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Partially Informed Stochastic Optimization via Zeroth-Order Residual Tracking

El Mahdi Chayti, Alireza Mirrokni, Martin Jaggi

arXiv:2610.13871v1cs.LG

TL;DR

Some optimization problems provide only part of the gradient, leaving the remainder to be inferred from function evaluations. PISO tracks that residual and centres zeroth-order observations on its current estimate. Across five settings, experiments report improved solution quality or convergence speed over baselines with comparable oracle access.

  • Problem

    In performative prediction, performative reinforcement learning, and bilevel optimization, part of the gradient is available while the remaining component can only be accessed through function evaluations or responses.

    Source: §1 · pp. 1–2

  • Method

    PISO combines the known gradient component with a running residual estimate and centres each zeroth-order observation on that estimate.

    Source: §2 · p. 3, §6 · p. 9

  • Results

    Across five settings, experiments demonstrate improved solution quality or convergence speed over baselines with comparable oracle access.

    Source: §6 · p. 10

  • Takeaways & Limitations

    The convergence guarantees capture gains from slowly varying residuals without requiring a uniform bound on the unavailable gradient component.

    Source: §6 · pp. 9–10

    The drift bound Lv depends on γ along the trajectory, so the identified drift-minimizing weight need not minimize Lv; the method also requires γ > 0.

    Source: §3.1 · p. 6

Abstract

from arXiv · show

Many optimization problems provide direct access to only part of the gradient, while the remaining component must be inferred from function evaluations. This structure arises in performative prediction, performative reinforcement learning, and bilevel optimization with a black-box lower level. We introduce Partially Informed Stochastic Optimization (PISO), which combines the available gradient component with a running estimate of the residual. By centring each zeroth-order observation on this estimate, PISO makes the variance due to random probing depend on the tracking error rather than the residual's magnitude. This yields a randomized sketch iteration with an explicit contraction mechanism and a principled interpretation of momentum. For smooth nonconvex objectives, we establish convergence guarantees that depend on the residual's variation and initial magnitude, without requiring a uniform bound on the unavailable gradient component. With independently sampled function evaluations and Gaussian or spherical probes, PISO attains leading function-evaluation complexities of \(O(d^{2}\varepsilon^{-6})\) under Lipschitz gradients and \(O(d^{2}\varepsilon^{-5})\) under an additional Lipschitz-Hessian assumption. These rates require only one probe direction, two function samples, and one known-component sample per update. The analysis also quantifies how batching trades observations for fewer deployments. Across five settings, PISO variants improve on the tested baselines with comparable oracle access in three, reach a comparable solution faster in bilevel routing, and perform best among methods without response-model derivatives in performative reinforcement learning.

1 INTRODUCTION

Many problems expose a computable gradient component but leave the residual accessible only through noisy function evaluations. PISO uses the known component while tracking that residual, extending one approach across performative prediction, performative reinforcement learning, and bilevel optimization.

1 INTRODUCTION

  • In performative prediction, the known component is the ordinary loss gradient under the current distribution, while the unavailable component captures how deployment changes that distribution.The paper also describes policy-response terms in performative reinforcement learning and a missing response Jacobian in bilevel problems.Source: §1 · pp. 1–2
  • In bilevel optimization, the leader can evaluate partial derivatives once followers respond, but the response Jacobian remains unavailable.The same structure occurs when a differentiable model is composed with a black-box simulator or proprietary subsystem.Source: §1 · pp. 1–2
  • PISO tracks the opaque gradient component and centres each zeroth-order observation on its running estimate, making the estimation error contract rather than restarting from scratch.The introduction presents this as a shared approach across three application areas.Source: §1 · p. 2
  • Earlier approaches either discard the known gradient component or estimate gradients or response Jacobians at the current iterate, with methods developed separately for individual areas.The cited examples include black-box zeroth-order methods, guided performative prediction methods, and bilevel methods that retain exact leader partial gradients.Source: §1 · p. 2
  • The paper's contributions include a unified convergence analysis and experiments across five settings, with PISO variants improving on tested baselines in several settings.The introduction also describes deployment-versus-observation trade-offs and a momentum sweep consistent with the predicted stability behavior.Source: §1 · p. 2

2 FROM GRADIENT ESTIMATION TO RESIDUAL TRACKING

PISO defines a weighted residual to track and estimates only its discrepancy from a running state. This centring yields a sketched error-feedback update whose contraction and momentum behavior can be analyzed explicitly.

