Adaptive Quantization in Generative Flow Networks for Probabilistic Sequential Prediction
Authors: Nadhir Hassen
Zhen Zhang
Johan W. Verjans
Venue: NeurIPS 2025 poster ()
Keywords: Generative Flow Networks
Probabilistic Forecasting
Quantization
Time Series
Sequential Prediction
Uncertainty Quantification
Primary area: deep_learning
TL;DR: Temporal Generative Flow Networks (Temporal GFNs) offer a novel approach to probabilistic time series forecasting by adapting GFN principles for continuous data.

Abstract: Probabilistic time series forecasting, essential in domains like healthcare and neuroscience, requires models capable of capturing uncertainty and intricate temporal dependencies. While deep learning has advanced forecasting, generating calibrated probability distributions over continuous future values remains challenging. We introduce Temporal Generative Flow Networks (Temporal GFNs), adapting Generative Flow Networks (GFNs) – a powerful framework for generating compositional objects – to this sequential prediction task. GFNs learn policies to construct objects (eg. forecast trajectories) step-by-step, sampling final objects proportionally to a reward signal. However, applying GFNs directly to continuous time series necessitates addressing their inherently discrete action spaces and ensuring differentiability. Our framework tackles this by representing time series segments as states and sequentially generating future values via quantized actions chosen by a forward policy. We introduce two key innovations: (1) An adaptive, curriculum-based quantization strategy that dynamically adjusts the number of discretization bins based on reward improvement and policy entropy, balancing precision and exploration throughout training. (2) A straight-through estimator mechanism enabling the forward policy to output both discrete (hard) samples for trajectory construction and continuous (soft) samples for stable gradient propagation. Training utilizes a trajectory balance loss objective, ensuring flow consistency, augmented by an entropy regularizer. We provide rigorous theoretical bounds on the quantization error's impact and the adaptive factor's range. We demonstrate how Temporal GFNs offer a principled way to leverage the structured generation capabilities of GFNs for probabilistic forecasting in continuous domains.

Adaptive Quantization in Generative Flow Networks for Probabilistic Sequential Prediction
Nadhir Hassen" University of Adelaide, AIML and Mila Quebec Al Institute
Zhen Zhang University of Adelaide, AIML
Johan Verjans
University of Adelaide, AIML
Abstract
Probabilistic time series forecasting, essential in domains like healthcare and roscience, requires models capable of capturing uncertainty and intricate temporal dependencies. While deep learning has advanced forecasting, generating calibrated probability distributions over continuous future values remains challenging. We introduce Temporal Generative Flow Networks (Temporal GFNs), adapting Gener- ative Flow Networks (GFNs) - powerful framework for generating compositional objects to this sequential prediction task. GFNs learn policies to construct ob- jects (eg. forecast trajectories) step-by-step, sampling final objects proportionally reward signal. However, applying GFNs directly to continuous time series necessitates addressing their inherently discrete action spaces and ensuring differen- tiability. Our framework tackles this by representing time series segments as states and sequentially generating future values via quantized actions chosen by forward policy. We introduce two key innovations: (1) adaptive, currieulum-basee quan- tization strategy that dynamically adjusts the number of discretization bins based on reward improvement and policy entropy, balancing precision and exploration throughout training.
(2) A straight-through estimator mechanism enabling the forward policy to output both discrete (hard) samples for trajectory construction and continuous (soft) samples for stable gradient propagation. Training utilizes a trajectory balance loss objective, ensuring flow consistency, augmented by an entropy regularizer. We provide rigorous theoretical bounds the quantization error's i impact and the adaptive factor's range. We demonstrate how Temporal GFNs offer principled way leverage the structured generation capabilities of GFNs for probabilistic forecasting in continuous domains.
1 Introduction
Time series forecasting- predicting future values based on historical observations- - fundamental informed decision-making across countless applications, from managing energy grids and financial portfolios to monitoring patient health and understanding brain activity [14] 77. In many critical domains, particularly healthcare (e.g., forecasting vital signs from Electronic Health Records EHRs) and neuroscience (c.g., predicting Electroencephalogram EEG signals), obtaining not just a single point prediction but a full probabilistic forecast paramount. Probabilistic forecasts quantify the inherent uncertainty in predictions, providing prediction intervals or complete density estimations, crucial for risk assessment and reliable decision support [00]. The advent of deep learning has
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39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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revolutionized time series forecasting. Models based on Recurrent Neural Networks (RNNs) 30 27
and Transformers [1 25 32] can capture complex temporal patterns from large datasets. These models often achieve probabilistic forecasts by parameterizing specific output distribution ge.g.. Gaussian (00) ) or predicting specific quantiles (34]. While effective, these approaches can impose potentially restrictive assumptions on the shape of the predictive distribution or suffer from issues like quantile crossing. More flexible generative models like Normalizing Flows (28) and Diffusion Models [9] have also been adapted, offering richer distributional modeling at potentially higher computational costs. Recently, Large Language Models (LLMs) have demonstrated remarkable zero-shot capabilities in various domains, including time series forecasting [2]. Frameworks like Chronos [ show that standard transformer architectures, trained on time series tokenized via simple scaling and quantization, can achieve state-of-the sof-the-art results. This highlights the power of sequence modeling and the potential of mapping continuous time series to a discrete vocabulary. However, these methods typically rely fixed quantization scheme decided priori. An alternative paradigm for structured generation is offered by Generative Flow Networks (GFNs) B G. GFNs learn policies to construct complex objects (like molecules or computational graphs) through sequence of discrete actions in a state space. The key idea is to sample terminal objects (completed structures) with a probability proportional to given reward function, effectively learning navigate the state space towards high-reward configurations. This focus on sampling from a reward-modulater distribution makes GFNs naturally suited for tasks where we want to generate diverse, high-quality samples, rather than just finding single optimal solution (as in reinforcement learning). This generative process seems well- oll--ali with time series forecasting: we can view a forecast trajectory as a compositional object built step-by-step, where the reward reflects the accuracy of the final forecast. However, GFNs traditionally operate in discrete state and action spaces. How can we adapt them to generate sequences of continuous time series values? Directly applying GFNs faces two primary hurdles: 1. Continuous Action Space: Time series values are continuous, but GFN policies typically output probabilities over discrete actions. 2. Differentiability: Selecting discrete action (even if representing continuous value via quantization) breaks the gradient flow required for training neural network policies.
In this work, we propose Temporal Generative Flow Networks (Temporal GFNs), a framework designed to overcome these challenges and harness GFNs for probabilistic continuous time series forecasting. Our approach leverages the strengths of GFNs structured sequential generation and reward-driver sampling while introducing mechanisms specifically tailored for the continuous nature of time series data. The main framework described in figure [
Temporal GFN Architecture
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Figure 1: Temporal GFN framework: the encoder conditions on the context window; forward policy samples actions via quantized bins to form the trajectory; a reward module scores it; Trajectory Balance drives learning; STE provides differentiability through discrete steps; and an adaptive K loop adjusts resolution as training stabilizes.
Our core contributions are:

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1. GFN Formulation for Probabilistic Forecasting: We frame forecasting as constructing trajectory in a GFN state space, where states are time series windows and actions append the next predicted (quantized) value. The GFN learns to sample trajectories whose associated rewards (forecast accuracy) match the target distribution.
2. Adaptive Curriculum-Base -ass Quantization: We move beyond fixed quantization by introducing a dynamic scheme. The number of quantization bins K is adjusted during training based on reward improvement and policy entropy, enabling the model learn effectively with adaptive precision.
3. Differentiable Discrete Actions via STE: We employ straight- thghuth estimator (STE) [3] to reconcile discrete action selection with gradient-based optimization. The forward policy generates both discrete samples for state transitions and continuous approximations for gradient propagation.
4. Trajectory Balance Learning: We train the framework using the Trajectory Balance (TB) loss (23), enforcing flow consistency across entire forecast trajectories, and include an entropy bonus encourage exploration.
5. Theoretical Guarantees:
We provide theoretical bounds on the policy gradient error intro- duced by our quantization and STE approach, and on the range of the adaptive quantization factor, ensuring controlled behavior.
The paper is structured as follows: Section 2 reviews related work in forecasting and Generative Flow Networks (GFNs). Section introduces the Temporal GFN framework, detailing its components like adaptive quantization, STE, and the training objective. Section 4 provides through mechanism- focused ablation studies and performance comparisons against benchmarks. Section 5 discusses the implications of our approach, and conelude with summary and future directions.
provides empirical validation
2
Background and Related Work
Our work builds upon concepts from probabilistic time series forecasting, Generative Flow Networks, and methods for handling continuous data in discrete frameworks.
2.1 Time Series Forecasting
Time series forecasting aims to predict future values EC+1C+H [xc+1,.... ".. TC+H] given histori- cal context F1.C I......, .c.. [c.]. We focus probabilistic forecasting, which involves predicting the conditional probability distribution p(rc+1-C++ [11:c). Classical methods like ARIMA and ETS [16] model time series components (trend, seasonality, noise) and often provide prediction intervals under specific statistical assumptions. Deep learning methods leverage architectures like RNNs 30 27 or Transformers 21 (5 to learn complex dependencies. Probabilistic outputs commonly achieved by parameterizing output distributions (30), predicting quantiles [4] or using generative models like Flows (28) Diffusion Models [1). L-M--ased forecasters 12 have
shown strong performance, often by tokenizing time series values into a discrete vocabulary similar natural language. Chronos M1, for instance, uses simple scaling and uniform quantization, demon- strating the effectiveness of applying standard language model architectures and training objectives (cross-entropy) to tokenized time series. While powerful, these often rely on fixed quantization and may require very large models. Our Temporal GFN differs by using a GFN objective and introducing adaptive quantization.
2.2 Generative Flow Networks (GFNs)
are a class of probabilistic generative models designed to learn policies for constructing objects 2 in compositional manner, such that the probability of sampling is proportional a given reward R(x) > 0. They operate on directed acyclic graph (DAG) where nodes represent states (partial or complete objects) and edges represent actions a leading from one state to another. GFNs excel at exploring complex, high-dimensional discrete spaces and sampling diverse, high-reward objects [17 36). Applying them to continuous domains, especially for sequential generation, remains an open challenge [20].
GFNs 3 O
3

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2.3 Discretization and Continuous Actions in Time Series Forecasting
Bridging continuous time series data with models often designed for discrete inputs is a central challenge in modern forecasting. Quantization, tokenization, which maps continuous values to a finite set of discrete symbols, is a widely adopted strategy. This has enabled the successful application of powerful sequence models, like Transformers, to time series by treating them as "sentences" of tokens [ or like amino acids in proteins navigating a continuous space to model discrete sequences [2] Recent advancements also explore more sophisticated tokenization schemes, such as sevelet-based approaches, to better capture multi-scale temporal features before feeding them to foundation models 24]. While these methods highlight the power of discretization, commonality is often the reliance on fixed quantization scheme (e.g., a predetermined number of bins or a fixed tokenization vocabulary). As discussed by Rabanser al. [26), such fixed schemes can be suboptimal: too few bins (coarse quantization) limit precision and lead to information loss, whereas too many bins (fine quantization) can create an overly sparse action space, hindering exploration and slowing down the learning process, particularly in early training phases. Our proposed adaptive quantization mechanism (Section(3) is designed to directly mitigate this by dynamically adjusting the quantization granularity. Furthermore, when discrete actions (like selecting a quantization bin) are part of the model's generative process, as in our Temporal GFN framework, ensuring differentiability Estimator (STE) 35] serves as a standard technique to enable gradient propagation through such discrete steps. We employ STE to allow our GFN's forward policy to make discrete action selections (choosing quantized values) while facilitating effective gradient-based learning of the policy network. Our work, therefore, uniquely combines the GFN paradigm with both adaptive quantization for optimized precision and STE for differentiaailityy, specifically targeting the challenges of probabilistic continuous time series forecasting.
for gradient- based training of neural policies is critical. The Straight-Througe
3 Methodology: Temporal Generative Flow Networks
In this section, we detail the proposed Temporal Generative Flow Network framework. We begin by mapping the probabilistic time series forecasting problem onto the Generative Flow Network paradigm. Subsequently, we describe the core components designed to handle continuous data within this framework: state representation, the adaptive quantization scheme coupled with the forward policy network, the mechanism for ensuring differentiability, and finally, the training objective based on Trajectory Balance.
3.1 Forecasting as Trajectory Generation in a GFN
Generative Flow Networks (GFNs) provide a powerful lens for learning to generate complex, struc- tured objects 3 They learn a policy to navigate a state space through a sequence of actions, ultimately sampling terminal objects (complete structures) with probabilities proportional to pre- defined reward signal. This compositional generation process, guided by reward, makes GFNs an appealing candidate for sequential prediction tasks like time series forecasting.
We conceptualize the generation of a forecast as constructing a trajectory r in a GFN state space.
States (s): A state 8; encapsulates the information available step 1 of the forecast generation. We choose to represent states as fixed-length sliding windows containing the most recent history and any forecast values generated so far. The process begins from an initial state 50. which corresponds to the observed historical context window, 8p (Iabss ..., TT1).
Actions (a): An action a, taken from state 8, corresponds to selecting the predicted value for the next time step, TobsT+H- As GFNs fundamentally operate over discrete action spaces, crucial step is mapping the inherently continuous space of possible time series values to a finite set of discrete actions. This is achieved through quantization, detailed in Section 3.2
State Transitions: Applying a chosen (discrete) action hard to state 8, results in the next state 8111- This is typically achieved by appending ahard to the window represented by 8t and removing the oldest value, effectively sliding the window forward: 8141 = (st [1]....,.. 1], )ard).

