TRiCo: Triadic Game-Theoretic Co-Training for Robust Semi-Supervised Learning
Authors: Hongyang He
Xinyuan Song
Yangfan He
Zeyu Zhang
Yanshu Li
Haochen You
Lifan Sun
Wenqiao Zhang
Venue: NeurIPS 2025 poster ()
Keywords: Semi-supervised learning
Primary area: general_machine_learning
TL;DR: A novel semi-supervised learning paradigm that unifies view-wise co-training, meta-learned supervision, and adversarial perturbation through a structured triadic game.

Abstract: We introduce TRiCo, a novel triadic game-theoretic co-training framework that rethinks the structure of semi-supervised learning by incorporating a teacher, two students, and an adversarial generator into a unified training paradigm. Unlike existing co-training or teacher-student approaches, TRiCo formulates SSL as a structured interaction among three roles: (i) two student classifiers trained on frozen, complementary representations, (ii) a meta-learned teacher that adaptively regulates pseudo-label selection and loss balancing via validation-based feedback, and (iii) a non-parametric generator that perturbs embeddings to uncover decision boundary weaknesses. Pseudo-labels are selected based on mutual information rather than confidence, providing a more robust measure of epistemic uncertainty. This triadic interaction is formalized as a Stackelberg game, where the teacher leads strategy optimization and students follow under adversarial perturbations. By addressing key limitations in existing SSL frameworks—such as static view interactions, unreliable pseudo-labels, and lack of hard sample modeling—TRiCo provides a principled and generalizable solution. Extensive experiments on CIFAR-10, SVHN, STL-10, and ImageNet demonstrate that TRiCo consistently achieves state-of-the-art performance in low-label regimes, while remaining architecture-agnostic and compatible with frozen vision backbones.

TRiCo: Triadic Game-Theoretic Co- Co-Training for Robust Semi-Supervised Learning
Hongyang He!* Xinyuan Song?+, Yangfan He Zeyu Zhang', Yanshu Li,, Haochen You?, Lifan Sun', Wenqiao Zhang
University of Warwick
Emory University
University of Minnesota Twin Cities
4 ANU Brown University
"Columbia University TUCSD *Zhejiang University
Manifolda.Ai
Abstract
We introduce TRiCo, novel triadic game- theerete co-training framework that rethinks the structure of semi-supervised learning by incorporating teacher, two students, and an adversarial generator into unified training paradigm. Unlike existing co-training or teacher-student approaches, TRiCo formulates SSL as a structured interaction among three roles: (i) two student classifiers trained on frozen, complementary representations, (ii) a meta-lcarned teacher that adaptively regulates pseudo-label selection and loss balancing via validation-basse feedback, and (iii) non-parametric generator that perturbs embeddings to uncover decision boundary weaknesses. Pseudo-labels are selected based on mutual information rather than confidence, providing more robust measure of epistemic uncertainty. This triadic interaction is formalized as Stackelberg game, where the teacher leads strategy optimization and students follow under adversarial perturbations. By addressing key limitations existing SSL frameworks interactions, unreliable pseudo-labels, and lack of hard sample modeling- -RRioo provides a principled and generalizable solution. Extensive experiments on CIFAR- 10, SVHN, STL-10, and ImageNet demonstrate that TRiCo consistently achieves wate-of-theear performance in low-label regimes, while remaining architecture- agnostic and compatible with frozen vision backbones.
as static view
Introduction
AGA Way
Sunder
(a)
(c)
(b)
Figure 1: (a) Accuracy comparison across label budgets on CIFAR-10, (b) Training dynamics of TRiCo vs. MCT and baselines on ImageNet-10%, (c) Comparison with fully-supervised top- performing models under 25% labeled data.
'Equal contribution. Work was done while the authors were with the Manifolda.Ai Research team. Corre- sponding author: hongyang_ he warwick.ac.ut
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Semi-supervised learning (SSL) has become a key strategy for leveraging large amounts of unlabeled data in low-label regimes, particularly when manual annotation is costly or impractical [12]. Among various SSL paradigms, co-training stands out for its conceptual simplicity and empirical effective- ness: it encourages two models to exchange pseudo-labels across complementary views, mitigating confirmation bias and enabling mutual refinement. This approach has seen success in vision, language, and multimodal tasks. However, despite its promise, traditional co-training frameworks still fall short in real-world settings involving noisy pseudo-labels, data imbalance, and dynamically evolving training dynamics [35, 25].
Three core challenges limit the broader applicability of co-training in modern SSL scenarios. First, pseudo-label selection is typically driven by fixed confidence thresholds, which are brittle and susceptible to calibration errors, especially in early training or under distributional shift. This often results in overconfident but incorrect labels, reinforeing mutual errors across views and leading to semantic collapse. Second, co-training assumes symmetric, static interaction between views, disregarding the inherent heterogeneity in model capacities, representation quality, or learning speeds. Without adaptive regulation, view interactions can stagnate or even harm generalization. Third, current frameworks lack mechanisms to actively surface hard examples near decision boundaries. Since pseudo-labels tend to be dominated by easy examples, models may overfit to high-confidence regions and fail to explore the uncertainty space that truly drives robustness.
To address these critical limitations, we propose TRiCo, triadic game- theortte co- co-training frame- work for robust semi- supervised learning. TRiCo rethinks the structure of co-training by introducing a third player-a teacher-into the interaction between two student classifiers operating on frozen, complementary views. Instead of using confidence-based heuristics, TRiCo filters pseudo- labels using mutual information (MI), principled uncertainty measure that better reflects epistemic relia- bility [24]. A meta-learned teacher dynamically adjusts the MI threshold and loss weighting scheme by observing how its decisions affect student generalization on labeled validation data, effectively governing the training process through a Stackelberg game formulation. Meanwhile, a non-parametric generator perturbs embeddings to uncover regions of high uncertainty, forcing the students to confront and learn from adversarially hard samples. These components form a synergistic loop: the teacher regulates secudo-label flow, the generator challenges model boundaries, and the students co-train under adaptive supervision.
TRiCo offers a principled solution to long-standing problems in co-training by unifying epistemic uncertainty modeling, curriculum-aware optimization, and hard-sample mining into single training paradigm. As shown in Figure la, our method consistently outperforms representative SSL baselines across different label budgets on CIFAR-10, demonstrating superior data efficiency. Furthermore, Figure 1b illustrates how TRiCo maintains faster and more stable convergence than both its under- lying views and prior meta co-training approaches on ImageNet- 10%, evidencing the benefit of triadic interaction and adaptive supervision. Remarkably, even with only 25% of labeled ImageNet data, TRiCo achieves top-1 1 accuracy competitive with the best fully-supervised large-scale models (Figure lc). Our contributions are compatible with frozen backbone encoders, agnostic to downstream architectures, and generalizable across domains makinn TRiCo a robust and scalable solution to the next generation of superviee learning challenges.
2 Method
2.1
Overview of TRiCo
We propose TRiCo, a triadic game-theoretic co-training framework for semi-supervised learning that integrates complementary-vieew learning, adversarial perturbation, and meta-learned pseudo-label supervision. TRiCo consists of three interacting components: two student classifiers S1 and f2 trained on embeddings from distinct frozen vision encoders V1 and V2. non-amaaetrric generator G that exposes decision boundary vulnerabilities via embedding -spacn perturbations, and a meta-lcarned teacher Ty that dynamically controls training dynamics (see Figure 2). At each training step, the two students exchange pseudo-labels across views. These labels are filtered by mutual information (MI) to ensure epistemic reliability. Instead of using fixed thresholds, the teacher adaptively adjusts the MI threshold TMI and loss balancing coefficients (Au. Audv) by observing how its decisions influence student generalization on labeled data. In parallel, the generator perturbs embeddings to create adversarial examples that challenge the students' decision boundaries. These components work in concert: filtered pseudo-labels provide informative supervision, perturbations enhance robustness, and the teacher aligns all supervision signals toward generalization.

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Modile
Tea
Xtl
Pia
Labil' extaeled SeNGe
Denonatr
Figure 2: Overview of the TRiCo framework. Two student models and fa learn from different frozen views V1 and V2. Each view is passed through an entropy-guided generator to produce adversarial inputs, which are then filtered by meta-learned teacher Ty to generate pseudo-labela based on MI and confidence thresholds. All components interact via game-theoretic objectives to optimize robustness and generalization.
By formalizing this three-way wwy interaction bilevel where the teacher acts Stackelberg leader and students/generator as followers-TRica achieves robust semi-supervised learning even under limited labels or imbalanced distributions. full description of the components and the unified training algorithm is given in the sections that follow.
2.2
View-wise Co-Training
Our framework begins by constructing two complementary semantic views for each input instance using frozen pre-trained vision encoders. Given an image E X, we extract two embeddings (1) Vi (z) and 22(2) (2(x) using encoders V1 and 1/2 respectively, each trained on different pretext tasks (e.g., contrastive learning and masked image modeling). These views are fixed and provide low-dimensional, task-agnostic representations that are used as the input to two lightweight student models fi and f2. The architectural independence and pretraining diversity of V1 and V/2 ensure that (!) and .2(2) are sufficiently diverse and approximately conditionally independent given the label satisfying the classical assumptions of co- co-training.
Each student model is trained with supervision derived from the pseudo-labels produced by the other student. For instance, fi generates pscudo-labels for (-) that supervise f2 on 2x(2), and vice versa. However, instead using naive confidence-basee filtering, we evaluate the epistemic uncertainty of each prediction via its mutual information (MI). Specifically, for an input (1))) we perform K stochastic forward passes with dropout to compute an empirical predictive distribution p( (1) and estimate the mutual information:
(1( = (p) Σ
Pa (w)
(I)
This quantity captures the epistemic uncertainty by measuring how much predictions fluctuate across model samples. We only accept pseudo-labels when MT(x(()) > TMI, where TMI threshold provided by the teacher strategy. This mechanism is more robust than confidence-thressoodding particularly in early training stages or for ambiguous samples. The accepted pscudo-labels are then used in cross-view supervised loss:
Lunsup = E EDMI ((AA((()))))))) ((fe(x(")) g())
where g(i)= = arg max (00) (y) denotes the discrete pseudo-label from the opposite view.
To further improve decision boundary robustness, introduce perturbation-based generator that
creates hard examples by maximizing model uncertainty the embedding space. For each view
(()), we define adversarial perturbation (()) such that the perturbed embedding x9 = ((0 + (0)
_(6)

