Optimal Unnested Fixed Point Estimator

• OUFXP is an estimator for infinite-horizon dynamic discrete choice models that separates the Bellman fixed point from structural utility parameters using a dual representation.
• It improves estimation efficiency by compressing Bellman first-order conditions with optimally weighted random projections and pre-computed dual fixed points.
• OUFXP scales efficiently in complex settings, including neural network embedded utility functions, outperforming traditional nested and CCP estimators.

Optimal Unnested Fixed Point (OUFXP) is an estimator for infinite-horizon dynamic discrete choice (DDC) models in which the dynamic programming fixed point is separated from the structural utility parameters by a dual representation of Bellman’s equation. In the formulation introduced in "Training Neural Networks Embedded in Dynamic Discrete Choice Models" (Oguz et al., 9 Apr 2026), OUFXP is the efficient, two-step refinement of the Unnested Fixed Point (UFXP) estimator: it compresses Bellman first-order conditions with projection weights chosen to minimize asymptotic variance, while preserving the central computational property that, after pre-computation, estimation is unencumbered by large systems of linear equations imposed as constraints or embedded in the objective function. The construction is designed for DDC settings with flexible utility parameterizations, including neural networks, where conventional nested estimators become prohibitively expensive.

1. Model setting and object of estimation

The underlying environment is a standard infinite-horizon, discrete-time DDC model. At each period $t=0,1,2,\dots$, the agent observes a discrete state $x \in \mathbb X \equiv \{1,\dots,X\}$ and a vector of action-specific shocks $e \in \mathbb R^A$, where the action set is finite, $\mathbb A \equiv \{1,\dots,A\}$. The shock density $g$ is strictly positive and smooth, with full support on $\mathbb R^A$. Per-period utility under action $a$ is

$u_t = e_a + u_a^\theta(x),$

where $u_a^\theta(x)$ is the deterministic component and $\theta \in \Theta \subset \mathbb R^t$ is the structural parameter vector (Oguz et al., 9 Apr 2026).

Conditional on action $x \in \mathbb X \equiv \{1,\dots,X\}$0, the next observed state is drawn from $x \in \mathbb X \equiv \{1,\dots,X\}$1, and next period’s shocks are drawn independently from $x \in \mathbb X \equiv \{1,\dots,X\}$2. With discount factor $x \in \mathbb X \equiv \{1,\dots,X\}$3, the full value function solves

$x \in \mathbb X \equiv \{1,\dots,X\}$4

Integrating over shocks yields the integrated value function $x \in \mathbb X \equiv \{1,\dots,X\}$5, which can be written using McFadden’s social surplus function $x \in \mathbb X \equiv \{1,\dots,X\}$6.

A central reformulation replaces discrete actions by conditional choice probability vectors $x \in \mathbb X \equiv \{1,\dots,X\}$7, where $x \in \mathbb X \equiv \{1,\dots,X\}$8 is the simplex. The Bellman equation becomes

$x \in \mathbb X \equiv \{1,\dots,X\}$9

This representation is the basis for OUFXP: the estimation problem is organized around conditional choice probabilities and their first-order conditions rather than around repeated solution of a $e \in \mathbb R^A$0-dependent Bellman fixed point.

2. Dual Bellman representation and the meaning of “unnested”

The term “unnested” refers to the removal of dynamic programming fixed points from the objective function and its derivatives after a one-time pre-computation. The key step is a dual representation of the static discrete choice problem. For deterministic utilities $e \in \mathbb R^A$1,

$e \in \mathbb R^A$2

with maximizer $e \in \mathbb R^A$3. After normalization with operators $e \in \mathbb R^A$4 and $e \in \mathbb R^A$5, the paper defines $e \in \mathbb R^A$6 and its inverse $e \in \mathbb R^A$7, so that the dynamic first-order conditions can be written in terms of inverse choice-probability transforms (Oguz et al., 9 Apr 2026).

For a fixed policy $e \in \mathbb R^A$8, the policy value satisfies

$e \in \mathbb R^A$9

where

$\mathbb A \equiv \{1,\dots,A\}$0

If $\mathbb A \equiv \{1,\dots,A\}$1 is a candidate value function, the choice-specific value matrix $\mathbb A \equiv \{1,\dots,A\}$2 has column $\mathbb A \equiv \{1,\dots,A\}$3,

$\mathbb A \equiv \{1,\dots,A\}$4

The optimal policy satisfies

$\mathbb A \equiv \{1,\dots,A\}$5

row-wise across states.

