DePICT: Decision-Preserving Interface for Constrained Downstream Tasks

1 INTRODUCTION

• Insensitive context can add sensing, transmission, storage, maintenance, licensing, and processing costs without changing the optimal feasible decision.The introduction frames interface selection as a way to avoid carrying such inputs through the pipeline.Source: §1 · p. 1
• The central question is which primitive context directions can change the constrained optimum, not simply which variables help prediction.A variable can appear explicitly in the optimization problem while leaving the optimum unchanged; removing a direction that moves the optimum can also discard unrecoverable information.Source: §1 · p. 1
• DePICT ranks primitive directions by local solution sensitivity across representative contexts and aggregates those sensitivities across an operating regime.It uses the differentiable objective and constraint parameterization, needs neither labeled context-decision pairs nor a pretrained predictor, and fixes the interface before predictor training.Source: §1 · p. 2
• A direction is irreducible when perturbing it moves the constrained optimum to first order, and KKT sensitivity gives an exact test for this relevance.Symbolically present directions can still be irrelevant because they do not reach the active KKT system or their effect is absorbed by multipliers.Source: §1 · p. 2
• Removing an irreducible direction induces first-order bias; interface rankings are stable across the operating regime within a certified radius.The paper tests its results on matched benchmarks satisfying its assumptions, controlled benchmarks, a Layer-2 diagnostic, and benchmarks that violate the assumptions.Source: §1 · p. 2

2.1 PROBLEM FORMULATION AND DECISION RELEVANCE

• Under LICQ, SOSC, strict complementarity, and smoothness, the primal-dual solution is differentiable; a coordinate is irreducible when its Jacobian column is nonzero somewhere in the operating regime.DePICT approximates the regime-level interface from representative contexts.Source: §2.1 · pp. 2–3

2.2 DECIDING RELEVANCE

• A direction is irrelevant exactly when it shifts no active margin and its stationarity perturbation lies in the active constraint-normal space, where multiplier adjustment absorbs it.Thus appearing in the active KKT system is necessary for relevance but not sufficient.Source: §2.2 · p. 3

2.3 SYMBOLIC SUPPORT VS. OPTIMIZER RELEVANCE

• Symbolic support upper-bounds optimizer relevance, and both inclusions may be strict because a variable can appear in the formulation without reaching the local sensitivity system.The full symbolic support records appearances in the formulation; active-set support records active-margin or stationarity shifts.Source: §2.3 · p. 4
• A positive calibration factor can appear in a binding budget constraint and the active KKT system yet leave the feasible set and optimum unchanged when the objective is independent of it.Its stationarity perturbation lies in the active constraint-normal space and is absorbed without primal motion.Source: §2.3 · p. 4
• With zero active-margin shift and zero mixed objective curvature, constraint scaling or proportional mixed partials can route stationarity perturbations through active constraint normals.Multiplier adjustment then absorbs the perturbation.Source: §2.3 · pp. 4–5
• When active constraints are affine and the objective has no mixed context-decision curvature, optimizer relevance reduces to active-margin dependence; a cost-only context perturbation in a fixed-constraint LP is irrelevant while its basis remains unchanged.The section contrasts this boundary case with changing constraint shape, which restores irreducibility.Source: §2.3 · p. 5

2.4 INTERFACE MISSPECIFICATION AND STABILITY

• If an interface removes an irreducible direction, every differentiable downstream decision map through that interface has a nonzero first-order mismatch at the affected context.Empirical risk minimization within that function class cannot remove this local structural bias.Source: §2.4 · p. 5

Stability and interface construction

• The TOP-m procedure solves the constrained program at representative contexts, computes sensitivity rankings, averages the rankings, and returns the top-m directions.The procedure also evaluates the active set and KKT quantities at each representative context.Source: Appendix 7.4 gives the proofs and a… · p. 6
• On a compact operating subset, the sensitivity ranking is locally Lipschitz, with its bound depending on function smoothness, active-set KKT conditioning, and subset geometry.A positive ranking gap certifies a neighborhood in which the top-m ranking is unchanged.Source: Appendix 7.4 gives the proofs and a… · p. 6
• Frequency filtering retains directions whose irreducibility persists across the operating regime, and the resulting interface sets are nested as the frequency threshold increases.The passage states Irr1(K) ⊆ Irrα(K) ⊆ Irr(M), with Irrα′(K) ⊆ Irrα(K) when α′ ≥ α.Source: Appendix 7.4 gives the proofs and a… · p. 6

