (Bogunovic et al., 2018).

• Optimization with Auxiliary Objectives: In evolutionary algorithms, one maximizes a target $t(x)$, aided by auxiliary objectives $h_i(x)$ which can transition from helpful to obstructive. Preservation constraints ensure the algorithm never loses its global best solution due to the selection of a currently obstructive helper (Petrova et al., 2017).
• Model Unlearning (Knowledge Distillation): A neural model $M$ is trained to simultaneously preserve its behavior on a set $D_r$ and delete its behavior on $D_f$ (nodes or edges to be forgotten) via a convex combination of distillation losses to a "preserver" and a "destroyer" model (Sinha et al., 2023).
• Gradient-driven Visual Attribution: Explanation masks are optimized so that their presence robustly preserves the classifier score and their deletion suppresses it, subject to geometric and area constraints (Karimzadeh et al., 3 Nov 2025).

2. Mathematical Frameworks and Objective Functions

Preserve/delete objectives typically fuse two competing loss components—one to maximize retention, another to effect deletion—often regulated by trade-off hyperparameters.

General Form

$$\text{Objective} = \alpha \cdot \text{Preservation Loss} + (1-\alpha) \cdot \text{Deletion Loss}$$

where $\alpha \in [0,1]$ tunes the extremity of preservation vs. deletion.

Examples

• Model Unlearning Distillation Losses:
  • Preservation: $$\mathrm{Loss}_r = L_\text{distill}(M(x;\varphi), M(x; \varphi^*))$$ for $x \in D_r$
  • Deletion: $$\mathrm{Loss}_f = L_\text{distill}(M(x;\varphi), N(x; \psi))$$ for $x \in D_f$
  • Total: $$\mathrm{Loss}_\text{total} = \alpha \cdot \mathrm{Loss}_r + (1 - \alpha) \cdot \mathrm{Loss}_f$$ (Sinha et al., 2023)
• Gradient Visual Masking (Extremal Contours):
  • Given scalar classifier $f$, mask $m$, original $x_0$, blurred $x̃$,

  $$x_p(m) = m \odot x_0 + (1 - m) \odot x̃\quad (\text{preserve})$$

  $$x_d(m) = (1-m) \odot x_0 + m \odot x̃\quad (\text{delete})$$

  $\ell_\text{ext}(m) = -f(x_p(m)) + f(x_d(m))$

  The total loss adds area and spectral regularization (Karimzadeh et al., 3 Nov 2025).

• RL Evolutionary Optimization:
  • Acceptance of new candidate $y'$ is allowed only if it both improves the chosen auxiliary objective $h(y') \geq h(y)$ and does not degrade the true target $t(y') \geq t(y)$, i.e.,

  $$\text{Accept if}\quad h(y') \geq h(y)\$$

  guaranteeing non-deletion of the extremal solution (Petrova et al., 2017).

3. Algorithmic Strategies

Explicit preservation/deletion requires algorithmic mechanisms for both optimization and stability:

RL-Based Evolutionary Algorithms

An RL controller selects among objectives. "Preserving the best" is enforced via additional acceptance checks:

while not optimal: select objective h via Q-learning propose offspring y' if h(y') ≥ h(y) and t(y') ≥ t(y): accept y'

Obstructive objectives are deactivated by their Q-value, while backtracking is prevented (Petrova et al., 2017).

Oblivious–Greedy for Set Maximization

The algorithm selects $\lceil\beta\tau\rceil$ highest-value singletons ("oblivious protection"), then greedily maximizes $g$ among the rest:

S₀ ← best singleton elements S₁ ← greedy over remaining S ← S₀ ∪ S₁

This ensures robustness against $\tau$ deletions and achieves constant-factor guarantees for general non-submodular objectives (Bogunovic et al., 2018).

Model Distillation for Unlearning

D2DGN updates the student model to approach the preserver on $D_r$ and the destroyer on $D_f$ by batchwise gradient descent:

for batch_r in D_r, batch_f in D_f: Loss_r = Distill(student(batch_r), preserver(batch_r)) Loss_f = Distill(student(batch_f), destroyer(batch_f)) Loss_total = α * Loss_r + (1–α) * Loss_f update parameters to descend Loss_total

KL and MSE losses are used over outputs and features, respectively (Sinha et al., 2023).

