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Learning Collective Dynamics with Differentiable Gaussian Representations
Jianxiang Ma, Mingfu Zhang, Xiaocui Yang, Yichen Gao, Junzhao Huang, Yuesong Hou
TL;DR
Aggregate counts combine heterogeneous responses, contact opportunities, and accumulated experience, making it challenging to learn both the population response distribution and the feedback process behind future behavior. DGD jointly learns these components, and its joint behavioral negative log-likelihood is lower than DeepAR-joint across four evaluation windows.
Problem
Aggregate counts combine nonlinear response heterogeneity, contact intensity, and experience-dependent dynamics, raising the question of how to learn the responses and feedback that generate future collective behavior.
Source: §1 · p. 1Method
DGD jointly learns a Gaussian response mixture, contact–behavior aggregation, and feedback recurrence from aggregate prediction errors.
Source: §1 · p. 2Results
Across all four windows, DGD has lower joint behavioral negative log-likelihood than DeepAR-joint, including 1.1232 versus 1.3466 under standard recommendation.
Source: §1 · p. 2Takeaways & Limitations
Across the evaluated windows and controlled experiments, the results support learning population representations and their feedback process from aggregate observations.
Source: §5 · p. 9
Abstract
from arXiv · showhide
Collective responses depend on individual differences, contact opportunities, and accumulated experience. Learning their dynamics from aggregate counts requires connecting a population's response distribution to both current observations and future behavior. We introduce Differentiable Gaussian Dynamics (DGD), which learns this connection through three components: a Gaussian mixture representing heterogeneous response propensities, differentiable aggregation of contact intensity and behavioral probabilities, and feedback recurrence that updates subsequent responses. Reparameterized integration and temporal recurrence let aggregate prediction errors jointly train the distribution, observation functions, and feedback parameters. On four windows from KuaiRand-Pure and Online Retail II, DGD achieves lower joint behavioral negative log-likelihood than a DeepAR adaptation with a joint-behavior head. In Retail 2010, its one-day behavioral-count MAE is 4.71 versus 6.88 for this adaptation. Learning the distribution reduces behavioral negative log-likelihood by 10.82% relative to a fixed Gaussian in KuaiRand's standard-recommendation window; removing feedback dynamics raises joint KL from 0.0340 to 0.2577 in a controlled experiment. These results establish the value of learning population representations and their feedback process from aggregate observations. Code is available at https://github.com/OranAi-Ltd/oransim.
1 Introduction
DGD addresses how aggregate event counts can reveal heterogeneous responses and feedback that shape future collective behavior. It combines a learned Gaussian response distribution, contact–behavior aggregation, and feedback recurrence, and outperforms a DeepAR adaptation on the reported behavioral prediction comparisons.
Motivation
- Aggregate counts combine response heterogeneity, contact intensity, and experience-dependent memory and fatigue, so learning requires linking population state to observations and subsequent state.Nonlinear behavioral probabilities mean populations with the same mean propensity can have different response rates.Source: §1 · p. 1
Approach
- DGD applies representation–observation learning to collective responses: counts across dates and observation units constrain a shared base response distribution, while feedback carries current responses into later predictions.The framing draws on differentiable image synthesis in 3D Gaussian Splatting [8].Source: §1 · p. 2
- DGD uses Gaussian components for latent response propensities, integrates contact intensity with behavioral probabilities, and feeds observed or predicted outcomes into later response states.Reparameterized integration and recurrence let aggregate prediction errors train the representation, observation functions, and feedback parameters jointly.Source: §1 · p. 2
Evaluation
- DGD has lower joint behavioral negative log-likelihood than DeepAR-joint in all four evaluation windows, including 1.1232 versus 1.3466 under standard recommendation.In both retail windows, it also has lower one-day behavioral-count MAE than DeepAR-joint and three historical-distribution baselines.Source: §1 · p. 2
2 Related Work
The related work connects DGD to heterogeneous-response models, aggregate dynamics learning, differentiable Gaussian representations, and population models for scenario evaluation. DGD combines these lines by jointly learning a response distribution and feedback transitions from aggregate observations.