3 EXPERIMENTS

• The evaluation proceeds from controlled recovery and decision quality to edge-regime transfer, Layer-2 annihilation, perturbation stability, and runtime scaling, with interfaces selected before predictor training.The controlled tasks use known parameter-map supports; Warcraft and FF25 test empirical transfer beyond the controlled setting.Source: §3 · p. 6, §3.1 · p. 7, §3.2 · p. 7

3.1 CONTROLLED VALIDATION

• F1 = 1.000: DePICT recovers the oracle interface on controlled knapsack and shortest path, achieving the best regret and objective value among reduced-interface methods.On shortest path it also obtains the highest path match; gaps narrow with MLP predictors, but DePICT remains best or competitive. Shortest-path sensitivity and evaluation use a regularized continuous flow formulation.Source: §3.1 · p. 7, Figure 1 · p. 7

3.2 REAL AND EDGE-REGIME BENCHMARKS

• On Warcraft, DePICT achieves the lowest regret under both predictor families; on FF25, it combines the lowest common-reference regret and turnover under the linear predictor with competitive Sharpe.On Warcraft, selector gaps nearly collapse under MLP predictors, consistent with nonlinear compensation among retained variables; the results indicate interface quality is most visible when predictor capacity is limited relative to context dimension.Source: §3.2 · pp. 7–8, Figure 2 · p. 7

3.3 LAYER-2 ANNIHILATION DIAGNOSTIC.

• DePICT retains all irreducible coordinates, excludes both nuisance classes, and reduces controlled knapsack linear-predictor regret to 0.009 versus 0.475 for the strongest non-DePICT baseline.On controlled knapsack it achieves perfect irreducible recall and support F1; on FF25 it has zero false inclusion for both nuisance classes, lowest regret (0.0054) and turnover (0.0468), and a Sharpe ratio of 0.550.Source: §3.3 · p. 8, Figure 3 · p. 8

3.4 STABILITY UNDER PERTURBATIONS

• For m = 60, Jaccard overlap stays above 0.979 at ϵ = 5 × 10−2 while regret remains close to zero, although exact top-m recovery decreases as perturbations grow.The study evaluates exact top-m recovery, Jaccard overlap, and regret across m ∈{20, 40, 60}.Source: §3.4 · pp. 8–9, Figure 4 · p. 9

3.5 RUNTIME SCALING WITH CONTEXT DIMENSION

• DePICT requires approximately 5.2-5.4 s across n1 = 200 to n1 = 2000, while LASSO rises from 1.79 s to 29.17 s and finite-difference ranking from 37.47 s to 372.78 s.With optimizer dimension held at nθ = 20, its KKT system size depends on the optimizer and active set, not ambient context dimension; some lightweight selectors are faster at small dimensions.Source: §3.5 · p. 9, Figure 5 · p. 9

• DePICT exactly recovers the decision-relevant interface on both controlled benchmarks and reduces linear-predictor regret from 0.475 to 0.009 while excluding nuisance classes.The conclusion also reports stable selected interfaces under context perturbations and effective transfer to Warcraft and FF25.Source: §4 · pp. 9–10
• The guarantees require stable active sets and KKT regularity; nonsmooth, discrete, and changing-active-set regimes remain future work.The analysis is local and first-order, and extending it to finite perturbations and higher-order guarantees remains open; DePICT also relies on second-order sensitivity information and linearized KKT solves.Source: §4 · p. 10

Decision-focused learning

• Decision-focused learning trains predictors for downstream decision quality, whereas DePICT asks which context components must be preserved before predictor training.DePICT uses the solution-sensitivity Jacobian to identify primitive coordinates whose columns vanish despite symbolic participation, treating vanishing solver gradients as relevance information rather than a training obstacle.Source: §5 · p. 13

