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Learning Not to Optimize: Physics-Informed Action-Space Reshaping for Intent-Based Network Control

2026 · arxiv_cs

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arXiv:2608.00908v1 [cs.NI] 2 Aug 2026

Learning Not to Optimize: Physics-Informed Action-Space Reshaping for Intent-Based Network Control Zuyuan Zhang

Vaneet Aggarwal

Tian Lan

The George Washington University zuyuan.zhang@gwu.edu

Purdue University vaneet@purdue.edu

The George Washington University tlan@gwu.edu

Abstract—Modern network policy control maps intent to sequential placement-control decisions. Bellman-style policy optimization primarily asks which action to optimize, while constraints are commonly handled through penalty, barrier, or Lagrangian mechanisms. We observe that before a value function can certify the best deployment, intermediate signals may already identify many candidates that should be excluded from further optimization. This motivates a complementary direction: Learning Not to Optimize. Before a value function is accurate enough to select the best placement-control decision, intermediate signals may already show that candidates are equivalent under state–intent relabeling (quotienting), lead to a uniformly worse future state (dominance), or violate executable network laws (residual screening). LNO-QRD uses these computed or learned signals as a shadow process to reshape the domain on which primal policy optimization is performed, thereby reducing the action space. We prove lossless quotienting and dominance under explicit equivariance and monotonicity conditions, bound frontier size and ranking cost, and quantify losses from approximate certificates and primal estimates. Experiments show that LNOQRD reduces small-instance candidates by 75.9% while retaining 90.8% near-oracle coverage and, on large instances, achieves the highest utility and intent satisfaction, the lowest hard-law violation and post-generation latency, and a 73.0% average reduction among candidate-based baselines. Index Terms—Network policy control, physics-informed learning, learning not to optimize, constrained reinforcement learning, constrained Markov decision process, action-space reduction, quotient action space, residual screening, dominance pruning.

I. I NTRODUCTION Modern programmable networks expose a concrete intentconditioned placement-control problem. A translated servicechain or network-slice intent must be realized through joint function placement, flow routing, resource allocation, and admission under time-varying capacity, queues, delay, reliability, isolation, and service dependencies [1]–[3]. This problem arises in network function virtualization, service function chaining, network slicing, mobile-edge computing, data-center virtualization, and SDN-based network management [4]–[10]. A standard abstraction is a sequential decision problem governed by Bellman optimality, in which a one-step deployment is evaluated together with its future state [11]. Existing learning-based controllers mainly improve which deployment is selected through policy learning, graph or

attention encoders, candidate ranking, and constrained policy updates [12]–[21]. This paper starts from a complementary observation: before the controller can certify the best deployment, it may already have enough information to exclude many candidates. Intermediate signals can reveal state–intent-equivalent deployments [22]–[26], candidates that leave a uniformly worse residual state under monotone dynamics [27]–[32], or violations of executable network laws. The information required to infer these network-physics-informed exclusion certificates can therefore be substantially weaker than the information required to establish optimality. Unlike penalty-, barrier-, and Lagrangian-based constrained RL, which enforce constraints within the policy objective or update [18], [33]–[38], LNOQRD runs as a shadow process that reshapes the candidate domain before expensive primal ranking. It converts intermediate primal-learning signals into residual, quotienting, and dominance certificates, thereby excluding candidates before exploration, value estimation, or ranking rather than merely changing their optimization scores. We instantiate this idea as Learning Not to Optimize via Quotienting, Residuals, and Dominance (LNO-QRD). For an intent-augmented state x = (s, I) and a generated candidate set CN (x), LNO-QRD constructs the reduced frontier λ I Bλ (x) = Topψ (C (x))/Γ Here quotienting ND F λ,I N x ε Kfr by the state-intent stabilizer Γx merges equivalent placementcontrol classes; FεI screens candidates by learned or computed violation residuals over resource-state, topology-flow, queue, reliability, and logical-intent laws; and NDλ,I removes dominated quotient classes under fixed-λ deployment dynamics. λ In the end, Topψ Kfr returns a reshaped frontier for the primal policy optimization under desired exploration and execution budget. During training and inference, LNO-QRD runs as a shadow process alongside primal policy learning: the primal learner provides residual estimates, structural signatures, and energy scores, while LNO-QRD converts these intermediate signals into certificates and feeds the reduced frontier back to the primal learner. We analyze theoretical properties of this closed-loop operation and provide rigorous guarantees. Theorems 1 and 2, together with Corollary 1, show that exact quotienting and dominance preserve at least one optimal generated executable deployment

under equivariant deployment dynamics and monotone fixed-λ values. Theorem 3 bounds the retained set by min{Kfr , αlaw (x)αnd (x)κΓ (x)|CN (x)|/|Γx |}, where αlaw , αnd , and κΓ denote the law-survival fraction, non-dominated fraction, and maximum stabilizer size; thus ranking cost falls with stronger screening, larger symmetry orbits, and a sparser non-dominated frontier when the saved scoring cost exceeds reduction overhead. Theorem 4 and Theorem 5 localize approximation loss to generation, screening, quotienting, dominance, budget truncation, stage slack, and critic error. Consequently, Theorem 6 gives the fixed-λ value gap (δ + 2ϵ + τ log Nmax )/(1 − γ), making the effects of frontier coverage, energy error, policy temperature, frontier size, and discounting explicit. We evaluate four placement-control families—service-graph deployment, network-slice realization, service-pipeline deployment, and dynamic drift—with joint placement, routing, allocation, and admission. Small instances use 8–10 nodes, 4–5 tasks, N = 64, and top-K = 16; large instances use 124–128 nodes, 10–14 tasks, N = 512, and top-K = 24, with traffic, queues, link rates, and reliability varying under drift. All candidate-based methods share the same generator. LNO-QRD reduces small-instance candidates by 75.9% while retaining 90.8% near-oracle coverage; on large instances, it attains the highest utility and intent satisfaction, the lowest hard-law violation and post-generation latency, and 73.0% average reduction among candidate-based baselines. II. I NTENT-C ONDITIONED D EPLOYMENT CMDP AND L AW R ESIDUAL S IGNALS This section defines only the finite-candidate control interface used by LNO-QRD. We do not model natural-language parsing; an upstream translator maps a request into a structured intent containing service dependencies, traffic demands, hard deployment laws, and optimization preferences [2], [3], [39]. Given the current network state and translated intent, the primal controller selects an executable high-utility deployment. The not-to-optimize layer uses the same candidate evaluation to remove generated candidates that are law-violating, dominated, or redundant before expensive primal ranking. Here “dual” means this auxiliary reduction role, not the Lagrangian dual of the CMDP. A. State, Intent, and Deployment Action At epoch t, the physical network is a directed graph Gt = (Vt , Et ). Let R = {cpu, mem, sto} be the resource types. Node and link features are ξv (t) = Cvr (t) r∈R ⊕ qv (t) ⊕ ρv (t) ⊕ αv (t), ζe (t) = Be (t), de (t), ρe (t), ηe (t) , Cvr

