StoBasket: Joint Upsell and Cross-Sell Optimization for Contribution Margin

Xin Heng

cs.AIOct 16, 2026

StoBasket: Joint Upsell and Cross-Sell Optimization for Contribution Margin

Authors: Xin Heng

Organizations: Tote AI United States

Abstract

Retail recommenders rank relevance, but retailers earn contribution margin: an upsell earns the margin difference over the item it replaces, whereas a cross-sell earns its full margin. We introduce joint stochastic basket editing, a formulation in which one policy selects replacement and addition offers for an existing basket, values them by expected incremental contribution margin, and couples them through displayed-offer, accepted-offer, and joint-acceptance response channels. To our knowledge, it is the first formulation in which upsell and cross-sell responses are coupled; prior separable treatments are its zero-coupling special case. In real-data-calibrated simulations on five public basket datasets, StoBasket captures essentially all attainable offer margin: within 0.5% in every default-price setting and within 0.7% when the true response lies outside its model family. Correct economic accounting is the broadest gain: margin-weighted popularity leaves up to 11% of attainable margin unrealized, and a published next-basket recommender leaves 11-52%. Joint selection adds 10.2% of attainable margin on Instacart, where premium tiers and add-ons are strongly linked. When offers pay only together, the best heuristics lose 64% while exact search stays within 0.5%. For a structured subclass, we characterize when staged policies fail and give an exact optimizer that solves every tested real basket in 0.13 ms median and at most 111 ms. Profit saturates long before prediction accuracy: a correctly structured predictor trained on 5,000 randomized sessions comes within 0.06% of perfect prediction, while a signal-free one loses 38%. The same estimates yield a decision-aware market-basket analysis whose cost does not grow with association order.

Figures & tables

Figure 1. Illustration only (constructed counterexample; not calibrated retail data). A deliberately chosen response law makes each singleton recommendation lose money while the joint set earns

+\

2.60$ . The figure teaches the staging trap; it does not claim that typical grocery upgrades are unprofitable alone.

Symbol	Meaning (example)
$B$	Basket at decision time (regular patties, regular buns)
$s$	Transaction context: prices, margins, inventory, store, time
$A$	Offer set, or policy (premium patties and cheddar)
$D$	Promotion terms; $D=\varnothing$ means ordinary prices
$\mathcal{A}(s;D)$	Permitted offer sets under budgets, conflicts, and stock
$Y$	Random shopper response: which offers are accepted

Table 1. Notation, with the running burger example.

Strategy	How it chooses offers	Strengths and weaknesses
1. Independent	Picks upsells as if no cross-sell were shown (maximizes $U(x)$ ) and cross-sells as if no upsell were shown (maximizes $\sum_{k\in C}\alpha_{k}$ ).	$+$ Fastest; two small separate problems; optimal when offers do not interact. $-$ Ignores all cross-action coupling; largest gap when coupling is strong.
2. Upsell-first	Keeps the independent upsells, then picks the best cross-sells given their summary $z(x)$ .	$+$ Cheap; captures the effect of upsells on cross-sells. $-$ Upsells are chosen blind to cross-sells, and a poor first stage is never revised.
3. Cross-sell-first	Keeps the independent cross-sells, then picks the best upsells given them.	$+$ Cheap; picks upsells that raise the value of the chosen cross-sells. $-$ Cross-sells are chosen blind to upsells; no revision.
4. Alternating best response	Starts from the better of strategies 2 and 3, then alternates $x\mid C$ and $C\mid x$ until neither step improves.	$+$ Fast and usually near-exact; within 0.14% of exact on Carbo-Loading. $-$ Stops at a local optimum; can stay at “show nothing” in the trap; no worst-case guarantee.
5. Greedy	Repeatedly adds the single offer with the largest positive gain until no offer helps.	$+$ Simple; needs no structure in the objective. $-$ Myopic: requires a profitable first step, so it fails in the trap and when pairs interact.
Exact ( StoBasket )	Enumerates candidate cross-sell sets by score geometry and optimizes upsells by tree-decomposition dynamic programming (Proposition 4.1 ).	$+$ Certified global optimum; captures profitable sets with unattractive parts. $-$ Needs fixed $r$ and bounded treewidth; more compute (0.13 ms median, 111 ms maximum on real baskets).