2.1 Heterogeneous Collective Responses
- Mixture choice models integrate conditional responses over continuous preferences or discrete classes, while Lu and Shimizu [10] combine random-coefficient integration, market totals, and count likelihoods for demand estimation.DGD instead uses a latent propensity shared by contact intensity and behavior, with historical outcomes updating the state for subsequent observations.Source: §2.1 · p. 2
2.2 Learning Dynamics from Population Observations
- Aggregate hidden Markov models [3], neural Hawkes models [11], world models, RecSim NG [6], GradABM [7], and differentiable agent-based simulation [12] provide foundations for learning dynamics from population observations.DGD jointly learns population distributions and feedback transitions through daily count and behavioral-pattern losses.Source: §2.2 · p. 2
2.3 Explicit Gaussian Representations
- Prior Gaussian work includes differentiable scene rendering [8], Gaussian world models [14], and ContactGaussian-WM [15]; DGD places Gaussian components in response-propensity space and integrates them into population counts.Response feedback updates the subsequent state.Source: §2.3 · p. 3
2.4 Population Models for Scenario Evaluation
- Prior OranSim work uses outcome predictions and population rollouts for marketing-scenario selection and persistent user histories for joint responses.DGD develops joint learning of response distributions, observation functions, and feedback transitions from aggregate predictive losses.Source: §2.4 · p. 3
3 Differentiable Gaussian Dynamics
DGD represents heterogeneous propensities with a Gaussian mixture, integrates contact intensity and behavior to predict aggregate observations, and updates the state through feedback. Aggregate count and behavioral-pattern losses train the population, observation, and transition components jointly.
3 Differentiable Gaussian Dynamics
- DGD links a population representation, contact–behavior aggregation, and feedback recurrence to predict current and future collective responses from aggregate observations.The prediction task covers creator streams and a whole retail store, with expected arrivals and joint behavior probabilities predicted from history and known future inputs.Source: §3 · p. 3
3.1 Gaussian Response Representation
- A learned Gaussian mixture models differences between and within response types, with mixture weights, means, and covariances shared across dates and translated by feedback shifts.Reparameterized integration nodes move with learned means and covariances, allowing prediction errors to train the representation.Source: §3.1 · p. 3
3.2 Differentiable Contact–Behavior Aggregation
- DGD multiplies contact intensity and behavioral probability at the same propensity before integration, so frequently reached population components can contribute disproportionately to recorded behavior.The normalized pattern intensity gives the probability that a recorded arrival exhibits that pattern, preserving the association between contact and response.Source: §3.2 · pp. 3–4
3.3 Feedback Recurrence
- Feedback recurrence updates response shifts, memory, fatigue, and arrival conditions using observed or predicted counts; learned retention factors control persistence.Memory and fatigue also enter behavioral logits directly, so current experience affects both the distribution and responses at a given propensity.Source: §3.3 · p. 5
3.4 Joint Learning from Aggregate Counts
- Daily arrival counts use a negative-binomial likelihood and conditional pattern frequencies use a multinomial likelihood, with observed- and expected-feedback branches training future updates.The retained computation graph lets current losses train mixture parameters through integration and later losses train feedback parameters through recurrence; the analysis also gives count-likelihood sufficiency and error-propagation results.Source: §3.4 · pp. 5–6
4 Experiments
Experiments evaluate DGD on four real-data windows and controlled tests of representation learning, feedback dynamics, and responses to new product contexts.