where is residual node resource, qv is queue pressure, Be is residual bandwidth, de is link delay, ρv , ρe ∈ (0, 1] are reliability values, and αv , ηe denote non-capacity node and link attributes. The physical state is st = (Gt , (ξv (t))v∈Vt , (ζe (t))e∈Et , Qt ) with Qt = (qv (t))v∈Vt .

A translated intent is It = (Ht , Dt , Ct , Ot ), where Ht = (Kt , Lt ) is a logical service graph, Dt contains task and traffic demands, Ct contains hard one-step deployment laws and service-level requirements, and Ot gives optimization preferences. Each task k ∈ Kt has demand rk = (rkr )r∈R , and each logical dependency ℓ = (i, j) ∈ Lt has traffic demand bℓ ≥ 0. We write the augmented state as xt = (st , It ) ∈ X , and omit t when clear. A deployment action is a = (M, F, Z, Aadm ). Here Mkv ∈ {0, 1} indicates whether task k is placed on node v, Aadm ∈ k |E| {0, 1} indicates admission, F = (f ℓ )ℓ∈L with f ℓ ∈ R+ is the r routing or flow realization, and zkv is the resource allocation. Let Araw (x) be the syntactically valid action domain induced by x. A raw action has valid dimensions and variable types but may still violate deployment laws; only law-satisfying actions are executable. B. Executable Network Laws Executability couples resource, flow, queue, reliability, and intent constraints. Each admitted task must be placed exactly once, and hard dependencies must be admission-consistent: X Mkv = Aadm , ∀k ∈ K, (1) k v∈V

Aadm = Aadm , i j

∀(i, j) ∈ Lhard ⊆ L.

(2)

adm

For any dependency ℓ = (i, j), let χℓ (A ) ∈ {0, 1} indicate whether the dependency is active. For hard dependencies, χℓ = Aadm = Aadm ; for optional or degraded dependencies, χℓ ≤ i j adm Ai and χℓ ≤ Aadm . Define ebℓ (a) = bℓ χℓ (Aadm ). j Allocation and capacity laws are

r zkv ≥ Mkv rkr ,

∀k ∈ K, v ∈ V, r ∈ R,

(3)

r zkv ≤ Mkv Cvr ,

∀k ∈ K, v ∈ V, r ∈ R,

(4)

r zkv ≤ Cvr ,

∀v ∈ V, r ∈ R,

(5)

∀e ∈ E.

(6)

k∈K

feℓ ≤ Be ,

ℓ∈L

For flow conservation, let B ∈ R|V |×|E| be the node-link incidence matrix with −1 at the tail and +1 at the head, and let mk = (Mkv )v∈V . For ℓ = (i, j), Bf ℓ = ebℓ (a)(mj − mi ), 0 ≤ f ℓ ≤ ebℓ (a), e

∀ℓ ∈ L,

(7)

∀ℓ ∈ L, e ∈ E.

(8)

Thus inactive dependencies carry zero flow, while active dependencies route bℓ units from the selected location of i to that of j. Intent-level hard laws are written compactly as D(a; x) ≤ D̄(I),

(9)

R(a; x) ≥ R̄(I),

(10)

hj (a; x) ≤ 0,

j = 1, . . . , mh ,

(11)

where D is end-to-end delay, R is deployment reliability, and hj represents isolation, affinity, anti-affinity, trust-domain,

jurisdiction, stage-ordering, admission, or policy-compliance laws. These functions may be instantiated by any executorcomputable model; for example, X X proc D(a; x) = Mkv Dkv (zkv , qv ) + ωℓe (a)de (12) k,v ℓ,e + Dqueue (a; x), X X log R(a; x) = Mkv log ρv + ψℓe (a) log ρe , (13) k,v

ℓ,e

shared by all compared candidate-based methods. The executable finite-candidate set is EN (x) = {a ∈ CN (x) : ϕ̄(x, a) = 0}.

(23)

To keep Bellman maximization defined even when no generated action is executable, define ( EN (x), EN (x) ̸= ∅, (24) EeN (x) = {⊥}, EN (x) = ∅,

where ωℓe = feℓ /ebℓ when ebℓ > 0 and 0 otherwise, and ψℓe is either path incidence or a continuous exposure weight. Queue evolution is (14) Qt+1 = Qt + Λt − µ(xt , at ) + ,

where ⊥ is an executor-level rejection, degradation, or infeasibility-declaration action with safe transition and fallback reward. If the intent permits controlled rejection, ⊥ can be implemented through admission control; otherwise it is a safe failure certificate.

where Λt is exogenous arrival and µ(xt , at ) is deploymentinduced service [40], [41].

D. Finite-Candidate CMDP

C. Law Residuals and Finite Candidates LNO-QRD converts executable laws into normalized residual coordinates. Let {cν (x, a) ≤ 0}ν∈J≤ collect scalar inequality laws and {eν (x, a) = 0}ν∈J= collect scalar equality laws, with vector laws treated coordinatewise. For scale σν > 0, ( [cν (x, a)]+ /σν , ν ∈ J≤ , ϕ̄ν (x, a) = (15) |eν (x, a)|/σν , ν ∈ J= . Let ϕ̄(x, a) collect these coordinates, grouped into placement, dependency, node, flow, link, queue, and intent residuals: ϕ̄(x, a) = (ϕ̄ν (x, a))ν∈J≤ ∪J= , Reslaw (x, a) = ∥ϕ̄(x, a)∥22 .

(16)

ϕ̄queue (x, a) =

ϕ̄intent (x, a) =

σvqueue

s.t.