2 FROM GRADIENT ESTIMATION TO RESIDUAL TRACKING

  • PISO tracks v(x) = ∇F(x) − γgK(x), which equals the unavailable component at γ = 1 and the full gradient at γ = 0.The method seeks a state m tracking v so that γ times the known component plus m estimates the full gradient.Source: §2 · p. 3
  • A single probe supplies a noisy scalar measurement along uk, which the score turns into a vector estimate; its second moment scales with dimension times the residual magnitude squared.The described multi-probe alternative requires Ω(d) probes per iteration to reduce that inflation.Source: §2 · p. 3
  • Rather than reconstructing v(xk) from scratch, PISO estimates only the discrepancy v(xk) − mk and adds the estimate back to mk.The resulting observation error scales with the tracking discrepancy rather than the magnitude of v(xk).Source: §2 · p. 3
  • PISO averages the residual estimate and known-component samples separately, while PISO2 reads out a mixture of the fresh observation and the tracked state.Averaging the known component reduces its injected variance, permitting one sample per iteration at the cost of a lag the step size must respect.Source: §2 · pp. 3–4
  • In the stationary exact-oracle case, the residual-state error follows a sketched gradient step on a quadratic, with contraction for 0 < β < 2/cd.The contraction bound is minimized at β = 1/cd; for Gaussian, sphere, and coordinate probes, the stated moment condition is exact.Source: §2 · p. 4
  • The momentum ρ encodes the sketch step size β = 1 − ρ, so ρ = 0 is not a no-momentum setting and can yield an expanding update.The residual iteration is neutral at ρ = 1 − 2/cd and expands below that threshold.Source: §2 · p. 4

3 CONVERGENCE

PISO’s convergence analysis depends on residual drift and initial magnitude rather than a uniform bound on the unavailable gradient component. It also compares oracle regimes and quantifies how batching trades observations for fewer deployments.

3 CONVERGENCE

  • For Gaussian and spherical probes, the smoothing identity gives bias bounds with χ1 = d and χ2 = d; coordinate probes have χ1 = d/2 and lack that identity.The paper attributes the known dimension gap for coordinate probes to this structural difference.Source: §3 · p. 5
  • The convergence analysis requires smoothness and oracle and probe-moment conditions, but no bounded gradients, bounded domain, convexity, PL condition, or uniform bound on ∥gU∥.Its bound depends on residual drift Lv and initial magnitude ∥v(x0)∥; only the mean finite-difference bias is undamped.Source: §3 · pp. 4–5

3.1 ORACLE REGIMES AND COMPLEXITY

  • Against the guided line, the dimension factor improves from d^4 to d^2; against the frontier, PISO matches d^2 with L replaced by Lv and one probe per update.The table notes that only its last two columns compare directly.Source: Table 1 · p. 6
  • With independent function evaluations, Gaussian and spherical probes give d^2 dimension dependence in both regimes, and leading terms are independent of batch sizes and σK.The regimes are Lipschitz gradients alone and Lipschitz gradients with an additional Lipschitz-Hessian assumption.Source: §3.1 · pp. 5–6
  • The weight γ minimizes per-step residual drift at γ⋆k when Ck > 0, but γ⋆k can be nonpositive and the trajectory-dependent minimizer need not minimize Lv.Moving toward γ = 1 can replace full-gradient drift with opaque-component drift, which may be smaller in performative prediction.Source: §3.1 · p. 6

3.2 DEPLOYMENTS, AND HOW THE BATCH SIZES CONTROL THEM

  • In performative problems, each query is an intervention, so deployment count and sample count are distinct resources.Each iteration deploys three decisions regardless of batch sizes.Source: §3.2 · p. 7
  • Increasing batches up to β1 leaves function evaluations flat while reducing deployments from Θ(d^2ε^-6) to Θ(d^2ε^-2), plus an additive known-component branch term.Going from β1 to β2 reduces deployments by another factor d at the cost of a factor d in observations.Source: §3.2 · p. 7
  • The deployment trade-off remains limited by a bias bound shown loose by a factor d on examples; a sharper general bound is left open.Multi-direction methods cannot make the same trade because their probes are distinct decisions that repeated sampling cannot replace.Source: §3.2 · p. 7

4 RELATED WORK

Prior work uses auxiliary or partial gradient information, zeroth-order estimation, and sketching, but often estimates the gradient or response Jacobian anew at each iterate. PISO instead tracks the residual, a template that does not require composite structure and extends to performative reinforcement learning.