4.1 Experimental Setup
- Across four windows, DGD uses two one-dimensional Gaussian components per cohort, seven Gauss–Hermite nodes per component, and jointly learns distribution, observation, and transition parameters.The windows comprise two KuaiRand-Pure exposure settings and two annual Online Retail II windows.Source: §4.1 · p. 6
4.2 Collective Response Prediction
- 1.1232 vs 1.3466 joint behavioral NLL under standard recommendation; DGD has lower joint NLL than DeepAR-joint in all four windows.DGD also has lower Brier scores and cohort-rate errors than recurrent comparators in each evaluated window.Source: §4.2 · p. 6, Table 1 · p. 7, Table 2 · p. 7
- 4.71 and 5.38 invoices/day: DGD’s one-day behavioral-count MAE versus DeepAR-joint’s 6.88 and 5.77 in the two Retail windows.DGD has lower one-day behavioral-count MAE than every listed baseline in both annual windows.Source: §4.2 · p. 7, Table 3 · p. 7
4.3 Learning the Population Representation
- 10.82%: learning DGD’s distribution reduces standard-recommendation NLL from 1.2594 to 1.1232 relative to a fixed Gaussian.Gaussian and matched discrete mixtures both improve predictions; the point representation removes within-cohort heterogeneity.Source: §4.3 · p. 7, Table 4 · p. 8
4.4 Learning Feedback for Future Responses
- 34.08 to 16.47 and 38.32 to 22.76 occurrences/day: recurrence lowers one-day and three-day behavioral-count MAE under random exposure.Both variants retain observed lag summaries, so the comparison evaluates propagating current responses into subsequent state.Source: §4.4 · p. 7, Table 5 · p. 8
- 0.0340 vs 0.2577 mean joint KL: DGD outperforms the no-dynamics variant in response recovery under a controlled generator.The generator has behavioral memory and response shifts; recurrence-based sequence losses recover cross-day response distributions from cohort counts.Source: §4.4 · p. 8, Table 6 · p. 8
4.5 Response Prediction under New Attributes and Prices
- 0.001994 to 0.001207 under the Gaussian generator and 0.020016 to 0.002542 under the discrete generator: learning reduces choice KL versus a fixed Gaussian.Test markets are held out from fitting and validation.Source: §4.5 · p. 8, Table 7 · p. 8
5 Conclusion
DGD jointly learns a Gaussian response representation, contact–behavior aggregation, and feedback recurrence from aggregate counts, improving joint behavioral prediction across four recommendation and retail windows.
5 Conclusion
- Across four recommendation and retail windows, DGD improves joint behavioral prediction over DeepAR-joint, with ablations supporting distribution learning and feedback contributions.Controlled experiments also recover cross-day dynamics and responses under new attributes and prices.Source: §5 · p. 9
Statement on the Use of AI Tools
The authors report that AI tools assisted with manuscript organization, language editing, and preparation of the LaTeX source and diagrams; the accompanying appendices detail computation, likelihoods, state updates, choice readout, and supporting analyses.
A Model Computation Details
- The aggregate model uses a Gaussian propensity representation and readouts for social-media exposures, retail invoices, and controlled product-choice experiments.The choice experiment fits distribution and choice-readout parameters independently from market counts.Source: A Model Computation Details · p. 11
A.1 Aggregate Observation Likelihoods
- Aggregate arrival counts use a negative-binomial likelihood, while behavioral-pattern counts conditional on arrivals use a multinomial likelihood.Real-data training fixes dispersion at 50; losses are normalized by events or unit–cohort cells and averaged over dates.Source: §A.1 · p. 11
A.2 State Updates
- State updates correct common arrival conditions and unit-specific offsets using log innovations, then center unit states to retain relative offsets.During forecasting, expected arrivals make the arrival innovation zero, while behavioral memory, fatigue, and response shifts continue to advance.Source: §A.2 · p. 11
A.3 Choice Response Readout
- The choice readout maps latent propensity to product utilities and purchase probabilities, including a zero-utility no-purchase option.Aggregate demand integrates choice probabilities over propensity distributions, weights by cohort size, and sums across cohorts.Source: §A.3 · pp. 11–12
B.1 Likelihood Sufficiency of Counts under Shared Conditions
- When events share a conditional distribution and are conditionally independent, behavioral counts preserve the event likelihood’s parameter gradients.Dependence among behaviors within an event remains represented by the joint pattern probability.Source: §B.1 · p. 12
B.2 Exponential Tilting of Gaussian Mixtures
- Exponential arrival intensity reweights Gaussian-mixture components and shifts their means; an intercept adjustment preserves aggregate outputs under the stated transformation.The equivalence applies under unclipped exponential intensity, shared arrival loading and dynamic shift, and adjustable intercepts.Source: §B.2 · pp. 12–13
C.1 Probabilistic Properties of Positive-Weight Quadrature
- Positive-weight quadrature yields normalized population probabilities and nonnegative expected behavior counts bounded by expected arrivals.Summing component contributions only reorders a finite sum.Source: §C.1 · p. 13
C.2 Quadrature Error Bounds
- Quadrature error in joint behavioral probabilities depends on errors in both the behavior-weighted intensity numerator and total-arrival denominator.The analysis also bounds average behavioral NLL and its gradient, accounting for parameter-dependent mixture weights and latent nodes.Source: §C.2 · pp. 13–14
C.3 Error Propagation and State Bounds under Expected Feedback
The appendix bounds expected-feedback state error by combining per-step approximation error with propagation of existing error, and describes conditions that keep states bounded.