Jj (π) = Eπ

" t=0 ∞ X

# t

γ gj (xt , at ) ≤ dj ,

j = 1, . . . , m,

t=0

(26)

(17)

By construction, ϕ̄(x, a) = 0 iff all encoded hard deployment laws are satisfied. Representative residual components are X [P z r − C r ]+ v k kv , (18) ϕ̄node (x, a) = node σ v,r v,r X [P f ℓ − Be ]+ ℓ e ϕ̄link (x, a) = , (19) σelink e X [Q b+ v (x, a) − Q̄v (I)]+

Under a selected action at ∈ EeN (xt ), the augmented state evolves by a Markov kernel PX (· | xt , at ). Let u(x, a) be bounded deployment utility and let gj (x, a) be bounded soft or long-term costs, such as energy, migration, service degradation, or long-term risk. These gj are distinct from the hard one-step laws encoded in ϕ̄(x, a). A deployed policy satisfies π(a | x) = π(a | s, I) with support on EeN (x). The discounted constrained problem is "∞ # X t max J0 (π) = Eπ γ u(xt , at ) , (25)

(20)

[D(a; x) − D̄(I)]+ [R̄(I) − R(a; x)]+ + σD σR m h X [hj (a; x)]+ + , (21) σjh j=1

b + (x, a) = [Q+Λ−µ(x, a)]+ and Q̄v (I) is the intentwhere Q conditioned one-step queue threshold at node v. Normalization prevents bandwidth, delay, queue length, or reliability units from dominating only because of scale. The raw domain Araw (x) may be mixed-integer and partially continuous [42], [43]. The online interface therefore uses a fixed generated candidate set CN (x) = {ai = Gi (x) : i = 1, . . . , N } ⊆ Araw (x),

(22)

with γ ∈ (0, 1). For a fixed multiplier λ ≥ 0, define rλ (x, a) = u(x, a) − λ⊤ g(x, a).

(27)

The fixed-λ Bellman quantities over the effective executable interface are Vλ⋆ (x) = max Q⋆λ (x, a),

(28)

a∈EeN (x)

Q⋆λ (x, a) = rλ (x, a) + γEx′ ∼PX (·|x,a) [Vλ⋆ (x′ )].

(29)

Thus the primal optimizer ranks executable candidates by value, while LNO-QRD uses additional signals from the same candidate evaluation—law residuals, queue residuals, dependency violations, and intent-preserving equivalence signatures—to decide which raw candidates should not be optimized separately. III. LNO-QRD: L EARNING N OT TO O PTIMIZE VIA Q UOTIENTING , R ESIDUALS , AND D OMINANCE This section constructs the not-to-optimize frontier used by LNO-QRD. Given the augmented state x = (s, I), the finite-candidate CMDP in Section II asks the primal controller to rank executable deployments. LNO-QRD inserts a dual reduction layer before this expensive ranking step: it does not choose the optimal candidate, but removes candidates that are

law-violating, state–intent equivalent, dominated, or outside the online budget. Given CN (x) = {ai }N i=1 , the reduction pipeline is CN (x) → Sε (x) → Qε (x) → Dλ (x) → Bλ (x), where the sets denote, respectively, residual-screened candidates, quotient classes, the non-dominated quotient frontier, and the final budgeted frontier. If no generated candidate is executable, the pipeline returns the fallback action ⊥ from Section II. The discarded set is Nλno (x) = CN (x) \ Bλ (x), and each discarded candidate receives a structural certificate rather than merely a low learned score. A. Residual Screening The first stage removes candidates that violate executable network laws. For each a ∈ CN (x), LNO-QRD evaluates the normalized residual ϕ̄(x, a) and scalar law residual Reslaw (x, a) from Section II. For tolerance ε ≥ 0, let FεI (CN (x)) = {a ∈ CN (x) : Reslaw (x, a) ≤ ε}. Strict executability uses ε = 0, while ε > 0 allows numerical margins. To keep the Bellman recursion defined even when all generated deployments violate hard laws, set Sε (x) =

( FεI (CN (x)), {⊥},

FεI (CN (x)) ̸= ∅, FεI (CN (x)) = ∅.

(30)

The fallback ⊥ is the executor-level rejection, degradation, or infeasibility-declaration action in (24), not an infeasible generated deployment. Hence exact residual screening removes only non-executable generated candidates; loss at this stage can only come from finite generation, numerical tolerance, or fallback when no executable candidate is generated. Operationally, ε is a screening tolerance rather than permission to deploy an invalid action. Before deployment, the executor recomputes the exact residual; a selected candidate with ϕ̄(x, a) ̸= 0 is rejected, repaired into an executable action, or mapped to ⊥. Thus, exact deployment remains supported on EeN (x), while errors introduced by approximate screening are charged to ∆res . B. State–Intent Quotienting Residual screening removes invalid candidates; quotienting removes redundant valid ones. If two deployments differ only by state–intent preserving relabelings of nodes, links, tasks, or dependencies, then the Bellman problem assigns them the same value. Ranking both separately only wastes primal optimization effort. Definition 1 (State–intent automorphism). Let Γ be a finite relabeling group acting on physical graphs, service graphs, intents, augmented states, and deployment actions. For x = (s, I), define the stabilizer Γx = {γ ∈ Γ : γx = x}, where each γ ∈ Γx preserves graph incidence, resource attributes, task demands, traffic demands, and intent constraints; also set γ⊥ = ⊥. For a, b ∈ Sε (x), a ∼x b ⇐⇒ ∃γ ∈ Γx such that b = γa. For finite U ⊆ Sε (x), U/Γx denotes the induced quotient set.

Using Definition 1, the quotient set is Qε (x) = Sε (x)/Γx . A deterministic map repx : Qε (x) → Sε (x) selects one representative per class, with repx ({⊥}) = ⊥ in the fallback case. Exact automorphism computation can be expensive on large attributed graphs. We therefore use exact quotienting when available and otherwise use deterministic state–intent signatures over resource, queue, reliability, domain, task-type, traffic, and path features. Exact quotienting is lossless; signature quotienting is conservative if it only splits exact orbits and approximate if it merges non-equivalent actions. The latter loss is accounted for in Section IV. Assumption 1 (Equivariant deployment interface). For every γ ∈ Γ, augmented state x, action a, measurable B ⊆ X , and multiplier λ ≥ 0, the executable interface, fixed-λ reward, and transition kernel satisfy EeN (γx) = γ EeN (x), rλ (γx, γa) = rλ (x, a), PX (B | γx, γa) = PX (γ −1 B | x, a). Theorem 1 (Lossless quotienting). Under Assumption 1, for every λ ≥ 0, Q⋆λ (γx, γa) = Q⋆λ (x, a),

∀γ ∈ Γ.