Table 2. The five heuristic strategies and the exact optimizer, all solving Eq. ( 3 ) with the same objective and sub-solvers. “Trap” refers to bundle-only profitability (Figure 1 ; Proposition 4.2 ).

Figure 2. Mean expected-margin gap between the exact joint optimum and each of the five heuristic strategies of Table 2 as the multiplier

\gamma

scales all generated upsell-to-cross-sell response terms. The same 60 synthetic instances are evaluated at each multiplier;

\gamma=0

removes cross-action coupling and

\gamma=1

restores the generated reference strength. All staged strategies are exact at zero coupling.

Market	Economic-policy gain	Joint-selection gain	Exact-search premium
	vs. margin-pop. (%)	vs. independent (%)	vs. portfolio (%)
Complete Journey	2.4	0.17	$\leq 0.09$
Ta Feng	1.5	$\approx 0$	$\leq 0.09$
Instacart	11.4	10.2	63.8 †
Online Retail II	1.0	0.34	$\approx 0$
Carbo-Loading	4.7	0.73	$\leq 0.14$

Table 3. Economic scorecard across five markets (S1, the in-family response truth of Section 6.1 ; percentage of oracle offer margin). Economic-policy gain is the learned exact policy over margin-weighted popularity in the base regime; it combines response learning and decision optimization rather than isolating accounting alone. Joint-selection gain uses EB shrinkage where available. Exact-search premium is the largest portfolio gap in the tested display-friction map through

\tau=3

for Complete Journey/Ta Feng and

\tau=2

for Instacart/Carbo-Loading; Online Retail II has only the base comparison. † : 63.8% uses EB shrinkage and default imputed prices; alternative price assumptions give 2–89%.

Data	Sim.	Oracle	Exact	Indep.	Upg.-first	Add-first	Greedy	Altern.	Margin-pop.	Assoc.
Complete Journey	S1	536	536	+3.5	+1.1	+2.6	+0.4	+0.0 †	+12.6	+534.2
	S2	509	507	+5.8	+1.3	+5.8	+2.1	+0.1	+13.2	+493.6
	S3	373	313	+2.4	+0.5	+1.7	+0.2	+0.0 †	+7.9	+308.3
Ta Feng	S1	20576	20570	+71.3	+25.8	+46.2	+10.0	+0.3	+300.7	+10746.4
	S2	19523	19498	+104.0	+52.9	+56.7	+21.7	+1.0	+296.2	+10651.8
	S3	14346	12069	+46.0	+10.0	+35.5	+4.7	-0.2 -	+170.2	+6371.9

Table 4. Exact joint policy minus each baseline, in expected incremental margin per 1,000 baskets (USD; NT$ for Ta Feng; Instacart prices imputed). Oracle and Exact are absolute values. All model-based policies use the same fitted model. † : 95% interval contains zero. - : baseline significantly better.

Figure 3. Exact joint policy minus each baseline as the receipt-calibrated cross-action channel effects are scaled by

\gamma

(in-family truth S1, Section 6.1 ), as a percentage of the oracle’s expected margin. Here

\gamma=0

removes the calibrated spillovers,

\gamma=0.5

halves them,

\gamma=1

is the reference calibration, and

\gamma=2

doubles them as a stress test.

Figure 4. Where exact optimization matters. Left: exact minus alternating best response (bars) and a post-hoc best-of-heuristics portfolio (ticks), as a percentage of oracle offer margin. Right: exact minus the portfolio under separate budgets as display cost increases;

\tau

is display cost divided by mean standalone add-on value and

\gamma

scales cross-action coupling. Weakly coupled Complete Journey and Ta Feng remain easy. Instacart enters a bundle-only-profitability regime in which all tested heuristics fail.