4 RELATED WORK

  • Prior methods using auxiliary information estimate a gradient or Jacobian at the current iterate, leaving a second moment of order cd redrawn each iteration.The related-work discussion spans performative prediction and mathematical programs with equilibrium constraints.Source: §4 · p. 7
  • Residual tracking makes direction error contract instead, needs no composite structure, and supports extension to performative reinforcement learning.The paper contrasts this with the O(d) leading term shared by Fischer & Paccagnan (2026)’s partial estimator and the black-box baseline.Source: §4 · p. 8
  • PISO’s zeroth-order machinery is standard, while its recursion is structurally related to EF21 error feedback and sketch-and-project methods.The paper notes that Huang et al. (2022) study zeroth-order momentum without identifying it as a sketch step size.Source: §4 · p. 8

5 EXPERIMENTS

Across five settings, PISO variants improve solution quality or convergence speed against baselines with comparable oracle access. Results include faster routing convergence to the PZOS endpoint, better security outcomes, and strong pricing, classification, and reinforcement-learning performance.

5 EXPERIMENTS

  • The evaluation covers five settings across performative prediction, performative reinforcement learning, and bilevel games, with published methods and vanilla policy gradient as baselines.The cycling variants use dependent probes, so the theory applies to them only heuristically.Source: §5 · p. 8
  • Across five settings, PISO improves solution quality or speed: cycling variants lead pricing, routing matches PZOS faster, security beats PZOS, and RL is best among methods that query the environment.On pricing, the cycling variants lead on all six instances; on routing, PISO and PZOS are indistinguishable at convergence.Source: §5 · p. 8, Figure 1 · p. 9, Figure 2 · p. 9
  • Every PISO variant attains strategic-classification test loss between 0.618 and 0.641, versus 0.821 for the best baseline, with accuracy higher by four to ten points.PISO variants remain flat as response strength increases, while methods that ignore gK degrade.Source: §5 · p. 8
  • The ρ sweep places the optimum at ρ = 0.92 across all four variants, near the predicted values 0.900 and 0.917; the components ablation finds neither component alone stable.With growing batches, ρK = 0 is already among the best choices.Source: §5 · p. 9

6 CONCLUSION

PISO tracks the residual and centres zeroth-order observations on its current estimate, supporting nonconvex convergence guarantees without a uniform bound on the unavailable gradient component. Experiments show improved solution quality or convergence speed over comparable-access baselines, while the analysis assumes a smooth objective.

6 CONCLUSION

  • PISO centres each zeroth-order observation on its residual estimate, making probing variance depend on tracking error rather than residual magnitude.The analysis gives nonconvex convergence guarantees using one probe direction per update, without a uniform bound on the unavailable gradient component.Source: §6 · pp. 9–10
  • Experiments across five settings show improved solution quality or convergence speed over baselines with comparable oracle access.Source: §6 · p. 10
  • The analysis assumes a smooth objective, although equilibrium response maps need not produce one.Extending residual tracking to Goldstein stationarity and analyzing dependent orthogonal cycling probes remain open directions.Source: §6 · p. 10

REPRODUCIBILITY STATEMENT

The reproducibility statement points to appendices for proofs, assumptions, protocols, baseline configurations, and hyperparameters. Appendix A restates assumptions and collects notation for reference.

REPRODUCIBILITY STATEMENT

A OVERVIEW OF THE APPENDIX AND SUMMARY OF NOTATION

B THE STRUCTURAL IDENTITY

The structural analysis expresses the state update through residual tracking and establishes when the randomized sketch contracts. It also contrasts centred and uncentred observations and characterizes the probe families used in the bounds.