(31)

Consequently, for fixed x and any γ ∈ Γx , Q⋆λ (x, γa) = Q⋆λ (x, a). Therefore, keeping one representative per exact orbit in Sε (x)/Γx preserves the best fixed-λ generated action value inside Sε (x). Proof sketch. Let Tλ be the fixed-λ Bellman optimality operator and define Qγ (x, a) = Q(γx, γa). Assumption 1 gives the operator equivariance Tλ (Qγ ) = (Tλ Q)γ . Since the discounted Bellman operator has a unique fixed point, (Q⋆λ )γ = Q⋆λ , which yields (31). If γ ∈ Γx , then γx = x, so all actions in the same orbit have identical optimal action value at x. Theorem 1 applies only to exact state–intent symmetries. Exact quotienting identifies actions that are value-equivalent under Assumption 1, whereas signature quotienting is approximate and its induced error must be quantified. C. Dominance Pruning Quotienting removes actions that are equivalent. Dominance pruning removes actions that are different but provably no better. The key point is dynamic: a candidate can be worse not only because it has smaller immediate Lagrangian reward, but also because it leaves a weaker residual network state for future decisions. Let ⪰X be a preorder over augmented states: x+ ⪰X x− means that x+ has no smaller residual capacity, no larger queue pressure, no worse reliability, and the same or no harder intent class than x− . Non-monotone coordinates for a specific deployment problem are excluded. For kernels at the same state, write PX (· | x, a) ⪰st PX (· | x, b) if E[ψ(x′ ) | x, a] ≥ E[ψ(x′ ) | x, b] for every bounded increasing ψ under ⪰X . Definition 2 (Deployment dominance). For fixed x and λ, action a dominates b, written a ⪰x,λ b, if rλ (x, a) ≥ rλ (x, b)

and PX (· | x, a) ⪰st PX (· | x, b). The strict relation a ≻x,λ b holds when at least one comparison is strict. Assumption 2 (Monotone fixed-λ value). For the fixed multiplier λ, the optimal value is increasing under ⪰X : x+ ⪰X x− =⇒ Vλ⋆ (x+ ) ≥ Vλ⋆ (x− ). This is the standard monotone dynamic-programming condition: additional residual capacity, lower queue pressure, and better reliability cannot reduce future deployment opportunities [11], [27]–[29]. Theorem 2 (Lossless dominance pruning). Under Assumption 2, if a ⪰x,λ b, then Q⋆λ (x, a) ≥ Q⋆λ (x, b). Therefore, a strictly dominated action cannot be the unique optimal deployment at x. Proof sketch. By the Bellman equation, Q⋆λ (x, a) Q⋆λ (x, b) = rλ (x, a) − rλ (x, b) + γ E[Vλ⋆ (x′ )

− |

x, a] − E[Vλ⋆ (x′ ) | x, b] . The reward difference is nonnegative by Definition 2. The expectation difference is nonnegative because Assumption 2 makes Vλ⋆ increasing and the transition under a stochastically dominates that under b. Hence Q⋆λ (x, a) ≥ Q⋆λ (x, b). Exact stochastic dominance is rarely available online. We therefore use a sufficient diagnostic certificate based thecostnormalized badness vector on e D(a;x), C (a;x), 1−R(a;x), Reslaw (x,a) b(x, a) = . −S node (x,a), −S link (x,a), Lpath (x,a), I logic (x,a) e is normalized delay, All coordinates are normalized. Here D C cost is deployment cost, S node and S link are residual node and link slacks, Lpath is routing burden, and I logic is the policy-logic residual; 1 − R and Reslaw measure unreliability and total law residual. Candidate b is pruned if some a satisfies b(x, a) ≤ b(x, b) coordinate-wise and is strictly smaller in at least one coordinate. This certificate is exact only when the badness-vector order implies Definition 2; otherwise its loss is charged to the dominance term in Section IV. For quotient classes, define ca ⪰qx,λ cb ⇐⇒ repx (ca ) ⪰x,λ repx (cb ). The non-dominated quotient frontier is Dλ (x) = NDλ,Io(Qε (x)) = n q ′ ′ c ∈ Qε (x) : ∄c ∈ Qε (x) such that c ≻x,λ c . Corollary 1 (Exact structural reduction). Suppose residual screening is exact and returns at least one executable generated candidate, quotienting is exact, and dominance is exact on the finite generated candidate set. Under Assumptions 1 and 2, reducing CN (x) to {repx (c) : c ∈ NDλ,I (S0 (x)/Γx )} preserves at least one optimal generated executable deployment. Proof sketch. Exact residual screening removes only nonexecutable generated candidates. Theorem 1 preserves the best value within each exact orbit, and Theorem 2 rules out strictly dominated representatives as unique optima. Finiteness then guarantees that at least one best generated executable deployment remains.

