Converge-based Quad-Balancing (CQB) in MiM-StocR

• The paper introduces CQB, a novel training-time mechanism that integrates gradient normalization, smoothing, and convergence-based adaptations to balance multi-task learning in stock recommendation.
• CQB balances four key aspects—gradient magnitudes, EMA smoothing, dynamic forgetting rates, and L2 regularization—by leveraging train-validation loss convergence to mitigate overfitting in volatile markets.
• Empirical results demonstrate that MiM-StocR with CQB outperforms standard MTL baselines, achieving superior IC and RankIC metrics while maintaining stable validation performance.

Converge-based Quad-Balancing (CQB) is a multi-objective optimization component introduced within the Momentum-integrated Multi-task Stock Recommendation framework (MiM-StocR) for stock recommendation (Wang et al., 5 Aug 2025). It is a training-time mechanism for two-task multi-task learning (MTL) that controls how task gradients are aggregated and how regularization is adapted during optimization. In MiM-StocR, the two tasks are one-day return ratio regression and momentum-line classification, and CQB addresses the problem that conventional MTL training can fail to balance trend prediction, ranking quality, and generalization under volatile stock-market conditions. The method is termed “Quad-Balancing” because it jointly balances four aspects: gradient magnitudes across tasks, gradient smoothing over time with exponential moving averages (EMA), convergence-based adaptation of the EMA forgetting rate, and convergence-based adaptation of L2 weight decay (Wang et al., 5 Aug 2025).

1. Definition within the MiM-StocR framework

CQB operates in a hard-parameter-sharing MTL setting with shared parameters $\theta$ and task-specific parameters $\psi_r$ and $\psi_c$ (Wang et al., 5 Aug 2025). The regression task predicts the one-day return ratio, while the classification task predicts a 5-level momentum line indicator with labels $0$–$4$. The regression loss is mean squared error, $\mathcal{L}_r = \text{MSE}$, and the classification loss is a composite objective: $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$ CQB does not alter these loss definitions; rather, it governs how the corresponding task gradients are smoothed, normalized, combined, and regularized during optimization (Wang et al., 5 Aug 2025).

MiM-StocR itself is organized into three stages: task preparation, multi-task learning, and multi-objective optimization. CQB belongs to the third stage. In this role it functions as a multi-task optimization, loss-reweighting, and scheduling mechanism for the shared parameters, rather than as an architectural change to the feature extractor or prediction heads (Wang et al., 5 Aug 2025). This division is significant because it isolates CQB as an optimizer-side contribution rather than a representational one.

The term “Converge-based” refers specifically to the fact that two of CQB’s balancing mechanisms depend explicitly on measured convergence and overfitting signals derived from training and validation losses. This makes convergence measurement central to the method’s design rather than a peripheral diagnostic (Wang et al., 5 Aug 2025).

2. Four balancing dimensions

CQB balances four aspects of optimization simultaneously (Wang et al., 5 Aug 2025). The first is multi-task gradient magnitude balancing. The second is gradient smoothing, which regulates the trade-off between noise and stability. The third is convergence-based control of the EMA forgetting rate on a per-task basis. The fourth is convergence-based control of L2 regularization strength.

Gradient smoothing begins with an EMA applied separately to the per-task gradients. For a generic gradient at iteration $\ell$,

$$\hat{g}_{\ell} = \beta \hat{g}_{\ell-1} + (1 - \beta) g_{\ell},$$

where $g_{\ell}$ is the mini-batch gradient, $\psi_r$0 is the previous EMA-smoothed gradient, $\psi_r$1 is the updated smoothed gradient, and $\psi_r$2 is the forgetting rate. CQB maintains separate smoothed gradients $\psi_r$3 and $\psi_r$4 for regression and classification, respectively (Wang et al., 5 Aug 2025).

To address magnitude imbalance between tasks, CQB normalizes the smoothed task gradients and rescales them using the maximum $\psi_r$5 norm among the tasks: $\psi_r$6 Here $\psi_r$7 is the aggregated gradient used to update the shared parameters $\psi_r$8 (Wang et al., 5 Aug 2025). The stated intuition is that each task’s gradient is first normalized to unit norm so that one task does not dominate purely because of scale, after which the combined direction is rescaled by the strongest task norm so that the resulting update remains comparable in magnitude to the most influential task.

The remaining two balancing dimensions are convergence-aware. CQB adapts the EMA forgetting rate based on per-task convergence indicators and adapts weight decay based on overall convergence. This makes the optimizer responsive not only to instantaneous gradients but also to epoch-level evidence of overfitting (Wang et al., 5 Aug 2025).