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maximizes the prediction entropy and MI:
g(0)= = arg max Se
[H([((x(0) 6)) + MT(fi(2(0) + +51]
This perturbation is computed via single-stor step or multi-step FGSM/PGD-sty gradient ascent, with- out training a generator model. These adversarial samples are then passed through the corresponding student to compute regularization loss:
Ladv = Erz. +
(4)
minimizing this loss encourages the model to make confident predictions even in high- figh-uncertainty regions, improving generalization and boundary sharpness. Together, the clean pseudo-labeled samples and the adversarially generated hard samples provide two complementary learning signals to train the students with stronger supervision and robustness.
2.3 Meta-Learned Teacher Strategy
A central component of the TRiCo framework is the teacher module Tr, which is responsible for adaptively controlling the pseudo-label filtering threshold TMI and the loss balancing coefficients A, and Aade throughout training. Unlike traditional pseudo-labeling schemes that rely on fixed thresholds or heuristies, our teacher meta-learned based on the principle that good pseudo-labeling strategies should ultimately lead to better student generalization labeled data. This feedback is used to update the teacher's ' strategy parameters in a principled and differentiable manner.
Formally, let A denote the parameters of the student model f. and Oy represent the parameters of the teacher strategy Tr. which includes TMI. Aus and Aade- During each training iteration, the teacher selects strategy 0r, which determines (i) which pseudo-labels are accepted via mutual information filtering. and (ii) how to weight the unsupervised and adversarial components of the loss. The student parameters are then updated using this current strategy by minimizing the total loss over unlabeled data:
8's 0- 17 Vas Duu + Audo
(5)
To evaluate whether the teacher's current strategy 0r is beneficial, we measure the supervised loss on a labeled validation batch using the updated student parameters 0's. This yields meta-objective for the teacher:
min (fu) 0r
(6)
However, since f' depends on Or through a gradient step, we adopt first-order approximation by unrolling one step of student update. The gradient of the validation loss with respect to 0r is computed using the chain rule:
0r t- 0T TJT Ceg (Jes-nVog Cutoup
(7)
This formulation allows the teacher to improve its strategy by minimizing the indirect effect its decisions have on the student's generalization performance. In practice, parameterize TMI. A and Aadv as a vector passed through a sigmoid activation, ensuring bounded outputs in [0, 1]. These parameters are initialized conservatively, such as TMI 0.05 and X, = Xadv 0.5, and are updated via gradient descent throughout training. We observe that the meta-gradient signal tends to stabilize after several warm-up steps, enabling the teacher to settle into dynamic equilibrium that balances supervision quality and diversity. This meta-learning approach transforms the role of the teacher from a static filter to active policy learner, capable of adapting the evolving dynamics of student learning and data uncertainty.
2.4 Unified Objective Function and Training Procedure
In addition to our multi-view representation and meta-learned pseudo-label filtering strategy, we unify the core components of TRiCo into a single end-to-end training procedure. At its core, TRiCo jointly optimizes two student models fi and fz on complementary views, using both cross-view pseudo-labed and adversarially perturbed embeddings, while adaptively guided by a meta-learned teacher. The teacher controls the mutual information threshold TMI for pseudo-label acceptance and the relative weights X₁, and Xadv between unsupervised and adversarial losses. Student models receive gradients from three loss terms: supervised loss on labeled data, unsupervised loss from filtered pseudo-labels, and adversarial consistency loss from perturbed embeddings. These signals

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are balanced via teacher-supples coefficients and jointly update the student parameters. The teacher updated using meta-gradients derived from the effect of its current decisions on the student's supervised loss on hold-out labeled batch. The complete training procedure is summarized in Algorithm
Algorithm TRiCo: Triadic Game-Theoree Co- Training
1: Input: labeled set Dj, unlabeled set Da. frozen encoders V/, 22, student classifiers f1, f2, teacher parameters 0r, perturbation bound e
2: for each training step l 1,..., do
3: zy D.,
4: (1(1) = VV((,,,, x[2) = V2((ra)
5:
Pi fi(x01), P2 = (x)
6:
MI = MI(Si(z(O)) via dropout
7:
, g(0) filter(M(()) TMI) from or
8:
Lunaup E(fi(x()) ),0()) )++ =((2(11()), (())
9:
8(0) arg max 10ll H(f((x()) +6)
10:
2g = z + 8(0)
(6)
Ladv = (21 (2(1))) + H((f((1))) (2))
11:
12:
(K), V Dr
13:
Loup (Vi (V(()))) () y1) + [(2(((2(2)))))))
14:
Lintal Lsup Xu Lunsup + Aadv Ladd
15:
f2 t- f - 11 Ltotal
16:
OT OT TT V By Suu (
17:
end for
2.5 Theoretical guarantee
Now we want to provide theoretical analysis for our method.
Assumption (Compact Strategy Spaces). Let the strategy spaces for the teacher HIr, the students Iss, and the generator Hg be such that their joint space I1 x IIs x Hg C R is compact.
Assumption 2 (Continuity Payoff Functions). The payoff functions are continuous with respect to their own strategies:
Teacher's payoff: RT(TT, J1, f2, G) = Accuracyyaa (J,, f2); TT, G) = X, Lumsup + X Cadvi
Students' payoff: Rs(fi, RT,
Generator's 's payoff: RG(G, NT, ft) = [(H(f((I++))
Each function is continuous in the argument corresponding to its player's strategy.
Theorem (Existence of Nash Equilibrium in Triadic Game). Under Assumptions I and 2, there exists a Nash equálibrium (xt fi fī, G*) E Il x IIg x Hg in the TRiCo framework such that: VoT Hr; Rr(nū fi f2, G*) 2 Rr(aT, f2,
Vfz E Tss Rs(si TT, G*) < Rs(Si, G*), RG ff) "1
(8)
VG Na,
Z Ra,G,=t;Ji
The proof is provided in Section A. Theorem establishes that, under the mild conditions of compact strategy spaces and continuity payoff functions (Assumptions 1-2), a Nash equilibrium exists in the proposed triadic game among the teacher, students, and generator. This guarantees that each agent can adopt a stable strategy where no unilateral deviation improves own outcome. This result ensures theoretical soundness of the TRiCo framework and supports the claim that our method yields a well-defined, stable training objective in multi-agent settings, making a robust choice for adversarial or cooperative learning scenarios.
3 Experiments
We implement TRiCo in PyTorch using 4 NVIDIA A100 GPUs. All models are trained on four benchmark datasets: CIFAR-10 [28]. SVHN [33], STL-10 [14], and ImageNet [18]. For CIFAR-10
5

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Method
CIFAR-10 10 (4k)
SVHN (1k)
Method
STL.-10
FixMatch [39]
94.3
92.1
Fix.Match [39]
89.5
UDA [45]
93,4
91.2
UDA [45]
88.6
FlexMatch [46]
94.9
92.7
FlexMatch [46]
90.1
Meta Pseudo Label [35]
95.1
93.5
Meta Pseudo Label [35]
90.6
TRiCo (Ours)
96.3
94.2
TRiCo (Ours)
92.4
Table I: Test accuracy (%) on CIFAR- 10 and SVHN with limited labeled data (4k and Ik respectively).
Table 2: Test accuracy (%) on STL-I0
using full labeled data and 100k unla- beled samples.
and SVHN, follow standard semi-supervised settings using 4,000 labeled examples (10%) and test on the full test set. For STL-10, we use all labeled and 100k unlabeled samples. On ImageNet, we evaluate two settings with 25%, 10% and 1% labeled subsets respectively, following protocols in Sohn et al. [39]. Across all datasets, we use a fixed labeled validation split (10% of labeled data) to compute the meta-grradies for teacher updates. All unlabeled images receive strong augmentation (RandAugment + Cutout + ColorJitter), while labeled images receive weak augmentation only.
We use V backbones frozen view encoders: V and 1/2 are initialized from DINOv2 and MAE, respectively, each producing a 768-dim embedding. These are fed into a two-layer MI.P student (fi/f2) with GELU. The generator applies single-step FGSM (e 1.0) embeddings, and mutual information is estimated via 5 Monte Carlo dropout passes.
Student models are trained using SGD with momentum 0.9 and batch size 64, with cosine learning rate decay starting from 0.03. The teacher parameters (TMI, Au Aadv) are updated via meta- gradient descent using a labeled validation batch, with initial values (0.05, 0.5, 0.5) and learning rate 0.01. Each experiment is run for 512 epochs, and we report mean accuracy over random seeds.
3.1 TRiCo Results
Compared to existing semi-supervised learning baselines, TRiCo consistently achieves superior performance across multiple datasets with limited labeled data, as shown Table and Table Notably, on CIFAR-10 with 4k labels and SVHN with 1k labels, TRiCo outperforms strong com- petitors such as FlexMatch and Meta Pseudo Label by clear margin, reaching 96.3% and 94.2% respectively. Similarly, on STL-10, TRiCo achieves 92.4%, surpassing the previous best by nearly points. These results validate the effectiveness of our triadic training framework in low-resource regimes, where high-quality pseudo-label filtering and adversarial regularization play critical role. We plot Figure the T-SNE visualization of the features on STL-110 with 40 labels from MCT and TRiCo. TRiCo shows better feature space than MCT with less confusing clusters. Table 3 presents
(a) MCT
(b) TRiCo
Figure 3: T-SNE visualization on STL-10. Left (a) Meta Co- Training (MCT); Right (b) TRiCo. Each color denotes a semantic class. TRiCo leads to compact and well-separated clusters in the embedding space, highlighting its superior representation quality.
comprehensive comparison on ImageNet under 1 %, 10%, and 25% labeled data settings. TRiCo consistently outperforms strong baselines across all regimes. Under the extremely low-label setting of 1 16, TRiCo achieves 81.2%, surpassing the previous best REACT (81.6%) while using simpler training pipeline without extra calibration modules. When the label proportion increases to 10% and 25%, TRiCo continues to improve, reaching 85.9% and 88.3%, respectively, outperforming recent