		StoBasket		TIFU-KNN		Margin-	Assoc.
Dataset	$\omega$	+ TIFU	anonymous	$\times$ margin	relevance	popularity	rules
Complete Journey	0	99.9	99.9	88.7	$-$ 10.8	97.6	$-$ 2.3
	0.5	99.7	99.6	88.8	$-$ 10.1	97.3	$-$ 1.7
	1	99.3	98.9	88.4	$-$ 9.2	96.7	$-$ 0.4
Ta Feng	0	100.0	100.0	48.2	37.0	98.5	47.3
	0.5	99.9	99.8	49.3	37.9	98.4	47.7

Table 5. Against prior art on shared benchmarks (E20; percentage of oracle offer margin, 3,000 test baskets per row). TIFU-KNN ( Hu et al., 2020 ) is run from the reference code of the next-basket benchmark study ( Li et al., 2023 ) with its published hyperparameters.

\omega

scales a personal-habit effect on add-on purchase rates;

\omega=0

is the reference in-family S1 truth (Section 6.1 ). StoBasket + TIFU adds the TIFU-KNN forecast as one regressor in the same response model; relevance fills every offer slot with TIFU-KNN’s top-scored items;

\times

margin reranks them by score times margin gain. Bold: best at the shown precision.

Figure 5. How much prediction quality matters (Instacart, exact optimizer held fixed). Left: as randomized logged sessions grow, prediction error in policy values (squares, dashed) falls slowly while decision regret (circles, solid) falls much faster. Right: decision regret for learned predictors versus a predictor fit to 50,000 sessions with shuffled outcomes (hatched). Percentages are of oracle offer margin; both axes in the left panel and the right panel’s vertical axis are logarithmic.

Prior work	What it already does	What StoBasket adds
Profit-aware recommendation ( De Biasio et al., 2024b ; Louca et al., 2019 ; De Biasio et al., 2024a )	Scores or trains rankings of individual products using provider profit, relevance, or a tradeoff between them.	Optimizes a set of edits to a basket already in progress. An upsell replaces an incumbent and earns only the margin difference; a cross-sell adds an item and earns incremental margin; their values can interact.
Association rules and market-basket analysis ( Agrawal et al., 1993 ; Singh et al., 2006 )	Finds products that co-occur, reports support/confidence/lift, and may rank upsell or cross-sell candidates by association, acceptance, revenue, or margin.	Uses the association report as an interpretable diagnostic, then adds signed premium-to-complement effects, shrinkage/placebos, action type, contribution margin, and a feasible joint offer decision.
Upgraded substitutes and co-purchases ( Ban et al., 2024 )	Places substitute upgrades and co-purchases in one business problem; under the studied response assumptions, the component optimizations separate.	Makes the upsell decision change cross-sell response and value through displayed, accepted, and jointly accepted channels; proves when separation is safe and when it can fail arbitrarily.
Next-basket recommendation ( Hu et al., 2020 ; Li et al., 2023 )	Forecasts which items a shopper will buy next from purchase history, with strong reproducible results on the dunnhumby, Ta Feng, and Instacart benchmarks.	Turns that forecast into margin by using it as a response feature, then deciding which replacement and addition offers to serve. On the same benchmarks it adds up to 6% as an input, whereas serving the forecast directly leaves 11–52% unrealized even after margin reranking (Table 5 ).
Exposure modeling ( Sato et al., 2020 )	Separates whether a user becomes aware of an item from whether the user prefers or purchases it, and models recommendation influence on exposure.	Places displayed-offer, accepted-offer, and joint-acceptance effects inside one expected-margin policy over replacements and additions.
Cross-category basket retrieval ( Acar, 2026 )	Retrieves complementary products for a checkout-adjacent recommendation surface and debiases transactional complement evidence.	Takes retrieved candidates as inputs, values displaced and incremental margin, and jointly chooses which replacement and addition offers to serve.

Table 6. How StoBasket advances representative foundations. Prior work supplies essential capabilities; StoBasket connects them in a joint post-basket decision over interacting replacement and addition offers.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Test	Instances	Failures
Forest plus pair moments, $r=2$	320	0
Nonempty interaction graph among those	289	–
Moment expansion vs. tree-joint law	120	0
Tree-decomposition DP vs. brute force	300	0
Treewidth solver, rational factors, $r\leq 2$	240	0
of which cyclic upgrade graphs	216	–

Appendix

Table 7. Reproducible finite-instance verification; exactness means agreement with exhaustive search on the enumerated instances, not a proof of all-instance correctness.