3. Convergence-aware adaptation

CQB measures convergence using a relative convergence rate $\psi_r$9 computed from the relationship between recent training-loss change and recent validation-loss change: $\psi_c$0 where

$\psi_c$1

with $\psi_c$2 (Wang et al., 5 Aug 2025). The method uses separate convergence indicators for the two tasks, $\psi_c$3 and $\psi_c$4, computed from the regression and classification train/validation losses.

The interpretation given is that when both training and validation losses decrease at similar rates, $\psi_c$5. When training loss continues to decrease but validation loss stops improving or increases, $\psi_c$6 drops, indicating overfitting (Wang et al., 5 Aug 2025). Practically, the method starts computing $\psi_c$7 only after sufficient loss history is available, from epoch 12 onward; before that, $\psi_c$8 is set to $\psi_c$9.

Task-specific forgetting rates are then defined as

$0$0

with base forgetting rate $0$1 and $0$2 the sigmoid function. In the two-task setting this yields

$0$3

These values replace the fixed $0$4 in the per-task EMA updates for all iterations within epoch $0$5 (Wang et al., 5 Aug 2025). The paper states that when overfitting occurs, the forgetting rate $0$6 monotonically increases with the decrease of $0$7, reducing the weight of the current gradient term $0$8 in the EMA equation and thereby mitigating overfitting.

The convergence signal also controls dynamic L2 regularization. CQB uses Adam with epoch-specific weight decay: $0$9 where the base weight decay is $4$0 and the mean is taken across tasks (Wang et al., 5 Aug 2025). The intended effect is that when validation performance deteriorates relative to training, the effective weight decay increases, imposing stronger shrinkage and reducing overfitting.

A plausible implication is that CQB couples local gradient dynamics and slower generalization diagnostics into a single control scheme. In the formulation given, the same convergence measurement governs both temporal smoothing and parameter regularization, allowing the optimizer to react to overfitting without changing the task losses themselves.

4. Role in parameter updates

CQB directly mediates updates of the shared parameters $4$1 in MiM-StocR. According to the appendix algorithm, the optimizer takes as inputs the learning rate $4$2, base $4$3, number of epochs $4$4, iterations per epoch $4$5, and losses $4$6, with $4$7, $4$8, and $4$9 initialized randomly (Wang et al., 5 Aug 2025).

At the beginning of each epoch $\mathcal{L}_r = \text{MSE}$0, the effective weight decay is set as

$\mathcal{L}_r = \text{MSE}$1

For each iteration $\mathcal{L}_r = \text{MSE}$2, CQB computes log-transformed shared-parameter gradients: $\mathcal{L}_r = \text{MSE}$3 It then updates the task-specific forgetting rates using the previous epoch’s convergence indicators: $\mathcal{L}_r = \text{MSE}$4 These are used to update the EMA-smoothed gradients $\mathcal{L}_r = \text{MSE}$5 and $\mathcal{L}_r = \text{MSE}$6, after which the balanced shared gradient $\mathcal{L}_r = \text{MSE}$7 is formed through the normalized max-norm aggregation described earlier (Wang et al., 5 Aug 2025).

The shared parameters are updated by

$\mathcal{L}_r = \text{MSE}$8

while the task-specific parameters are updated independently by their own log-loss gradients: $\mathcal{L}_r = \text{MSE}$9

$$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$0

At the end of each epoch, the model is evaluated on the training and validation sets for both tasks, and the convergence indicators are updated from the recorded loss histories (Wang et al., 5 Aug 2025).

This training loop makes clear that CQB affects the shared representation rather than the task heads. The task heads remain task-local, whereas CQB is the coordination layer through which the tasks influence shared learning.

5. Relation to momentum integration and ranking loss

MiM-StocR combines regression with a momentum-line classification task designed to improve short-term trend capture and stock ranking (Wang et al., 5 Aug 2025). In the task-preparation stage, the regression label is the one-day return ratio,

$$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$1

and the classification label is a 5-level momentum line indicator derived from short-term momentum $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$2 across a window. The classification task also incorporates Adaptive-k ApproxNDCG, with

$$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$3

and adaptive cutoff

$$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$4

where the threshold is set to 20% of stock-pool size (Wang et al., 5 Aug 2025).

The paper explicitly states that the momentum-line classification task and Adaptive-k ApproxNDCG define a relatively complex and noisy classification loss that can be prone to overfitting, especially on the validation set. CQB is intended to counter this by using the classification convergence indicator $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$5 to slow learning from the classification gradient when overfitting begins, while also increasing weight decay to regularize the model more strongly (Wang et al., 5 Aug 2025). The same logic applies to the regression task through $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$6.