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Trajectories (T): A complete forecast trajectory т = (su. $1...... ST is formed by sequen- tially applying actions 00:...,;r. ar according learned policy, spanning the desired forecast horizon TV.
Reward (R(T)): reward function assigns a scalar value to each completed trajectory, quantifying the desirability quality of the generated forecast (e-g., based on its accuracy against the true future values). The GFN's objective is learn policy that samples trajectories T such that the sampling probability P(r) x (xp(R(r)).
The main technical challenges lie in adapting the discrete action mechanism of GFNs to the continuous nature time series values and ensuring the entire process remains differentiable for effective training using gradient-based optimization.
3.2 Forward Policy, Adaptive Quantization, and Differentiability
The core generative component is the forward policy Pr(ag)s;) which learns the probability dis- tribution over possible next actions given the current state. To manage the continuous action space and enable learning. combine quantization, Transformer-bacee policy network, adaptive mechanism for quantization granularity, and the Straight- Through Estimator (STE).
3.2.1 Quantization and the Forward Policy Network
To bridge the gap between continuous time series values and the GFN's discrete action space, we employ quantization. We map continuous values within range [Umins Umax] to a of K discrete bins, represented by their centers:
k (Umax Umin), k =,,,...,4 K.
gk Umin
(I)
K
These K bin centers (g1:..., (K constitute the discrete action space A available to the GFN at any given state st. The forward policy network Pr(a: qu (t) computes the probability of selecting action given the current state window Recognizing the success of Transformer architectures [32 in capturing long-range dependencies in sequential data, we use Transformer encoder as the backbone of our policy network. The state window St, sequence of T real values, is processed by the Transformer encoder, yielding a greddddmensional representation h2 that summarizes the relevant historical context. This representation is then projected by linear output layer to produce logits over the possible discrete actions: logits, Wrh, + bp, where logits, RR Applying the softmax function to these logits yields the policy distribution:
exp(logits,,
Pp(a: = qu(s;) = Σラ exp(logits,,)
This network learns to predict the probability of the next value falling into each of the K quantization bins, based on the preceding time series pattern encoded in st.
3.2.2 Adaptive Quantization: Curriculum for Precision
Using a fixed number of bins K throughout training presents a dilemma common in discretization tasks [26). A small K leads high quantization error, limiting forecast precision, while large K creates a vast, sparse action space that can significantly slow down learning and hinder exploration, especially in the initial phases when the policy is still poorly defined. To overcome this, introduce an adaptive quantization scheme that treats the number of bins as a parameter that evolves during training, akin curriculum learning strategy (4). The model starts with a relatively small K (coarse granularity), making the initial exploration task simpler. As training progresses, K is dynamically adjusted based on the model's 's performance and confidence. Specifically, after warmup period Ewarmpp, we compute an adaptive update factor at each epoch e. This factor depends on the recent reward improvement AR, R. Re-s (average reward difference over 8 epochs) and the average normalized forward policy entropy H. E [0, 1]:
max(0, ARe)
7) =
(1 (-H
Contidence Signal
Improvement Signal
5

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Here, A > 0 controls the sensitivity of adaptation, and >> 0 is target reward improvement threshold.
The Improvement Signal becomes positive the model's 's reward gain below the threshold
c. Slow improvement suggests the current quantization might be too coarse, thus pushing
Ne 1 to encourage increasing K.
The Confidence Signal is large when entropy He is low (i.e., the policy is highly confident "peaked"). High confidence might indicate premature convergence or insufficient exploration. Increasing K in this refines the action space, potentially revealing finer structures and encouraging further exploration.
The number of bins for the next epoch Ke is updated multiplicatively, bounded by maximum K
Ke = min Mmax;
[K-- -]=])-
(4)
If Ke changes value, the size of the final linear layer (Wr, F. by-) in the forward policy network (and the backward network, if learned) must be adjusted to match the new number of output bins.
To ensure training stability when the number of bins K2 is increased, a simple random re- re-initializaaton of the output layer would be catastrophic, erasing previously learned knowledge, Instead, we employ weight-reuse strategy. The weights and biases in the linear layer corresponding to the pre-existing bins are preserved. The weights and biases for the newly added bins are initialized to near-zero values. This method allows the policy to retain its learned distribution over the existing action space while cautiously exploring the new, finer-grae aner-grained actions, thus preventing catastrophic forgetting and ensuring smooth curriculum.
3.2.3 Ensuring Differentiability: The Straight-Through Estimator
critical issue arises when selecting the discrete action at = qs from the policy Pp(+(s) to update the state. Operations like sampling taking the argmax non-differentiable, which would prevent gradients from flowing back from the final loss calculation to the parameters of the policy network (Wp. by and the Transformer encoder). To circumvent this, we employ the Straight-Through Estimator (STE) technique [3]] The core idea is to use the discrete value in the forward pass for computations that require it (like state transitions) but to continuous upproximation for the backward pass (gradient computation). Our forward policy generates two outputs at each step t:
Hard Sample ) The discrete quantized value selected for constructing the trajectory. This is typically the most likely action: at hard = 4argmxx, Pp(ay=auss)) This hurd value is used update the state
8t-1-
Soft Sample (n A continuous, differentiable proxy for the chosen action, computed as the expectation of the action under the current policy distribution: a. =
Lk=1 > Pr
@k (t). During backpropagation, the gradient is computed with respect a This gradi- ent is then passed straight through the non-differentiable argmax/sampling operation that produced hard effectively using the gradient of the continuous expectation aoof update the parameters that generated the policy (=(+(s))-
This STE mechanism allows maintain the integrity of the discrete state transitions inherent the GFN formulation while enabling end-to-end training with gradient descent.
3.3 Learning Objective: Trajectory Balance and Exploration
GFNs are trained by enforcing flow consistency conditions. We adopt the Trajectory Balance (TB) loss 223,, which offers robust credit assignment, particularly for long sequences.
3.3.1 Backward Policy
The TB objective necessitates backward policy Pn(st- 18t), modeling the probability of transition- ing backward from state 8, to a potential predecessor state 8t-1- We consider two implementations:
1.
Uniform Policy: Assumes all valid predecessor states are equally likely. Given the fixed action space size K, this simplifies to log but ignores temporal structure.
P PP(#--11a0 = logg K. This parameter-free

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2. Learned Policy: Parameterizes PB using a neural network that takes the rep- resentations of the involved states (h,-1,h)) as input, e.g., log Pn(st-1(ss) log softmax(Wgconct(hh. + bB). This allows PB to adapt but increases model size.
3.3.2 Reward Function
The reward function R(T) guides the GFN towards generating desirable trajectories. For forecasting, desirability equates to accuracy. We define the reward based on the discrepancy between the generated forecast sequence
(Z1,... ................................ 2T (derived from the sequence of soft samples at and the ground truth future sequence = (y1, , We use an exponential function of the negative normalized Mean Squared Error (MSE):
(2) y/
R(T) exp
(5)
T
(mmax
Umun )
where 3 > 0 controls the reward scaling. Higher values of create a sharper reward landscape, more strongly penalizing deviations from the ground truth.
3.3.3 Trajectory Balance Loss with Entropy Regularization
The TB loss [23 equates the flow along trajectory based forward probabilities and the initial flow Z (a learnable parameter representing the partition function or total flow) with the flow based on backward probabilities and the terminal reward R(T)). The squared difference forms the loss for single trajectory:
T'-1
T
CTB(T)
log Z +
log (5
log PB PB($(-1(s)) - log R(T))
(6)
t=0
Log Reward
Forward Log-Prob
Backward Log-Prob
The overall TB loss is the expectation over trajectories sampled from the forward policy: LTB (CTB
To mitigate the risk of the policy becoming prematurely deterministic and to encourage broader exploration of the action space, incorporate entropy bonus into the learning objective (cquivalent to subtracting an entropy penalty from the loss). The entropy of the forward pol- icy at state is: H H(PF(-|se)) = - -i Pr(a: = qu|s;) log Pr(a: = (.ls)). The final loss function balances the TB objective with the average entropy across the trajectory: L
ET" H(PF((1s))) where Aentropy 0 is a hyperparameter control- ling the weight of the entropy regularization. Öptimizing this final loss trains the forward policy Pr. the backward policy PB (if learned), and the partition function Z, summarize this section with Algorithm A and we provide theorical analysis our methodology Appendix
LTB Ae
Experiments
In this section, we present a comprehensive empirical evaluation of the Temporal Generative Flow Network (Temporal GFN) framework. Our experimental investigation is twofold. First (Section 4.1). we assess the performance of optimized Temporal GFN configurations against broad range of established baselines on standard forecasting benchmarks and challenging related datasets. Through- out this section, we explicitly link empirical observations to the mathematical concepts introduced Appendix B Detailed experimental setup common to both studies is described in Appendix D Second (Section 4.2) we conduct mechanism-focusee analyses and ablations using synthetic data and standard benchmarks validate the theoretical underpinnings of our approach, particularly concerning adaptive quantization and the Straight-Through Estimator (STE).

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4.1 Benchmark Performance Comparison
Datasets, Training Strategy and Evaluation Metries. We train and evaluate Temporal GFN on large and diverse collection of publicly available time series datasets, comprchensively gathered and curated, mirroring the scale used recent foundation model studies [T] the dataset, the training strategy, evaluation tasks and metrics detailed in Appendix [[] We evaluate optimized Temporal GFN configurations against relevant baselines across the defined benchmarks.
4.1.1 Comparison with RI. and MCMC Baselines (Benchmark I)
Table 1 summarizes the aggregated relative performance on Benchmark against PPO, SAC, and MCMC baselines. Temporal GFN (Adaptive K20) shows substantial improvements (23-35%) across all metrics compared to these methods. 'hee Learned Policy variant further improves probabilistic metrics (CRPS/WQLL). This highlights the benefits of the GFN framework tailored for forecasting over generic sequential decision-making approaches shown in table !
Table I: Aggregated Relative Performance vs. RL/MCMC Benchmark
Method
CRPS (L)
WQL (4)
MASE (4)
Avg. Improv. (%)
Temporal GFN (Adaptive K20)
0.1674
0.2273
0.8938
Temporal GFN (Fixed K20)
0.1781
0.2401
0.9363
-5.9%
Temporal GFN (Learned Pol. Adapt K10)
0.1453
0.2137
1.0345
+6.0%*
PPO
0.2299
0.3013
1.1759
-25.1%
SAC
0.2423
0.3227
1.2491
-29.5%
MCMC
0.2673
0.3522
1.3523
-35.5%
Avg. Improv. for Learned Policy calculated based CRPS/WQL Adaptive K20.
4.1.2
Comparison with SOTA Time Series Forecasters (Benchmark I & II)
Table 2 presents aggregated performance against SOTA deep learning models, evaluated on both Benchmark (In-Domain) and Benchmark II (Zero-Shot) where applicable. Temporal GFN demon- strates highly competitive performance, often leading in probabilistic metrics (CRPS, WQL) and particularly excelling in zero-shot scenarios and on metrics sensitive distribution shape like Cal- ibration Error and Multimodality Score (evaluated on extended/healthcare benchmarks, details in Appendix H). Qualitative examples in Appendix (Figunex 2T) ability to capture complex dynamics and multimodality.
visually support Temporal GFN's
Table 2: Aggregated Relative Performance vs. SOTA Models (In-Domain Zero-Shot).
Model
Eval Setting
CRPS (L)
WQL (1)
MASE (L)
Calib. Err. (L)
Multimod. (f)
Temporal GFN
ZeootShot/ In-Domain
0.1542
0.2158
0.9378
0.0348
0.8762
Lag-Llama
Zero-Shot/ Pretrained
0.1675
0.2401
0.9532
0.0623
0.7234
Chronos (Base)
Zero-Shot Pretrained
0.1731
0.2534
L.0233
0.0571
0.6957
MOREI (Base)
Task- Specific Trained
0.1859
0.2678
1.0871
0.0745
0.6182
TimeFM
Tusk-Specific Trained
0.2076
0.2846
L.1258
0.1183
0.3012
Temporal GFN (Leamed Pol.)
In-Domain
0.1453
0.2137
1.0345
Main Temporal GFN results reflect strong performance across settings. Calib. Multimod. primarily from healthcare/extended henchmarks. Task-Specific models trained per dataset. l.camed Policy results from Table 3
Table 3: Ablation Study Configurations and Main Results.
Experiment Config
Quantization
Start K
Policy Type
CRPS
WQL
MASE
Fixed K=10
fixed
10
uniform
0.1546
0.2346
1.1946
Fixed K=20
fixed
20
uniform
0.1921
0.2521
0.9721
Adaptive K=10
adaptive
10
uniform
0,1688
0.2408
1.1048
Adaptive K=20
adaptive
20
uniform
0.1845
0.2385
0.9532
Learned Policy (Adapt K10)
adaptive
10
learned
0.1453
0.2137
1.0345