C.3 Error Propagation and State Bounds under Expected Feedback
- The finite-horizon state-error bound accumulates per-step approximation errors, each propagated by subsequent update Lipschitz factors.When each factor is at most L < 1 and per-step error at most ε, the bound is L^T E_0 + ε(1 − L^T)/(1 − L).Source: §C.3 · p. 14
- With initial memory, fatigue, and reference rates in [0, 1] and nonnegative effort and cohort sizes, expected-feedback memory and fatigue remain in [0, 1].Expected arrival innovations are zero, so common and centered unit-specific states decay by their retention factors.Source: §C.3 · p. 14
- The bound separates decay of the initial shift from accumulated daily feedback along the deterministic expected-feedback path.Its recurrence-error constants depend jointly on retention factors, feedback coefficients, and observation-readout sensitivity.Source: §C.3 · p. 14
D.1 Real-Data Preprocessing
The real-data windows use specified preprocessing and time splits for Retail and KuaiRand; the table caption identifies their training, validation, and test statistics.
D.1 Real-Data Preprocessing
- The real-data table reports training, validation, and test statistics, with events defined as exposures for KuaiRand and retained invoices for Retail.The caption does not provide numerical table entries in the supplied passage.Source: Table 8 · p. 14
- Retail preprocessing retains 39,434 positive-price merchandise invoices after removing duplicates, cancellations, invalid quantities or prices, non-merchandise entries, and invoice conflicts.Annual windows use preceding December for warm-up, January–February for training, March 1–14 for validation, and remaining March dates for testing.Source: §D.1 · p. 14
- KuaiRand dates are recalculated in Asia/Shanghai time, with separate warm-up, training, validation, and test periods for standard recommendation and random exposure.Cohorts are fixed before training using historical click-rate and exposure-volume groups, with new users assigned to a ninth cohort.Source: §D.1 · p. 15
- Aggregate histories use previously released pattern counts, whereas event-level models additionally use individual histories preceding exposures.Compared models within each task share the same exponentially moving count summaries.Source: §D.1 · p. 15
D.2 Model Evaluation Setup
The appendix describes the DeepAR-joint baseline, forecast and scoring protocols, and supplementary evaluation settings. It also specifies numerical comparisons of matched representations and controlled choice and dynamics experiments.
D.2 Model Evaluation Setup
- The supplementary full-projection GRU comparison was conducted after some original test results were available, with epochs selected on validation data and no subsequent tuning.The appendix reports that its test predictions followed model selection.Source: §D.2 · pp. 15–16
- Behavioral-pattern NLL is conditional on a recorded exposure or retained invoice, and behavioral-count models are compared by MAE at identical forecast origins and endpoints.Multistep behavioral evaluation freezes cohort context at the origin; longer horizons have fewer eligible origins within a fixed window.Source: §D.2 · p. 15
- DeepAR-joint retains DeepAR’s shared LSTM, count scaling, negative-binomial likelihood, and sampled autoregressive prediction, while adding a joint-mark head.The head uses state-dependent logits and learned lower-triangular mark dependencies; validation selects the epoch and hidden width.Source: §D.2 · p. 15
- At each test origin, DeepAR uses only preceding observations, freezes non-calendar inputs, and integrates 2,048 sampled count histories for multistep forecasts.After scoring, the observed count enters the recurrent history for the next origin; one-day NLL is arrival-weighted, while behavioral-count MAE averages daily absolute errors.Source: §D.2 · p. 16
D.3 Numerical Comparisons
- Representation comparisons match trainable distribution-parameter counts, while quadrature comparisons hold fitted parameters fixed and vary integration precision.Real-data Gaussian and discrete representations use two Gaussian components and three support points, respectively; training uses seven nodes, and reevaluation uses 25 nodes.Source: §D.3 · p. 16
- The controlled choice experiment uses a two-dimensional, two-component Gaussian mixture, while the controlled dynamics experiment uses a one-dimensional, two-component mixture.The choice optimization trains distribution and readout parameters; the dynamics experiment specifies nine quadrature nodes, dispersion 80, and early-stopping patience 25.Source: §D.3 · p. 16
E.1 Historical Joint Distributions
The appendix defines historical joint-distribution baselines and behavioral scoring procedures. It also specifies how co-occurrence rates and forecasts at different horizons are evaluated.