This interaction is important because CQB is not presented as a generic task-balancing heuristic detached from the financial setting. Rather, it is motivated by the need to stabilize joint optimization when one task emphasizes ranking-aware classification and the other focuses on scalar return prediction under highly non-stationary time-series conditions (Wang et al., 5 Aug 2025). This suggests that CQB’s principal significance within MiM-StocR lies in enabling the simultaneous use of momentum-derived supervision and ranking-sensitive objectives without allowing either to destabilize shared training.

6. Comparison with other multi-task balancing methods

The paper compares MiM-StocR with CQB against several MTL baselines: Equal Weighting (EW), DB-MTL (Dual-Balancing) ascribed to Lin et al. 2023, and CAGrad (Conflict-Averse Gradient Descent) ascribed to Liu et al. 2021 (Wang et al., 5 Aug 2025). EW uses simple average task weighting. DB-MTL normalizes gradients and includes a more elaborate balancing mechanism. CAGrad seeks a common descent direction that reduces gradient conflict.

The principal distinction drawn is that prior methods mainly rely on training-side quantities such as loss magnitude, gradient norms, or gradient conflicts, whereas CQB explicitly incorporates validation performance to detect overfitting and steer optimization (Wang et al., 5 Aug 2025). The method also differs in using a dynamic per-task EMA forgetting rate rather than fixed smoothing or per-task learning-rate control. In addition, it jointly controls gradient smoothing and weight decay using the same convergence signal, while many prior methods focus only on task weighting or gradient combination.

The paper therefore locates CQB’s novelty less in gradient normalization alone, which it describes as related to existing work, and more in the convergence-based adaptation of EMA and regularization. In this framing, the “converge-based” element is the distinctive feature (Wang et al., 5 Aug 2025).

A common misconception would be to treat CQB as merely another gradient-normalization rule. The formal description does not support that narrower reading. Normalization is only one of four balancing dimensions, and the method’s defining mechanism is the use of convergence and overfitting indicators from training-versus-validation dynamics to modify optimization behavior.

7. Empirical evidence, implementation, and limitations

Empirically, MiM-StocR, which includes CQB, is reported to outperform STL, EW, DB-MTL, and CAGrad on SEE50, CSI100, and CSI300 using LSTM, GATs, and HIST backbones (Wang et al., 5 Aug 2025). On CSI300 with HIST, the reported values are: STL with IC 0.0672 and RankIC 0.0630; DB-MTL with IC 0.0631 and RankIC 0.0599; CAGrad with IC 0.0638 and RankIC 0.0611; and MiM-StocR with IC 0.0667 and RankIC 0.0633. The paper characterizes these as best or tied for best results in that setting.

The ablation study on HIST with CSI300 isolates two CQB components. Without $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$7 balancing, but retaining L2 balancing, the model achieves IC 0.0656 and RankIC 0.0619. Without L2 balancing, but retaining $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$8 balancing, it achieves IC 0.0665 and RankIC 0.0625. The full MiM-StocR with CQB achieves IC 0.0667 and RankIC 0.0633 (Wang et al., 5 Aug 2025). The reported interpretation is that removing either adaptive forgetting-rate control or adaptive weight decay degrades performance.

The paper also cites Figure 1 as evidence on overfitting dynamics. EW, DB-MTL, and CAGrad reportedly show a significant rebound in validation and test losses after about 25 epochs while training loss continues to decrease, whereas CQB keeps validation and test losses much flatter after that point (Wang et al., 5 Aug 2025). This is presented as direct evidence that the convergence-based mechanisms mitigate overfitting.

For reproduction, the implementation details given are specific. The optimizer is Adam with dynamic weight decay. The learning rate is $$\mathcal{L}_c = 0.5\,\mathcal{L}_{\text{cross entropy}} + 0.5\,\mathcal{L}_{ndcg}.$$9. The base forgetting rate is $\ell$0. The EMA is initialized with $\ell$1. The convergence window uses $\ell$2, with $\ell$3 computed from epoch 12 onward and set to 1 before that. The base decay is $\ell$4. Training runs for up to 100 epochs with early stopping if the best validation score is not improved for 30 epochs. Batch sizes are not explicitly specified, only that standard values consistent across baselines are used (Wang et al., 5 Aug 2025).

The paper does not list explicit failure modes, but it indicates several limitations and directions. CQB is tailored to stock time series where distribution shifts and overfitting are severe. It uses validation data within the optimization loop, which is straightforward in offline backtesting but potentially more complex in online or streaming settings. It is described and evaluated only in a two-task design, although the paper states that extension to more tasks is conceptually straightforward. It also notes that alternative convergence metrics could be explored and that CQB is model-agnostic enough to be tried in other domains where overfitting and task imbalance are problematic (Wang et al., 5 Aug 2025). A plausible implication is that the method’s transfer beyond stock recommendation depends less on architecture than on whether similar train/validation divergence provides a useful control signal for optimization.