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4.2 Analysis of Quantization Mechanism
We investigate the quantitative relationship between quantization resolution (number of bins, K), the resulting approximation error, and the implications for policy learning and performance. Table summarizes the configurations and main results for this ablation study.
Quantization Error Bound Validation: Theorem B. established that the policy gradient error depends linearly on the quantization error We empirically verify the relationship between K and c. Figure 2 (bottom left) plots the Mean Squared Error (MSE), serving as a proxy for e², against K. As predicted, increasing K significantly reduces the quantization error, consistent with the premise that finer discretization leads smaller e. The relationship between error reduction and reward gain explored in Figure 3 The left panel shows a strong negative correlation between quantization error and achievable reward, while the right panel shows that increasing K yields diminishing returns in reward improvement, suggesting trade-off that motivates adaptive approaches.. We visually confirms this reduction in approximation error across the data distribution for increasing K in Appendix Figure
Continuu
Cuantred
Quantoed Secipeekeii
Quarrired Lundscape
C5P
PM
CSR
LOW
LCS
Numte Sra
Paeeder11
Figure 2: Gradient flow fields: Continuous (left) vs. Quantized with STE (right, K=16), validating STE.
Adaptive Quantization Dynamics and Bounded Updates: The adaptive mechanism (Section 3.2.2
aims optimize K based on reward improvement (ARR) and policy entropy (He), as governed by Eq. 3 Theorem ?? guarantees the update factor ne is bounded: < 7e S 1 + 2A. This prevents uncontrolled explosion or collapse of K. In Appendix []] figure (left panel, purple/blue lines for Adaptive Quantization) shows the coupled evolution of reward and entropy, which drive Tje- The resulting non-um non-uniform bin distribution learned by adaptive quantization (Figure 5 Adaptive panel) demonstrates its ability to allocate bins effectively compared to fixed uniform quantization (Uniform panel). This optimized allocation minimizes the effective e in relevant regions, leading to superior performance observed in benchmark comparisons (Tables
Quantization and Policy Representation: In Figure 6 Appendix visualizes how the choice of K impacts the policy's 's representation of the underlying continuous energy landscape. A low K provides a very coarse approximation, while higher values capture more detail but remain discrete. The adaptive mechanism, by optimizing bin placement, aims to create quantized landscape that best preserves the essential features (modes, relative probabilities) of the continuous landscape within the constraints of K bins, facilitating more accurate policy learning.

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Entropy-Error Trade-off Quantification: In Figure 7 Appendix quantifies the trade-off inherent in selecting K. plots Normalized MSE (proxy for quantization error e) against Normalized Entropy (uniformity of bin usage). As K increases (indicated by color/size), error decreases, but entropy might slightly decrease from its maximum if adaptation leads to concentrated bins. This plot empirically illustrates the optimization landscape navigated by the adaptive mechanism.
Rebetionetipp betwwen aorrizziionn Eroor and Reward
Fflerd of Bin Cruani Rlooerd and MBF
1.0
K-10. 10
0.95
MSE-0.002
0.12
0.-
0.10
0.85
96
Avorage Roward
0.08
Mean Scuardd Emor
Reweed
0.00
Avp Reward
MIEE
0.36
02-
0.34
9.-
0.70
0.52
0.50
-00
02
0.6
0.8
1.2
40
Quentiontion Firzs
Number Binoe (K)
Figure 3: Reward and Entropy dynamics over training and their relationship for different configura- tions.
Discussion
While results demonstrate the effectiveness of Temporal GFNs, particularly in probabilistic accuracy and multimodality capture, it is important to contextualize its limitations. The sequential, step-by-step generation of trajectories, fundamental to the GFN paradigm, can be more computa- tionally intensive during training compared to models like standard Transformers or LLMs that generate forecasts in a single forward pass. Furthermore, the performance is contingent designed reward function R(T). Although the exponential MSE reward used in our work is effective for accuracy, designing rewards for more complex or domain-specific objectives (c.g., penalizing physiologically implausible trajectories in healthcare) remains a critical and potentially challenging modeling decision.
well-
Adaptive quantization. Our methodology (Section robustly applies GFNs to continuous data by dynamically balancing precision and exploration. Governed by reward and entropy signals (Eq. 3) and stabilized by theoretical bounds (Theorem B.2 Appendix it acts curriculum. While alternative data-driven quantization strategics (c.g., quantile-based bins) were explored in 19, particularly for skewed data by reducing quantization error more aggressively in dense regions (Figure 9 leading to CRPS improvements in Figure 10), it introduces significant implementation complexity (e.g.. estimating quantiles dynamically learning boundaries) and potentially higher training overhead (15% reported increase). Our adaptive uniform spacing approach provides a strong balance, achieving competitive performance (Tables ( and adapting the number of bins effectively based on learning dynamics (Figure IT) without the full complexity of learning non-uniform boundaries, representing a practical and performant design choice. The reduction in achieved through adaptation directly contributes to improved gradient accuracy (Theorem C. Eq.
effective learning
9).
Gradient flow. For enabling gradient flow through discrete quantization choices, we employ the While STE a widely adopted and practically effective method, alternatives like Gumbel-Soffma (GS) relaxation [8] offer different approach to managing
Straight-Thron Through Estimator (STE) 5
10

---

Quanizatioo with K=5 bins (MSE-0.1273, Normalzed Entrapy=0.7879)
Continous Quantized
Value
80
100
Sample Index (sorled)
Quantization with K=10 bins IMS=+0.0322 Normalized (ttroyy-0.229)
Continuous
Quantized
Value
100
Sample Index (sorled)
Quantization with K-20 bins (MSE-0.0102. Normalzed Entropy=0.8618)
Continucus Quantized
Valuue
100
Sample Index (sorted)
Entrapy=0.80322
Continucus
Vilue
40
80
80
100
Sample Index (sorted)
Figure 4: Quantization of sorted continuous data points. Visualizes the reduction in MSE (red area) as K increases from to 50.
differentiability in discrete action spaces via 2ạ softmax((fx. + gg)/ Ttemp). +gg))/Teemp) Our comparative analysis highlights the nuanced trade-offs. Figure [2] shows that GS can lead to faster initial convergence in terms of training loss, potentially due its inherently smoother gradient signals early in training (high Ttemp). However, this initial advantage can sometimes plateau, with STE achieving slightly lower final loss in some configurations. key factor is gradient stability; Figure g iillustrates that while STE maintains relatively stable gradient variance throughout training, GS, despite generally lower average variance, can exhibit occasional spikes, particularly if temperature annealing is not perfectly tuned. Furthermore, as detailed in Figure [3] GS introduces notable overhead: approximately 35% increased compute time, 41 % higher memory usage, and significant 70% rise in perceived implementation complexity due to temperature annealing and managing the continuous relaxation. Considering these factors, Figure [4] summarizes our practical guidance: STE is prefemred for its simplicity, resource efficiency, and stability, especially for smaller models or complex distributions where its biased but consistent gradient proves effective.
Entropy Regularization. The combination of the Trajectory Balance objective with entropy regularization (L) proves highly effective. TB loss facilitates credit assignment, while the entropy bonus explicitly manages the exploration-exploitatioo trade-off, crucial for learning diverse and well-calibrated distributions. Removing the entropy bonus leads to significantly poorer final rewards (Figure [5), highlighting role preventing mode collapse and ensuring sufficient exploration (Figures [6 17
11

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Ideal Distribution
Unfform Quantization
Entropy: 1.00
Entropy: 0.58
0.2
0.25
Probadiit
Probabllty
0.15
0.13
0.05
0.05
1
0
Bin Irdex
Bin Indix
Adaptive Quartiization
Without STE
Entropy: 1.00
Entropy:
025
0.25
Ridbablity
Probab
0.15
8.15
0.5
0.05
Bin Index
Indixdox
Figure 5: Comparison of learned bin probability distributions (K=10). Adaptive Quantization learns a non-uniform distribution, contrasting with Fixed Uniform. Without STE, the policy fails to learn the target (Ideal). Entropy values reflect uniformity.
Conclusion
Temporal Generative Flow Networks (Temporal GFNs) offer novel approach to probabilistic time series forecasting by adapting GFN principles for continuous data. Key to our framework is the framing of forecasting as reward-driven trajectory sampling, which facilitates exploration and learn- ing distributions over diverse futures, potentially yielding more robust probabilistic forecasts than standard methods. We successfully address the challenges continuous data through adaptive quan- Seation--provddngg dynamic precision control curriculum and overcoming fixed discretization limitations and and the (STE), which enables learning by bridging discrete state transitions with differ- entiable gradient updates. Trained via Trajectory Balance for effective credit assignment and entropy regularization for managed exploration, Temporal GFNs learn to sample accurate and diverse forecast trajectories. Theoretical bounds affirm the stability of adaptive mechanism and quantify the quantization error impact. Compared to alternatives like LLM-based forecasters [4] [29]. Temporal GFN provides adaptive precision and distinct flw-matching learning paradigm, while its unique sequential decision-making generative process offers advantages for complex dependencies and multimodality over standard probabilistic models. Future work will focus on multivariate extensions, extensive empirical validation, and exploring alternative GFN objectives.
Multivariate Extension. A key direction for future work is the extension of Temporal GFNs to the multivariate setting. While this paper focused on the univariate case to rigorously establish the core framework, the approach is not fundamentally limited. The state representation 8₁ can be expanded from vector to matrix representing window of multivariate observations, and the action a, can become a vector of quantized values for each dimension. The primary challenge lies in designing a policy network Pr that efficiently models the joint distribution action space. This could be achieved through an autoregressive factorization of the output dimensions or by learning a parameterized covariance structure, offering rich area for future research.
the multivariate
Irregular Sampling. Furthermore, the current implementation assumes regularly sampled time series due to its use of fixed-size sliding window. handle the irregularly sampled data common in domains like healthcare (EHRs) or finance, the framework can be adapted. The state representation
12

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could be augmented to include not just observation values but also the time elapsed between them, for instance, by feeding a sequence of (value, time_delta) tuples into the Transformer encoder. This would allow the policy network to explicitly learn time-dependent dynamics, making the model robust irregular sampling and significantly broadening its applicability.
13

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A Training Algorithm
Algorithm I Temporal GFN Training Procedure
Require: Time series dataset D, initial Ko, max maxx adaptive params X, e, 6, Ewarmup. E, hyper-
parameters B, Aentropy. 1: Initialize 0 (Pp. opt. PB. Z). K +- Kg. Init past rewards.
2: for epoch = to E do
3: e 2 Ewarmup then
Adaptive Quantization Update
4:
Compute avg. reward Re, avg. entropy He.
5: ARe &- Rs Re-8-
6:
Te 1 X(ma((0,
ARc)/ H.)) (Bq.
7:
Knew +- min( Kmax, [nn)) (Eq. 4
8:
if K new + K then
9:
K + Knew. Adjust policy network output layers for K bins.
end if end if
11:
for
each batch (x.
do Sy < Extract context window from I.
Trajectory Sampling and Loss Computation f- {so]. log pium &- Hsum O, generated seq
13:
[. 8t 80-
15:
for / = 0 to T 1 do
Forward Pass
h4 f- Transformer((s))
17:
(-(()): +- (Wft + by)-
18:
dz f- Aurgmar, Pr(a==u
hard
>> Hard sample for state transition
19:
at -k=1 96: Pr = Qk (8))
Soft sample for reward/gradient
20:
log PF t- log Paum log( Pp (a st)).
21:
Hamm f- Hsum + H(Pp(-1ss)),
22:
8141 concat((t :), ahard
Update state
23:
Append 8841 to Append agott to generated_seg 8t + 81+1-
24:
end for
25:
Compute log pa + Et log (
Backward Pass (or use uniform)
Compute R(T) using generated seq and target y (Eq. 5p.
27:
Compute (T) + (log 2 + log mmwn log Prum log R(T))²
>> TB Loss Final Loss
Compute batch loss Louch + LIB Aentropy (Huum TT).
Compute gradients V Caachhh
b Gradients flow via soff
30:
Update using gradients.
Parameter Update
end for
32: end for
B Theoretical Analysis
To better understand the behavior and stability of our Temporal GFN framework, provide theoreti- cal analysis on two crucial aspects: the impact of our quantization and STE approximation on the learning signal, and the bounds governing the adaptive quantization mechanism.
B.1 Quantization Error Bound on Policy Gradient
The core learning process relies on policy gradients. However, our framework uses discrete actions (ahard) for state transitions while calculating gradients via continuous approximations (aşoft) using the STE. This introduces potential discrepancy between the true gradient of the expected reward and the gradient actually used for updates. It is crucial to understand if this approximation significantly corrupts the learning signal. Theorem B. provides a bound on this error, showing that it depends on the inherent quantization eтror and the STE approximation quality.
Theorem B.1 (Quantization Error Bound). Assume the reward function is LR-Lipschitz w.E.t its input sequence and V log PF has bounded variance. Let e max action actian be the
17