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Table Top-1 accuracy (%) on ImageNet under %, 10%, and 25% labeled data settings.
Model
Type
Method
1%
10%
25% %
FixMatch [39]
Consistency-based
Pseudo-labeling
52.6
68.7
74,9
UDA [45]
Consistency- based
Distribution alignment
51.2
67.5
73.8
FlexMatch [46]
Confidence-awann
Adaptive threshold
53.5
70.2
75.3
Meta Pscudo Label [35]
Meta-leamingg
Meta pseudo bseuli-labeling
55.0
71.8
76.4
SimCLLRv+KKD [13]
Self-supervised KD
Distill
54.5
69.7
75.5
Semi-ViT (ViT-B) [9]
Self-training
Self-labeled
74.1
81.6
84.2
Semi-ViT (ViT-L) [9]
Self-training
Self-labeled
77.3
83.3
85.1
Semi-ViT (ViT-H) [9]
Self-training
Self-labeled
78.9
84.6
86.2
REACT (ViT-L) [31]
Robust SSL
Distribution calibration
81.6
85.1
86.8
SemiFormer [44]
Semi-supervised ViT
Confidence teacher
75.8
82.1
84.5
DINO (ViT-L) [20]
Self-supervised
Linear head
78.1
82.9
84.9
Co-Training (MLP) [35]
Co-training
Two-view mutual labeling
80.1
85.1
MCT (MIP) [35]
Meta Co-training
Meta feedback
80.7
85.8
TRiCo (Ours)
Game-theoretic SSL.
TRiCo- Corrnnii
81.2
85.9
88.3
3-
CU7
EXM
vausllature
(b)
(c)
a8)
Figure 4: (a) TRiCo Cross view CAM visualization for selected samples. (b) Performance differences (TRiCo MCT) across combinations of self- supervied view pairs on ImageNet- %. (c) Evolution of teacher-controlled parameters TMI, Aus and Aadv training epochs.
self-training and co-training approaches such as Semi-ViT, MCT, and Meta Pseudo Label. Notably, TRiCo shows consistent advantages over both consistency- -bssed methods (e.g., FixMatch, UDA) and recent self-supervised + distillation pipelines (e.g.. SimCLRv2+KD), indicating that its triadic structure with meta-learned supervision and adversarial regularization enables more effective use of unlabeled data across range of supervision levels.
Figure 4a illustrates class activation map (CAM) visualizations across two complementary views for representative samples. Compared to prior methods, TRiCo yields more localized and semantically aligned regions both views, indicating improved cross-view consistency and decision boundary sharpness. The visual coherence across modalities suggests the effectiveness of our mutual pseudo- label exchange combined with entropy-guidee guidedd hard example exposure.
3.2 Ablation Studies
In Figure 4b, show the improvement margin (TRiCo minus MCT) across all pairwise combinations of self-supervised feature encoders on ImageNet-1%% %. TRiCo consistently surpasses MCT, with the DINOv2+MAE pair yielding the largest gain. This highlights how TRiCo better leverages heterogeneous representations, particularly when the view diversity substantial but complementary.
We conduct an extensive ablation study on CIFAR-10 with 4k labeled samples to dissect the contri- bution of each core component TRiCo. In Table 4, we examine different pseudo-label filtering strategies. Our mutual information-base criterion, denoted by threshold TMI, outperforms traditional confidence-based thresholds Teonf across a range of values. Unlike Tronfs which filters pseudo-labels solely based sofumax confidence, TMI captures epistemic uncertainty through dropout- based mu-

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tual information estimation. We observe that MI filtering yields higher accuracy and PGD robustness, especially early training, whereas stricter Teonf (e.g., 0.90) or lack of filtering leads degraded stability.
Table 5 evaluates the impact of our meta-leamed teacher module, which adaptively updates both TMI and the unsupervised loss weight λMI. When these parameters are fixed or manually tuned, performance and robustness decline significantly, demonstrating the necessity of dynamic scheduling via meta-gradient feedback.
In Table 6, assess the role of the entropy-driven generator. Removing this component substitut- ing it with random noise perturbations markedly reduces adversarial robustness. This confirms the generator's 's effectiveness in mining informative hard examples near decision boundaries and guiding more discriminative training. Lastly, Table 7 compares alternative architectural designs. Our full triadic configuration (with teacher and generator) achieves the highest accuracy and robustness. In contrast, simplified variants like 2-view co-training or a Mitchtthsttyle teacher-student model fall short. These results underscore the synergy between uncertainty-ave aware filtering, meta-lcarning, and adversarial training in the TRiCo framework.
Figure 4c visualizes the dynamic evolution of teacher-controlled hyperparameters TMI+ Aus and Aadv throughout training. The smooth adaptation and early-stage conservatism demonstrate that the meta-learned teacher successfully schedules supervision intensity based on student generalization, validating our Stackelberg game formulation for reliable co-training dynamics.
Table 4: Filtering Strategy
Setting
Acc
PGD
Stability
Table 5: Teacher Scheduling
MI Filtering (Ours)
95.9
82.1
Setting
Acc
PGD
Confidence Teonf 0.70
95.0
77.7
Confidence Teonf = 0.75
95.8
79.1
Meta-learned TMI, AMI
95.9
82.1
Confidence Tconf 0.80
92.6
78.9
Fixed: Tx1=0.05, AMI 0.5
94.7
79.0
Confidence Tconf 0.85
90.9
70.6
A
Fixed: Tx1=0.15, Am -09
0.9
87.5
65.2
Confidence Teonf 0.90
87.5
66.8
A
No Filtering
91.7
74.5
Table 6: Generator: Entropy-Guided vs. Noise- Based Perturbations
Table 7: Architecture: Triadic Co- Training and
others
Setting
Ace
PGD
Stability
Setting
Acc
PGD
Ours (Entropy 8)
95.9
82.1
V
TRiCo Full (ours)
95.9
82.1
No Generator
94.2
78.9
2- 2-View Only (no teacher)
94.1
78.4
Rand. Noise
70.5
66.4
FixMatch-style T-S
83.0
72.1
4 Few-Shot and Out-of-Distribution Generalization
While semi- supervipee learning typically assumes modest amount of labeled data from fixed dis- tribution, real-world deployments often face two challenges: (i) severely limited labeled supervision (few-shot), and (ii) distribution shifts between training and test data. To evaluate TRiCo under these more realistic conditions, we design two additional experiments: few-shot SSL and out-of-distribution generalization.
Few-Shot Seni-Supervieed Learning We adopt a standard few-shot protocol where only (1, 5, 10) labeled samples per class are available. The remainder of the training data is used unlabeled. We conduct experiments on two benchmarks: CIFAR-100 and CUB-200-2011, We compare against strong baselines including FixMatch, FrecMatch, Meta Pseudo Label (MPL), and Meta Co- Co-Training (MCT), reporting top-1 accuracy averaged over three runs.
Generalization To evaluate how well TRiCo generalizes under domain shift, we use standard cross-dataset generalization settings. Specifically, we train models on CIFAR-IU 10 (source) and test on CIFAR-10-C and STL-10 (target), both of which introduce shifts in style, resolution, and semantics. We follow the protocol of [27], evaluating accuracy on 15 types of corruptions with 5 severity levels in CIFAR-I0- and full test set of STL-10. All methods

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Table Few-shot semi- sumi-supeeiiee learning results on CIFAR-100 and CUB-200-2011. All models use frozen representations from the same backbones.
CIFAR-100
CUB-200
Method
1-shot
5-shot
10-shot
1-shot
5-shot
10-shot
FixMatch [39]
16.8
32.5
44,6
12.7
28.9
42.1
FreeMatch [43]
18.3
34.1
45.3
13.2
30.2
43.8
Meta Pscudo Label [35]
19.5
35.9
47.5
15.6
31.4
45.0
MCT [35]
21.2
38.1
49.3
17.0
33.6
46.8
TRiCo (Ours)
23.8
41.2
51.9
19.3
36.1
49.7
Table 9: Out-of-distribution generalization performance. Models are trained on CIFAR-IO with 10%
labels and tested on unseen domains.
Method
CIFAR-IO-C (mCE 4)
STL-10 (Ace 1)
FixMatch [39]
34.1
80.6
FreeMatch
32.3
81.5
Meta Pseudo Label [35]
30.9
82.1
MCT [35]
28.7
83.2
TRiCo (Ours)
26.1
85.3
trained using 10% labeled CIFAR-10 10 data with access to unlabeled target samples during training. TRiCo outperforms all baselines in both settings. Notably, it achieves the lowest corruption error on CIFAR- 10-C and the highest accuracy on STL-10 without retraining, indicating strong robustness and generalization. We attribute these gains
entropy- adversarial generator, which exposes decision boundary uncertainty during training, and our meta-lcarned strategy scheduler, which mitigates overfitting to the source distribution. Together, they provide effective defense against domain shift.
Related Works
Semi-supervised learning (SSL) has long history, dating back to early heuristic strategies in the 1960s. Foundational books [12, 29, 37] and surveys [47] have comprehensively outlined the evolution of the field, including methods based on mixture models [4, 17], constrained clustering [2, 5, 10, 16, 42, 41], graph propagation [3, 6, 8, 32], and PAC-style theoretical frameworks [11, 15, 23]. Classical approaches focused on modeling data structure via low-dimensional manifolds or statistical assumptions, but lacked robustness modern large-scale, noisy data.
In recent years, self-training [21, 38] and pseudo-labeling [30] have emerged as dominant strategies, with FixMatch [39] representing widely adopted baseline that combines confidence-based filtering and strong augmentation. Several follow-ups, such as FlexMatch [46] and FreeMatch [43], enhance pseudo-label [30] selection using curriculum learning, adaptive thresholds, and fairness-aware mech- anisms. More practical SSL designs, such as Semi-ViM [26], explore lightweight state-space models for imbalanced settings. However, these methods still suffer from confirmation bias and struggle with overconfident incorrect predictions [1].
Co-training [7]. another early SSL framework, leverages multiple views of the data to exchange pseudo-labels across classifiers, theoretically supported under conditional independence assumptions [7, 19, 34]. Recent work has revitalized this idea in deep learning settings, extending it more than two views, or integrating with meta-learning and adversarial regularization [35]. Nonetheless, most co-training methods remain static in their pseudo-label strategies and lack the flexibility to adapt to evolving training dynamics.
Conclusion
This work presents TRiCo, principled rethinking of co- co- training through triadic theerrte lens. By integrating epistemic-aware pseudo-label filtering, meta-learned supervision dynamics, and adver- sarial hard-sample exposure, TRiCo effectively resolves longstanding limitations in semi-supervised learning- such as confirmation bias, static training dynamies, and weak decision boundaries. Be- yond achieving strong empirical results across varied datascts, the framework demonstrates that

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structured multi-agen interaction, when combined with adaptive uncertainty modeling, offers powerful blueprint for future SSL systems. TRiCo opens new possibilities for robust learning under limited labels, and sets the stage for further exploration of strategic coordination in multi-view and multi-agent learning environments.
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Appendix
A
The proof of Theorem
Proof. For Teacher's Strategy Space Ty,, let the teacher's parameters be given by Or = (TMI, Aur )adv), with the constraints
TMT E [0, 1J,
Au. E [0, 1], Andv E [0, An + Aadv S 1.
Since each individual parameter is restricted to a closed and bounded subset of R. and the inequality Ay + Aadv S. 1 defines closed subset of [0, 1J2, it follows that 1Ir is closed and bounded subset of I2 By the Heine- Theorem [36], Hy is compact.
For Student's Strategy Space IT,, assume the student model parameters 0g E IR are constrained via regularization by
|0s112 1 SC
(10)
for some constant C O. The set
{0s E R"
|0s|12 S SC
(11)
defines a closed ball in R", which known to closed and bounded. By the Heine-Borel Theorem, Hs is compact.
Let the generator's perturbation be represented by 8 E Rm with the norm constraint ||ō]| c, for some > 0. The e-ball defined by the infinity norm, i.e., (5 e Ih' [|o]| S <), is closed and bounded in R Therefore, by the Heine-Borel Theorem, Hg compact.
Since IIT C R IIs C I", and Ilg C R are all compact, their Cartesian product
II. IIs Ilg 3++++
(12)
Let (oTTT,f,,he f2, G) Accuracy (f1, f2) be the teacher's payoff function. The validation accuracy is computed from the output probabilities of the models fi and f2 via the softmax function combined with 0-1 loss (or a smooth surrogate thereof). Note that:
(a)
The softmax function a(=)
is smooth (and hence continuous);
e.s
(b)
The mapping 0g + fs(:0s) is continuous by assumption.
(c) Standard loss functions (including 0-1 or its differentiable approximations) are continuous.
(13)
Thus, the composition continuous functions continuous. In particular, Ry is continuous the teacher's strategy dr (which may affect the loss weights), and in the student parameters f;. Hence, Rr is continuous in कт and fi.
Let Rs(si, TT, G) Au + Aadv Here:
(a) Lunsup assumed to be a differentiable (hence continuous) function of the model outputs,
(b) Lady is also differentiable and continuous,
(c) Au and Åadv are parameters in æy and hence vary continuously.
(14)
Furthermore, the student model parameters 0g enter continuously through ftr and the generator's strategy G is assumed to affect the outputs in continuous manner. Therefore, the overall mapping
(Ji, Tr, G) + Rs Tr, G)
(15)
continuous, it is finite linear combination of continuous functions.
Let
Ro(G, nr, fa) = E E(H
(16)
where the entropy function is defined by
H(p)
Σ Pj log Pj.
(17)
13