E.1 Historical Joint Distributions
- Historical baselines use training-period counts, the latest day’s pattern counts, or exponentially smoothed daily counts, with smoothing settings selected on validation joint NLL.Smoothed history includes zero-arrival dates; multistep predictions hold the distribution and arrival forecast fixed at the origin.Source: §E.1 · p. 16
E.2 Probabilities, Behavioral Rates, and Counts 𝑦𝐶𝑐𝑡(𝑦), 𝑌𝑐𝑡ℎ= Í
- Behavioral scoring reports event-weighted joint NLL and Brier scores, and aggregates observed, full-predicted, and conditional-predicted counts across cells.Conditional rates use observed arrival volumes, whereas full count predictions use predicted arrivals.Source: §E.2 · p. 17
- Count-error scoring retains zero-arrival dates; rate scoring excludes dates only when the corresponding denominator is zero, and cohort scoring weights cohorts equally.Cohort-level scoring first pools observation units within each cohort.Source: §E.2 · p. 17
E.3 Co-occurrence Rates and Forecast Horizons
- Co-occurrence MAE evaluates the probability that two behaviors occur in one arrival, while multistep scoring pairs each origin’s forecast with the observation at the matching endpoint.One-day scores use 10, 7, 17, and 17 origins for standard recommendation, random exposure, Retail 2010, and Retail 2011, respectively.Source: §E.3 · p. 17
F Numerical Evaluations of Aggregate Learning
The numerical evaluations check count-likelihood equivalence, gradient propagation, and the effects of quadrature accuracy. They also test the predicted equivalence of Gaussian parameterizations under exponential tilting.
F Numerical Evaluations of Aggregate Learning
- The event and count objectives differ numerically by zero up to a parameter-independent constant, with maximum gradient difference 5.55 × 10−17.Central differences agree with automatic differentiation for mixture weights, means, and Cholesky parameters; later-day outputs have nonzero gradients with respect to retention, observation-correction, and feedback parameters.Source: F Numerical Evaluations of Aggregate… · p. 17
- With 7, 25, and 61 quadrature nodes, maximum absolute differences in mean test NLL are 9.65 × 10−4, 1.01 × 10−5, and 4.36 × 10−8, respectively.Across six sets of fitted Gaussian parameters, the two parameterizations’ predictions converge as integration accuracy increases, consistent with analytic equivalence under exponential tilting.Source: F Numerical Evaluations of Aggregate… · p. 17
- Across 30 KuaiRand Gaussian models, the maximum absolute test-NLL change from seven to 25 quadrature nodes is 0.001747; for standard-recommendation DGD models, it is 0.000251.Across 18 Retail models, the maximum change is 1.62 × 10−5.Source: F Numerical Evaluations of Aggregate… · p. 18
G Joint Training and Detailed Computation
The training procedure separates daily state progression from parameter optimization and jointly updates DGD’s population, observation, and transition parameters. Appendix H presents daily retail behavioral-count forecasts and describes their plotted categories and seed averaging.
G Joint Training and Detailed Computation
- Algorithm 1 applies an observation operator to predict arrivals and behavioral counts, then a transition operator updates state using feedback counts and predicted arrivals.The observation operator comprises Equations (1)–(6), and Figure 1 expands the node-level computation and two feedback branches.Source: G Joint Training and Detailed… · p. 19
- Each training epoch resets the daily state, accumulates behavioral and count losses, advances state using observed feedback, and backpropagates through quadrature and the full sequence.The population, observation, and transition parameters are updated jointly using Adam, with saved parameters and stopping epoch selected on validation data.Source: G Joint Training and Detailed… · p. 19
H Daily Retail Behavioral-Count Forecasts
- Figure 2 plots daily behavioral-count forecasts for both retail windows, with columns for invoice counts in the high-value, high-quantity, and many-SKU categories.Curves average predictions across three training seeds; Table 3 averages errors computed separately for each seed.Source: Figure 2 · p. 20