Differentiable Contour Optimization

Mask m is parameterized by truncated Fourier series subject to area and smoothness constraints, optimized over the extremal objective:

for t in range(T): compute mask via σ(τ[Contour–Radius–Polar–Distance]) form x_p, x_d compute scores and loss ℓ_ext regularize area, spectral norms backpropagate and optimize parameters

The approach enforces compact, interpretable regions (Karimzadeh et al., 3 Nov 2025).

4. Theoretical Guarantees and Bounds

Extremal objectives enable rigorous bounds on retained performance or avoided loss:

• Matrix Extremal Numbers: For $A$ with at most $t$ ones per row,

$\ex(n,A) \leq C n^{2 - 1/t + o(1)}$

confirming tightness for broad families of patterns and aligning with combinatorial conjectures (Janzer et al., 7 Mar 2024).

• Robust Maximization: Oblivious–Greedy achieves for monotone set functions (with submodularity ratio γ, bipartite ratio θ, inverse curvature $\check{\alpha}$, superadditivity $\check{\nu}$) a guarantee

$$f(S \setminus E^*) \geq \alpha(\gamma, \theta, \check{\nu}, \check{\alpha}, c) \cdot g(\text{OPT}_{k−\tau})$$

with $\alpha$ constant in linear regime $\tau = c k$ (Bogunovic et al., 2018).

• Evolutionary RL Robustness: The modified EA+RL always preserves the best-found solution and retains RLS's asymptotic runtime bounds even under arbitrary switch-points (transition of auxiliaries from helpful to obstructive) (Petrova et al., 2017).
• Distillation Unlearning: D2DGN achieves retained AUC on D_r within 0.6% of training-from-scratch, consistency on D_f within 0.1%, and strong empirical deletion guarantees (Sinha et al., 2023).

5. Practical Applications and Empirical Outcomes

Preserve/delete objectives are central to problems spanning privacy, interpretability, and robustness:

• Privacy Compliance: D2DGN supports "the right to be forgotten" in GNNs by enabling efficient, targeted forgetting without retraining, matching state-of-the-art with reduced computational cost (Sinha et al., 2023).
• Interpretable AI: Extremal Contours produce compact, smooth visual explanations invariant to fragmentation and adversarial masking, outperforming dense per-pixel masking in both fidelity and stability, particularly on self-supervised vision transformers (Karimzadeh et al., 3 Nov 2025).
• Robust Feature Selection: Oblivious–Greedy sustains high post-deletion utility in support selection and GP variance reduction, outperforming naive greedy and stochastic approaches in synthetic and real datasets (Bogunovic et al., 2018).
• Combinatorial Pattern Avoidance: Tight upper bounds on matrix extremal numbers and their alignment with graph-theoretic Turán-type theorems enable broader application in ordered and unordered pattern-avoidance problems (Janzer et al., 7 Mar 2024).
• Dynamic Optimization: Evolutionary optimization with RL-based dynamic objective selection and best-solution preservation outperforms traditional evolutionary and single-objective search in variable non-stationary landscapes (Petrova et al., 2017).

6. Special Cases and Extensions

The preserve/delete construct subsumes various established frameworks:

• Acyclic and Permutation Matrices: For matrices with $H_A$ a forest or a permutation pattern, specialty bounds (e.g., Marcus–Tardos $O(n)$ for permutation matrices) are recovered or generalized (Janzer et al., 7 Mar 2024).
• Ordered Graphs: The matrix containment framework extends directly to extremal bounds for ordered bipartite graphs by translation to biadjacency matrices and vice versa (Janzer et al., 7 Mar 2024).
• Multi-Object Visual Attribution: The contour optimization framework generalizes to multi-contour regions, enabling simultaneous localization and attribution for multiple targets in an image (Karimzadeh et al., 3 Nov 2025).

A plausible implication is the general adaptability of preserve/delete objectives to any setting where robust selection, privacy-preserving deletion, attribution compactness, and avoidance of adversarial loss interact, given that formal objective functions, acceptance criteria, or combinatorial bounds can represent the relevant trade-offs.

7. Significance and Unification Across Domains

The extremal preserve/delete objective establishes a principled foundation for simultaneous retention and controlled deletion across optimization, learning, combinatorics, and explainability. Its rigorous mathematical characterization and robust algorithmic implementations unify diverse approaches by explicit encoding of what must be preserved and what must be safely eliminated, yielding provable guarantees and strong empirical performance in practical domains. The paradigm subsumes well-studied problems in pattern avoidance, robust optimization under deletions, dynamic auxiliary-guided search, privacy-centric blocking, and interpretable model attribution, offering tight results and transparent control mechanisms for complex systems.