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max quantization enror. Let = V Vogg V log Pf be the difference in score functions due to STE. The error between the true policy gradient [E[R and the estimated gradient VE(R] is bounded by:
VE[R
VE[RI| - L (c+ [[18
where I. max(In B log Pr|l [[
Proof. (The proof is given in the appendix C following the decomposition in Eq. 13 and applying Lipschitz continuity and controlled by the quantization resolution (e) and the fidelity of the STE approximation (measured by 8). If quantization is fine (c small) and STE provides a reasonable gradient proxy (Ø small in expectation), the learning signal remains informative.
The bound demonstrates that the gradient error is
B.2 Adaptive Update Factor Bound
The adaptive update factor (Eq. 3 controls the change in the number of bins K. It is important to ensure this factor remains within reasonable bounds guarantee stability.
Theorem B.2 (Adaptive Update Factor Bound). Let the adaptive update factor ne be defined in Eq.
where the reward improvement ARe E [0,e] (assuming reward is non-decreasing or improvement capped by and the normalized entropy He [0, Then, Ve bounded
1 ?e 11
when AR. (maximum improvement considered) and He (maximum entropy). Conversely, TBe = + 2A when ARe = (no improvement) and H = 0 (minimum entropy/maximum confidence)-
Specifically, 1/e
Ho)) Let T, max(0 R..) and Proof. The adaptive factor is m) = 1+ max(O,e- ARc) + (1 (1-Ha)) T- = He).
Since AR, E [0, c]:
If AR, c, then AR. 0, so maax(0,0) 0.
If AR, 0, then 6 ARe 6, so Ti max(0.c)
For intermediate values 0 < ARe < e, we have < AR <, thus 0 < Ti <
Therefore, Ti E 10.
Since the normalized entropy H. E [0, 1]:
If H. then T/
0.
If H, then T3 - 1. For intermediate values 0 < He < 1, we have <T <
Therefore, T2 E [0, 1].
The sum inside the parenthesis is Ti + T2. Since Ti E [0, 1] and Ta E [0, 1]. the sum S E [0, 2]. The adaptive factor is 7). 1 + XS. Since S E [0, 2] and A > 0. we have XS E [0, 2*). Therefore,
7e = 1 XS E [1, 11,1+21 222
The minimum value = 1 is achieved when S = 0, which requires T; 0 and T. = 0. This happens when AR. = e and H. = The maximum value 7/- + 2A achieved when S = which requires Ti = 1 and T2 1. This happens when AR, = 0 and H, 0. This completes the proof. The theorem shows that the multiplicative update factor is bounded, preventing excessively large or small changes in K in single step.
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C Quantization Error Bound
The use of discrete hard samples (ahard in the state transition and trajectory construction, while using soft samples (a for gradient computation via STE, introduces a discrepancy the policy gradient estimates. We aim bound the error in the policy gradient due this approximation.
Let R(x)) be the true reward based on continuous action and R((rar)) be the reward obtained when using the discrete action Thard- Let Pr be policy producing continuous outputs (before quantization) and PF be the policy over discrete bins. Let VE R be the true policy gradient and [E[R] be the gradient obtained using our STE approach (gradient through (oons reward based on Phard 9sott).
Theorem C.1 (Quantization Error Bound), Assume the reward function R LR-Lipschitz with respect to its input sequence and that the score function V log PF hus bounded variance. Let denote the maximum quantization error, Le., ||actioncominuouss -cciion S 6. Let V log PF- V log PF represent the difference in score functions due to the STE approximation (gradient computed via Qoot vs. actual discrete choice Thart). The error in the policy gradient estimate bounded by:
VE - VE[RII S (c+ E[[0|111
(c
E[ is a constant depending on the Lipschitz constant of the reward and the moments of the reward and score function.
where L max E log PF|1T,, V
Proof. We decompose the gradient difference:
VE[R] [R[A]
B(RV log Pr]
ERRV log Pr]
(10)
B[RV log PF]
EIRV log Pel + ERRV log Pr] E[ log Pp]
(11)
= E[(R R log PP E log Pp V log Pr)
(12)
-[(RR R log Pr E[
(13)
Term
Term
Bounding Term A: By the Lipschitz property of R. we have IR [t] < LRR| - < LRe, where
I represents the sequence generated with continuous actions and with quantized actions. Using Cauchy-Schwarc
Term All || R) V log Pell
$ E R| V log Prl
LReBil| log Prll]
< LREV B[[ V log Pr[1a]. (Jensen's inequality)
Bounding Term B: Using Cauchy Mauchy-Schwarzaaee again:
Term B/ = [[[]]]]
< [[ ||
S v R 5
(We use R in the definition of as might not be directly computable in the context of PF).
Combining the bounds for Term A and Term B using the triangle incquality on Eq. B3
IV R
VE[R]] < Term All + Term B]
s Lnc v El V log 1P1
/
1 B[ (e+ v
max{LR E log Pr 11,
I. (e [E[1|1121
19

---

This completes the proof. The bound shows that the gradient error depends linearly on the quantization error and the square root of the expected squared norm of the score function difference induced by the STE.
D Experimental Setup for Benchmark Comparaison
Datasets: We utilize a range of datasets for evaluation. Core ablation studies are performed on benchmark datasets commonly used in time series forecasting (e.g.. Electricity, Traffic, ETTml, ETTh1. We also present results on healthcaresppecffic datasets (MIMIC-III Vital Signs) to evaluate performance in our target application domains.
Training Strategy. Temporal GFN models, particularly those intended for zero-shot evaluation, undergo a large-scale training phase on the combined "Pre-training only" and "In-domain (Benchmark I)" datasets. The context length T (input window size) for sequences sampled during training is set to 512, matching common Transformer setups, and the prediction length T" generated during training rollouts is set 64, ensuring coverage for various downstream evaluation horizons H. Training employs the Trajectory Balance objective (SectionB) with default entropy regularization (Aentropy = 0.01) and adaptive quantization enabled (typically starting K = 20, max K 128) unless evaluating fixed-K ablations. To enhance robustness and data diversity, we incorporate data augmentation: each training sequence is generated either from the original datasets or with probability 0.9 from TSMixup set (convex combinations of different time series, adapted from (37) and with probability 0.1 from a synthetic dataset generated via Gaussian Processes with randomly combined kernels (yntnelSynth, similar I). Training typically proceeds for 10,000 optimization steps on multi-GPU hardware.
Evaluation Tasks and Metries, For both in-domain (Benchmark I) and zero-shot (Benchmark II) evaluations, we use the final observations of each time series a held-out test set (H is task-specific, see Appendix Db. We compute the primary metrics: WQL. [10] on 9 levels ( 0.9]. MASE [15] using the median prediction, and CRPS [i To aggregate metries across diverse datasets and provide fair comparisons, we compute each model's 's score divided by the score of a baseline model (here, Seasonal Naive), yielding relative scores. These relative scores are then aggregated across all datasets within the benchmark using the geometric mean, which provides robust aggregation insensitive to outlier datasets and the choice of baseline.
Following standard practices in probabilistic forecasting
10
we evaluate models using:
CRPS (Continuous Ranked Probability Score): Assesses the overall accuracy and calibration of the full predictive distribution. Lower is better.
WQL (Weighted Quantile Loss, T {0.1, ..., 0.9)): Assesses calibration across different quantiles. Lower is better.
MASE (Mean Absolute Scaled Error); Evaluates point forecast accuracy (median) relative
a naive seasonal baseline (15] Lower better.
Calibration Error & Multimodality Score: Used specifically for healthcare/mulliinddal analysis (lower/higher better, respectively).
Trajectory / Action Diversity: Quantifies the variety in generated forecast trajectories chosen actions.
Baselines: We compare Temporal GFN configurations against:
GFN Ablations: Fixed K (K=10, K=20), Adaptive K (start K=10, start K=20), Learned Backward Policy vs. Uniform.
RL/Sampling Methods: PPO 31 SAC 13
and a generic MCMC approach.
SOTA Forecasters: Lag-Llama
Chronos
MOREI 22 and a decoder-only Trans-
former (TimeFM) [8].
Implementation Details: The core policy network is a Transformer encoder 32 Default hyperpa-
rameters include Aentropy 0.01. Adaptive quantization parameters are tuned. We train typically
for 10,000 steps. Baselines are run using publicly available implementations or standard libraries where possible.
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Training Startegy The adaptive quantization mechanism's is sensitivity to training dynamics is pri- marily managed by the hyperparameter 6, the number of epochs over which reward improvement is averaged (Eq. 1). This parameter's role is to smooth the reward signal A preventing volatile updates to the number of bins that might arise from high-variance, batch-level rewards. We observed during development that settings with smaller batch sizes or higher learning rates, which typically yield noisier gradient and reward signals, benefit from a larger ô (c.g., d = 10) to ensure stability. For the experiments reported in this paper, we found 8 5 to provide a good balance between responsiveness and stability across our primary configurations. The adaptation is governed by À 0.1 and 0.02, which we found to be robust across datasets.
E Ablation study of Temporal GFN
This section presents empirical results designed to validate the theoretical underpinnings and ana- lyze the behavior of the proposed Temporal Generative Flow Network (Temporal GFN) framework. Through controlled experiments and ablations, we investigate the quantitative impact of the core mechanisms adaptive quantization and the Brraight-Througe Estimator (STE) on learning dy- namics, representation accuracy, and optimization efficacy. We explicitly connect these empirical findings the mathematical concepts developed in Section [ including the Quantization Error Bound, Adaptive Update Factor Bound, and the role of entropy in the Trajectory Balance objective.
E.1
Validation of the Straight-Through Estimator (STE)
STE is crucial for enabling gradient- based optimization with discrete actions. We validate its effectiveness empirically.
B. bounded the gradient error based on e and the STE approximation quality 8. Figure 22 provides a visual validation by comparing the continuous gradient field with the STE-computed gradient field on the quantized landscape. The preservation of gradient directions and relative magnitudes, particularly the flow towards energy minima, demonstrates that STE provides a sufficiently accurate approximation for effective optimization (8 functionally small).
Gradient Flow Approximation: Theorem
Optimization Trajectory: Figure 18 shows that optimization using STE gradients on the quantized landscape successfully converges to the same low-energy regions as optimization on the continuous landscape. The energy plots (right panel) confirm convergence, although dynamics differ slightly. This provides empirical proof that the gradient signal provided by STE, despite being an approximation bounded by Theorem B.T. is effective for guiding the policy towards optimal (high-reward) states.
Necessity of STE (Ablation): Removing STE entirely breaks the gradient path. Figure 5 ( (With- out STE panel) shows the policy completely fails to learn the target distribution without gradient information. Figure 15 confirms this, with the "No STE" configuration achieving minimal reward, demonstrating its critical role.
F Analysis of GFN Learning Objective and Dynamies
We examine the role of the Trajectory Balance loss and entropy regularization.
Reward Components and TB Loss: Figure 20 illustrates the components contributing to the overall reward signal driving learning, implicitly linked to minimizing the TB loss (Eq. ( Successful training maximizes the total reward, which requires balancing prediction accuracy, achieving flow consistency (implicitly rewarded), and benefiting from the entropy bonus.
Entropy Regularization and Exploration: The entropy term H(Pp(-Is()) in L section 3.3.3
encourages exploration. Figure [T shows the typical inverse relationship between reward and entropy during convergence. Critically, comparing the "Adaptive Quantization" curves (with Xentropy = 0.01) to "No Entropy Regularization" reveals that regularization helps maintain higher entropy, particularly later in training. Figure 15 confirms that removing this regularization ("No Entropy Regularization' teal line) leads to suboptimal final reward (0.55 vs 0.74 for full model), empirically demonstrating the benefit of the entropy term for balancing exploration and cxploitation, leading to better overall
21

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solutions. This is further supported by sustained high levels of unique actions ratio (Figure 17) and trajectory diversity (Figure T6p throughout training when entropy regularization is present.
G
GFN Calibration
A key advantage of Temporal GFN is its inherent capacity for strong calibration. Unlike methods relying solely on quantile regression or implicit uncertainty mechanisms (like dropout), which often exhibit systematic over- or under-confidence Chronos, and MOREI, Temporal GFN's calibration curve aligns closely with the ideal diagonal. This superior intrinsic calibration arises from the GFN objective itself (learning P(T) R(T) via Trajectory Balance, Eq. ( which naturally encourages modeling the full distribution of high-reward outcomes, leading to more reliable uncertainty estimates directly from sampled trajectories. While post-hoc techniques like Conformal Prediction (CP) [33] can further enhance calibration and provide formal coverage guarantees for any model (Figure 23 24), Temporal GFN provides significantly better-calibrated starting point, making subsequent recalibration potentially more effective or less necessary compared inherently miscalibrated models.
seen in Figure 19 for baselines like LagLlama,
22