	Complete Journey	Ta Feng	Instacart
Source	84.51° via completejourney , CC0	Taiwanese supermarket, 2000–01	2017 public release
Train / test baskets	102,604 / 37,434	88,527 / 31,051	400,000 (sample) / 131,209
Split	weeks 1–39 / 40–53	Nov–Jan / Feb	prior / final orders
Margin	assumed rates, 3 scenarios	real cost	imputed prices
Context	store, weekday, hour	residence area, weekday	weekday, hour
Premium tier	national vs. private label	price above subclass median	organic

Appendix

Table 8. Datasets. Splits are chronological; customer identifiers are not used.

Data	Setting	Oracle	Exact regret	$-$ Indep.	$-$ Upg.-first	$-$ Altern.
Complete Journey	$\gamma=0.0$	528	0.2	-0.0	-0.0	-0.0
Complete Journey	$\gamma=0.5$	531	0.2	1.1	0.3	0.0
Complete Journey	$\gamma=2.0$	550	0.7	9.1	3.3	0.1
Ta Feng	$\gamma=0.0$	20022	4.9	-1.0	-0.9	0.0
Ta Feng	$\gamma=0.5$	20283	4.3	25.7	8.7	0.0
Ta Feng	$\gamma=2.0$	21217	12.4	185.3	76.1	2.0

Appendix

Table 9. Secondary experiments (S1 unless stated): oracle expected margin and exact-policy regret per 1,000 baskets, and the exact policy’s margin over three baselines. Currency as in Table 4 .

Data	Truth	Oracle	Exact regret	$-$ Indep.	$-$ Upg.-first	$-$ Altern.	$-$ Portfolio	$-$ Margin-pop.
Online Retail II	S1, $\gamma=1$	6615	4.0	22.3	5.5	0.0	0.0	68.2
Online Retail II	S2, $\gamma=1$	5992	6.3	28.3	9.6	0.2	0.2	91.6
Online Retail II	S3, $\gamma=1$	4663	893.9	31.8	3.7	0.0	-0.0	56.6
Online Retail II	S1, $\gamma=0$	6575	4.2	-0.2	-0.1	0.0	0.0	45.1
Online Retail II	S1, $\gamma=2$	6693	5.3	47.5	13.6	0.2	0.2	96.5
Complete Journey	S1, Dominick’s margins	439	0.3	2.5	0.6	0.0	0.0	9.8

Appendix

Table 10. External-data checks: Online Retail II (GBP per 1,000 invoices), Complete Journey under Dominick’s margin rates, and Carbo-Loading (USD per 1,000 baskets). Exact regret and exact minus each baseline are absolute amounts.

Data	Setting	Oracle	Upsell share	Coupling value	Exact $-$ independent
		per 1,000	% of oracle	per 1,000 (% of oracle)	% of oracle	% of coupling value
Complete Journey	S1	536	88	11.2 (2)	0.65	31
Complete Journey	S2	509	84	28.4 (6)	1.14	20
Ta Feng	S1	20576	88	600.8 (3)	0.35	12
Ta Feng	S2	19523	85	1144.2 (6)	0.53	9
Instacart	S1	474	33	159.2 (34)	11.28	34

Appendix

Table 11. Market opportunity and method effectiveness. Coupling value is the oracle policy’s value under the stated channels minus the value of that same policy with

\gamma=0

; it overlaps add-on value and is not an additional additive category. The final column normalizes the exact-minus-independent gain by this coupling value.

Data	Setting	Oracle	Exact regret	$-$ Indep.	$-$ Greedy	$-$ Altern.	$-$ Portfolio	Differs
		per 1,000	% of oracle margin					% baskets
Complete Journey	Separate budgets (batch 1)	536	0.06	0.65	0.08	0.00	–	0.7
	Display cost $\tau=0.5$	525	0.06	0.66	0.08	0.01	0.01	0.9
	Display cost $\tau=1$	513	0.07	0.70	0.06	0.00	0.00	0.5
	Display cost $\tau=2$	492	0.07	0.89	0.05	0.01	0.01	0.8
	Budgets $L=K=3$	582	0.06	0.57	0.04	0.00	0.00	0.5

Appendix

Table 12. Where exact optimization matters (S1). “Portfolio” is the post-hoc best-of-heuristics baseline evaluated by the fitted model; “Differs” is the share of baskets on which exact and alternating choose different policies. Exact regret and exact-minus-baseline columns are percentages of oracle offer margin.