B THE STRUCTURAL IDENTITY

  • The state recursion isolates known-component sampling error as a mean-zero term, while the tracking-error definition is indexed so it is measurable before the current iteration’s draws.Source: B THE STRUCTURAL IDENTITY · p. 15
  • The sketch contracts in mean square exactly when 0 < b < 2/cd, with the strongest contraction at b = 1/cd.Source: B THE STRUCTURAL IDENTITY · p. 16

B.1 WHAT CENTRING BUYS

  • Centring makes the sketch act on tracking error rather than the residual; the uncentred variant instead incurs a residual-magnitude-dependent complexity factor.For the uncentred variant, driving its stationary tracking error below ε² requires β ≲ ε²/(cdG_U²).Source: §B.1 · p. 16

C THE PROBE DISTRIBUTION

  • For Gaussian probes cd = d + 2; for sphere and coordinate probes cd = d, and the Gaussian and sphere contraction bounds are exact.Source: C THE PROBE DISTRIBUTION · p. 16
  • After rescaling κ=d, sphere and Gaussian probes have the same χ1 and χ2, so their bounds agree in order; coordinate probes lose a factor d in χ1 because (D4) fails.For coordinate probes, the direct bound is χ1=d/2 under L-smoothness.Source: C THE PROBE DISTRIBUTION · pp. 16–17

D THE FINITE-DIFFERENCE BIAS

The finite-difference bias has distinct mean and fluctuation scales, and its mean can accumulate across momentum steps. The tracking analysis uses this distinction to control the error recursion.

D THE FINITE-DIFFERENCE BIAS

  • The mean bias is undamped across the momentum window, while mean-zero bias fluctuations are damped by β.The mean accumulates coherently over roughly 1/β steps, unlike fluctuation noise.Source: D THE FINITE-DIFFERENCE BIAS · p. 17
  • For Gaussian and sphere probes, the sharp bound on the weighted mean bias is proportional to dLµk, or dQµk² when the Hessian is Q-Lipschitz.These bounds are stated for the random probe families.Source: D THE FINITE-DIFFERENCE BIAS · p. 18
  • Sharp Gaussian and sphere bounds reduce the generic bias bound by Θ(d), permitting β = Θ(1/d) instead of Θ(1/d²).The resulting minimum deployment count is Θ(dε^-2), rather than Θ(d²ε^-2).Source: D THE FINITE-DIFFERENCE BIAS · p. 18

E THE TRACKING LEMMA

  • The tracking recursion keeps the bias term with the estimate so its cross term can be evaluated exactly, leaving the mean bias as the undamped quantity.The second moment of the bias instead receives a factor β.Source: E THE TRACKING LEMMA · p. 20

F PROOF OF THEOREM 2

The proof combines a descent bound with control of the tracking error. The complexity analysis then distinguishes shared-randomness and independent-randomization oracle regimes.

F PROOF OF THEOREM 2

  • The descent lemma bounds expected objective decrease by the gradient norm and update magnitude, with a tracking-error penalty.It applies when η ≤ 1/(2L).Source: F PROOF OF THEOREM 2 · p. 21

G COMPLEXITY

  • The sharp complexity route permits β up to Θ(1/d), while the generic bias route restricts it to Θ(1/d²), increasing deployment requirements.The stated minimum deployment counts are Θ(dε^-2) and Θ(d²ε^-2), respectively.Source: G COMPLEXITY · p. 22

G.1 REGIME A: SHARED RANDOMNESS

  • With shared randomness and smooth sampled functions, the finite-difference bias terms vanish as µk approaches zero.The shared-randomness regime applies when randomness in F is exogenous to the decision.Source: §G.1 · p. 23

G.2 REGIME B: INDEPENDENT RANDOMIZATION

  • With independent function evaluations, the leading function-evaluation count is Θ(C1∆v/ε⁶) and is independent of nF within the stated range; beyond it, the count grows linearly with nF.The count is N = 2nF T, and the known-oracle constraint caps the range.Source: §G.2 · p. 25

G.3 REGIME C: DETERMINISTIC ORACLE WITH TOLERANCE

A deterministic oracle tolerance creates a non-vanishing finite-difference error and an accuracy floor. The complexity discussion separates residual smoothness from other terms, while initialization choices affect the known-component state.