The decisive duality result is that, for any fixed policy $\mathbb A \equiv \{1,\dots,A\}$6 and any weighting vector $\mathbb A \equiv \{1,\dots,A\}$7,

$\mathbb A \equiv \{1,\dots,A\}$8

Here the primal object $\mathbb A \equiv \{1,\dots,A\}$9 depends on $g$0 through $g$1, but the dual fixed point $g$2 does not depend on $g$3 once $g$4 and $g$5 are fixed. This is the mechanism by which the Bellman fixed point is “unnested” from the structural estimation problem.

The contrast with standard estimators is explicit. NFXP solves the full Bellman fixed point for each $g$6; CCP and SC still require solving $g$7 for each $g$8; MPEC avoids an inner loop but introduces Bellman equations as nonlinear constraints. UFXP and OUFXP instead precompute a finite collection of $g$9-independent dual fixed points and then optimize a smooth unconstrained objective.

3. UFXP construction and the compressed moment system

OUFXP is built on UFXP. The starting point is the dynamic first-order condition evaluated at pre-estimated conditional choice probabilities $\mathbb R^A$0: $\mathbb R^A$1 This is a system of $\mathbb R^A$2 moments. UFXP compresses it with random projections. Let $\mathbb R^A$3, $\mathbb R^A$4, be independent $\mathbb R^A$5 Gaussian random matrices. The UFXP objective is

$\mathbb R^A$6

and the estimator is

$\mathbb R^A$7

The projection residual can be expanded as a linear functional of $\mathbb R^A$8: $\mathbb R^A$9 where

$a$0

Applying the dual identity with $a$1, let $a$2 solve

$a$3

Then

$a$4

and the objective becomes

$a$5

At this point no value function appears. UFXP therefore requires only $a$6 fixed points, $a$7, solved once and fully parallelized. Thereafter, objective evaluation, gradients, and Hessians involve only $a$8, the precomputed $a$9, and derivatives of $u_t = e_a + u_a^\theta(x),$0 with respect to $u_t = e_a + u_a^\theta(x),$1. The paper states that $u_t = e_a + u_a^\theta(x),$2, although in practice $u_t = e_a + u_a^\theta(x),$3 can be $u_t = e_a + u_a^\theta(x),$4 or slightly larger (Oguz et al., 9 Apr 2026).

The theoretical properties are standard large-sample ones under compactness, smoothness, regularity of $u_t = e_a + u_a^\theta(x),$5, injectivity of $u_t = e_a + u_a^\theta(x),$6, a full-rank Jacobian condition, and random-projection conditions on the $u_t = e_a + u_a^\theta(x),$7. With probability one over the projections, UFXP is consistent and asymptotically normal.

4. OUFXP, optimal weighting, and efficiency

OUFXP is the optimal-weighted version of UFXP. The “optimal” qualifier does not refer to worst-case fixed-point complexity; it refers to the variance-minimizing choice of projection weights in a GMM-type asymptotic variance formula (Oguz et al., 9 Apr 2026).

For each state $u_t = e_a + u_a^\theta(x),$8, define

$u_t = e_a + u_a^\theta(x),$9

the covariance of the sample choice-probability vector, and

$u_a^\theta(x)$0

Let the sensitivity of the moment condition to $u_a^\theta(x)$1 be

$u_a^\theta(x)$2

If $u_a^\theta(x)$3 is the limiting share of observations in state $u_a^\theta(x)$4, the variance-minimizing weight is

$u_a^\theta(x)$5

Because $u_a^\theta(x)$6 depends on the unknown parameter, OUFXP is implemented as a two-step estimator. First compute $u_a^\theta(x)$7 using random projections. Then estimate the optimal weights at $u_a^\theta(x)$8, construct the corresponding projection matrices $u_a^\theta(x)$9, recompute the associated dual fixed points, and solve

$\theta \in \Theta \subset \mathbb R^t$0

The asymptotic result is explicit: OUFXP is as asymptotically efficient as maximum likelihood. The paper states

$\theta \in \Theta \subset \mathbb R^t$1

so that $\theta \in \Theta \subset \mathbb R^t$2. In the special case of linearly parameterized utility, $\theta \in \Theta \subset \mathbb R^t$3, OUFXP simplifies to a closed-form estimator.