	Exact $-$ portfolio, % of oracle				Exact regret, % of oracle
Price assumption	$\tau=0$	$\tau=1$	$\tau=1.5$	$\tau=2$	$\tau=0$	$\tau=1$	$\tau=1.5$	$\tau=2$
Organic $+25\%$ (main)	0.08	0.44	16.13	65.12	0.02	0.05	0.14	0.50
Organic $+10\%$	0.11	3.02	70.58	89.31	0.02	0.09	0.54	3.31
Organic $+40\%$	0.04	0.13	1.53	5.78	0.03	0.04	0.09	0.16
Item price noise (s.d. 0.3)	0.03	0.09	0.69	2.00	0.04	0.06	0.08	0.14
Margin rate 18%	0.05	0.44	16.13	65.12	0.02	0.05	0.14	0.50

Appendix

Table 13. Instacart price sensitivity (E16, S1,

\gamma=1

). Entries are percentages of oracle offer margin. The portfolio is the post-hoc best fitted-value heuristic. Margin-rate arms rescale both margins and the display-cost unit and are therefore decision-invariant.

Data	Setting	Exact $-$ independent, per 1,000		Exact $-$ portfolio/altern., % of oracle
		fixed shrinkage	EB shrinkage	fixed	EB
Complete Journey	S1	3.5	0.9	0.00	0.00
Complete Journey	S2	5.8	2.2	0.01	0.00
Complete Journey	S3	2.4	0.9	0.01	-0.00
Complete Journey	S1, $\gamma=0.5$	1.1	0.2	0.00	-0.00
Complete Journey	S1, $\gamma=2$	9.1	2.8	0.02	0.00

Appendix

Table 14. Fixed versus empirical-Bayes coupling shrinkage (E17). Absolute exact-minus-independent values use dataset currency per 1,000 baskets; portfolio/alternating gaps are percentages of oracle offer margin.

Bundle rebate: loss of rebate-blind exact, per 1,000 baskets
	Complete Journey	Ta Feng	Instacart
$\rho=0.25$	0.0	0.8	1.5
$\rho=0.5$	0.1	3.3	7.2
$\rho=1.0$	0.6	19.9	34.0
Stress test: exact regret, % of oracle (base $\to$ rich model)
cannibalization + fatigue	17.5 $\to$ 0.10	17.3 $\to$ 0.08	10.6 $\to$ 0.08

Appendix

Table 15. Robustness tests. Rebate losses are in dataset currency per 1,000 baskets; stress-test values and log-size regret are percentages of oracle offer margin.

Data	Median $G$	90th pct. $G$	Cyclic	Cyclic, $G\geq 8$	Width $\leq 2$
Complete Journey	6	22	50%	94%	66%
Ta Feng	4	12	22%	68%	93%
Instacart	4	8	68%	100%	54%

Appendix

Table 16. Structure of real upsell interaction graphs (

G

: items with an eligible upgrade).

Data	$L$	$\tau$	$G\geq 8$	DP = brute	DP time (median / p99)	DP faster	$-$ Greedy	$-$ Altern.	$-$ Portfolio
			% baskets	checked		% baskets	% of oracle
Complete Journey	2	0	34	887/887	3.2 ms / 33.2 s	0.5	0.10	0.00	0.00
Complete Journey	4	0	34	887/887	3.1 ms / 38.3 s	27.1	0.09	-0.00	-0.00
Complete Journey	4	1	34	887/887	3.0 ms / 37.8 s	27.5	0.08	0.00	-0.01
Ta Feng	2	0	23	900/900	1.8 ms / 0.5 s	0.6	0.09	0.01	0.01
Ta Feng	4	0	23	900/900	2.1 ms / 0.5 s	36.4	0.07	0.00	0.00

Appendix

Table 17. Tree-decomposition dynamic programming on real interaction graphs (E15). Runtime is median/p99; baseline gaps are percentages of oracle offer margin. “DP faster” compares measured runtimes on each basket.