G.3 REGIME C: DETERMINISTIC ORACLE WITH TOLERANCE

  • A deterministic oracle error is not mean-zero and does not shrink with resampling, so the probe radius cannot tend to zero and convergence stalls at an accuracy floor.The floor has bounds 240δtolLςχ1 under L-smoothness and 135(δtolς)^(4/3)(Qχ2)^(2/3) with a Q-Lipschitz Hessian.Source: §G.3 · p. 26

G.4 WHERE L, Lv AND LK ENTER

  • The leading zeroth-order complexity uses Lv rather than L, while L remains in a lower-order finite-difference-bias term and Q governs the corresponding Hessian-smoothness term.The initialization term depends on ∥v(x0)∥² rather than ∥∇F(x0)∥².Source: §G.4 · p. 27

H INITIALIZING THE KNOWN-COMPONENT STATE

  • A single-sample warm start uses nK samples at x0 without adding a deployment; batching that warm start once can remove its term from leading order.The batched warm start cost is negligible relative to the total sample count described in the passage.Source: H INITIALIZING THE KNOWN-COMPONENT STATE · p. 28

I.1 DISCUSSION OF THE PRESCRIPTIONS

The prescriptions assign distinct roles to sketch momentum, known-component averaging, the control-variate weight, and the readout gain. Their tuning depends on dimension, oracle noise, and residual drift; the probe analysis explains why one probe per update can suffice.

I.1 DISCUSSION OF THE PRESCRIPTIONS

  • Known-component momentum ρK trades variance O(βKγ²σ²K) against lag, while its contraction has no dimension-dependent stability window.Balancing variance and lag gives βK = Θ(ε²/(γ²σ²K)).Source: §I.1 · p. 29
  • γ is a control-variate coefficient on gradient increments, with a per-step drift minimizer that need not minimize the trajectory-wide smoothness bound Lv.The per-step minimizer depends on increment correlation; γ also changes the tracked residual and the trajectory.Source: §I.1 · pp. 29, 31
  • The sketch step β = 1 − ρ contracts for β in (0, 2/cd) and is optimal at β = 1/cd.The contraction factor is minimized at that step size.Source: §I.1 · pp. 28–29
  • ρ = 0 sets β = 1 and can produce an expanding map, so the sketch momentum must be scaled with dimension.The momentum-free comparison is the uncentred estimator, not PISO with ρ = 0.Source: §I.1 · pp. 28–29
  • PISO2 requires λ → 0 as ε → 0 because a fixed readout gain leaves an irreducible fresh-observation variance floor.As observation noise grows, the optimal readout switches off and PISO2 reduces to PISO.Source: §I.1 · pp. 28–29
  • The readout gain acts like a scalar Kalman gain, balancing correctable signal against sketch inflation and observation noise.Its denominator can be approximated from the running mean-squared innovation, so the gain is adaptive rather than exactly observable.Source: §I.1 · pp. 28, 30

J PROBES PER UPDATE

  • PISO uses one probe per update because contraction averages sketch variance across iterations, whereas multi-direction methods require Ω(d) queries before each update.In performative settings, those queries are distinct deployed decisions.Source: J PROBES PER UPDATE · p. 31

K.1 DEPLOYMENTS, THE BINDING COST IN PERFORMATIVE PREDICTION

The section treats deployments as a distinct cost from observations and analyzes how batching changes the number of iterations and the observation–deployment trade-off. Two caps limit the step-size increase, leaving a gap in the attainable deployment count.

K.1 DEPLOYMENTS, THE BINDING COST IN PERFORMATIVE PREDICTION

  • Increasing nF and nK permits larger β and βK, reducing iterations and deployments, but deployments remain 3T regardless of batch sizes.In Regime B, each queried decision is a deployment, while batching collects more samples at those decisions.Source: §K.1 · pp. 31–32
  • The step-size caps are β1 = 1/(9d²) for Gaussian probes and β2 = 1/(3d); increasing β past β1 raises observations linearly in β.At β2, the section reports Θ(d³∆vε^-6) observations and Θ(d∆vε^-2) deployments.Source: §K.1 · p. 32
  • The minimum deployment count without increasing observations is Θ(d²∆vε^-2), while Θ(d∆vε^-2) deployments cost a factor d in observations.The section attributes the gap between the caps to looseness in a generic bound and leaves a sharper bound open.Source: §K.1 · p. 32
  • The known-component batch size interpolates between minimizing known-component samples and minimizing deployments; at the deployment-optimal end, βK = Θ(1).The known-component average is then close to inactive.Source: §K.1 · p. 32
  • At β1, PISO uses a factor ε^-4 fewer deployments than the multi-direction method for the same observations; at β2, it uses a factor d ε^-4 fewer at a factor d more observations.The comparison is with the multi-direction method described in the section.Source: §K.1 · p. 32

L ADDITIONAL EXPERIMENTAL RESULTS

In strategic classification, PISO variants outperform the reported baselines across response strengths, while performative RL results favor PISO among methods that query the environment. The section also specifies the pricing setup used for its ablations.