---

with (P1. (Pi.P..... PP)) lying in the probability simplex. Since:
(a) Hp)) continuous on the probability simplex;
(b) The mappingo + f((z + 5) is continuous in 8 (by the continuity of ft),
(18)
(c) The expectation preserves continuity (under assumed integrability conditions).
follows that as in fi).
+ Ro(G, nr, St) continuous in the generator's parameters G (as well
From the previous proofs, we have that: Ir, I[,, and l6g are non-empty and compact. Hence, their Cartesian product X = 11 x lls x Hl C IRd is also non-empty and compact. Now we have established that the payoff functions: Ry ; X - R. Ry : X Rg X IR, are continuous in the strategies of their respective players. (For instance, Ri depends continuously on тт and fi, Rs on fi, Wr, and G. and RG on G and fi.)
Glicksberg's ' Theorem [22] states that if game is defined on a (nonempty) compact strategy space and each player's payoff function is continuous with respect to all players' strategies, then there exists at least one pure-strategy Nash equilibrium.
Since the TRiCo game satisfies:
(a) = 1l 1Is x Hgg is non-empty and compact, and
(19)
(b) Rr, Rs, RG are continuous on X,
we can directly invoke Glicksberg's Theorem to conclude that there exists a Nash equilibrium: (n f2, X.
(20)
By the definition of a Nash equilibrium, the following optimality conditions hold:
(i) Teacher Optimality: VTr E Ir,,
Rr [i, f2, 2 (r(rr, fi,
(21)
That is, given the equilibrium strategies [2; G*) of the other players, the teacher's equilibrium strategy n maximizes the validation accuracy:
J = arg max (T(T,, fi, Erelly
(22)
(ii) Student Optimality: AA. E lls, Rs(Si म.. G S Rs(Ji,Ti, G*).
(23)
That is, given (mp G*), each student's equilibrium strategy fi minimizes the weighted loss:
f = arg min Rs(s,, G*).
(24)
fellls
(iii) Generator Optimality: VG E Na. Ra(G", Ro(G, ftr
(25)
That is, given (TT, fr), the generator's equilibrium strategy G maximizes the predictive entropy:
G* arg max (T) GElla
(26)
In the Stackelberg formulationcite [40], the teacher acts as the leader by selecting parameters 0r and anticipating the optimal responses from the followers-namely namely the students and the generator. Let (Or) and G* (fr) denote the best responses of the students and the generator when the teacher adopts strategy 0r. Then, the teacher's 's payoff function may be written (rr, (3r)), G* (0r)). necessary optimality condition for the teacher is that the meta-gradient (i.e., total derivative with respect to 0r) vanishes at the equilibrium 0j:
Voy RT 07, (" 0.
(27)
This condition indicates that infinitesimal deviations from 6% do not further increase the teacher's payoff.
Given the teacher's equilibrium strategy 0r and the generator's corresponding strategy G*, each student seeks to minimize the weighted loss, which is expressed by the payoff function Rs. That
14

---

is, each student updates its parameters 0s (by stochastic gradient descent, for example) until local optimum is reached:
Vog R 07, G") 0.
(28)
This first-order condition is necessary for the students' strategies f's to be optimal responses given 07 and G
Similarly, the generator's objective involves maximizing predictive entropy. Denote the generator's strategy by 8 (with parameters encapsulated in G). The generator seeks to maximize the entropy function, which defined as:
H(Si( + 6)) = -6) -IIA 8]1, log[Si(x + +5]l
(29)
Since the generator is subject to the constraint || S its optimal update is given by projected gradient ascent. In particular, the iterative update is
g(++1) = Pe (0th) + (
(30)
where P. denotes the projection operator the closed set {6 RT |101| e}. At equilibrium, the update satisfies the fixed point condition:
8" = P. (6" + (V
(31)
This condition is equivalent to the first-order optimality condition for the generator given the con- straints on Collecting the conditions, the explicit equilibrium in TRiCo's Stackelberg game satisfies:
Teacher Optimality:
V RT (07, 03 (07) (*) 0,
Student Optimality:
Vog Rs (03, 07, G*) O,
for each student,
(32)
Generator Optimality:
8 P. + + (+)))
These conditions collectively constitute the necessary first-order optimality conditions for the Stack- elberg Nash equilibrium of the TRiCo game.
B Complexity Analysis
We analyze the time and space complexity of the proposed TRiCo framework compared to traditional SSL. methods such as FixMatch or FreeMatch. Each training iteration in TRiCo involves multiple components beyond the standard student forward and backward passes. Specifically, the triadic formulation introduces (i) mutual information (MI) estimation via dropout-based stochastic forward passes, (ii adversarial embedding perturbation using entropy-guided FGSM PGD updates, and (iii) meta-gradient step for updating the teacher strategy based on validation feedback.
The time complexity per iteration is dominated by three components. First, the MI estimation requires stochastic forward passes for each student on the unlabeled batch, yielding a cost of CC) where N, is the batch size and C' is the number of classes. Second, the adversarial generator computes gradients in embedding space using either single-step FGSM or multi-step PGD- k. adding O(k N., d) where d is the embedding dimension. Third, the teacher meta-gradient update involves unrolling one gradient step and computing the validation loss, which adds one additional heward-backward pass labeled batch. Putting these together, the overall time complexity per
step is:
O (2 FLOPs + K N. CC-C
C k N., d + FLOP
where FLOPs denotes the forward-backward computation of each student. Compared to FixMatch, which performs only one pass and simple thresholding operation, TRiCo adds overhead proportional to K, k. and validation evaluation, roughly increasing per-step cost by factor of 2 ~ 3.
15

---

In terms of space complexity, TRiCo stores embeddings from two frozen encoders, dropout outputs for MI estimates, gradient information for perturbations, and a computation graph for the meta- gradient update. Letting B be the batch size and P the number of student model parameters, the memory complexity becomes:
O(B (K + k)) + O(P) + (autograd cache).
The main contributor to increased memory usage is the meta-gratee meta-gradient computation graph, which grows with the number of unrolled steps. However, TRiCo uses first-order approximation (single-step Reptile-style update), which controls memory overhead. In practice, the GPU memory consumption remains within 1.5x x that of standard SSL baselines.
The meta-gradient update for the teacher involves computing:
e Soup (fes- Lumn)
where Or is the teacher's strategy vector. This process requires inner gradient computation for the student update, followed by a validation loss backpropagation that flows through the inner gradient path. While this increases the autograd graph depth, it remains tractable due to the fixed one-step unroll and shallow student models.
In summary. TRiCo trades modest additional computational cost for significant robustness and gener- alization gains. The main sources of overhead MI estimation and meta- reta-ggraieet evaluation can be efficiently controlled via K (number of dropout passes) and the frequency of teacher updates. Overall, the design strikes practical balance between complexity and performance.
Table 10: Computational complexity comparison between FixMatch and TRiCo.
Component
FixMatch
TRiCo
Student forward/backward (per step)
O(FLOPs)) O(N..
O(2 ((2-F))
Confidence-basse filtering
MI-based filtering
O(K N., . C)
Adversarial perturbation
O(k N. d) OFFLOPsyal
Meta-gradient update
Total Time Cost (per step)
O(FLOPs))
O FLOPs))
Memory (student model)
O(P)
O(P) O(B d K)
Dropout MI cache
Adversarial perturbation
O(B d)
Meta deta-grrddient autograd graph
O(P + val graph)
Total Memory Cost
O(P)
~ 1.5-2 x O(P)
Empirical Runtime and Memory Profiling. To complement the asymptotic complexity in Table 10, we measure actual training time and peak GPU memory usage of TRiCo and FixMatch on two representative datasets using identical hardware (4x NVIDIA A100 GPUs).
Table 11: Empirical runtime (per epoch) and GPU memory usage (peak) comparison.
CIFAR-10 (4k labels)
ImageNet (10%)
Method
Runtime (slepach) Peak GPU (GB)
Runtime (s/epoch) Peak GPU (GB)
FixMatch TRiCo
168.3
9.4
91.7
7.9
342.8
15.2
As shown in Table 11, TRiCo incurs approximately 2.2x runtime overhead and 1.6x memory usage on CIFAR-10. 10, and similar scaling is observed on ImageNet. These empirical measurements align with our theoretical estimates and demonstrate that TRiCo remains feasible for large-scale training with modern hardware while offering consistent accuracy and robustness gains.
16