D. Budgeted Composite-Priority Frontier The final stage enforces the online inference budget. It is not a feasibility certificate and is lossless only when the retained budget includes a near-optimal representative. Let S(x, a) = bub (x, a) be the learned upper S node (x, a) + S link (x, a), let U utility score, and take all weights w· ≥ 0. For finite repreλ sentative set U , let Topψ Kfr (U ) return the Kfr candidates with smallest diagnostic priority, using deterministic tie-breaking: e x) + wc C cost (a; x) + ψλ (x, a) = wϕ ∥ϕ̄(x, a)∥1 + wd D(a; bub (x, a). wr (1 − R(a; x)) − ws S(x, a) + wp Lpath (a; x) − wu U ψλ The budgeted frontier is Bλ (x) = TopK ({repx (c) : c ∈ Dλ (x)}). If top-K removes a high-value candidate, the loss is accounted for by ∆K in Section IV. E. Frontier Size and Ranking Speedup The previous subsections define what LNO-QRD removes. We now state the resulting reduction in expensive primal ranking calls. This is the formal speedup result for the dual not-to-optimize layer: the primal scorer is applied only to the retained frontier, while residuals, quotienting, and dominance use cheaper structural information produced during candidate evaluation. Assume that FεI (CN (x)) is nonempty, finite, and invariant under Γx . Let Stabx (a) = {γ ∈ Γx : γa = a}, κΓ (x) = maxa∈FεI (CN (x)) | Stabx (a)|, and αlaw (x) = |FεI (CN (x))|/|CN (x)|, αnd (x) = |Dλ (x)|/|Qε (x)|. Set κλ (x) = αlaw (x)αnd (x)κΓ (x). Theorem 3 (Frontier complexity and ranking speedup). For every state–intent pair x = (s, I) with nonempty FεI (CN (x)), κλ (x) |CN (x)| . (32) |Bλ (x)| ≤ min K, |Γx | Suppose evaluating the primal ranking model on one candidate costs cE , and let Tred (x) be the online cost of residual evaluation, quotienting, dominance testing, and top-K selection after candidates have been generated. Raw primal ranking costs Traw (x) = cE |CN (x)|. Ranking after LNO-QRD costs n o λ (x) at most TLNO (x) ≤ Tred (x) + cE min Kfr , κ|Γ |C (x)| . N x| Thus, LNO-QRD reduces online ranking cost whenever κλ (x) Tred (x) < cE |CN (x)| − min Kfr , |CN (x)| . |Γx | (33) If FεI (CN (x)) = ∅, then Bλ (x) = {⊥} and no expensive primal ranking over generated deployments is required. Proof sketch. For any feasible generated candidate, orbit– stabilizer gives |[a]x | = |Γx |/| Stabx (a)| ≥ |Γx |/κΓ (x). Since FεI (CN (x)) is Γx -invariant, |Qε (x)| ≤ I κΓ (x)|Fε (CN (x))|/|Γx |. Substituting the retained law fraction αlaw and non-dominated fraction αnd , then applying the top-K cap, gives (32). The cost statement follows because the expensive primal scorer is evaluated only on Bλ (x). Corollary 2 (Predictive compression bound). Suppose that, over a class of states, residual screening keeps at most a fraction plaw of generated candidates, non-dominated filtering

keeps at most a fraction pnd of quotient classes, and every action stabilizer has size at most s. Then plaw pnd s |Bλ (x)| ≤ min Kfr , |CN (x)| . (34) |Γx | Corollary 2 characterizes predictive compression through the residual feasibility rate, quotient redundancy rate, nondominated frontier fraction, final ranking budget, and measured reduction overhead. F. Intent-Conditioned Deployment Energy After LNO-QRD constructs Bλ (x), the primal learner ranks only this frontier using the deployment energy Eθ (x, a) = bθ (x, a) + λ⊤ G b θ (x, a) + ηlaw Reslaw (x, a), where C bθ is a C b learned negative utility-to-go proxy, Gθ estimates discounted soft CMDP cost, and the hard-law residual remains an executor-facing signal. The reduced Boltzmann policy is exp(−Eθ (x, a)/τ ) , b∈Bλ (x) exp(−Eθ (x, b)/τ )

πθ (a | x) = P

a ∈ Bλ (x).

(35) At evaluation, the controller may take at ∈ arg mina∈Bλ (x) Eθ (x, a). Here τ > 0 is the policy temperature and ηlaw ≥ 0 controls residual-aware energy shaping. Consequently, the energy model focuses on value differences within the reduced frontier rather than repeatedly distinguishing candidates already excluded by explicit structural certificates. The energy model uses permutation-invariant or equivariant encoders for candidate sets, physical graphs, and service graphs [14]–[17], [44]. It is trained to respect the network symmetry group through Eθ (γx, γa) = Eθ (x, a) for γ ∈ Γx . The learning objective augments PPO with value fitting and three structured regularizers: a soft-cost regression term Lg , a certified-order ranking term Lord , and a symmetry-consistency term Linv . The resulting objective is L = LPPO + cv LV + βg Lg + βord Lord + βinv Linv − π 2 b βH E[H(πθ (· P| x))]. Here, Lg = E[∥Gθ (x, a)− G (x, a)∥2 ], Lord = E a≻b [m + Eθ (x, a) − Eθ (x, b)]+ , and Linv = Eγ∈Γx [(Eθ (γx, γa) − Eθ (x, a))2 ]. IV. P ERFORMANCE G UARANTEES This section accounts for what is lost after LNO-QRD reduces the generated candidate set. Exact residual screening, exact quotienting, and exact dominance are lossless in the senses proved above, while practical LNO-QRD may incur loss from finite generation, tolerance, signature quotienting, approximate dominance, top-K truncation, and learned energies. We first decompose frontier loss by stage, then bound the effect of intermediate primal estimates, and finally convert frontier coverage plus energy approximation into fixed-λ policy performance. Throughout, λ ≥ 0 is fixed; multiplier learning can be handled by standard primal–dual updates around this analysis [33]. Let Abench (x) be a benchmark action domain containing CN (x) ∪ {⊥} on which Q⋆λ (x, a) is defined. It may be either the generated interface with fallback or a larger executable

Algorithm 1 LNO-QRD decision step Require: state–intent pair x, candidate budget N , frontier budget Kfr , parameters θ, multiplier λ 1: C ← G ENERATE C ANDIDATES(x, N ) 2: S ← {a ∈ C : Reslaw (x, a) ≤ ε} 3: if S = ∅ then 4: B ← {⊥} 5: else 6: Q ← Q UOTIENT(S, Γx ) using exact or signature-based state–intent canonicalization 7: D ← N ON D OMINATED(Q, λ) using exact or certified approximate dominance λ 8: B ← Topψ K ({repx (c) : c ∈ D}) 9: end if 10: Choose at ∼ πθ (· | x) on B during training, or at ∈ arg mina∈B Eθ (x, a) during execution 11: Revalidate or repair at under the exact executor laws; map an unsuccessful repair to ⊥ 12: Execute the resulting action and update θ, λ from reward, soft costs, residuals, and next state Ensure: frontier Bλ (x) = B and not-to-optimize set Nλno (x) = C \ B

oracle domain, in which case the first loss term is the finitegeneration gap. A. Stagewise Frontier Error Accounting The frontier is constructed through nonempty sets U0 (x) = Abench (x), U1 (x) = CN (x) ∪ {⊥}, U2 (x) = Sε (x), U3 (x) = {repx (c) : c ∈ Qε (x)}, U4 (x) = {repx (c) : c ∈ Dλ (x)}, U5 (x) = Bλ (x). Here U1 is the generated interface with fallback, U2 is residual-screened, U3 is quotient-representative, U4 is non-dominated, and U5 is budgeted. If screening returns {⊥}, then U3 = U4 = U5 = {⊥}. For i = 1, . . . , 5, define the stage loss ∆i (x) =

max a∈Ui−1 (x)