Budget	Method	Median	95th pct.	99th pct.	Worst	$<$ 100 ms
$L=2$	Always enumerate	0.4 ms	10.2 ms	16.0 ms	29.6 ms	100.0%
	Always tree-decomposition DP	2.5 ms	1.1 s	57.8 s	132.6 s	87.9%
	Hybrid (predicted faster method)	0.4 ms	10.2 ms	16.0 ms	29.6 ms	100.0%
	Best of both per basket	0.4 ms	10.2 ms	16.0 ms	29.6 ms	100.0%
	Vectorized enumeration (E15c)	0.1 ms	0.2 ms	0.2 ms	0.4 ms	100.0%
$L=4$	Always enumerate	0.9 ms	1.1 s	2.0 s	3.0 s	85.8%

Appendix

Table 18. Per-basket solve time of exact methods on held-out seeds 1–2 (E15b, E15c). The scalar hybrid predicts enumeration or DP runtime from seed 0; “Best of both” chooses with hindsight; DP instances above the width cap are unfinished, so its worst time is over completed instances. Vectorized enumeration evaluates the same policy set and objective with array operations and finishes every test basket.

Predictor	Value RMSE	Oracle-policy	Margin regret	Regret per 1,000
	% oracle	agreement	% oracle	baskets
Oracle truth	0.00	100.0%	0.000	0.00
Learned, $N=100$	17.20	54.8%	4.627	21.31
Learned, $N=250$	11.27	75.3%	1.106	5.09
Learned, $N=500$	6.34	83.9%	0.371	1.71
Learned, $N=1$ k	5.09	84.8%	0.299	1.38

Appendix

Table 19. Prediction quality with the exact optimizer held fixed (E19, Instacart S1). Value RMSE is the error in predicted policy values; oracle-policy agreement is the share of baskets on which the chosen offer set equals the oracle’s.

Premium anchor	Cross-sell receipt cell	Premium	Regular	Policy effect	Relative rate
Pancake Mixes	syrups: Aunt Jemima	11.3%	3.3%	+8.0 pp	3.43 $\times$
Pasta Sauce	pasta: Barilla	8.2%	4.8%	+3.5 pp	1.73 $\times$
Pancake Mixes	syrups: Mrs Butterworth	4.7%	1.7%	+2.9 pp	2.71 $\times$
Pancake Mixes	syrups: Private Label	9.4%	28.4%	-15.0 pp	0.33 $\times$
Pasta Sauce	pasta: Private Label	24.7%	37.5%	-12.8 pp	0.66 $\times$
Pancake Mixes	pasta: Private Label	8.5%	14.1%	-5.6 pp	0.60 $\times$

Appendix

Table 20. Post-hoc decision-aware market-basket diagnostic from Carbo-Loading training receipts. “Premium” and “Regular” are conditional co-purchase rates given the anchor tier; policy effect is the empirical-Bayes-shrunk difference used by the calibrated simulator, clipped to

\pm 15

percentage points. Relative rate is premium divided by regular. Like all market-basket statistics, these are observational associations.

Market	Quantity at $10^{6}$ baskets	$q=2$	$q=3$	$q=4$
Instacart	Classical candidate itemsets	$1.1\times 10^{9}$	$1.7\times 10^{13}$	$2.0\times 10^{17}$
	Classical counting updates	$7.4\times 10^{7}$	$4.9\times 10^{8}$	$3.1\times 10^{9}$
	Typed contrasts estimated	17,956	17,956	17,956
	Typed counting updates	$7.6\times 10^{7}$	$7.6\times 10^{7}$	$7.6\times 10^{7}$
Complete Journey	Classical candidate itemsets	$2.3\times 10^{9}$	$5.1\times 10^{13}$	$8.5\times 10^{17}$
	Classical counting updates	$1.3\times 10^{8}$	$1.7\times 10^{9}$	$2.4\times 10^{10}$

Appendix

Table 21. Classical itemset mining versus typed contrasts at

10^{6}

baskets, from each market’s observed catalog and basket sizes. A counting update adds one to the tally of one combination found in one basket; classical mining performs

\binom{b_{n}}{q}

updates for a basket of

b_{n}

items, typed contrasts

a_{n}m_{n}

regardless of

q

. Candidate itemsets do not depend on the number of baskets; counting updates do.