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H
Multimodality Metric and GEN Advantage
In time series forecasting, particularly in domains like healthcare or complex system modeling, future outcomes may not follow a single path but could diverge into multiple plausible scenarios (modes). Effectively capturing this inherent multimodality crucial for robust recissio-mmaknng This appendix details the composite metric used to quantify a model's 's ability to represent multimodal predictive distributions and analyzes how the GFlowNet framework is inherently suited to excel at this task.
H.1 Definition of the Multimodality Score
The Multimodalitty Score used in our evaluations is a composite measure designed to assess several facets of how well predicted distribution P (derived from forecast samples) matches true (poten- tially multimodal) distribution Q. It incorporates the following components, typically averaged over relevant forecast horizons:
H.1.1 Mode Count Accuracy
This measures whether the model identifies the correct number of distinct modes (peaks) in the distribution. It penalizes both missing true modes and detecting spurious ones.
min( (atttetee Namee)
Mode Acc.-
exp(-> max (0, Nestected
Nuue));
(14)
Nuwe
False Positive Penalty
where Nettested and Nime are the number of detected and true modes respectively, and À (e.g., 0.5) controls the penalty strength for detecting extra modes. A value of 1 indicates perfect mode count agreement (Nattected Niru). Nune 0 is handled special case.
H.1.2 Jensen-Shanoon (JS) Divergence
IS divergence quantifies the similarity between the overall shapes of the predicted distribution P and the true distribution Q. is a symmetrized and smoothed version of the Kullback-Leibler (KL) divergence.
[S(I)) 1 DKL(PIIM) _ (KL(QIIM),
(15)
where M (P+ is the mixture distribution, and (xI(()) E, P(i) log Qल) Lower JS divergence indicates higher similarity between the distributions. and Q are typically derived from binned densities or empirical distributions from samples.
P(t)
H.1.3 Mode Proportion Error
This assesses whether the model assigns the correct probability mass to each identified mode.
NEw
Mode Prop. Err.
propinee Prre
prop pred
(16)
N
where (rop(1) is the probability mass associated with the i-th true mode (and its corresponding predicted mode, if found). Calculation requires identifying modes and estimating the probability mass associated with each (e.g., by integrating density between valleys). Lower error indicates better allocation of probability mass.
H.1.4 Wasserstein Distance
The W-Wasserstein distance (Earth Mover's Distance) measures the cost of transforming one distri- bution into another. For ID distributions, it can be efficiently computed using the inverse CDFs Fp! Fo or sorted samples:
(( (1
Wi(P,Q) -
F (z)
F - Vool.
(17)
23

---

where ( M(e) are the i-th sorted samples from P and Q. It captures differences in location and shape. Lower distance is better.
These components are often combined into a single score or reported individually to provide a nuanced assessment of multimodality capture.
H.2 Multimodality Algorithm
The practical calculation of the multimodallity score involves several steps, summarized in Algorithm This typically requires estimating probability density functions from samples, identifying modes within these PDFs, and comparing properties of the predicted modes and distribution against the true ones.
2
Algorithm 2 Pseudo-Code of Multimodality Score Calculation
Require: Forecast samples Sp = {I1, 4N }. True distribution samples Sq = {y11 ... yM (or analytical form Q). Horizon indices Himerest, Density estimation parameters (bandwidth h, range Xange: resolution Ngnu). Peak detection threshold Ademiity
1: Initialize aggregate metrics ugg +- (Mode Acc.: O, JS Div.: 0, Mode Prop. Err.: 0, Wass. Dist.: 0}
2: for each horizon index h E Hinterest do
3: Extract horizon samples SPA < (aiih J
SQnr + (UgA) Isolate data for current
horizon
4: Estimate predicted PDF (z)) using KDE on Sp. over Xrange with Ngri points, & Density
from forecast
5:
Estimate true PDF (x using KDE SQ,4 (or use analytical Q) over Xrange-
Density
from ground truth
6:
Detect predicted modes Modesp +- [(1kk dx)3 } by finding peaks in Pk(x) ddeniiyy Find
peaks in predicted PDF
7:
Ndetected |Modespl
8:
Detect true modes Modesq + {(uk) dz)} by finding peaks in (x >
Odensity
>> Find
peaks in true PDF
9:
Ninu + |Modesol
10:
Calculate Mode Count Accuracy mace using Eq.
Compare number of modes
Normalize estimated densities Ph < Pa/ Σ Pr. Qn Qn/ Σ Qu. b Ensure densities sum
to
1
Calculate JS Divergence mis < JS(P,) Qn)) using Eq.
Similarity of overall shapes
Estimate predicted mode proportions (prop (k) pred) from M Modesp, (x)). Requires robust area calculation
13:
Estimate true mode proportions {prop from Modeso, Qn (z))
Mass under each true
mode
15:
Calculate Mode Proportion Error Mprop using Eq. 16
Accuracy of probability mass per
mode
16:
Calculate Wasserstein Distance Mlwss f WS SQ,a) using Eq.
17
P. Distaacce between
sample sets
17:
Store horizon metrics M₁ &- {mace: Mys Mpopp: Mwass
18: Update aggregate metrics Magg Magg Mn
19:
end for
20:
Average aggregate metrics Minal Magg/ H Himeersst D Overall multimodality score components
21:
return Minal
H.3 How GFlowNets deal with Multimodality
The GFlowNet framework possesses inherent properties that make it particularly well-suited for capturing multimodal distributions, explaining the superior performance observed empirically (e.g., T).
Sampling Proportional to Reward: forward policy PF such that the probability of sampling complete trajectory terminating in state
The fundamental objective of a GFlowNet to learn

---

sy (representing a full forecast) is proportional to its reward R(sy):
P(T ending in 8y) x R(sy)-
(18)
If the true underlying process generating the time series can lead to multiple distinct future scenarios (modes), and our reward function R((T) correctly assigns high values to forecasts matching any of these true scenarios, the GFN will naturally learn to sample trajectories corresponding to all high- reward modes. Unlike maximum likelihood estimation which might focus on the single most likely mode, or simple regression which might average modes, the GFN objective explicitly encourages exploration and sampling from the entire distribution defined by R(()).
Trajectory Balance and Flow Consistency: The Trajectory Balance loss (Eq. 6 enforces consistency across the entire state space. It ensures that the flow generated by PF matches the flow dictated by the reward R and the backward policy Pg. For a distribution modes (peaks), the TB objective forces the learned flow F(s) and policy Pp to correctly distribute probability mass across the different pathways leading to these modes. the policy were to ignore a significant mode, the flow equations would be violated, leading to high TB loss. Training implicitly pushes the policy to allocate appropriate probability flow towards all high-reward regions of the state space.
with multiple
The entropy bonus term -Dentropy (P') in the loss function L (section. 3.3.3) explicitly encourages the forward policy to maintain diversity and single mode (mode collapse). By penalizing overly confident (low- entropy) policies, the training process incentivizes the GFN to keep exploring different trajectories, making more likely discover and represent multiple distinct modes present in the reward landscape (and thus, the true data distribution if the reward is well-specified). The empirical results in Figure [T and [3] support this, showing that entropy regularization leads to better overall reward, likely by preventing mode collapse.
Exploration via Entropy Regularization:
avoid premature convergence
Adaptive Quantization's Role: Our adaptive quantization mechanism further enhances multi- modality capture. By dynamically allocating more bins (higher resolution) around regions corre- sponding to emerging or established modes (as suggested by Figure 3). it allows the GFN policy Pr to represent the shape and separation of these modes with greater fidelity than a fixed, uniform quan- tization scheme might permit. This synergy between the GFN's s distributional learning objective and the adaptive representation capacity allows Temporal GFN to effectively model complex. multimodal predictive distributions.
That is, the combination of swaad-proporrtie sampling, flow consistency constraints enforced by the TB loss, explicit exploration encouraged by entropy regularization, and enhanced representational capacity from adaptive quantization provides a strong theoretical and practical foundation for Tem- poral GFN's superior performance in capturing multimodality compared traditional forecasting approaches.
25

---

I Datasets
These datasets span multiple domains (e.g., energy, finance, healtheare, web traffic, transport, nature) and exhibit varied properties regarding time series length, sampling frequencies (ranging from minutes to yearly), seasonality, trend complexity, noise levels, and potential multimodality. For structured evaluation and analysis of generalization capabilities, these datasets are divided into three distinct categories:
Pre-training only: A large set of 13 diverse datasets (approx. 795k series) used exclusively for the initial large-scale training phase of Temporal GFN models intended for zero-shot evaluation. These datasets not used during evaluation.
In-domain (Benchmark I): A collection of 15 datasets (approx. 97k series) that are included in the training corpus. The final segment (forecast horizon H) of the time series in this benchmark is held out for evaluating the model's performance on familiar data distributions and tasks.
Zero-shot (Benchmark II): A of 27 datasets (approx. 190k series) that the model during any training phase. These are used solely for evaluating the model's 's ability generalize its learned forecasting capabilities to entirely new time series and domains without any task-specific fine-tuning.
seen by
26

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Table 4: Details of datasets used for experiments, adapted from Ansari al. [ partitioned according to their use in training and evaluation for Temporal GFN models. Dataset
Domain
Freq. Num. Series
Series Length
Prediction Length (H)
min
avg
max
Pretraining-only
Solar (5 Min.)
energy 5min
5166
105,120
105, 120
105,120
Solar (Hourly)
energy
lh
5166
8760
8760
8760
Spanish Energy and Weather
energy
lh
66
35,064
35,064
35,064
Taxi (Hourly)
transport
lh
2428
734
739
744
USHCN
nature
1D
6090
5906
38,653
59,283
Weatherbench (Daily)
nature
1D
225,280
14,609
14,609
14,610
Weatherbench (Hourly)
nature
1h
225,280
350,633
350,639
350,640
Weatherbench (Weekly)
nature
IW
225,280
2087
2087
2087
Wiki Daily (100k)
web
ID
100,000
2741
2741
2741
Wind Farms (Hourly)
energy
1h
337
1715
8514
8784
In-domain evaluation (Benchmark 1)
Electricity (15 Min.)
energy
15min
370
16,032
113,341
140.256
Electricity (Hourly)
energy
1h
321
26,304
26,304
26,304
24
Electricity (Weekly)
energy
1W
321
156
156
156
8
KDD Cup 2018 London Smart Meters
nature 1h
270
9504
10,897
10,920
48
energy
30min
5560
288
29,951
39,648
48
M4 (Daily)
various ID
4227
107
2371
9933
14
M4 (Hourly)
various Ih
414
748
901
1008
48
Pedestrian Counts
transport
1h
66
576
47,459
96,424
48
Rideshare
transport
1h
2340
541
541
541
Taxi (30 Min.)
transport 30min
2428
1469
1478
1488
48
Temperature
nature 1D
32,072
725
725
725
30
Uber TLC (Daily)
transport
ID
262
181
181
181
7
Uber TLC (Hourly)
transport
1h
262
4344
4344
4344
24
Zero-shot evaluation (Benchmark II)
Australian Electricity
energy
30min
230,736
231,052
232,272
48
CIF 2016
banking
1M
72
28
98
120
12
Car Parts
retail
1M
2674
51
51
Covid Deaths
healthcare
ID
266
212
212
212
30
Dominick
retail
100,014
201
296
399
ERCOT Load
energy
1h
8
154,854
154,854
154,854
24
ETT (15 Min.)
energy
15min
14
69,680
69,680
69,680
ETT (Hourly)
energy
lh
14
17,420
17,420
17,420
24
Exchange Rate
finance
8
7588
7588
7588
30
FRED-MD
economic
IM
107
728
728
728
12
Hospital
healthcare
1M
767
84
84
84
12
MI (Monthly)
various
IM
617
48
90
150
18
M3 (Monthly)
various
1M
1428
66
117
144
18
M4 (Quarterly)
various
3M
24,000
24
100
874
8
M5
retail
ID
30,490
124
1562
1969
28
NN5 (Daily)
finance
ID
111
791
791
791
56
NN5 (Weekly)
finance
IW
111
113
113
113
8
Tourism (Monthly)
various
IM
366
91
298
333
24
Tourism (Qaatterly)
various
IQ
427
30
99
130
8
Tourism (Yearly)
various
IY
518
11
24
47
Traffic
transport
1h
862
17,544
17,544
17,544
24
Weather
nature ID
3010
1332
14,296
65.981
Zero-shot evaluation (Benchmark Healthcare)
MIMIC-III Heart Rate
healthcare 1h
500
100
240
720
48
MIMIC-II SpO2
healthcare
1h
500
100
240
720
48
MIMIC-III Resp. Rate
healthcare
1h
450
100
230
680
24
27

---

List of Figures
Continacus Policy Energy
Quanlized Palicy Energy Landscape
Mr-i- A
Madie
McsiC
Made
Que thadion proeerve
Parerate Dimanaion
Parsmale Dimeneor
Figure 6: Continuous policy energy landscape (left) vs. its quantized approximation (right, example for fixed K) on synthetic data. Adaptive quantization dynamically optimizes bin placement to better capture the modes (A, B, C).
brttopp
10-
Nurtarre of Gand DC
Nomssbee (ighe
8--
Devert Rewaut 6:00
K05
Reward 040
0.00
0.64
0,00
Figure 7: Trade-off between bin distribution entropy (uniformity) and quantization accuracy (Nor- malized ) for different numbers of bins (K). Color/size indicates K. Increasing K generally improves accuracy (reduces error) but may slightly decrease entropy if adaptation concentrates bins.
Note Normalized MSE axis: The displayed MSE values were transformed for visualization via (MSE/ max(MSE)). Therefore, higher values on this axis correspond to lower original MSE (better accuracy), facilitating comparison where higher values are generally desirable across axes.
28

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Gradient Variance Throughout Training
STE
Gumbel-Softmax
Gradient Variance (lower is better)
oocatsional varience spikes
STE achieves stable gradients
Training Progress (Epochs)
Figure 8: Gradient variance: STE (green) vs. Gumbel-Softmax (orange). STE is more stable.
Data-Driven vs. Uniform Quantization Bins
100
Quantization Error
Reduction: 38%
Fequency
Dala-digg 27
Unitomm Bl
Uniform Bins
Data
20
Value
Figure 9: Visualization comparing uniform vs. data- binning strategies.
29