Q⋆λ (x, a) − max Q⋆λ (x, a). a∈Ui (x)

(36)

Write ∆gen = ∆1 , ∆res P5= ∆2 , ∆q = ∆3 , ∆dom = ∆4 , ∆K = ∆5 , and δfr (x) = i=1 ∆i (x). Theorem 4 (Frontier error accounting). For every state–intent pair x = (s, I) with nonempty Ui (x), max a∈Abench (x)

Q⋆λ (x, a) − max Q⋆λ (x, a) = δfr (x). a∈Bλ (x)

(37)

Moreover, exact residual screening contributes no loss from feasible generated candidates, exact quotienting gives ∆q (x) = 0, exact dominance gives ∆dom (x) = 0, and a budget rule that retains a near-best representative gives small ∆K (x). Proof sketch. The identity is the telescoping sum of the five differences in (36). Exact residual screening removes only non-executable generated candidates, exact quotienting gives ∆q (x) = 0 by Theorem 1, and exact dominance gives ∆dom (x) = 0 by Theorem 2. The remaining terms correspond

to finite generation, tolerance or fallback effects, and budget truncation. Accordingly, any frontier-quality loss is isolated in a named stage term, motivating the reported reduction gaps, near-oracle coverage, violation, top-K loss, scalability, and overhead. B. Effect of Intermediate Primal Estimation Error Approximate stages may rely on critic estimates, negativeenergy proxies, learned soft-cost predictions, or state–intent embeddings. The next result bounds how such intermediate primal errors affect frontier loss. b θ (x, a) denote the estimate used by approximate Let Q quotienting, dominance, or top-K selection. Assume that, on all candidates entering the reduction pipeline, b θ (x, a) − Q⋆ (x, a) ≤ ϵQ . Q λ

(38)

For a reduction stage Ui−1 (x) → Ui (x), define the estimated slack ηi (x) by max a∈Ui−1 (x)

b θ (x, a) − max Q b θ (x, a) ≤ ηi (x). Q a∈Ui (x)

(39)

A zero value of ηi (x) means that the stage keeps an estimated best candidate. A positive value allows approximate screening, approximate signatures, approximate dominance, or top-K truncation. Theorem 5 (Intermediate-signal error bound). Assume (38). For any stage Ui−1 (x) → Ui (x) satisfying (39), ∆i (x) ≤ ηi (x) + 2ϵQ .

(40)

Consequently, if the approximate reduction stages i = 2, 3, 4, 5 satisfy (39), then δfr (x) ≤ ∆gen (x) +

5 X

ηi (x) + 8ϵQ .

(41)

i=2

Exact lossless stages contribute zero to the corresponding true loss ∆i (x); their estimated slack ηi (x) need not vanish. Proof sketch. Fix a stage Ui−1 → Ui . Let a⋆ ∈ argmaxa∈Ui−1 (x) Q⋆λ (x, a), ā ∈ ⋆ ⋆ b argmaxa∈Ui (x) Qθ (x, a). By (38), Qλ (x, a ) ≤ b θ (x, a⋆ ) + ϵQ ≤ maxa∈U (x) Q b θ (x, a) + ϵQ . By Q i−1 b θ (x, a) ≤ Q b θ (x, ā) + ηi (x). Again using (39), maxa∈Ui−1 (x) Q b θ (x, ā) ≤ Q⋆ (x, ā) + ϵQ ≤ maxa∈U (x) Q⋆ (x, a) + ϵQ . (38), Q i λ λ Combining the three inequalities gives (40). Summing the four approximate-stage bounds and adding the generation term gives (41). This theorem identifies the explicit error channel created by learning not to optimize. If the primal estimates are accurate and each approximate reduction stage keeps a candidate with near-best estimated value, the true frontier loss remains small. If a signature quotient or approximate dominance rule removes a high-value candidate, the loss appears in the corresponding ηi (x) term.

C. Reduced-Policy Performance Bound We now convert frontier coverage and energy approximation into a value bound. Let Nmax = supx |Bλ (x)| < ∞; by Theorem 3, Nmax ≤ Kfr when an executable candidate is found and Nmax = 1 under fallback. Assumption 3 (Uniform reduced-frontier coverage). There exists δ ≥ 0 such that, for every state x, maxa∈Abench (x) Q⋆λ (x, a) − maxa∈Bλ (x) Q⋆λ (x, a) ≤ δ. Theorem 4 decomposes δ by reduction stage, and Theorem 5 bounds the contribution of approximate primal signals. Assumption 4 (Uniform energy approximation). There exists ϵ ≥ 0 such that, for every x and a ∈ Bλ (x), | − Eθ (x, a) − Q⋆λ (x, a)| ≤ ϵ. Theorem 6 (Approximate reduced-policy performance). Under Assumptions 3 and 4, the Boltzmann policy (35) satisfies, for every initial state x, Vλ⋆ (x) − Vλπθ (x) ≤

δ + 2ϵ + τ log Nmax . 1−γ

(42)

b θ (x, a) = −Eθ (x, a). AsProof sketch. Fix x and write Q sumption 3 bounds the loss from restricting the benchmark b θ − Q⋆ | ≤ ϵ domain to Bλ (x) by δ. Assumption 4 gives |Q λ on the reduced frontier. Since (35) is the softmax policy over b θ , the log-sum-exp bound over at most Nmax actions gives Q b θ (x, a) − Ea∼π (·|x) [Q b θ (x, a)] ≤ τ log Nmax . maxa∈Bλ (x) Q θ Using the uniform approximation once for the maximum and once for the expectation yields a one-step Bellman suboptimality of at most δ+2ϵ+τ log Nmax . The discounted performancedifference inequality then gives (42). Corollary 3 (Exact-structure limit). If the generator contains an optimal benchmark action, residual screening does not remove feasible optimal actions, quotienting and dominance are exact, top-K preserves an optimal non-dominated representative, and the energy represents −Q⋆λ exactly on the reduced frontier, then δ = ϵ = 0. A deterministic zerotemperature policy over Bλ (x) is optimal for the fixed-λ MDP. Proof sketch. The stated exact conditions make the frontier loss zero by Theorem 4 and set ϵ = 0 in Assumption 4. Taking the deterministic zero-temperature limit removes the entropy term in Theorem 6. V. E XPERIMENTS This section evaluates whether LNO-QRD improves intentconditioned network deployment by learning what not to optimize. The controlled benchmarks test two claims: Theorem 3 predicts that quotienting, residual screening, dominance, and the top-K budget reduce the candidates exposed to primal ranking when saved scoring cost exceeds dual overhead; Theorem 4 and Theorem 5 predict that quality loss can be traced to generation, screening, quotienting, dominance, budget truncation, or intermediate primal-estimation error. We therefore report utility, reduction, near-oracle coverage, reduction–gap