Data	Setting	Comparison	Gap per 1,000	Baskets/arm (offer log)	Baskets/arm (till)
Complete Journey	default	vs independent	3.9	1.4M	67.1M
Complete Journey	default	vs upgrade-first	1.4	11.3M	550.2M
Complete Journey	default	vs alternating	0.0	30943.0M	1497174.2M
Complete Journey	default	vs no offers	530.5	37	3,510
Complete Journey	shared budget $T=3$	vs independent	121.1	1,295	69k
Complete Journey	shared budget $T=3$	vs upgrade-first	9.1	266k	12.2M

Appendix

Table 22. Two-arm field-test power under calibrated effects: baskets per arm for 80% power at 5% two-sided size. “Offer log” uses offer-attributable margin from logged exposure and acceptance; “till” uses total basket margin. M denotes million and k thousand.

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CASE: Cadence-Aware Set Encoding for Large-Scale Next Basket Repurchase Recommendation

Repurchase behavior is a primary signal in large-scale retail recommendation, particularly in categories with frequent replenishment: many items in a user's next basket were previously purchased, and their timing follows stable, item-specific cadences. Yet most next basket repurchase recommendation models represent history as a sequence of discrete basket events indexed by visit order, which cannot explicitly model elapsed calendar time or update item rankings as days pass between purchases. We present CASE (Cadence-Aware Set Encoding) for next basket repurchase recommendation, which decouples item-level cadence learning from cross-item interaction, enabling explicit calendar-time modeling while remaining production-scalable. CASE represents each item's purchase history as a calendar-time signal over a fixed horizon, applies shared multi-scale temporal convolutions to capture recurring rhythms, and uses induced set attention to model cross-item dependencies with sub-quadratic complexity, allowing efficient batch inference at scale. Across three public benchmarks and a proprietary dataset, CASE consistently improves precision, recall, and NDCG at multiple cutoffs compared to strong next basket recommendation baselines. In a production-scale evaluation with tens of millions of users and a large item catalog, CASE achieves up to 8.6% relative precision lift and 9.9% relative recall lift at top-5, showing that scalable cadence-aware modeling yields measurable gains in both benchmark and industrial settings.

Yanan Cao, Ashish Ranjan, Sinduja Subramaniam +3

Walmart Global Tech, Sunnyvale, CA, USA

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Sep 26, 2026

Learnable Randomization as Commitment Against Adaptive Optimizers

A pricing page can walk the posted price up to the last amount a buyer still accepts, a recommender can hold back a better item for a barely acceptable promoted one, and a classifier can shift its boundary once applicants change their features. The system predicts the response and then picks the menu that serves its own objective, so the surplus above the user's cutoff is taken. Playing the single best action publishes that cutoff, while noise on actions the user would never take throws away payoff and teaches the platform that a worse menu is still acceptable. We study unpredictable near-optimal policies (UNOP), which mix uniformly on near-best actions that remain individually rational. The mixture is a commitment about the response. On a finite price grid, when the best sure-demand price strictly out-earns the randomized band, a seller who already knows the curve posts below the band, and the purchase that occurs is deterministic. Knowing that curve is not the same as predicting the next draw. The mixture can be learned and the optimizer can match its best response, while the user's payoff stays higher because the mixture changes which action is targeted. In pricing and in policy-aware recommendation this leaves more surplus than greedy play when the platform optimizes against the curve and more than one action is acceptable. The gain goes away under quality ranking, a singleton near-optimal set, a wrong utility estimate, or a short-horizon explorer. That is also where mixing should be turned off if the other side is trying to cooperate.

Zihan Deng, Chuanzhi Xu, Xiaozhen Zhong +2

The University of Hong Kong · The University of Sydney · University of Electronic Science and Technology of China +1

Policy Optimization Stackelberg Games Dynamic Pricing

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