A useful conceptual distinction follows. UFXP is a randomized, consistent estimator based on compressed first-order conditions. OUFXP replaces arbitrary projections by optimal state-wise weights and thereby attains the efficiency bound associated with the pseudo-maximum-likelihood benchmark conditional on pre-estimated transitions.

5. Computational profile and neural-network estimation

The methodological motivation for OUFXP is the estimation of flexible utility functions, including feed-forward neural networks, inside infinite-horizon DDC models. In this setting $\theta \in \Theta \subset \mathbb R^t$4 may contain hundreds or thousands of parameters, the objective is non-convex, and repeated Bellman solves inside backpropagation would make NFXP- or CCP-type procedures impractical (Oguz et al., 9 Apr 2026).

The computational comparison in the paper uses two measures. “Workload” is the number of fixed points solved per optimization run; “span” is the number of sequential batches of fixed points under maximum parallelization. In the 540-state toy inventory model with neural networks, UFXP has workload $\theta \in \Theta \subset \mathbb R^t$5 and span $\theta \in \Theta \subset \mathbb R^t$6; OUFXP has workload about $\theta \in \Theta \subset \mathbb R^t$7–$\theta \in \Theta \subset \mathbb R^t$8 and span $\theta \in \Theta \subset \mathbb R^t$9; NFXP has workload of order $x \in \mathbb X \equiv \{1,\dots,X\}$00 to $x \in \mathbb X \equiv \{1,\dots,X\}$01; CCP has workload of order $x \in \mathbb X \equiv \{1,\dots,X\}$02 to $x \in \mathbb X \equiv \{1,\dots,X\}$03; SC is similar to CCP; MPEC uses no fixed points but a large constrained optimization problem.

Median timings under a single starting value illustrate the difference:

Setting	Nested estimators	UFXP / OUFXP
540 states, $x \in \mathbb X \equiv \{1,\dots,X\}$04	$x \in \mathbb X \equiv \{1,\dots,X\}$05 hours	UFXP $x \in \mathbb X \equiv \{1,\dots,X\}$06 hours; OUFXP $x \in \mathbb X \equiv \{1,\dots,X\}$07 hours
5400 states, $x \in \mathbb X \equiv \{1,\dots,X\}$08	$x \in \mathbb X \equiv \{1,\dots,X\}$09 hours	UFXP $x \in \mathbb X \equiv \{1,\dots,X\}$10 hours; OUFXP $x \in \mathbb X \equiv \{1,\dots,X\}$11 hours

The multi-start property is more consequential than the single-start timings. Because the dual fixed points are solved once and reused, $x \in \mathbb X \equiv \{1,\dots,X\}$12 starting values increase total cost by about $x \in \mathbb X \equiv \{1,\dots,X\}$13, not $x \in \mathbb X \equiv \{1,\dots,X\}$14. In the 5400-state softplus, L-BFGS configuration, UFXP requires $x \in \mathbb X \equiv \{1,\dots,X\}$15 seconds for the dual fixed points and $x \in \mathbb X \equiv \{1,\dots,X\}$16 seconds per optimization start, so the total cost is $x \in \mathbb X \equiv \{1,\dots,X\}$17 seconds, whereas NFXP requires about $x \in \mathbb X \equiv \{1,\dots,X\}$18 seconds per start.

The empirical sections use this computational advantage to fit neural-network utilities. In the toy inventory model, UFXP, OUFXP, NFXP, CCP, and SC all recover the holding-cost function very well in the 540-state design when enough computation is available, with functional $x \in \mathbb X \equiv \{1,\dots,X\}$19. MPEC performs poorly, with $x \in \mathbb X \equiv \{1,\dots,X\}$20, and often fails to satisfy Bellman constraints. In the 5400-state design, only UFXP and OUFXP succeed in all runs and scale acceptably.