Task-specific data and feature relevance

• DePICT studies primitive context directions preserving the local solution map of a nonlinear constrained problem, rather than dataset sufficiency for linear programs or features useful for prediction.Its optimizer-induced relevance notion uses local KKT sensitivity rather than global polyhedral geometry.Source: §5 · p. 13

Feature-selection baselines

• Feature-selection baselines select by prediction utility, sparsity, or heuristic and local perturbation signals, while DePICT defines relevance through the downstream optimizer’s KKT sensitivity.The approach accounts for constraint geometry when identifying context directions needed to preserve the constrained optimum.Source: §5 · p. 13

Positioning

• Prior work addresses predictor training, task-sufficient data in linear decision problems, and parametric optimum variation; the remaining gap is which primitive context directions preserve a constrained task’s local optimum.DePICT introduces decision-preserving interfaces and characterizes them through local KKT sensitivity.Source: §5 · pp. 13–14

• At each representative context, DePICT computes the solution-sensitivity Jacobian J(ct) = ∂θ∗(ct)/∂ct by solving a linearized active-set KKT system.When stronger closed-form conditions hold, the Jacobian can equivalently be computed through the geometric decomposition J = P ⊥GA −ΦCA; the saddle-system form avoids explicitly forming H−1.Source: §6 · pp. 14–15
• Coordinate scores are Jacobian-column norms, and DePICT aggregates them across representative contexts before selecting a TOP-m, FREQ-FILTERED, or STRICT-UNION interface.The persistence frequency records the fraction of contexts where a coordinate exceeds the sensitivity threshold; the modes trade off fixed budget, suppression of unstable directions, and conservative coverage.Source: §6 · pp. 14–15
• Each exact scoring context requires one linearized KKT solve of dimension nθ + |At|, and the factorization can be reused across multiple right-hand sides.The method supports matrix-free or approximate and amortized sensitivity scoring; for shortest-path benchmarks, sensitivities use a strongly convex regularized flow QP rather than differentiating the discrete argmin.Source: §6 · p. 14

7.1 DERIVATION OF THE SOLUTION-SENSITIVITY JACOBIAN

• DePICT’s solution-sensitivity Jacobian separates feasibility tracking from constraint-geometry rotation, and its TOP-m construction averages coordinate sensitivities across representative contexts.The local differentiability argument relies on LICQ, SOSC, and strict complementarity; when the objective is independent of context, the objective mixed-curvature term vanishes.Source: §7.1 · pp. 15–17

7.2.1 SYMBOLIC SUPPORT AS A STRICT UPPER BOUND

• Optimizer-relevant directions are contained in symbolic support, but support can also include directions whose constraint effects are absorbed by multiplier adjustment and leave the optimizer unchanged.Regime-level irrelevance holds exactly when the active-margin derivative and the tangent component of the stationarity derivative vanish throughout the regime; the excess support has two layers.Source: §7.2.1 · pp. 17–18

7.2.2 AFFINE COLLAPSE

• Corollary 11 identifies an affine-collapse case when active constraints are affine in decisions and context and the considered directions have no mixed context–decision curvature.Source: §7.2.2 · p. 20

7.2.3 STRUCTURAL CHARACTERISATION OF LAYER-2 ANNIHILATION

• Layer-2 annihilation holds for positively scaled active constraints, including a single active constraint of that form, but can fail when context changes constraint shape rather than scale.In the supplied counterexample, the active-margin derivative is zero while the stationarity derivative lies outside the constraint-normal space, so the direction remains irreducible.Source: §7.2.3 · p. 20

7.2.4 FEATURE-MAP NECESSITY AND STRUCTURAL WARM-START ERROR

• Omitting an irreducible primitive direction from a smooth feature map forces a data-independent first-order decision error wherever that direction changes the optimizer.The stated lower bound is ∥(h ◦ψ)(c + tej) − θ∗(c + tej)∥ ≥ |t|∥vj(c)∥ − O(t^2) = Θ(|t|).Source: §7.2.4 · p. 21