Service graph

Small/Large DAG service graph Placement–routing coupling Small/Large Latency, isolation, Multi-tenant contention reliability Large Ordered pipeline Unseen service-graph DAG transfer Large Time-varying Queue, demand, link-rate, intents and reliability variation

Network slice Service pipeline Dynamic drift

Intent

Main stressor

TABLE II S MALL - INSTANCE EXPLANATORY RESULTS . VALUES ARE MEAN ± STANDARD DEVIATION .

0.10

0.6 0.4 0.2

0.09 K=8

0.08 0.07 K=16

0.06

0.0

ate

did

otie

ly nt

otie

nc ina om

O LN

ron

-QR

tie

Coverage

OracleGap

Violation

Raw candidates 0.011 ± 0.015 0.986 ± 0.013 0.021 ± 0.007 0.268 ± 0.010 Quotient only 0.338 ± 0.025 0.957 ± 0.015 0.036 ± 0.008 0.196 ± 0.010 Quotient + dominance 0.579 ± 0.026 0.934 ± 0.016 0.056 ± 0.007 0.123 ± 0.010 Full LNO-QRD 0.759 ± 0.025 0.908 ± 0.016 0.076 ± 0.008 0.064 ± 0.010

behavior, violation, scalability, transfer, ablation, and postgeneration latency. A. Experimental Setup We evaluate four benchmark families: service-graph deployment, network-slice realization, service-pipeline deployment, and dynamic drift [5], [6], [9]. Small instances use |V | ∈ {8, 10}, |K| ∈ {4, 5}, N = 64, and frontier budget Kfr = 16 over 10 seeds for oracle diagnostics. Large instances use |V | ∈ [124, 128], |K| ∈ [10, 14], N = 512, and frontier budget Kfr = 24; dynamic drift varies traffic, queues, link rates, and reliability. All candidate-based methods share the same CN (x) generator, including capacity-, load-, locality-, trust, delay-, bandwidth-, reliability-aware, balanced, randomized, and repair-based candidates. Hence any gain must come from excluding infeasible, equivalent, dominated, or unnecessary candidates, not from a stronger generator. B. Baselines and Metrics We compare LNO-QRD with six baselines: a smallinstance MILP/CP oracle [42], [43], Greedy, MLP-PPO [45], GNN-PPO [14], [15], Transformer candidate ranking [16], [17], and P ENALTY-RL [18], [33]. The ablations remove quotienting, dominance pruning, law residuals, order supervision, or invariance supervision. Utility is the deployment reward after evaluating delay, reliability, queue, and resource terms; IntentSat is normalized to [0, 1]; Violation is the normalized pre-revalidation residual of the selected proposal; Reduction is 1 − |Bλ (x)|/|CN (x)|. OracleGap is measured against the small-instance oracle or a large-pool proxy, Coverage is the preserved fraction of nearoracle deployments, GenGap is the unseen-intent/topology transfer gap, and Time is post-generation latency for candidate evaluation, reduction, and ranking. C. Small-Instance Explanatory Results Small certifiable instances test the error-accounting view in Theorem 4. Raw candidates preserve high oracle coverage but contain many law-violating deployments. Quotienting reduces

K=64

0.04

0.2

0.4

0.6

0.8

Frontier compression

(a) Candidate reduction. (b) Reduction–gap tradeoff. Fig. 1. Small-instance frontier diagnostics. LNO-QRD progressively shrinks the candidate set while keeping the oracle gap controlled.

Oracle gap

Reduction

K=24 K=32

0.05

Oracle gap Compression

0.10

Method

K=4

0.11

0.8

0.08

0.6

0.06

0.4

Compression

Scale

1.0

Oracle gap

Scenario

Reduction ratio 1 − ||/||

TABLE I B ENCHMARK SETTINGS .

0.2

0.04 20

Frontier budget K

Fig. 2. Budget-induced reduction–gap tradeoff. A larger frontier budget lowers ∆K but exposes more candidates to primal scoring.

the set by 33.8% with 95.7% coverage; adding dominance raises reduction to 57.9%; full LNO-QRD reaches 75.9% reduction and the lowest violation while retaining 90.8% nearoracle coverage. Thus most candidates are removed before primal ranking, while the high-value frontier remains largely covered. D. Large-Scale Deployment Results Large-scale results test whether frontier reduction improves policy quality and online latency. Across four large benchmark families, LNO-QRD obtains the best utility, IntentSat, violation, reduction, and post-generation time in Table III. Relative to Transformer, utility increases from 3.642 to 3.973 and latency drops from 29.327 ms to 7.008 ms; relative to P ENALTY-RL, violation decreases from 0.291 to 0.191. With average reduction 0.730, the primal learner ranks only about 27% of the raw candidate pool. E. Feasibility and Law-Residual Analysis Residual diagnostics explain the feasibility gain. P ENALTYRL reduces violations through training penalties, whereas LNO-QRD uses law residuals as pre-ranking not-to-optimize certificates. Mean violation drops from 0.291 for P ENALTYRL to 0.191 for LNO-QRD, with lower placement, nodecapacity, flow, link-capacity, queue, and intent residuals in Fig. 4. F. Scalability and Dual-Learning Overhead Scalability and overhead results test the speedup condition in Theorem 3 and (33). As scale grows, the retained-frontier ratio |Bλ (x)|/|CN (x)| remains substantially below one, so the primal scorer is applied to a compact subset rather than the full candidate pool. The overhead decomposition further shows that residual evaluation, quotienting, dominance testing, and

TABLE III L ARGE - SCALE DEPLOYMENT RESULTS . VALUES ARE MEAN ± STANDARD DEVIATION OVER FOUR LARGE BENCHMARK FAMILIES , THREE SEEDS , AND ALL TEST EPISODES . Method