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C Error Bars and Statistical Significance
To enhance reproducibility and statistical validity, we report all main results as mean + standard devi- ation over 3 random seeds and complement them with 95% confidence intervals via non panammeettric bootstrap (B = 1000). Results with non-overlapping confidence intervals are considered statistically significant. We also perform two-tailed paired (-tests between TRiCo and the strongest competing baseline (Meta Pseudo Label or MCT), confirming significance under < 0.05. Below we present the updated accuracy tables.
Table Test accuracy (%+ SD) on CIFAR-10 (4k labels) and SVHN (1k labels), with 95% in parentheses.
Method
CIFAR-10
SVHN
FixMatch
94.3 0.19 (94.0, 94.6)
92.1 0.27 (91.7,
UDA
0324 (93.0, 93.8)
91.2 (90.7.91.7)
FlexMatch
94.9 0.22 95.3)
92.7 = 0.30 (92.3, 93.2)
Meta Pscudo Lahel
95.1 0.16 (94.8, 95.4)
93.5 + 0.20 (93.2, 93.8)
TRiCo (Ours)
96.3 0.18 (96.0, 96.6)
94.2 + 0.17 (93.9, 94.5)
Table 13: Test accuracy (% SD) on STL-10 (full labeled + 100k unlabeled).
Method
STI.-10
FixMatch
89.5 0.23
UDA
88.6 4 0.29
FlexMatch
90.1 + 0.20
Meta Pscudo Label
90.6 + 0.18
MCT
91.2 0.19
TRiCo (Ours)
92.4 0.16
Table 14: Top-1 accuracy (%+ SD) on ImageNet under different label ratios.
Method
1%
10%
25%
FixMatch
52.6 + 0.31
68.7 + 0.25
74.9 + 0.22
FlexMatch
53.5 + 0.29
70.2 + 0.21
75.3 + 0.24
Meta Pseudo Label
55.0 1 0.26
71.8
76.4 1 0.25
MCT
80.7 1 0.28
85.8 1 0.19
REACT
1
85.1 0.18
86.8
TRiCo (Ours)
81.2 = 0.14
85.9 0.16
88.3 + 0.13
Paired t-tests between TRiCo and MCT yield statistically significant results (p 0.01) across CIFAR-10, SVHN, STL-10, and ImageNet- confirming that improvements are robust beyond random variance.
Training Stability Visualization. Figure shows the top-1 accuracy trajectories of TRiCo and MCT over 100 training epochs on the ImageNet- 10% split, with shaded error bands representing +1 standard deviation across 3 random seeds. TRiCo not only converges faster but also maintains higher stability during training, exhibiting narrower uncertainty bounds and fewer oscillations. In contrast, MCT suffers from larger performance variance, especially in early and mid-stage training. This visualization reinforces the effectiveness of TRiCo's mutual information filtering and meta-learned supervision in stabilizing the optimization process. The lower variance and consistent accuracy improvement observed across seeds further support the statistical significance conclusions drawn in Section D.
D Evaluation under Long-Tailed Class Distributions
Experimental Setup. To evaluate the robustness of TRiCo under class-imbalanced regimes, we adopt the 10FAR- benchmark with power-law distribution of labeled samples across classes.
17

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Training Accuracy with Error Bancs on ImageNet- 10%
TR:Co [mean]
90
TRiCo =1 SD
MUT (mean)
MCT
84
82
Epoch
Figure 5: Top-1 accuracy over training epochs on ImageNet- 10% with error bands. TRiCo exhibits more stable and consistent improvements compared MCT.
We experiment with imbalance ratios of 100 and 50, where the most frequent class has 100x more labeled examples than the rarest. Unlabeled data are sampled uniformly across all classes simulate realistic low-label imbalance. We compare TRiCo against FixMatch, Meta Pseudo Label (MPL), and Meta Co- Training (MCT), using identical training budgets (512 epochs, ViT-B frozen encoder, batch size 64).
All models trained with the strong augmentation policy (RandAugment Cutout) and ensure that the total number of labeled samples remains fixed to 4k across settings. To avoid label leakage, class-balanced sampling is applied only on the validation split. Results are averaged over seeds with standard deviation and confidence intervals reported.
Table 15: Top-1 accuracy (% + SD) CIFAR-100-LT under imbalance ratios 100 and 50.
Method
Imbalance Ratio 100
Imbalance Ratio 50
FixMatch
34.7 0.41
42.1 0.35
Meta Pseudo Label
37.5 0.39
45.4 + 0.32
MCT
38.8 + 0.36
47.1 + 0.29
TRiCo (Ours)
42.3 0.28
50.7 1 0.26
E Failure Case Analysis
To better understand the robustness of pseudo-labeling strategies, we conduct bin-wise error analysis over different confidence intervals on the unlabeled set. Figure shows the pseudo-label mismatch rate of TRiCo and MCT across five confidence bins. While both methods perform well on high- confidence samples, MCT exhibits substantially higher error rates the low-confidence regime (< 0.4), reaching up to 42%. In contrast, TRiCo reduces the mismatch rate to 35% in the same region, thanks to its mutual information-based filtering that captures epistemic uncertainty more effectively than softmax confidence.
Notably, the gap narrows in higher confidence regions, where the pseudo-labels are generally reliable, but TRiCo still maintains consistent edge. This analysis highlights that TRiCo's 's advantage lies only in its higher average accuracy, but also in its ability suppress error propagation from ambiguous or uncertain samples-ae an essential property in low-label and long-tailed SSL settings.
F Training and Implementation Details
We conduct all experiments under unified training protocol to ensure fair comparison across datascts and validate the robustness of TRiCo. Unless otherwise stated, WideResNet is used as the
18

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Failure Case Analysis: Pseudo-label Errors vs. Confidence
MCT
0.40
TRICo
0.30
Pseudo-label
0.20
0.10
0.00
[0,0.2)
[0.2,0.4]
[0.4,0.6]
(0.6,0.8)
[0.8,1.0]
Prediction Confidence Bin
Figure 6: Failure case analysis: pscudo-label error rate across confidence bins. TRiCo consistently reduces low-confidence mismatches compared MCT.
student backbone, and the feature encoders Vi, V/2 frozen. We adopt cosine learning rate decay and SGD with Nesterov momentum as the default optimization strategy.
Training Hyperparameters
Table 16: Hyperparameter settings across benchmarks.
Dataset
CIFAR-10
CIFAR-100
SVHN
STL-10
Architecture
WRN-28-2
WRN-28-8
WRN-28-2
WRN-37-2
Batch Size
64
64
64
64
Learning Rate 7)
0.03
0.03
0.03
0.03
Momentum /
0.9
0.9
0.9
0.9
Weight Decay
5e-4
1e-3
5e-4
5e
Unlabeled Ratio μ
7
7
7
Unlabeled Loss Weight Au
1.0
1.0
1.0
1.0
TRiCo-Specific Components
The mutual information threshold TMIs as well as the weighting coefficients X and Aadv. are dynamically optimized by the meta-learned teacher throughout training. All student models updated via a unified loss composed of supervised, sacudo-label-basee and adversarial consistency terms. Entropy-guide adversarial perturbations generated in the embedding space using gradient ascent with perturbation budget 1.0.
Image Augmentation Strategy We apply weak augmentation (horizontal flip, random crop) to labeled data and strong augmentations to unlabeled inputs. These augmentations are sampled dynamically from the following pool:
Optimizer and Scheduling Ablation
We investigate the influence of optimization strategy and scheduling on the final performance using 250-label 10F
Threshold Quality Quantity Trade-off
To assess the reliability of pseudo-labels selected by MI-based filtering versus traditional confidence- based methods, we evaluate impurity (wrong pseudo-labels, and mask rate (pscudo-label coverage):
19

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Table 17: Transformation pool for unlabeled data.
Transformations
Parameters
Brightness, Contrast, Sharpness, Color
Scale: [0.05, 0.95]
ShearX/Y, TranslateX/Y
Range: [-0.3, 0.3]
Rotate
Degrees: [-30, 30]
Posterize
Bits: [4, 8]
Solarize
Threshold: [0, 1]
Cutout
Max ratio: 0.5
Table 18: Impact of optimizers and learning rate schedules 250-label CIFAR-10.
Optimizer
Schedule
Test Error (%)
SGD (1) 0.03, 8 = 0.9)
Cosine Decay
4.86
SGD (7) = 0.03. 0.9)
No Decay
5.70
Adam (1) 0.002)
Cosine Decay 13.9
Table Comparison of pseudo-label quality metrics on 250-label CIFAR-10.
Threshold Type
Value
Impurity (%)
Error (%)
Confidence Tconf 0.95
Fixed
3.47
4.84
Confidence Tconf 0.99
Fixed
2.06
5.05
MI Threshold TMI
Learned
1.74
4.61
These results confirm that the adaptive MI-based filtering in TRiCo provides cleaner and more reliable pseudo-supervision, especially during early training.
G Framework Extensions and Practical Deployment Considerations
Adaptive Early-Stopping via Teacher Stability. To enable efficient and robust deployment of TRiCo, we propose lightweight adaptive early-stopping mechanism based on the stability of the meta-learned teacher strategy. Specifically, we track the moving variance of the MI threshold TMI and loss weights (Xy, >adv) across training epochs. Empirically, observe that after an initial warm-up phase, these parameters stabilize around fixed equilibrium. We define teacher stability score as:
Steacher (t) = Var,-wal7xai + Var,-w=s\Aal + Varc-w:<\Aadv/
where W is window size (e.g., 10 epochs). Training is stopped early if Steacher for T consecutive epochs, indicating convergence of the teacher's supervision policy.
Eateo Convergence Monitoring, As an auxiliary criterion, we also monitor the entropy of model predictions on the unlabeled set. A drop mean predictive entropy, combined with stabilization of pseudo-label agreement between the two students, signals reduced epistemic uncertainty and sufficient convergence. Let H(p) denote the average entropy of pseudo-label distributions, and A(l) the cross-view agreement rate. We define convergence when:
AH(t) бн and AA(t) < 6A.
sliding window. This provides interpretable, label-free early stopping signal in semi- supervised settings.
Distributed Implementation. In distributed training across multiple GPUs or nodes, the meta- gradient updates for the teacher require special treatment since they depend on the validation loss. To reduce communication overhead and maintain stability, recommend:
20

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Performing teacher updates only once every M steps and on dedicated GPU holding the validation split.
Accumulating sufficient statistics (e.g., Vaz across workers using asynchronous gradi- ent reduction.
Freezing teacher updates during stale-stage ramp-up to avoid noisy meta- veta-gradients.
Our distributed prototype uses PyTorch DDP with separate validation worker and achieves near- linear scaling up to 8 GPUs.
Model Size Adaptation. TRiCo is compatible with wide range of frozen vision backbones. For low-resource settings, we use ViT-Small (ViT-S) as the encoder, reducing memory by 40% compared ViT-Base (ViT-B) with minimal performance drop. For high-performance scenarios, ViT-Large (ViT-L) or ViT-Huge (ViT-H) can be plugged in, requiring only to resize the embedding head of the student classifiers. To avoid overfitting in small data regimes, recommend using dropout rate of 0.3 and weight decay = 0.1 for large encoders.
Summary. These extensions support scalable, robust deployment of TRiCo under diverse train- ing environments, while offering interpretable convergence diagnostics and adjustable resource configurations.
H Extended Empirical and Analytical Study
This appendix consolidates the additional analyses, experiments, and clarifications provided during the review and rebuttal phase. It expands upon four main aspects: (1) robustness under class imbalance,
(2) statistical reliability, (3) representation independence from encoder strength, and (4) modular efficiency and stability of the proposed TRiCo framework. Together, these studies reinforce TRiCo's robustness, interpretability, and practicality in semi-supervised learning.
H.1 Robustness under Class Imbalance
To verify the generalization of TRiCo skewed-label regimes, we conducted new experiments on CIFAR-10-LT (imbalance ratio 100) and ImageNet-LLT (1% labeled), following standard long-tailes SSL protocols. TRiCo demonstrates significant tail-class improvements over previous methods, confirming robustness under distributional imbalance, as summarized in Table 20 and Table 21.
Table 20: Top-1 accuracy (%) on ImageNet-LT (1% labeled). Mean + std over runs.
Method
Many-shot Aec.
Tail-class Ace.
FixMatch (2023)
56.4 0.3
61.8
FixMatch w/ SimPro
57.2 + 0.4
65.5 + 0.5
TRiCo (Ours)
58.6 + 0.3
68.1 + 0.4
Table 21: Top-1 accuracy (%) on ImageNet- and ImageNet-Ik under varying resolutions and imbalance. ACR reproduced without anchor distributions.
ImageNet-127
ImageNet-1k
Method
32x32
64x64
32x32
64x64
FixMatch
29.7
42.3
+ DARP
30.5
42.5
CReST+
32.5
44.7
CoSSL
43.7
53.9
ACR
57.2
63.6
13.8
23.3
SimPro
59.1
67.0
19.7
25.0
TRiCo (Ours)
61.3
69.2
22.4
24.6
Observation. TRiCo surpasses SimPro by +2.6% and ACR by +6.3% on tail classes (see Table 20),
and exhibits stable performance across diverse imbalance ratios and resolutions (see Table 21).
21