L ADDITIONAL EXPERIMENTAL RESULTS

  • In performative RL, PISO and PISO2 lead among environment-querying methods; PISO’s occupancy change falls from about 10^-1 early to about 10^-2 late.The authors note that larger early policy steps may explain part of PISO’s advantage over vanilla policy gradient, but do not attribute the gap to the estimator alone.Source: L ADDITIONAL EXPERIMENTAL RESULTS · p. 32
  • On Figure 3, PISO variants reach training loss of about 0.63 at every τ, while the strongest baseline plateaus near 0.80.The methods that ignore the computable component degrade sharply as τ increases.Source: Figure 3 · p. 33
  • All four PISO variants dominate every baseline on test loss and accuracy at every strategic-response strength, with roughly 0.2 lower loss and 4–10 more accuracy points.The two methods that ignore the computable component degrade as response strength grows, while PISO variants remain essentially flat.Source: Table 3 · p. 33

L.1 ABLATIONS

  • The ablations use the pricing problem with d = 10 and batches starting at 30 and increasing by 2 per iteration, leaving nK large throughout.The section notes that the large known-component batch matters for Figure 9.Source: §L.1 · p. 33

L.2 EXPERIMENTAL HYPERPARAMETERS

The appendix reports experiment-specific hyperparameter selections and conventions for PISO variants and baselines. Baseline settings generally follow their references, except that zeroth-order baselines are tuned independently for performative RL.

L.2.5 BILEVEL SECURITY GAME

  • For the security game, the normalized objective comparison spans dimensions 100–300 and three iteration counts; PISO and PZOS remain separated from guided baselines, which barely improve after 50 iterations.A norm cap is applied to residual observations and momentum as a safeguard against unusually large observations.Source: Figure 5 · p. 34, §L.2.5 · p. 40

L.2.4 BILEVEL ROUTING GAME

  • PZOS matches Gaussian PISO per routing instance at every horizon, while Gaussian PISO outperforms ZOS, GZO-NS, and GZO-HS, especially at small horizons and dimensions.The comparison uses points above the diagonal to indicate better maximization objectives for PISO.Source: Figure 6 · p. 35

L.2.1 MULTIPRODUCT PRICING

  • Pricing uses a two-stage grid search; the selected Gaussian PISO settings include γ = 2.00, ρ = 0.92, and fK = 0.50, with a batch increasing from 30 by 2 per update.The same settings are used for Cycle PISO, with cycle length 10; PISO2 uses λ = 0.25.Source: §L.2.1 · pp. 35–36

L.2.2 STRATEGIC CLASSIFICATION

  • The grid searches r0, η0, ρ, fK and, for specified variants, L or λ—not γ; nevertheless, the interior γ optimum is 1 or 2 depending on variant, unresolved precisely by the coarse grid.Performance decreases monotonically over λ ∈ {0, 0.25, 0.5, 0.75, 1}, with λ = 1 far worse than λ = 0. At ρ = 0.92 and cd ∈ {10, 12}, the high-signal-to-noise optimum λ⋆ is 0.004 and 0.022, both below the smallest nonzero grid point, so the grid cannot separate λ⋆ from 0.Source: §L.2.2 · p. 37, Figure 11 · p. 38

L.2.3 PERFORMATIVE REINFORCEMENT LEARNING

  • For performative RL, PISO tuning varies γ, ρ, ρK, and the known-sample fraction; the selected Gaussian PISO2 configuration uses total batch 100 and one probe direction per update.Among methods operating under the paper’s information setting, PISO and PISO2 make the fastest early progress; GZO-NS closes part of the gap later.Source: §L.2 · p. 34, §L.2.3 · p. 39

L.2.4 BILEVEL ROUTING GAME

  • Routing experiments search residual momentum and, for PISO2, mixing weight; the effective cycle length is capped at the problem dimension.The selected Gaussian PISO configuration uses step size 0.70/d, γ = 1.00, and ρ = 0.98.Source: §L.2.4 · p. 40

L.2.6 PRICING ABLATIONS COMMON SETTINGS

  • The pricing ablations use 100 simulations, 5,000 training samples, and one direction per step, with central differences and independent coupling for both PISO variants.Cycle panels replace the default Gaussian direction with an orthogonal cycle.Source: §L.2.6 · p. 41
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