---

Performance Improvements
95%
Higher
better
90%
90
75%
70%
Improvement Percentage (%)
60
45%
40
16%
-20
-40
-60
Calibration
Multimodal Preservation (%)
Reduction (%)
Improvement (%)
Stabillyy (%)
Temporal (Baseline) Conformal Prediction
Data-driven Quantization
Combined Approach
MOREI
Gumbel-Softmax
LagLiama
Figure 10: Performance improvements across metrics when implementing enhancements.
Entrupy
Resrad Entrayy
Adopthve
(Reward)
0.9
Adsptive Quantiction tEvenopy
Flood Quartization (Rowart)
Ermmpy Regileitstion
Floed Chaeriowkeon (Bobroy)
No Ertiopy Rogula regulaization
We bathow
08
8.8
80
0.2
0.9
Trair Epocha
Entropy
Figure 11: Left: Coupled evolution of Reward and Entropy over training epochs for Adaptive (purple/blue) vs. Fixed (teal) Quantization and No Entropy Regularization (green/lime). Right: Reward vs. Entropy scatter plot, illustrating the operating regimes.
30

---

Training Convergence: STE vs. Gumbel-Softmax
Temporal GFN-S STE
Temporal GFN- GS
Loss (lower is better)
convergence
G9 plateau effect
by
20
40
60
Training Epochs
Figure Training loss: STE (green) vs. Gumbel-Softmax (red). shows faster initial convergence.
Implementation Resource Requirements
STE
Gumbel-Soolmaa
2.0
Relative Cost (lower better)
+70%
1.5
+41%
+35%
Baseline
Baseline
Baseline
1.0
0.5
Compute Time
Memory Usage
Implementation Complexity
Figure 13: Relative resource requirements: STE (baseline) vs. Gumbel-Softmax.
31

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Practical Approach Selection Guide
Model Type
Better Approach
Small models 10M params)
STE
Complex distributions
STE
Very large models
Gumbel-Soflmax
Resource-constrainne
STE
Implementation simplicity
STE
STE is preferred for most practical applications
Figure 14: Practical guide for selecting STE vs. Gumbel-Softmax.
Reward Evolution During Training
Ful Model
trapy
Adaptive
A WMM Amn
No Trepecoory Belence
No STE
M
Adsoove Doann 0.53
No Entroyy Regulerzzaton 0.56
Averaee Reward
0.4
Trauectory Dalance: 0.37
No 0.27
0.2
0.-
Training Epochs
Figure 15: Average reward evolution ablating key components. near-zero reward, highlighting its necessity for learning.
"No STE" (lime green) shows
32

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Trquectoy Diveriity va. Trai Staps
231
230
Confgunation
227
Fbad 10-10
10-2
Aslestive 80-10
Febazaive K--2
2.20
Louanoe Police
Figure 16: Trajectory diversity metric over training steps, showing sustained diversity.
Unique Actiona Ratio va. Training Stops
0.88
0.04
0.00
Confguuratoo
0.80
Flodd Ki-t0
Foodd K::00
Adaeftion K--10
Adaiption K--00
Leamned Pollcy
Tranning Stopa
Figure 17: Ratio of unique actions utilized during training, indicating exploration breadth.
Opt rncortio Palls (Ceotinzuu)
Dytimitation Fattss (Cuontired k-1i))
Energ During Optimimatiion
Figure 18: Optimization paths: Continuous (left) vs. Quantized with STE (middle). Energy during
optimization (right) shows convergence for both continuous (solid) and quantized (dashed) paths.
33

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Calibration Curves for Different Forecasting Models
1,0
0.5
Empinical Frequency
0.6
Pertect Collbration
Temporal
Lagt.ama
Chronos
MOREI
000
02
04
0.6
0.5
1.0
Nominal Probability
Figure 19: Calibration curves for different forecasting models.
Components
Training
1.0
Base Reward
Noy MMM.mmmm /m
Flow Consistoncy Bonus
Bonus
Prodiction Accuracy
Reword Component Volue
0.8
Total Reward
0.6
0.0
00
100
125
Epochs
Individual Reward Componem Evolution
0.5
Mr Aog wp Ln Aww Mng www. Mmm
Base Reward
Consisioncy Bonus
Entoopy Banua
Prediction Accunacy
0.4
Component Value
0.1-
0.-
125
Training Epochs
Figure 20: Breakdown and evolution of reward components during training.
34

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Multimodality Metrics Compartson
Distributional Accuracy (Lower is Better)
Multimodality Performance Metrics
Calb ation Score
Kay Strengths Tampora GRr multmnopo dctection
63
Sextstrclien
(ighest cellbaaton Score
Lagilara
NOVE
Oode
Mode
Mudi Connaeg
Trro
VOREI
Comtinuous
Pybbabiiiy
Oronce
Lagiiama
GIN
Lkg.
Elergy
MOREI
00
02
0.4
0.8
Figure 21: Multimodality performance metries comparing forecasting methods. The figure presents three visualizations of comparative metrics. Left: Radar chart showing five key multimodality metrics across all methods, with Temporal GFN (green) consistently outperforming other approaches across all dimensions. Top right: Bar chart of distributional accuracy metrics (CRPS and Energy Score) where lower values indicate better performance; Temporal GFN achieves the lowest scores. Bottom right: Horizontal bar chart of Mode Coverage showing Temporal GFN's superior ability (92%) to identify and assign appropriate probability true modes compared to other methods..
(k:16)
Figure 22: Gradient flow fields: Continuous (left) vs. Quantized with STE (right, K=16). STE effectively approximates the true gradient directions.
35

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OPN
Cleoria (74.% improvement)
(50 7% irpisement)
(70 improvement
0.
Emprrool Frequency
0.2
MOREI (07 3% ieproweentt
TrraFM (80 Sh, improwemerd
0.2
0.4
Nominal Probability
Figure 23: Effect of applying recalibration methods.
Conformal Prediction Calibration Improvement
1.0
Error Reduction: 72.1%
0.8
Empirical Frequency
0.6
0.4
0.2
Perfect Calibration
Temporal GFN
Temporal CP
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Nominal Probability
Figure 24: Temporal GFN calibration improvement with Conformal Prediction.
36

---

Comparison of Key Metrics Across Model Configurations
0.248:066
CRPS Score (lower is better)
Calibration Error (lower better)
C,228
Training Time (lower is better)
Normalized Metric Valuc
0.155
52
0.6
45
0.133
40
0.125
0444
DO43
2012
0.0
Temporal GFN
TempGFN CP
TampGFN DDQ
TempGFN GS
TampGFN Combined
LagLlama
MOREI
Figure 25: Comparison of key metrics across different model configurations.
37

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K
Broader Impacts
The development of Temporal Generative Flow Networks (Temporal GFNs) for probabilistic time series forecasting, while primarily an advancement in machine learning methodology. carries spectrum of potential societal impacts that merit careful consideration. The most significant positive impacts stem from the potential to enhance decision-makingg in critical domains where understanding uncertainty is paramount. healthcare, for instance, the ability of Temporal GFNs to provide more accurate and well-calibrated probabilistic forecasts for patient vital signs physiological signals could lead earlier detection of adverse events, optimized resource allocation in intensive care units, and more personalized treatment strategies. The framework's 'ss capacity capture multimodality is particularly valuable here, as patients may exhibit distinct future health trajectories. Similar benefits extend to finance, where reliable probabilistic forecasts can improve risk management and portfolio optimization; to energy systems, where better predictions of demand and renewable generation can enhance grid stability and efficiency; and to climate science, where nuanced forecasts with uncertainty can better inform policy. Beyond specific applications, by offering a new way to learn distributions over complex sequential behaviors, Temporal GFNs might also contribute to deeper scientific understanding of the underlying dynamics in diverse systems.
38

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
The abstract and introduction (Section 1) clearly state the core contributions regarding the Temporal GFN framework, adaptive quantization, STE for differentiability. TB loss, and theoretical bounds, which are then supported by the methodology, theory, and experimental sections.
Justification:
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: The Discussion and Conclusion 6 touches upon trade-off's (e.g., STE bias, complexity of data-driven quantization vs. adaptive uniform). A dedicated "Limita- tions" subsection within the Discussion or Conclusion are elaborated on assumptions (e.g.. univariate focus, specific GFN objective) and potential areas for improvement.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
Justification: Theoretical results (Theorems C.I and B. are presented in Section B As-
sumptions (e.g.. Lipschitz reward, bounded variance) are stated, and full proofs are provided in Appendix B
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification: Section and Appendix [ detail the datasets, evaluation strategy, baseline implementations, and Temporal GFN hyperparameters (context length, prediction length, training steps, adaptive parameters, optimizer details). The core methodology, including the algorithm, is described in Appendix A
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, described in supplemental material?
Answer: [Yes]
Justification: We plan to release the source code for Temporal GFNs and scripts for ex- periments upon publication. Publicly available benchmark datasets are used and cited. Instructions for data access and preprocessing will be provided in the supplementary mate- rial/repository README.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Answer: [Yes]
Justification: Section provide details on data splits (Pre-training, Benchmark Bench- mark II), evaluation horizons, model hyperparameters (e.g.. Transformer architecture, GFN parameters, adaptive quantization settings), optimizer (Adam), learning rates, and training duration.
39

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7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: Main benchmark results for Temporal GFN variants are reported as averages over 3 random seeds (Section D). While explicit emror bars statistical tests present in all tables/tigures in the main text, the multi-seed averaging addresses variability, Appendix figures (c.g., gradient variance) sometimes show error bars.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification: Appendix D mentions training on multi-Gi UPU hardware and typical training duration (e.g.. 10,000 steps). The (previously discussed but removed) computational ef- ficiency table provided typical training times per configuration. This information will be consolidated in the appendix.
9. Code of ethies
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics
Answer: [Yes]
Justification: The research aims to advance probabilistic forecasting with a novel method. Standard benchmarks and synthetic data are primarily used. For any healthcare data, ethical considerations regarding data privacy and responsible use would be paramount (to be detailed further if specific private datasets used and not just MIMIC-III which has its usage agreements).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [Yes]
Positive im- pacts include improved decision-making in healthcare and other critical domains via better uncertainty quantification. Potential negative impacts could relate to misuse applied to sensitive personal data without safeguards, or over-reliance model predictions. Mitigation strategies like emphasizing model calibration and responsible deployment are discussed.
Justification: A Broader Impacts section has been included in Appendix K
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The proposed Temporal GFN model itself, while generative model for time series, does not inherently pose the same high risk for direct misuse large-scale image/text generators trained on vast uncurated web data. Benchmark datasets used are standard public datasets. If applied sensitive healthcare data, access would be governed by existing data use agreements.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
40

---

Publicly available benchmark datasets (e.g., ETT, M-series, MIMIC-III are cited with their original sources. Baselines like Chronos, Lag-Llama are cited. Any open- source code leveraged (e.g., for GFN components or evaluation metries) will be credited in the implementation details or acknowledgments, and licenses respected.
Justification:
13. New assets
Question: Are assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
The primary contribution is new modeling framework and methodology. If code is released, will be accompanied by README and usage instructions. No new large-scale datasets foundational models are being introduced as assets themselves.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: This research does not involve crowdsourcing or direct research with human subjects in terms of data collection for this study. Any healtheare data used (e.g.. MIMIC-III is pre-existing, de-identified, and accessed under established ethical protocols and data use agreements.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
The research uses pre-existing, de-identified datasets (like MIMIC-III, public benchmarks) synthetic data. No new human subject data was collected for this study, thus direct IRB approval for this specific work was not required beyond the original approvals for the datasets themselves.
Justification:
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMS if it is an important, original, or non-standard component of the methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality the research, declaration is required.
Answer: [No]
Justification: LLMs were not used as a core component of the proposed Temporal GFN methodology or experimental research.
41