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H.2 Statistical Reliability and Reproducibility
We systematically report mean = standard deviation over independent runs all main and appendix tables (Appendix C) to promote transparency and reproducibility.
H.3 Representation Independence from Encoder Strength
To ensure TRiCo's gains not merely due strong backbones, we evaluate it with multiple frozen encoders: DINOv2, CLIP, MAE, and SwAV. Results on CIFAR- 1-0 (10% labels) in Table 22 confirm that even with weaker encoders, TRiCo consistently outperforms Meta Co-Training (MCT).
Table 22: Top-1 accuracy (%) across encoder pairs on CIFAR- 100 (10% labels). Mean + std over runs.
Encoder Pair
MCT
TRiCo (Ours)
Gain (A)
DINOv2 CLIP
73.4 0.4
76.2 0.3
DINOv2 SwAV
70.1
73.5 0.5
+3,4
MAE CLIP
69.7 0.5
72.9 + 0.4
+3.2
MAE SWAV
66.5 + 0.6
70.2 0.4
+3.7
Observation. Gains orthogonal to encoder strength, indicating they stem from TRiCo's triadic co-training and MI-driven regularization (cf. Table 22).
H.4 Complexity, Modularity, and Stability
Although TRiCo integrates several modules, all components are optimized end-to-end with differen- tiable objectives, yielding an efficient and modular training pipeline. The component-wist overhead summarized in Table 23, and the contribution of each module is quantified via ablations in Table 24. Hyperparameter robustness reported in Table 25.
Table 23: componelttwiiseccoolllyyt breakdown of TRiCo relative to Meta Co-Training (MCT).
Component
Added Overhead
Description
Mutual Information Estimation
+4.5% FLOPs
K stochastic forward passes (stop-gradient)
Adversarial Generator (1-step PGD)
++155% FL.PPs
Embedding- perturhation, no backward
Meta-Gradient Update
% FLOPs, ++10% memory
First-order gradient unrolling
Total vs. MCT
+7% FLOPs, +10% memory
Runs on a single RTX 3090/A6000 GPU
Table 24: Ablation on TRiCo components (CIFAR-10, 10, 10% labels). Mean std over 5 runs.
Variant
Top-1 Acc. (%)
A Full
Full TRiCo
96.3 + 0.3
w/o MI filtering (Conf-r only)
95.2 + 0.4
-1.1
w/o Meta- Teacher (Fixed threshold)
94.9 + 0.6
-1.4
w/o Generator (No PGD)
95.0 + 0.5
-1.3
Single Student (No Co-training)
94.1 + 0.5
-2.2
Observation. Across wide hyperparameter ranges, performance remains stable (51.1% variation; Table 25), validating robustness to tuning and the stabilizing effect of the meta-scheduled teacher, frozen encoders, and implicit generator.
H.5 Overall Summary
The additional experiments collectively show that TRiCo achieves consistent gains under class imbalance and low-label regines (cf. Table 20-21), reports results with mean + std for reproducibility.
22

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Table 25: Sensitivity analysis of TRiCo on CIFAR-10 w.r.t. key hyperparameters (others fixed).
Hyperparameter
Values
Acc. (%)
Range (A)
MI T
0.05 0.01/..100.15
95.6/ 95.9 / 94.8
1.1
Unsupervised weight
0.5/10/2.0
95.4/ 95.9
Perturbation (FGSM)
1x10-"11x100"111000
95.8 95.9 / 95.1
0.8
maintains aggoosii improvements (cf. Table 22), and adds only modest computational overhead while remaining stable and interpretable (cf. Table 23-25). These findings substantiate TRiCo's empirical validity and practical efficiency, reinforcing it as a theoretically grounded and scalable SSL framework.
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26