7.3.1 LIPSCHITZ CONTINUITY OF THE SOLUTION-SENSITIVITY JACOBIAN

• Under A1–A2, the solution-sensitivity Jacobian is Lipschitz on compact regimes, and a positive top-m sensitivity gap on a connected regime makes the Topm set constant.The appendix uses this regularity to justify local ranking stability and frequency-filtered interfaces across an operating regime.Source: §7.3.1 · p. 21, §7.3.2 · p. 21, §7.3 · p. 21

7.3.3 FREQUENCY FILTERING

• Frequency filtering retains directions whose irreducibility frequency meets its threshold, recovering positive-volume irreducibility when the threshold does not exceed that frequency.Source: §7.3.3 · p. 22

7.3.4 CONNECTION TO STABLE TOP-m SETS

• With a uniform positive sensitivity gap on a connected compact regime, the top-m set is stable; Irr1(K) equals that set exactly when every outside coordinate has frequency below one.Equivalently, each coordinate outside Topm has a positive-measure zero set on K.Source: §7.3.4 · p. 23
• Frequency filtering recovers the stable Top-m set when all coordinates in Top-m meet the threshold and all coordinates outside it fall strictly below it.Exact recovery by frequency-one filtering instead requires every coordinate outside Top-m to have frequency below one.Source: §7.3.4 · p. 23

7.3.5 WORKED EXAMPLE

• In the worked example, x2 is absent from the active constraint, while x3 is Top1 throughout a certified neighborhood around c0.The top coordinate depends on comparing x1 with the stated threshold; the spectral gap vanishes at equality.Source: §7.3.5 · pp. 23–24

7.4 INTERFACE MISSPECIFICATION AND QUANTITATIVE CONSEQUENCES

• An interface that removes an irreducible direction forces every downstream decision map through it to have the wrong first-order response in that direction.The oracle response is nonzero while the interface-mediated response is zero.Source: §7.4 · p. 24, §7.4.1 · p. 24

7.4.1 PROOFS OF THE MAIN-TEXT STATEMENTS

• Because every predictor in the interface representation class has zero first-order response in a removed irreducible direction, empirical risk minimization cannot eliminate the mismatch.The mismatch follows from the representation class itself, not from the choice of predictor within that class.Source: §7.4.1 · p. 24

7.4.2 A LOCAL QUANTITATIVE CONSEQUENCE

• For sufficiently small perturbations, discarding an irreducible direction induces linear error, even when the predictor matches the oracle at the base context.This local bound requires bounded second derivatives; population-level risk lower bounds require further assumptions on the context distribution and are left outside the main theorem.Source: §7.4.2 · pp. 24–25

8.0.1 BENCHMARKS

• Controlled KKT benchmarks provide oracle supports, while real and edge-regime tasks test the interface principle without guaranteed oracle labels or exact smooth KKT assumptions.Controlled knapsack and shortest-path tasks test support recovery and decision quality; Warcraft is a solver-mediated transfer benchmark, not a direct test of exact support recovery.Source: §8.0.1 · p. 25, §8.0.2 · p. 27

8.0.2 PREDICTOR TRAINING PROTOCOL

• Selectors fix interfaces before shared downstream training; DePICT uses optimizer sensitivities, and experiments use continuous formulations rather than discrete argmins.The reported knapsack and path experiments use fractional knapsack or regularized flow QPs.Source: §8.0.2 · pp. 25–26

8.0.3 DATA SPLITS AND PROTOCOL

• Data splits, model selection, and test evaluation are matched across methods, with test metrics computed once after final selection.Experiments repeat across deterministic seeds where applicable and match randomized baselines within each seed.Source: §8.0.3 · p. 27

8.0.4 PREDICTOR FAMILIES

• Linear predictors expose interface quality, whereas MLPs can exploit nonlinear correlations and narrow selector gaps.The authors note this compensation is especially relevant in nonsmooth or solver-mediated benchmarks.Source: §8.0.4 · p. 27

8.0.5 COMPARED METHODS

• Comparisons span statistical, model-based, wrapper, and search selectors; DePICT ranks optimizer-side sensitivity, with variants testing regime-level aggregation.The ablations include top-m, strict-union, and frequency-filtered Jacobian rankings.Source: §8.0.5 · pp. 28–29