Reviews:
[{"review_id": "wJeZzInkgo", "reviewer": "Reviewer_5JJR", "summary": "The authors provide a new method for augmenting Generative Flow Networks to model timeseries data that addresses the challenge in adapting the discrete action space for continuous variables as seen in many time series. This method uses a dynamic bin-size strategy where bin widths are learned during training through a metric that incorporates model reward and action space entropy. The authors test this method on a range of clinical data sets, comparing to other baselines.", "questions": "How do batch size and learning rate affect the reward improvement over the \\delta epoch and therefore the magnitude of the bin update? Was there a sensitivity analysis done for these variables?\n\nLine 225-226 (new linear layer for new bin number): Is this adjustment be done with weight re-use in some way? Or is this a new randomly initialized layer?\n\nMedical data often consists of time-series that are irregularly and informatively sampled. Can the authors explain whether this method captures these aspects of the medical data?", "limitations": "The authors discuss limitations largely around design choices and ablation studies within the body of work presented here. They could expand on these to broader limitations of the method in general compared to other methods of timeseries modeling.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nFixed-bin discretization is a major limitation of other timeseries modeling methods as mentioned by the authors. The ability to adapt this discretization in a reward driven manner is an appealing characteristic. The paper is clear to read.\n\nWeaknesses:\nThis method appears to be limited to single-variate timeseries models. While this is an important contribution, this limitation should be discussed (or if the reviewer is incorrect, perhaps the distinction can be expanded upon). Often multiple variables are collected together and share mutual information.", "was_revised": true, "final_justification": "The authors have provided thoughtful responses and for the additions to key sections of the manuscript. I will keep my original rating.", "rebuttal": "Dear Reviewer 5JJR,\n\nWe sincerely thank the reviewer for their positive assessment and valuable, constructive feedback. We are very encouraged that the reviewer found our work to be a \"technically solid paper\" and appreciated the clarity and novelty of our reward-driven adaptive discretization approach.\n\nWe have carefully revised our manuscript to address every point raised. We believe the changes have significantly improved the paper's clarity, scope, and discussion of limitations. Below are our point-by-point responses.\n\n### **Regarding Weaknesses:**\n\n**1. Weakness: Limitation to Single-Variate Time Series**\n\nWe agree that the current paper focuses on univariate time series and that this is an important point to discuss. This was a deliberate methodological choice to first rigorously establish the novel core components of our framework—adaptive quantization and the application of GFNs to continuous sequential prediction—in a clear and controlled setting.\n\nHowever, the Temporal GFN framework is not fundamentally limited to the univariate case. We have now expanded the **Conclusion and Future Work (Section 6)** to explicitly discuss the extension to multivariate forecasting. In short:\n\n*   The state representation `s_t` can naturally be extended from a vector to a matrix representing a window of multivariate observations.\n*   The action `a_t` would become a vector of quantized values, one for each time series dimension.\n*   The forward policy network `P_F` would then learn to output a joint probability distribution over this multivariate action space. This could be implemented, for example, by using an autoregressive factorization of the output dimensions to capture inter-series dependencies at each step, or by making a conditional independence assumption for a simpler model.\n\nWe have added this discussion to our manuscript and thank the reviewer for prompting this important clarification.\n\n**2. Question 1: Sensitivity to Batch Size, Learning Rate, and Delta (`δ`)**\n\nYou raised a very crucial point about the sensitivity of the adaptive bin update mechanism to hyperparameters. The update factor `η_e` is indeed influenced by the stability of the reward improvement signal `ΔR_e` (Eq. 1), which in turn depends on training dynamics.\n\n*   **Role of `δ`:** The parameter `δ` (the number of epochs for averaging reward, line 214) is designed specifically to smooth out the high-variance, batch-level reward signals. A larger `δ` provides a more stable estimate of reward improvement, making the bin updates less reactive to training noise.\n*   **Interaction:** We observed during development that smaller batch sizes and higher learning rates, which lead to noisier reward signals, benefit from a slightly larger `δ` to prevent premature or erratic changes to the number of bins `K`.\n*   **Our Findings:** While we did not include an exhaustive sensitivity analysis in the original submission due to space constraints, we found the mechanism to be robust within standard hyperparameter ranges. We have now added a more detailed discussion of these dynamics and our hyperparameter choices in **Appendix E (Experimental Setup)**, clarifying these interactions for better reproducibility.\n\n**3. Question 2: New Linear Layer Initialization**\n\nYou have correctly identified a critical implementation detail regarding the adjustment of the policy network's output layer when the number of bins `K` changes. Simply re-initializing the layer randomly would indeed be catastrophic to the learning process.\n\nWe employ a weight-reuse strategy to ensure training stability. When the number of bins `K_e` is increased, the weights and biases in the linear layer corresponding to the existing bins are preserved. The new weights and biases for the newly added bins are initialized to near-zero values. This allows the model to retain its learned policy over the existing action space while cautiously beginning to explore the new, finer-grained actions.\n\nWe apologize for this lack of clarity in the original text and have updated **Section 3.2.2 (lines 225-226)** to explicitly state:\n*\"...the size of the final linear layer (WF, bF) [...] must be adjusted. To maintain training stability, we preserve the existing weights for the original bins and initialize the weights for the newly added bins to near-zero values, preventing catastrophic forgetting.\"*\n\n**4. Question 3: Handling of Irregularly Sampled Data**\n\nThis is an excellent question, as irregular sampling is a key challenge in many domains, especially healthcare. The current implementation of Temporal GFNs, which uses a fixed-size sliding window, does assume regularly sampled time series.\n\nWe acknowledge this as a limitation of the current work and have added it to our discussion in **Section 6 (Conclusion and Future Work)**. We also outline a clear path forward:\n\nThe framework could be extended to handle irregular data by modifying the state representation. Instead of just a sequence of values, the input to the Transformer encoder could be a sequence of `(value, time_delta)` tuples, where `time_delta` is the time elapsed since the previous observation. This would allow the policy network to learn time-dependent patterns, making it robust to irregular sampling. This is a significant and exciting direction for future research.\n\n**5. Limitations Discussion**\n\nWe appreciate the reviewer's suggestion to expand our discussion of limitations beyond internal design choices. In our revised manuscript, we have broadened **Section 5 (Discussion)** and **Section 6 (Conclusion)** to better contextualize Temporal GFNs within the broader landscape of time series models (e.g., Diffusion and LLM-based models). This includes a discussion of:\n\n*   **Computational Trade-offs:** The sequential generation of trajectories in GFNs can be more computationally intensive at training time than models that produce forecasts in a single forward pass.\n*   **Reward Function Sensitivity:** The performance of our model is contingent on a well-specified reward function. While our MSE-based reward is effective, designing reward functions for more complex, domain-specific objectives remains a key modeling choice.\n*   **Scope:** We now more clearly frame the univariate focus and regular sampling assumption as limitations of the current study that motivate future work.\n\n**6. Paper Formatting Concerns**\n\nWe sincerely apologize for the inconsistencies in figure citations and any confusion this caused. We thank the reviewer for their careful reading.\n\n*   **Figure Numbering:** We have meticulously reviewed the manuscript and corrected all figure references to be consistent and explicit (e.g., \"Figure 10 in Appendix K\").\n*   **Figures in Appendix:** The decision to place all figures in the appendix was made to comply with the NeurIPS page limit. We have ensured that the main text sufficiently summarizes the key findings from each figure, allowing the appendix to serve for detailed inspection.\n\nWe are confident that these revisions, guided by the reviewer's insightful feedback, have substantially improved the quality and clarity of our paper. We thank the reviewer again for their time and expertise. We hope our responses and the updated manuscript will merit an even stronger evaluation of our work."}, {"review_id": "Mo5HCg3THZ", "reviewer": "Reviewer_o3UX", "summary": "This paper extends generative flow networks for continuous-value time series forecasting. It does so by framing a forecast as a trajectory in the GFN state space, where future values are quantized into discrete actions. The number of bins are in this case not fixed but are adapted during training, leading to a more flexible quantization procedure. This is enabled by a straight-through estimator that allows gradients to flow through discrete choices.", "questions": "Q1: Can the authors comment on the overall computational overhead at inference time and how it compares to other approaches? The comparison in Figure 12 is indeed very helpful, can this be expanded to the baselines used in Section 4?\n\nQ2: Could the authors please elaborate on the choice of baselines in the experimental section, particularly in e.g. Table 2? Why were these baselines chosen and would it makes sense to e.g. include the Diffusion / Flow models mentioned in the Related Work?", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n\n- The idea to introduce GFNs for continuous forecasting is interesting and makes sense, especially from the point of reward-proportional sampling to achieve better sampling diversity. This is the first application of GFNs to continuous time forecasting, making this approach novel.\n- The paper provides a solid empirical evaluation that demonstrates the effectiveness of the proposed approach. In particular, the paper does a good job at empirically supporting the theoretical claims.\n- The adaptive quantization scheme is well-motivated and definitely interesting. While additionally making the bin boundaries adaptive is compelling, it seems it is sufficient to have just the number of bins adapt dynamically.\n\n## Weaknesses\n\n- Using the exponential of a normalized MSE as the reward might lack flexibility or e.g. long horizon forecasting, switching to CRPS or other loss types would likely require re-deriving the reward scaling factor\n- While the authors explicitly state that a multivariate extension of the proposed approach is left for future work, I do think a limited discussion / execution of this is important for this work. Most real forecasting tasks (power grids, healthcare dataset) are multivariate in nature.", "was_revised": true, "final_justification": "All concerns from the initial review have been discussed and resolved in the rebuttal and I recommend acceptance.", "rebuttal": "Dear Reviewer o3UX,\n\nWe sincerely thank you for your time and for providing a thoughtful and constructive review of our work. We are encouraged that you found the application of GFNs to continuous forecasting interesting and our empirical results solid. We were particularly grateful for your low confidence score, as it signals a valuable opportunity for us to clarify key aspects of our work. We hope to address your concerns below and demonstrate the soundness of our approach.\n\n### **Regarding Weaknesses:**\n\n**1. Flexibility of the Reward Function:**\n\nYou raised a very pertinent point about the flexibility of our chosen reward function—the exponential of the negative normalized MSE. We agree that a single reward function may not be optimal for all forecasting tasks, such as those evaluated by CRPS or in long-horizon scenarios.\n\n*   **Modularity of the Framework:** We would like to clarify that our Temporal GFN framework is **not** fundamentally tied to the NMSE-based reward. We chose this function because it provides a smooth, non-negative reward signal (a requirement for GFNs) that effectively guides the model to minimize large errors. However, the reward function is a modular component.\n*   **Adaptability to Other Metrics (e.g., CRPS):** Our framework can readily incorporate other standard forecasting metrics. For a negatively oriented score like CRPS (where lower is better), a valid reward function could be formulated, for instance, as `R(τ) = exp(-α * CRPS(τ))` or `R(τ) = 1 / (1 + CRPS(τ))`. [1,2] While the scaling factor `α` (or `β` in our paper) would indeed need to be tuned depending on the metric and dataset, this is a standard hyperparameter tuning step common to reward-driven methods[3].\n\nIn our final version, we will revise Section 3.3.2 to explicitly state the modularity of the reward function and discuss how other metrics like CRPS can be integrated.\n\n**2. Discussion on Multivariate Extension:**\n\nWe acknowledge your point that multivariate forecasting is a crucial and highly relevant problem. Our decision to focus on the univariate case was deliberate, aimed at providing a clear and rigorous introduction of the core contributions—the adaptive quantization scheme and the application of the GFN trajectory balance objective to forecasting—without the confounding complexities of high-dimensional action spaces.\n\nAs you noted, we identify this as a key area for future work in our conclusion (Section 6, line 397). A promising direction for this extension, which we will add to our discussion, is to adapt the action space. At each time step, an action could be a vector of quantized values for all dimensions, or the model could learn a policy that sequentially generates a value for each dimension before proceeding to the next time step. We believe this represents a significant research direction that builds upon our foundational work and merits its own thorough investigation.[4,5,6]\n\n### **Responses to Questions:**\n\n**Q1: Computational Overhead at Inference:**\n\nThis is an excellent question. Figure 12 focuses on the *training* overhead of different components of our method, and we appreciate the suggestion to elaborate on *inference* overhead compared to baselines.\n\nAt inference, Temporal GFN generates a forecast trajectory by sequentially sampling `T'` future values (where `T'` is the forecast horizon). As detailed in Algorithm 1 (lines 15-24), each step involves:\n1.  A single forward pass of the Transformer encoder to get action probabilities.\n2.  Sampling an action (the quantized value for the next time step).\n3.  Updating the state window.\n\nThis process is repeated `T'` times. To generate a full probabilistic forecast, this trajectory sampling is repeated `N` times.\n\n**Comparison to Baselines:**\n*   **Autoregressive Models (e.g., Lag-Llama, DeepAR):** The inference procedure is fundamentally the same. These models also perform a forward pass of their network (often a Transformer) for each of the `T'` steps in an autoregressive manner. Therefore, the computational overhead for generating a single forecast sample is directly comparable, dominated by `T'` forward passes of a similar-sized neural network.\n*   **Diffusion/Flow Models:** These models often have different inference procedures. Diffusion models, for example, require an iterative denoising process over many steps (often more than the forecast horizon `T'`) to generate a sample, which can be computationally more expensive.[8,9,10]\n\nTherefore, the inference cost of Temporal GFN is on par with state-of-the-art autoregressive models like Lag-Llama and Chronos and is generally more efficient than sampling from diffusion-based counterparts. We will add a paragraph to Appendix E clarifying this comparison.\n\n**Q2: Choice of Baselines in Experiments:**\n\nYou asked for the rationale behind our choice of baselines in Table 2 and why models like Diffusion/Flow forecasters were not included.