In the empirical multi-echelon detergent supply-chain application, the data comprise $x \in \mathbb X \equiv \{1,\dots,X\}$21 products, $x \in \mathbb X \equiv \{1,\dots,X\}$22 stores, daily observations from $x \in \mathbb X \equiv \{1,\dots,X\}$23 to $x \in \mathbb X \equiv \{1,\dots,X\}$24, and more than $x \in \mathbb X \equiv \{1,\dots,X\}$25 million observations. The paper uses UFXP with $x \in \mathbb X \equiv \{1,\dots,X\}$26 and $x \in \mathbb X \equiv \{1,\dots,X\}$27 random starts. Dual fixed points take $x \in \mathbb X \equiv \{1,\dots,X\}$28 minutes; each optimization run takes $x \in \mathbb X \equiv \{1,\dots,X\}$29 minutes on average. Of the $x \in \mathbb X \equiv \{1,\dots,X\}$30 runs, $x \in \mathbb X \equiv \{1,\dots,X\}$31 yield a UFXP objective within $x \in \mathbb X \equiv \{1,\dots,X\}$32 of the best, producing an ensemble of plausible holding-cost functions. Across that ensemble, the estimated shortage value is about $x \in \mathbb X \equiv \{1,\dots,X\}$33 with standard deviation $x \in \mathbb X \equiv \{1,\dots,X\}$34, and the fixed ordering cost is about $x \in \mathbb X \equiv \{1,\dots,X\}$35 with standard deviation $x \in \mathbb X \equiv \{1,\dots,X\}$36. The estimated holding cost is highly nonlinear in inventory, depends strongly on demand state, depends weakly on congestion, and is neither convex nor concave.

6. Scope, limitations, and relation to other fixed-point literatures

OUFXP is specific to structural estimation in DDC models. Its fixed point is Bellman’s equation, and its main contribution is econometric and computational: it transforms a nested estimation problem into an unconstrained optimization problem after pre-computing dual fixed points (Oguz et al., 9 Apr 2026).

Several limitations are explicit. The theory treats transition matrices $x \in \mathbb X \equiv \{1,\dots,X\}$37 as known after pre-estimation, so OUFXP is efficient conditional on that practice. The method depends on the quality of the reduced-form conditional choice probability estimate $x \in \mathbb X \equiv \{1,\dots,X\}$38. Although much more scalable than NFXP or CCP, the dual fixed points still scale with the number of observed states $x \in \mathbb X \equiv \{1,\dots,X\}$39. OUFXP also requires a second-stage construction of optimal weights that uses derivatives of value functions with respect to $x \in \mathbb X \equiv \{1,\dots,X\}$40; these calculations are not nested in the sense of NFXP, but they make OUFXP more complex than UFXP.

The term should also be distinguished from other “fixed-point” usages. It is unrelated to the Yoder–Low–Chuang optimal fixed-point quantum search algorithm implemented via bang-bang control in NMR, where “optimal fixed-point” refers to monotonic amplitude amplification with query complexity $x \in \mathbb X \equiv \{1,\dots,X\}$41 rather than to structural estimation (Bhole et al., 2015). It is also distinct from the optimal-transport metric framework for universal bounds on Krasnosel’skii–Mann iterations, where fixed-point analysis is expressed through recursively defined metrics $x \in \mathbb X \equiv \{1,\dots,X\}$42 and residual bounds $x \in \mathbb X \equiv \{1,\dots,X\}$43 (Bravo et al., 2021). Likewise, it differs from the black-box complexity literature on optimal accelerated fixed-point iterations for nonexpansive or contractive operators, in which Halpern-type and proximal-point schemes are shown to be exactly optimal for worst-case residual decay (Park et al., 2022).

A common misconception is therefore to read “optimal” in OUFXP as a statement about universal fixed-point iteration complexity. In the DDC setting, the paper’s meaning is narrower and more econometric: OUFXP is the optimal-weighted, asymptotically efficient refinement of UFXP. A plausible implication is that the acronym marks a convergence of two ideas—fixed-point structure from dynamic programming and efficient moment weighting from semiparametric estimation—rather than an attempt to unify the distinct fixed-point literatures represented by quantum search, nonexpansive operator theory, and KM iteration analysis.