8.0.6 FAIRNESS OF COMPARISON

• Post-selection comparisons hold predictor, split, training protocol, solver, and shared target budgets constant to isolate interface quality.Methods with native subset sizes report their realized interface dimensions.Source: §8.0.6 · p. 29

8.0.7 EVALUATION METRICS

• Held-out evaluation covers regret, feasibility, routing path match, and portfolio metrics; oracle-support recovery and Layer-1/2 false-inclusion rates apply where labels exist.Higher Sharpe and path match are better; lower regret, turnover, feasibility violation, and false-inclusion rates are better.Source: §8.0.7 · pp. 29–30

8.1.3 CONTROLLED CONTEXTUAL KNAPSACK

• In controlled knapsack, nuisance coordinates may be absent from the active KKT system or enter constraints but have effects absorbed by multiplier adjustment.The continuous formulation supplies a local KKT system for computing optimizer sensitivity.Source: §8.1 · p. 31, §8.1.3 · p. 33

8.1 CONTROLLED BENCHMARK VALIDATION

• Known differentiable task maps and oracle supports let controlled benchmarks test interface recovery and downstream decision quality separately.The benchmarks use fractional knapsack and shortest-path flow, with sparse context coordinates determining task parameters.Source: §8.1.1 · p. 31, §8.1 · p. 31

8.1.1 CONTROLLED BENCHMARK VALIDATION

• DePICT recovers the oracle interface exactly while preserving downstream decision quality under matched linear and MLP predictors.Table 1 reports averages over 10 independent seeds.Source: Table 1 · p. 32

8.1.2 DEPICT SCORING IN THE CONTROLLED BENCHMARKS

• DePICT scores each primitive coordinate by the first-order motion of the optimal decision under its perturbation, then aggregates scores across representative contexts.Persistence variants distinguish sensitivity that recurs across the operating regime from sensitivity at a single context.Source: §8.1.2 · pp. 32–33

8.1.3 CONTROLLED CONTEXTUAL KNAPSACK

• The knapsack interface targets coordinates that move the optimal allocation, rather than coordinates that merely predict item value.Source: §8.1.3 · p. 34

8.1.4 CONTROLLED CONTEXTUAL SHORTEST PATH

• For shortest-path scoring, DePICT differentiates a strongly convex regularized flow problem because the discrete path map switches discontinuously at path boundaries.The regularized flow problem is a sensitivity instrument, not a claim that the discrete argmin is differentiable or identical to the QP.Source: §8.1.4 · pp. 35–36

8.1.5 MATCHED POST-SELECTION EVALUATION

• Selectors are compared with the same data splits, interface budget, predictor family, target, training objective, and downstream solver.This matched protocol isolates interface quality from predictor capacity and solver choice.Source: §8.1.5 · p. 36

8.1.6 EVALUATION METRICS

• Support recovery measures agreement with oracle primitive supports, while post-selection regret measures decision quality after a separately trained predictor uses the selected interface.The evaluation also reports feasibility and flow agreement; flow agreement is coordinatewise agreement of continuous flows, not exact discrete path match.Source: §8.1.6 · p. 37, §8.1.7 · p. 38

8.1.7 WHAT THE CONTROLLED EXPERIMENTS ESTABLISH

• Controlled experiments separately test whether sensitivity recovers known relevant coordinates, whether those coordinates preserve decision quality, and whether the approach applies to a combinatorial task.For shortest path, interface construction differentiates a regularized continuous flow problem, not the discrete argmin.Source: §8.1.7 · p. 38

8.1.8 REAL-DATA AND EDGE-REGIME STRESS TESTS

• Warcraft and FF25 provide no oracle decision-preserving support, so their stress tests assess compact interfaces and decision quality rather than support recovery.They examine solver-mediated pipelines with learned context-to-task mappings, nonsmooth behavior, changing active constraints, or real observational data.Source: §8.1.8 · p. 38

8.1.9 WARCRAFT TERRAIN ROUTING

• Warcraft represents each terrain cell with RGB context and uses regularized flow sensitivity to rank coordinates for downstream routing evaluation.The differentiable continuous flow model stabilizes local sensitivity; the authors do not claim the discrete shortest-path map is differentiable.Source: §8.1.9 · pp. 38–39