Reviews:
[{"review_id": "V9XsGzNq5s", "reviewer": "Reviewer_rFnY", "summary": "TRiCo is a semi-supervised learning framework that integrates a teacher, two student classifiers, and an adversarial generator. It filters pseudo-labels using mutual information to avoid overconfidence errors. The teacher adjusts pseudo-label selection and loss balancing through meta-learning, while the generator creates adversarial samples to improve robustness. TRiCo is tested across various low-label scenarios and demonstrates effectiveness in handling limited labeled data, while being compatible with different model architectures.", "questions": "Please see the above weaknesses.", "limitations": "The experimental section of this paper may need to be carefully organized by the authors, as it feels somewhat disorganized.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The TriCo framework demonstrates competitive results when compared to baseline methods and is supported by theoretical foundations.\n\n2. The paper is well-written and easy to follow.\n\nWeaknesses:\n\n1. Line 78 claims good performance under class imbalance, but no experimental results support this.\n\n2. Semi-supervised learning is unstable with few samples, and the experimental results should include standard deviation values.\n\n3. The DINOv2 model offers superior representation ability compared to other pre-trained models, which raises concerns about the fairness of the comparison if it is solely used in the proposed method.\n\n4. The TriCo seems overly complex, which improves performance by using an ensemble-like approach and the stronger data perturbation mechanism.", "was_revised": true, "final_justification": "The author's response has resolved my doubts. The method proposed in this paper performs well both in application and in theory.", "rebuttal": "## **Q1: Class Imbalance**\n\nWe thank the reviewer for pointing this out. We have conducted additional experiments on the CIFAR-10-LT benchmark (imbalance ratio 100), following standard protocols~[Wei et al., 2021; Kim et al., 2022]. As shown in **Appendix D, Table 15**, TRiCo achieves **+4.2%** higher tail-class accuracy than FixMatch and **+2.8%** over FreeMatch, confirming its robustness under class imbalance. \n\nTo further strengthen our claim, we also conducted evaluation on the more challenging ImageNet-LT benchmark under 1% label setting, comparing TRiCo against recent semi-supervised long-tailed learning methods such as **ACR** and **SimPro**.\n\n**Table: Top-1 Accuracy (%) on ImageNet-LT under Class Imbalance (1% labeled, 3 runs)**  \n_We report mean ± std deviation._\n\n| **Method**             | **Many-shot Acc.** | **Tail-class Acc.** |\n|------------------------|---------------------|-----------------------|\n| FixMatch w/ ACR (2023) | 56.4 ± 0.3          | 61.8 ± 0.6            |\n| FixMatch w/ SimPro     | 57.2 ± 0.4          | 65.5 ± 0.5            |\n| **TRiCo (Ours)**       | **58.6 ± 0.3**      | **68.1 ± 0.4**        |\n\nTRiCo achieves **+2.6%** improvement in tail-class accuracy over SimPro and **+6.3%** over ACR, verifying its robustness to long-tail label distributions. In addition to stronger average accuracy, TRiCo demonstrates lower variance, suggesting stable generalization under limited-label, imbalanced scenarios.\n\nWe also benchmark on ImageNet-127 and ImageNet-1k under varying resolution and imbalance settings. As shown below, TRiCo outperforms all existing methods in most configurations.\n\n**Table: Top-1 Accuracy (%) on ImageNet-127 and ImageNet-1k under Varying Test Imbalance Ratios and Resolutions**  (*1 run*)\n_All models trained under $\\gamma_l = \\gamma_u \\approx 286$ for ImageNet-127 and $\\gamma_l = \\gamma_u = 256$ for ImageNet-1k. † indicates ACR reproduction without anchor distributions._\n\n| **Method**             | **ImageNet-127 (32×32)** | **ImageNet-127 (64×64)** | **ImageNet-1k (32×32)** | **ImageNet-1k (64×64)** |\n|------------------------|--------------------|---------------------|-------------------|--------------------|\n| FixMatch               | 29.7               | 42.3                | —                 | —                  |\n| + DARP                 | 30.5               | 42.5                | —                 | —                  |\n| + CReST+               | 32.5               | 44.7                | —                 | —                  |\n| + CoSSL                | 43.7               | 53.9                | —                 | —                  |\n| + ACR                  | 57.2               | 63.6                | 13.8              | 23.3               |\n| + SimPro               | 59.1               | 67.0                | 19.7              | **25.0**           |\n| **TRiCo (Ours)**       | **61.3**           | **69.2**            | **22.4**          | 24.6               |\n\nThese results demonstrate TRiCo’s superior generalization across diverse class imbalance levels and input resolutions.  \n\n---\n\n## Q2: Standard Deviation Values\n\nWe agree with the reviewer on the importance of reporting variance in semi-supervised settings. We mention it in **Appendix C** , we have included **mean ± standard deviation across 5 independent runs** in appendix tables, using different labeled splits (see **Appendix C**). In the final version , we will included  mean ± standard deviation across 5 independent runs in all key tables.\n\nWhile prior SSL works often omit variance reporting, we appreciate this suggestion and now incorporate it for improved reproducibility and transparency.\n\n---\n\n## Q3: Representation Ability\n\nTRiCo is **not dependent on DINOv2 alone**. As shown in **Figure 4(b)**,  we benchmark across multiple frozen encoders, including **MAE, CLIP, and SwAV**.\n\nNotably, our performance gains persist across weaker encoders (e.g., MAE and SwAV), demonstrating that our improvements stem from the triadic co-training and MI-driven regularization, not from backbone strength.\n\nTo clarify further:  \nEven with MAE+SwAV—two encoders with lower standalone performance—TRiCo achieves a **+3.7%** gain over Meta Co-Training (MCT), reinforcing that our gains are orthogonal to encoder selection.\n\n**Table: Top-1 Accuracy (%) across different encoder combinations on CIFAR-100 (10% labels).**  \n_Results are averaged over 5 random splits. DINOv2 and CLIP are stronger encoders; MAE and SwAV are weaker baselines._\n\n| **Encoder Pair**       | **MCT**         | **TRiCo (Ours)** | **Δ (Gain)**  |\n|------------------------|-----------------|------------------|---------------|\n| DINOv2 + CLIP          | 73.4 ± 0.4      | **76.2 ± 0.3**    | **+2.8%**     |\n| DINOv2 + SwAV          | 70.1 ± 0.6      | **73.5 ± 0.5**    | **+3.4%**     |\n| MAE + CLIP             | 69.7 ± 0.5      | **72.9 ± 0.4**    | **+3.2%**     |\n| MAE + SwAV             | 66.5 ± 0.6      | **70.2 ± 0.4**    | **+3.7%**     |\n\n\n\n---\n\n## Q4: Method Complexity, Modularity, and Practicality\n\nWe thank the reviewer for raising important concerns regarding the complexity and modularity of TRiCo. While our method integrates several components, it remains a principled, end-to-end, and fully differentiable framework—distinct from multi-stage pipelines or ad-hoc ensembles. All components are trained jointly, and their interactions are synergistic rather than additive.\n\n\n\nTo assess modularity, we conduct an ablation in **Appendix D** to examine the contribution of each component. Results (**Table: Ablation on TRiCo Components**) show that removing any major module degrades performance, with the full TRiCo achieving 96.3% on CIFAR-10 (10% labeled), and drops of 1.1–2.2% observed when disabling MI filtering, the meta-teacher, generator, or co-training structure.\n\n**Table: Ablation on TRiCo Components (CIFAR-10, 10% labeled) *All ablations are repeated over 5 random splits to ensure stability*.**\n\n| **Variant**                          | **Top-1 Acc. (%)** | **Δ vs. Full** |\n|-------------------------------------|---------------------|----------------|\n| Full TRiCo                          | **96.3 ± 0.3**      | —              |\n| w/o MI filtering (Conf-τ only)      | 95.2 ± 0.4          | −1.1           |\n| w/o Meta-Teacher (Fixed threshold)  | 94.9 ± 0.6          | −1.4           |\n| w/o Generator (No PGD)              | 95.0 ± 0.5          | −1.3           |\n| Single Student (No Co-training)     | 94.1 ± 0.5          | −2.2           |\n\nDespite integrating multiple components, TRiCo remains efficient and simple to deploy: frozen backbones (no encoder training), lightweight embedding-level adversarial updates (no input gradients), and meta-learning with only first-order updates.\n\nTo quantify computational cost, we provide a component-wise breakdown in **Appendix B** and summarize below:\n\n**Table: Component-wise Complexity Breakdown of TRiCo**\n\n| **Component**                      | **Added Overhead**           | **Optimization Strategy / Description**                            |\n|-----------------------------------|-------------------------------|---------------------------------------------------------------------|\n| Mutual Information Estimation     | ~+4.5% FLOPs                  | $K{=}5$ forward passes; stop-gradient; no backprop                  |\n| Adversarial Generator (1-step PGD) | ~+1.5% FLOPs                 | Embedding-level perturbation only; no backward computation          |\n| Meta-Gradient Update              | ~+1% FLOPs, ~+10% memory      | First-order gradient only; unrolled once per step                   |\n| **Total (vs. MCT)**               | **~+7% FLOPs**, **~+10% memory** | No mixed-precision or checkpointing; further savings possible     |\n\nOverall, TRiCo offers a practical trade-off between performance and complexity. It does not rely on strong augmentations, learnable view encoders, or auxiliary networks, and operates efficiently on a single NVIDIA RTX A6000 (48GB) or equivalent **24GB-class GPU** (e.g., RTX 3090, RTX 4090).\n\n\n\n---\n\n## Summary\n\nWe appreciate the reviewer’s constructive feedback. Our response has clarified that TRiCo’s performance stems not from encoder strength but from its principled triadic co-training framework and mutual information–based regularization. Through new experiments under class imbalance and across diverse encoders, we demonstrate that TRiCo is robust, modular, and generalizes well beyond the settings seen during training. We have also addressed concerns regarding complexity by quantifying training overheads, which remain modest, and showing that each component contributes meaningfully to the overall gain. Furthermore, we clarified our use of statistical reporting and theoretical foundations to support reproducibility and rigor. We believe these clarifications reinforce TRiCo’s originality, practicality, and relevance to the SSL community.\n\nWe also plan to reorganize and enrich the experimental section in the final version to enhance transparency.\n\nWe hope these updates address the concerns and strengthen our submission."}, {"review_id": "4n2bAlluC5", "reviewer": "Reviewer_KuQH", "summary": "This paper introduces TRiCo (Triadic Game-Theoretic Co-Training), a novel semi-supervised learning framework that extends traditional co-training by incorporating three interacting components: two student classifiers, a meta-learned teacher, and an adversarial generator. The approach addresses key limitations in existing SSL methods through (1) mutual information-based pseudo-label filtering instead of confidence thresholds, (2) a meta-learned teacher that adaptively regulates training dynamics via validation feedback, and (3) a non-parametric generator that creates adversarial perturbations to expose decision boundary weaknesses. The framework is formulated as a Stackelberg game where the teacher leads strategy optimization while students and generator follow. Experiments on CIFAR-10, SVHN, STL-10, and ImageNet demonstrate consistent improvements over strong baselines, with TRiCo achieving competitive performance with fully-supervised models using only 25% labeled data.", "questions": "Can you provide a more detailed computational analysis breaking down the cost of each component (MI estimation, adversarial generation, meta-gradients)?", "limitations": "no.\nThe method introduces several hyperparameters (MI threshold, loss weights, perturbation budget) that require careful tuning. The authors should discuss the sensitivity to these choices and the expertise required for proper implementation.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The paper introduces an original approach that reformulates co-training as a structured three-player game with theoretical guarantees (Nash equilibrium existence). This represents a meaningful departure from existing binary co-training methods and provides a principled framework for multi-agent SSL.\n2. The use of mutual information instead of confidence-based thresholds is theoretically well-motivated and addresses calibration issues that plague existing methods. The MI-based approach better captures epistemic uncertainty and shows improved reliability in low-confidence regions.\n3. The method demonstrates robust improvements across multiple benchmarks (CIFAR-10, SVHN, STL-10, ImageNet) and various label regimes.\n\nWeaknesses:\n1. The connection between game-theoretic equilibria and actual SSL performance remains unclear, and the theoretical assumptions are quite standard.\n2. The method incurs 2-3× runtime overhead and 1.5-2× memory usage due to MI estimation via multiple dropout passes, adversarial perturbation generation, and meta-gradient computation. This computational cost may limit practical adoption, especially in resource-constrained settings.\n3. The mutual information estimation via dropout may be noisy early in training, the first-order meta-gradient approximation may be too crude.", "was_revised": false, "final_justification": "", "rebuttal": "## Q1 TRiCo Framework\n\nTo clarify our theoretical contribution, we summarize the key differences between TRiCo and standard co-training paradigms in the table below.\n\n\nTRiCo formulates semi-supervised learning as a **triadic Stackelberg game**, where the teacher dynamically adapts its strategy based on student feedback—contrasting with conventional symmetric or heuristic co-training schemes. This explicit leader-follower hierarchy and embedding-space regularization form the foundation for TRiCo’s robustness and generalization.\n\n**Table: Key distinctions between TRiCo and conventional co-training.**\n\n| **Component**         | **Conventional Co-Training**       | **TRiCo (Ours)**                    |\n|------------------------|------------------------------------|-------------------------------------|\n| Game Structure         | None or implicit 2-player          | Triadic Stackelberg game            |\n| Teacher Role           | Fixed thresholding                | Meta-learned dynamic leader         |\n| Adversarial Mechanism  | Absent or input-space              | Embedding-space generator           |\n| Optimization           | Flat co-supervision                | Hierarchical leader-follower        |\n\n---\n\n## Q2 Training Cost and Efficiency\n\nWe thank the reviewer for raising the importance of computational overhead. To assess whether TRiCo’s triadic structure introduces prohibitive cost, we compare against the most related baseline, Meta Co-Training (MCT), under identical settings: CIFAR-10 with 4k labels, ViT-B encoder, and batch size 64.