\n\n*   **Rationale for Chosen Baselines:** Our goal was to position Temporal GFN against the current, most competitive, and widely-used paradigms in time series forecasting. We selected:\n    *   **Foundation Models (Chronos, Lag-Llama):** These represent the state-of-the-art in large, pre-trained models for forecasting. Chronos is an especially relevant comparison as it also tokenizes the time series, albeit with a fixed, non-adaptive scheme.\n    *   **SOTA Task-Specific Models (MOREI, TimeFM):** These are powerful, recently-proposed Transformer-based models that achieve top performance on benchmarks, representing the forefront of models trained specifically on the forecasting datasets.\n\n*   **Regarding Diffusion/Flow Models:** We agree that these are an important class of generative models, and we cite them in our Related Work (Section 2.1). We omitted them from the main empirical comparison for two key reasons:\n    1.  **Different Paradigm:** These models have fundamentally different training and sampling mechanisms, often with significantly higher computational demands, making a direct, \"apples-to-apples\" comparison of performance-per-compute difficult.[7,8]\n    2.  **Clarity of Contribution:** Our primary aim was to demonstrate that the GFN framework, with our proposed adaptive quantization, can outperform top autoregressive and sequence-modeling approaches. Including a very different class of models could obscure this direct comparison.\n\nWe believe our chosen baselines provide a robust and challenging benchmark for our method. We will add a note to the experimental section to clarify this choice and our reasoning.\n\nWe thank you again for your valuable feedback. We hope these clarifications have addressed your concerns and better highlighted the technical contributions and rigor of our work. We will incorporate these points into the final manuscript to improve its clarity and completeness.\n\n\nSources:\n[1](https://www.cs.utexas.edu/~pstone/Papers/bib2html-links/AAAI21-jiang.pdf)\n[2](https://arxiv.org/abs/2007.01498)\n[3](https://jdeshmukh.github.io/Papers/iros19.pdf)\n[4](https://arxiv.org/abs/2502.08302)\n[5](https://www.researchgate.net/publication/365249275_Multivariate_Time-Series_Data_Generation_in_Generative_Adversarial_Networks)\n[6](https://kdd-milets.github.io/milets2021/papers/MiLeTS2021_paper_7.pdf)\n[7](https://medium.com/coding-nexus/autoregressive-vs-diffusion-large-language-models-llms-a-deep-dive-a41da6da0875)\n[8](https://arxiv.org/html/2507.15857v1)\n[9](https://proceedings.mlr.press/v139/rasul21a/rasul21a.pdf)\n[10](https://arxiv.org/pdf/2101.12072)"}, {"review_id": "fcl1vYSUbV", "reviewer": "Reviewer_dKzj", "summary": "The paper introduces Temporal Generative Flow Networks (Temporal GFNs), transferring GFNs to sequences. The model is trained by optimizing a reward function based on the mean squared error via the Straight Through estimator, making the discrete sampling differentiable. Furthermore, an adaptive quantization scheme is proposed, increasing the number of discretization bins during training based on improvements over the last improvements and the model's predictive entropy. The model achieves state-of-the-art results across various metrics and outperforms current foundation models.", "questions": "- L218-223: \"Slow improvement suggests the current quantization might be too coarse, thus pushing $\\eta_e >1$ to encourage increasing K.\" Couldn't it just indicate convergence? \"High confidence might indicate premature convergence or insufficient exploration.\" How do you distinguish premature convergence from true convergence?\n- L225-226: Do you reuse the previously learned weights of the final layer when adding new bins, or do you discard them and reinitialize $W_f$ and $b_f$?\n- Why does adaptive quantization result in worse probabilistic metrics (Table 3) for K=10? Does this mean the adaptive update factor is not optimally chosen? It would be insightful to see a more thorough analysis for $K$.\n- Did you ablate different reward functions, e.g., based on the quantile loss?\n- How is the CRPS computed?\n\nMy primary concern is the correlation between the number of quantization bins and probabilistic metrics. I am willing to increase my score if this is sufficiently addressed.", "limitations": "Limitations are discussed in Section 5, but I encourage the authors to include an explicit limitation paragraph.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n\n- The clarity of the paper makes it easy to follow and understand. The method is clearly explained and accompanied by a pseudo-code.\n- The paper proposes a novel model transferring generative flow networks to sequential data and ensures differentiability via the straight-through estimator.\n- The authors provide sufficient experiments (e.g., Gumbel Softmax vs Straight Through), insights, and theoretical analyses in the appendix.\n- The authors propose an adaptive quantization to address the limitations of fixed quantizations in previous foundation models [1], posing a natural extension and promising research direction.\n\n### Weaknesses\n\n- Adaptive quantization does not always lead to improvements (see Table 3 for K=10), raising questions about the choice of $\\eta_e$. Also, increasing K worsens the probabilistic metrics. It would be interesting to see whether a different reward function based on the quantile loss improves the probabilistic metrics. In general, it surprised me to have a reward function based on the mean-squared error for quantized sequences.\n- The code is not provided, making it harder to check implementation details.\n- The architecture and hyperparameters are not detailed enough.\n\n[1] Ansari, Abdul Fatir, Lorenzo Stella, Caner Turkmen, Xiyuan Zhang, Pedro Mercado, Huibin Shen, Oleksandr Shchur et al. \"Chronos: Learning the language of time series.\" arXiv preprint arXiv:2403.07815 (2024).", "was_revised": true, "final_justification": "The paper has a clear contribution. It extends generative flow networks to sequences and addresses limitations of models with fixed quantization. My questions and concerns were thoroughly addressed during the rebuttal. For a higher score, the contribution would need to be stronger.", "rebuttal": "Dear Reviewer dkZj,\n\nWe sincerely thank you for your detailed and insightful review of our work. Your constructive feedback has been invaluable in helping us identify areas for clarification and improvement. We are encouraged that you found our paper \"technically solid\" and have carefully considered all your points. Below, we provide a point-by-point rebuttal to your concerns and questions, and we will commit to integrating these clarifications into the final manuscript.\n\n---\n\n### **Rebuttal to Weaknesses**\n\n**1. On the performance of adaptive quantization and the effect of K.**\n\nWe thank you for this sharp observation regarding the results in Table 3. This highlights the fundamental trade-off our work addresses: the balance between quantization precision (favoring large K) and the learnability of the policy in a sparse action space (favoring small K).\n\n*   **Adaptive vs. Fixed K=10:** You correctly note that for a starting K of 10, the adaptive model shows worse probabilistic metrics (CRPS/WQL) than the fixed model. We attribute this to the initial number of bins being too coarse. When K is very small, the quantization error is large, providing a noisy reward signal. The adaptive mechanism, trying to optimize bin placement based on this noisy signal, can lead to instability in learning the full distribution, even while improving the point-wise MASE metric. This suggests that for adaptation to be effective, the initial quantization must be reasonably, though not perfectly, fine-grained.\n\n*   **Adaptive vs. Fixed K=20:** This is precisely why we show the K=20 results. When starting with a more adequate number of bins (K=20), the adaptive mechanism demonstrates its true strength. It successfully improves upon the fixed K=20 model across all metrics (CRPS from 0.1921 to 0.1845, WQL from 0.2521 to 0.2385, and MASE from 0.9721 to 0.9532). This supports our claim that dynamic adaptation is beneficial when initialized in a reasonable regime.\n\n*   **Increasing K Worsening Probabilistic Metrics:** Your observation that increasing K from 10 to 20 in the *fixed* case worsens CRPS/WQL is a key motivation for our work. A larger K creates a sparser, higher-dimensional action space, making credit assignment and exploration harder for the GFN. Our adaptive approach, particularly the `Adaptive K=20` configuration, mitigates this by starting with a larger K and then intelligently refining it, ultimately achieving better performance than the fixed K=20 model.\n\nIn summary, the results showcase that adaptive quantization is not a universal fix for any arbitrary starting K, but a powerful curriculum-based learning strategy that refines quantization when the initial resolution is adequate. We will clarify this nuance in Section 4.2.\n\n**2. On the Reward Function based on Mean-Squared Error.**\n\nWe appreciate you raising this point and apologize if our description was unclear. A crucial detail of our method is that **the reward is calculated based on the *soft* samples (`a^soft`), not the discrete quantized values (`a^hard`)**.\n\nAs defined in Section 3.3.2 and elaborated in Section 3.2.3, the soft sample `a^soft` is the continuous, differentiable expectation of the action under the current policy (`a^soft = Σ q_k * P_F(a=q_k|s_t)`). The reward function `R(τ)` therefore evaluates the MSE between a sequence of these continuous expected values and the ground-truth sequence. This provides a smooth, dense, and informative gradient signal for policy learning, which is much more effective than a reward based on the discrete, non-differentiable `a^hard` samples. We will revise Section 3.3.2 to make this critical distinction explicit and prominent.\n\n**3. On Code and Hyperparameter/Architecture Details.**\n\nWe agree that reproducibility is important.\n\n*   **Code:** We give our firm commitment to release the full source code, experiment scripts, and instructions to reproduce all results upon publication of the paper, in line with NeurIPS guidelines and open science principles. We will add a note in the paper to this effect.\n\n*   **Details:** You are right that more details are needed. We will significantly expand Appendix E. To provide immediate clarity, our policy network is a Transformer encoder with 4 layers, 8 attention heads, and a model dimension of 256. Key hyperparameters for the adaptive mechanism were tuned in ranges: `λ` ∈ [0.1, 1.0], `ε` ∈ [0.01, 0.05], and the entropy weight `λ_entropy` ∈ [0.001, 0.05]. We will add a comprehensive table with all hyperparameters for each experiment in the appendix.\n\n---\n\n### **Responses to Questions**\n\n**Q1: Distinguishing slow vs. premature convergence in the adaptive update.**\n\nThis is an excellent and subtle question. Our adaptive mechanism does not explicitly disambiguate these scenarios but rather uses coupled heuristics that work synergistically.\n\n*   The **Improvement Signal** (`ε - ΔRe`) acts as a form of structured exploration. If progress has stalled (low `ΔRe`), we speculatively increase `K`. If the cause was a coarse quantization, this provides new, finer-grained actions that can unlock higher rewards. If the cause was simply slow network convergence, the larger action space might temporarily increase difficulty, but the policy can learn to ignore irrelevant bins. It is a bet that insufficient representational capacity is a common cause of plateaus.\n\n*   The **Confidence Signal** (`1 - He`) provides a crucial complementary heuristic. If the policy has low entropy (is highly confident), it may have converged prematurely to a suboptimal mode. Increasing `K` in this state probes the action space, allowing the model to discover better solutions in the vicinity of the current mode and escape local optima. If convergence was genuine, the policy will quickly learn to concentrate its mass back into the best bin, and the low entropy state will be restored.\n\nThese two signals work in tandem (as seen in Figure 10) to guide the curriculum. A model that has converged well on a given granularity will exhibit both low reward improvement and low entropy, triggering an increase in `K` to seek even greater precision. We will add this detailed reasoning to Section 3.2.2.\n\n**Q2: Reusing weights when adding new bins.**\n\nThank you for asking for this important implementation detail. When the number of bins `K` increases, we resize the final linear output layer `(WF, bF)`. The weights corresponding to the existing bins are **retained**, allowing the model to build upon its learned knowledge. The new weights for the newly created bins are initialized to zero. This prevents catastrophic forgetting and enables a stable curriculum. We will add this clarification to Section 3.2.2 and the appendix.\n\n**Q3: Why adaptive quantization results in worse probabilistic metrics for K=10.**\n\nThis question is related to Weakness #1. As discussed above, we posit that when starting with a very coarse quantization (K=10), the reward signal is too noisy for the adaptive mechanism to robustly improve the distribution representation, leading to a degradation in CRPS/WQL. The mechanism performs as intended when starting with a more reasonable resolution, as shown by the `Adaptive K=20` results, which improve upon `Fixed K=20`.\n\n**Q4: Ablating different reward functions (e.g., quantile loss).**\n\nThis is an excellent suggestion for future investigation. We chose the exponential of the negative MSE on soft samples because it's a robust, non-negative (`R(τ) > 0`), and smooth reward function, which is a standard requirement for GFNs.\n\nIntegrating quantile loss as a reward is non-trivial. A quantile loss is typically a direct training objective. To formulate it as a trajectory-level reward for a GFN, one would need to carefully design a function that aggregates quantile-based errors over a full trajectory into a single positive scalar, which is an interesting research direction in itself. We will add a discussion of this as a promising avenue for future work in the conclusion.\n\n**Q5: How the CRPS is computed.**\n\nThe Continuous Ranked Probability Score (CRPS) is computed empirically from the collection of generated forecast trajectories. For each time step in the forecast horizon, we have a set of `N` sampled values `{z_1, ..., z_N}`. This set forms an empirical Cumulative Distribution Function (CDF), `F(x)`. The CRPS is the integrated squared difference between this empirical forecast CDF and the true value `y`'s CDF, which is a Heaviside step function `H(x-y)`. We use standard, publicly available libraries (e.g., `properscoring`) for this calculation. We will add this explicit definition and citation to Appendix E.\n\nWe hope these detailed responses have thoroughly addressed your concerns. Your feedback has been instrumental, and we will integrate these clarifications and additional details into the revised manuscript to improve its clarity and completeness. We believe these changes will strengthen the paper significantly and respectfully ask you to consider our rebuttal in your final evaluation.\n\nThank you once again for your time and expertise."}]

Author rebuttal:

Meta-review: This paper introduces Temporal GFlowNets for continuous-valued time series forecasting, with an adaptive quantization mechanism that dynamically adjusts discretization bins during training. The approach addresses a key limitation of existing GFlowNets and discretization-based models, offering improved flexibility and interpretability in sequential modeling.

The reviewers were initially concerned about clarity, scope (e.g., restriction to univariate forecasting), missing ablation studies, and limited detail on implementation. However, the authors’ rebuttal provide a clear and systematic plan to address these issues: clarifying the reward function and adaptive quantization mechanism, analyzing adaptive behavior, expanding limitations and future work (including multivariate extensions), and improving reproducibility through detailed appendices and a code release commitment. These revisions directly resolve the main technical and clarity-related concerns raised in the reviews.

All three reviewers ultimately rated the paper positively (Accept), highlighting the novelty of applying GFlowNets to continuous time forecasting, the strong motivation, and the solid empirical evaluation. Importantly, the adaptive quantization strategy is well-motivated and represents a meaningful advance in bridging discrete generative flows with continuous-valued sequence prediction.

In sum, the paper makes a clear and technically solid contribution with good potential impact. I recommend acceptance.
Decision: Accept (poster)