8.1.10 FF25 CONSTRAINED PORTFOLIO ALLOCATION

• The 25-portfolio context uses trailing return means (1/3/6/12 months), volatilities (3/6/12 months), momentum (1/6/12 months), and Fama–French factor means (3/6/12 months); covariance uses preceding 60 months.The reduced-interface and full-context models are passed independently through the same constrained portfolio optimizer over the chronological test period, each carrying forward its own previous portfolio weights.Source: §8.1.10 · pp. 40–41

8.1.11 INTERPRETATION OF THE REAL-DATA EXPERIMENTS

• Warcraft and FF25 test transfer beyond the controlled setting, but the authors do not claim globally minimal interfaces or extend formal guarantees beyond their stated assumptions.The empirical comparison is at matched interface budgets under the same downstream prediction and optimization pipeline.Source: §8.1.11 · p. 41

8.1.12 ADDITIONAL RESULTS AND ABLATIONS

• The extended experiments cover controlled-benchmark budget sweeps, selector-isolated results, portfolio-allocation tables, and ablations of aggregation, instance filtering, and predictor family.These tests examine robustness to budget choice, aggregation rule, representative-context filtering, and predictor family.Source: §8.1.12 · p. 41

8.1.13 SEED VARIABILITY ACROSS BENCHMARKS

• Across controlled knapsack, controlled shortest path, Warcraft, and FF25, DePICT attains the lowest mean regret among compared reduced-interface methods.On Warcraft, linear-predictor regret is 8.2974 versus 8.8871 for the nearest baseline; on FF25 at m = 40, linear common-reference regret is 0.0095. FF25 results describe a regret-Sharpe-turnover tradeoff, not uniform metric leadership.Source: Figure 7 · p. 42, §8.1.13 · p. 42

8.1.14 IMPLEMENTATION DETAILS

• Methods use matched data splits and evaluation protocols within each seed; Table 2 reports task-specific solver-mediated metrics, including FF25 results at matched budget m = 40.The metrics differ by task: Warcraft includes path match, while FF25 includes realized Sharpe and turnover.Source: §8.1.14 · p. 42, Table 2 · p. 43

8.1.15 COMPUTATIONAL COST AND SCALABILITY

• Exact scoring requires a one-time second-order KKT sensitivity computation at interface-design time; scalable variants are proposed, but their systematic empirical evaluation is future work.A dense implementation factorizes a dK × dK saddle matrix per representative context; alternatives include iterative matrix-free solves and randomized adjoint probing.Source: §8.1.15 · pp. 43–45

8.1.16 LAYER-2 ANNIHILATION DIAGNOSTIC

• DePICT separates optimizer-relevant from Layer-2 nuisance directions on controlled knapsack and FF25, achieving lower regret but not uniformly best financial metrics.On knapsack, support F1 is 1.000 and linear-predictor regret is 0.009 versus 0.475 for PFMM. On FF25, regret is 0.0054 versus 0.0127 for the full interface and 0.0213 for the best reduced-interface baseline; Sharpe is not maximized.Source: §8.1.16 · p. 45, Table 4 · p. 46

8.1.17 STABILITY PERTURBATION PROTOCOL

• Under local context perturbations, exact top-m identity decreases while Jaccard overlap remains high and regret inflation stays near zero.The results indicate decision stability even when the exact ranking changes.Source: Figure 9 · p. 47, §8.1.17 · p. 47

8.2 DISCUSSION: INTERFACE SELECTION, ADAPTATION, AND UNLEARNING

• Before or after unlearning, edits to nuisance directions while preserving the DePICT interface should leave the downstream decision stable, whereas removing an irreducible direction should cause structural task degradation.The discussion frames this as a proposed diagnostic use.Source: §8.2 · p. 47
• The discussion connects decision-preserving interfaces with privacy and membership auditing, where gradient- or feature-drift signals can indicate whether trained-example evidence remains.Source: §8.2 · p. 47
• Future work could test whether targeted unlearning removes undesirable behavior while DePICT checks whether necessary decision-relevant information remains in embodied and edge-deployed systems.The proposed check is intended to help distinguish genuine forgetting from behavior that is merely hidden.Source: §8.2 · p. 47