Utility

IntentSat

Violation

Reduction

Time (ms)

4.0

LNO-QRD

3.25 3.00 2.75 GNN-PPO Greedy MLP-PPO

2.50 2.25 0

Penalty-RL LNO-QRD Transformer

100

150

Retained ratio |λ|/|N|

3.50

Deployment utility

3.75

3.8 Transformer

3.6

Penalty-RL

GNN-PPO

3.4 MLP-PPO

3.2 Greedy

3.0 0.20

0.25

0.30

0.35

0.40

0.45

0.50

Hard-law violation

Training step

(a) Learning curves. (b) Utility–violation tradeoff. Fig. 3. Large-scale deployment performance. LNO-QRD improves utility while shifting the tradeoff toward lower violation. 0.5

place node flow

link queue intent

0.3 0.2 0.1

0.4

0.3

dy ee

P P-P

N-P

nsfo Tra

y-R

alt

n Pe

O -Q

0.4 256 cand.

384 cand.

512 cand.

768 cand.

1024 cand. 1280 cand.

0.2

128

160

Dominance testing Top-K selection Reduced energy ranking

192 32

Problem scale |V|

s 64

s 96

8n 12

(a) Frontier ratio vs. scale. (b) Overhead decomposition. Fig. 5. Scalability and overhead. Panel (a) shows the retained-frontier ratio as scale grows. Panel (b) decomposes post-generation latency and compares the reduced pipeline with raw full-candidate ranking.

Variant

0.2

0.0

0.6

Raw full ranking Residual evaluation Quotienting

TABLE IV A BLATION STUDY ON THE TWO LARGE STRUCTURAL BENCHMARKS . VALUES ARE MEAN ± STANDARD DEVIATION .

0.5

Violation

Mean residual

0.4

0.8

0.0

200

Raw candidate pool LNO-QRD retained frontier

1.0

4.00

Post-generation latency (ms)

Greedy 3.001 ± 0.073 0.277 ± 0.059 0.500 ± 0.024 0.010 ± 0.014 10.537 ± 0.401 MLP-PPO 3.305 ± 0.073 0.374 ± 0.062 0.431 ± 0.022 0.010 ± 0.015 22.896 ± 0.463 GNN-PPO 3.548 ± 0.074 0.486 ± 0.059 0.355 ± 0.021 0.011 ± 0.015 25.186 ± 0.488 Transformer 3.642 ± 0.070 0.525 ± 0.058 0.327 ± 0.020 0.009 ± 0.014 29.327 ± 0.488 P ENALTY-RL 3.523 ± 0.072 0.579 ± 0.056 0.291 ± 0.019 0.009 ± 0.014 24.029 ± 0.448 LNO-QRD 3.973 ± 0.071 0.724 ± 0.056 0.191 ± 0.017 0.730 ± 0.025 7.008 ± 0.459

PO P-P

N-P GN

Tra

nsfo

-RL lty

na Pe

O -Q

(a) Law-residual profile. (b) Violation spread. Fig. 4. Feasibility diagnostics. LNO-QRD reduces multiple residual coordinates and lowers violation spread across large scenarios.

top-K selection are outweighed by the saved full-candidate energy-ranking cost. G. Transfer Across Intents and Topologies For transfer, we train on service-graph or network-slice deployments and test on unseen large benchmark families. We report aggregate behavior to save space: LNO-QRD achieves the highest average transfer utility, 3.908, compared with 3.577 for Transformer and 3.490 for GNN-PPO. Same-family transfer is easier, while dynamic drift and unseen service pipelines produce larger gaps; nevertheless, residual, quotient, and dominance certificates transfer better than fixed candidateindex policies because they depend on network laws and state– intent structure. H. Ablation Study The ablation study isolates the certificates. Removing quotienting weakens utility and compression, confirming the role of state–intent equivalence. Removing dominance causes the largest reduction loss, showing that it excludes many feasible but noncompetitive candidates. Removing residual screening gives the largest feasibility degradation, while removing order or invariance losses weakens energy ranking on the reduced frontier.

Utility

Violation

Reduction

GenGap

Full 4.003 ± 0.080 0.187 ± 0.016 0.731 ± 0.026 0.168 ± 0.016 NoQuotient 3.886 ± 0.077 0.206 ± 0.018 0.569 ± 0.025 0.175 ± 0.016 NoDominance 3.935 ± 0.082 0.195 ± 0.017 0.402 ± 0.025 0.172 ± 0.015 NoResidual 3.805 ± 0.079 0.276 ± 0.019 0.599 ± 0.025 0.183 ± 0.015 NoOrderLoss 3.904 ± 0.078 0.213 ± 0.018 0.685 ± 0.026 0.176 ± 0.016 NoInvLoss 3.914 ± 0.080 0.208 ± 0.018 0.701 ± 0.025 0.174 ± 0.016

Overall, the experiments support the central claim: LNOQRD reduces small-instance candidates by 75.9% while retaining 90.8% near-oracle coverage, and achieves the best large-scale utility, violation, reduction, and ranking latency. Scalability, transfer, and ablation results attribute these gains to residual screening, quotienting, dominance, and structural energy learning rather than to the shared generator. VI. C ONCLUSION This paper studied learning not to optimize as a complementary counterpart to constrained network policy optimization. LNO-QRD uses residual screening, state–intent quotienting, and dominance pruning to remove infeasible, redundant, and dominated candidates before expensive primal ranking, so the learner optimizes only over a budgeted frontier. We proved lossless reduction conditions, a frontier-size and rankingspeedup bound, and stagewise error accounts for approximate certificates and intermediate learning signals. Experiments on controlled deployment benchmarks show that this dual layer preserves near-oracle candidates on small instances and improves large-scale utility, violation, transfer, and ranking latency. The main conclusion is that intermediate network-law information can shape not only how actions are ranked, but also which actions should not be optimized.

Normalized score

1.0

Utility Feasibility

Compression Generalization

Latency

0.8 0.6 0.4 0.2 0.0 LN

O-

n nti

e oti

u oQ

esi

ing en

ina

D /o

l ua

re Sc

oss

rde

O /o

ari Inv

ce an

ss Lo

Fig. 6. Ablation summary. Residual screening mainly improves feasibility, quotienting removes state–intent redundancy, dominance gives the largest compression, and order/invariance losses stabilize reduced-frontier ranking.

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