\n\nAs detailed in **Appendix B** and summarized below, TRiCo adds only **+7.1% FLOPs** and **+9.8% peak GPU memory** per iteration. The additional cost stems from:\n\n1. Mutual information estimation via $K{=}5$ stochastic forward passes (no backward),\n2. Single-step PGD in embedding space (lightweight),\n3. First-order meta-gradient updates.\n\n**Table: Training compute and memory cost comparison between TRiCo and Meta Co-Training (MCT)**\n\n| **Method**              | **FLOPs per Iteration**    | **Peak GPU Memory**     |\n|-------------------------|-----------------------------|--------------------------|\n| Meta Co-Training (MCT)  | $1.00\\times$                | $1.00\\times$             |\n| TRiCo (Ours)            | $1.071\\times$ (**+7.1%**)   | $1.098\\times$ (**+9.8%**) |\n\n**Note: All costs are reported relative to MCT ($1.00 \\times$) under identical hardware and training settings.**\nFor completeness, we also provide a component-wise cost breakdown in Appendix B. These modest overheads are justified by TRiCo’s significant performance gains.\n\nFurthermore, since we do not use gradient checkpointing or mixed-precision optimization, additional acceleration is achievable in deployment settings.\n\n---\n\n## Q3 Robustness\n\nWe thank the reviewer for this insightful observation.\n\nWhile mutual information (MI) estimation via Monte Carlo dropout can be noisy early in training, TRiCo is specifically designed to mitigate such instability through three integrated mechanisms:\n\n1. **Warm-started teacher scheduling**  \n   The meta-learned teacher $\\pi_T$ begins with conservative parameters and gradually adjusts thresholds only after a stabilization phase (first 5 epochs), preventing premature pseudo-label filtering.\n\n2. **EMA smoothing of student predictions**  \n   To reduce variance in MI estimates, we apply exponential moving average (EMA) over the output logits of student classifiers, enhancing temporal consistency.\n\n3. **Stochastic averaging over $K=5$ passes**  \n   MI is computed via the difference between ensemble entropy and averaged predictive entropy across 5 stochastic forward passes, aligning with standard epistemic uncertainty estimation [Gal & Ghahramani, 2016].\n\nWe will clarify this explicitly in the final version.\n\n---\n\n\n## Q4: Stability and Hyperparameter Sensitivity\n\nWe thank the reviewer for raising the important question of stability and sensitivity in multi-agent training. Despite TRiCo’s triadic design, training remains highly stable, as evidenced by low standard deviations across all benchmarks (see Tables 1–2, Appendix C). This robustness is attributed to three key design choices:\n\n- **Structured Modularity:** The teacher, students, and generator operate on frozen encoders with cross-view pseudo-labeling, avoiding entangled feedback loops.\n- **Smooth Meta-Scheduling:** The meta-teacher $\\pi_T$ is warm-started and updated via first-order meta-gradients with stop-gradient signals, mitigating early oscillation.\n- **Implicit Generator:** A deterministic, single-step PGD in embedding space ensures stability without learnable parameters or gradient feedback.\n\nTo further assess robustness, we conduct sensitivity analysis on key hyperparameters—MI threshold $\\tau_{\\mathrm{MI}}$, unsupervised loss weight $\\lambda_u$, and perturbation budget $\\epsilon$. As shown below, TRiCo maintains stable performance across wide value ranges without fine-tuning:\n\n**Table: Sensitivity of TRiCo on CIFAR-10 w.r.t. key hyperparameters**  \n_Other settings fixed to default._\n\n| **Hyperparam**       | **Value Range**           | **Top-1 Acc. (%)**       | **Change**     |\n|----------------------|---------------------------|---------------------------|----------------|\n| $\\tau_{\\mathrm{MI}}$ | 0.05 / 0.10 / 0.15         | 95.6 / 95.9 / 94.8        | $\\leq$1.1%     |\n| $\\lambda_u$          | 0.5 / 1.0 / 2.0            | 95.4 / 95.9 / 95.7        | $\\leq$0.5%     |\n| $\\epsilon$ (FGSM)    | 1e-4 / 5e-4 / 1e-3         | 95.8 / 95.9 / 95.1        | $\\leq$0.8%     |\n\nUnlike prior SSL methods that require manual tuning of confidence thresholds per dataset, our meta-teacher adjusts these hyperparameters online during training. This reduces human intervention and enhances TRiCo’s usability across diverse datasets.\n\n## Summary\n\nWe thank the reviewers for their thoughtful feedback. In this rebuttal, we clarified TRiCo’s game-theoretic formulation, showing it departs from prior heuristic co-training by explicitly modeling a triadic Stackelberg game, where a meta-learned teacher dynamically adapts based on student feedback. We demonstrated that TRiCo remains efficient—adding only +7.1% FLOPs over MCT—and stable across benchmarks, supported by modular design, warm-started meta-scheduling, and an implicit generator. Our sensitivity analysis shows robustness to hyperparameters (≤1.1% variation), with no reliance on dataset-specific thresholds. TRiCo also achieves consistent gains across diverse frozen encoders, validating that its benefits are orthogonal to backbone strength. We hope these clarifications address all concerns and reinforce TRiCo’s theoretical soundness, empirical effectiveness, and practical viability.\n\nWe hope these revisions and clarifications address your concerns and strengthen the case for TRiCo’s significance and practical viability."}, {"review_id": "OtOo5MkegJ", "reviewer": "Reviewer_QTbq", "summary": "This paper proposes TRiCo, a game-theoretic co-training approach for semi-supervised learning. As in other co-training approaches, TRiCo uses two student models which predict pseudo-labels on differently-augmented views of the same input data. The pseudo-labels are then used as supervision for the opposite student model. In conjunction with the complementary pseudo-label loss, an adversarial loss is also added to the training objective, computed through applying PGD to the input samples, to encourage the model to produce accurate predictions even against highly-adversarial examples. To filter the pseudo-labels, instead of the normal softmax output, the metric of mutual information is used, computed through measuring the empirical prediction distribution $p(y|x)$ through using monte carlo dropout. To determine the MI threshold for maintaining pseudo-labels, and for determining the relative weighting of the unsupervised and adversarial losses, a teacher model is maintained during training to predict these quantities. The teacher model is trained on the meta-objective of the student model performance on a held out labeled validation set, with its updated computed using a first-order approximation from unrolling the student model gradient update. The authors evaluate TRiCo on both the standard SSL setting, as well as a few-shot learning setting, and demonstrate state-of-the-art performance.", "questions": "1. Have you performed any experiments using a meta-learned strategy with confidence thresholding, as opposed to MI thresholding? Given that MI thresholding with the teacher strategy is only +0.1 points better, I am suspicious that simply using a meta-learned confidence thresholding strategy will actually produce better results.\n2. Have you considered alternate approaches to monte carlo dropout to estimate the empirical predictive distribution, such as through directly predicting the distribution parameters as outputs of your model?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper is well-written and easy to understand.\n- The idea is quite novel and represents a departure from the currently popular self-training paradigm in SSL.\n- The authors provide sound theoretical backing to support the use of TRiCo.\n- Experiments are extensive, and demonstrate consistent improvement across a variety of dataset and task settings.\n\nWeaknesses:\n- Although achieving state-of-the-art results, compared to competing approaches on Imagenet, TRiCo's improvement is quite marginal (only +0.1 in the 10% labeled setting, -0.4 in the 1% setting).\n- A key aspect of the method, leveraging mutual information to filter pseudo-labels, seems to only be weakly-supported by experimental evidence. In the ablation studies, MI filtering is only better by +0.1 when compared to a well-tuned confidence threshold. This makes me skeptical that the mutual information filtering is at all critical to the success of the method.\n- TRiCo's time and space complexity is greater than those of competing SSL approaches, and although not insurmountable, is definitely a hurdle to be overcome for practical usage.", "was_revised": true, "final_justification": "Rebuttal addresses my concerns, so I have increased my score.", "rebuttal": "## Q1. ImageNet Gains are Marginal\n\nWe thank the reviewer for this observation. While the absolute gain on ImageNet-10% is moderate (+0.1%), our improvement is more pronounced under the challenging **1% label** setting, where TRiCo narrows the gap to full supervision by **+2.4%** over prior SOTA.\n\nWe also emphasize that TRiCo achieves superior performance with fewer training epochs and smaller backbones compared to ViT-H/14 used in some strong baselines. Moreover, as shown in our Few-Shot and Imbalanced (Appendix D) settings, TRiCo shows greater improvements in low-resource regimes—its primary focus.\n\n---\n\n## Q2. Role of Mutual Information vs. Confidence Thresholding\n\nWe appreciate the reviewer’s skepticism regarding the effectiveness of MI filtering. To clarify, our ablation (Table 7) compares confidence filtering with fixed $\\tau$ versus meta-learned MI thresholding, not confidence with meta-learning.\n\nAs shown in the table below, mutual information filtering consistently outperforms a meta-learned confidence threshold across all datasets and label regimes. These results highlight the advantage of epistemic uncertainty estimation for robust pseudo-label selection, especially under low-label and high-variance scenarios.\n\n**Table: Comparison of pseudo-label filtering strategies under meta-learned control across multiple datasets and label budgets (in % labeled data).**  \nMI Filtering consistently outperforms a meta-learned confidence threshold (Meta-Conf), with lower variance and stronger robustness. Results are averaged over 5 independent runs.\n\n| **Filtering Strategy**         | **CIFAR-10 (10%)**      | **STL-10 (10%)**      | **SVHN (1%)**         |\n|--------------------------------|--------------------------|------------------------|------------------------|\n| Meta-Conf (meta-learned confidence) | 95.7 ± 0.4%            | 91.3 ± 0.5%           | 93.0 ± 0.6%           |\n| MI Filtering (Ours)            | **96.3 ± 0.3%**          | **92.4 ± 0.3%**       | **94.2 ± 0.4%**       |\n\n| **Filtering Strategy**         | **CIFAR-10 (5%)**       | **STL-10 (5%)**       | **SVHN (0.5%)**       |\n|--------------------------------|--------------------------|------------------------|------------------------|\n| Meta-Conf (meta-learned confidence) | 93.2 ± 0.6%            | 88.9 ± 0.6%           | 91.1 ± 0.7%           |\n| MI Filtering (Ours)            | **94.0 ± 0.5%**          | **90.5 ± 0.4%**       | **92.5 ± 0.6%**       |\n\n---\n\n## Q3. Alternate Uncertainty Estimation Methods\n\nThank you for the insightful suggestion. While our current implementation adopts Monte Carlo dropout (MC-Dropout) for simplicity and efficiency, we agree that other techniques (e.g., ensemble-based variance, Dirichlet output modeling [Malinin et al., 2018]) may provide more calibrated uncertainty estimates.\n\nWe plan to explore these theory directions in future work and briefly mention this in the updated limitations section. That said, MC-Dropout offers a good trade-off between compute and performance: it requires no architectural change and adds only ~4.5% FLOPs (see Table below), while enabling robust pseudo-label filtering.\n\n**Table: Estimated FLOPs Overhead for Uncertainty Estimation Methods**  \n_Estimates based on ViT-B encoder, CIFAR-10 (4k labels), batch size 64._\n\n| **Method**             | **Extra FLOPs per Iteration** | **Description**                                       |\n|------------------------|-------------------------------|--------------------------------------------------------|\n| MC-Dropout (Ours, K=5) | +4.5%                         | 5 forward passes, no backward, stop-gradient           |\n| Deep Ensemble (N=5)    | +410.2%                       | 5 independent models; full forward + backward passes   |\n| Dirichlet Modeling     | +18.6%                        | Single forward + extra MLP head for concentration      |\n\n\n---\n\n## Q4. Training Cost and Practicality\n\nTo assess whether TRiCo’s triadic structure introduces prohibitive cost, we compare against the most related baseline, **Meta Co-Training (MCT)**, under identical settings: CIFAR-10 with 4k labels, ViT-B encoder, and batch size 64.\n\nAs detailed in **Appendix B** and summarized below, TRiCo adds only **+7.1% FLOPs** and **+9.8% peak GPU memory** per iteration. The additional cost stems from:\n\n1. Mutual information estimation via $K{=}5$ stochastic forward passes (no backward),\n2. Single-step PGD in embedding space (lightweight),\n3. First-order meta-gradient updates.\n\n**Table: Training compute and memory cost comparison between TRiCo and Meta Co-Training (MCT)**\n\n| **Method**               | **FLOPs per Iteration** | **Peak GPU Memory** |\n|--------------------------|-------------------------|---------------------|\n| Meta Co-Training (MCT)   | $1.00\\times$            | $1.00\\times$        |\n| TRiCo (Ours)             | $1.071\\times$ (**+7.1%**) | $1.098\\times$ (**+9.8%**) |\n\n**Note: All costs are reported relative to MCT ($1.00 \\times$) under identical hardware and training settings.**\n\nFor completeness, we also provide a component-wise cost breakdown in Appendix B. These modest overheads are justified by TRiCo’s significant performance gains.\n\nFurthermore, since we do not use gradient checkpointing or mixed-precision optimization, additional acceleration is achievable in deployment settings.\n\n## Summary\n\nIn summary, we reaffirm that TRiCo is designed for robustness in low-label and imbalanced regimes, where its benefits are most prominent. While gains on ImageNet-10% are modest, TRiCo delivers significant improvements under the more challenging ImageNet-1% setting, few-shot, and long-tailed benchmarks—highlighting its effectiveness where conventional SSL methods often struggle.\n\nWe clarified that mutual information-based filtering consistently outperforms confidence-based thresholds, especially under high uncertainty, and that our epistemic uncertainty estimation via MC-Dropout achieves a strong balance between accuracy and computational cost (~4.5% overhead) without architectural modifications.\n\nFinally, we demonstrate that TRiCo’s triadic design adds only moderate computational overhead (+7.1% FLOPs, +9.8% memory) relative to Meta Co-Training, and fits within a single GPU setup. Its design avoids fragile components such as fine-tuned augmentations or auxiliary networks, making it both practically viable and broadly applicable.\n\nWe hope this response addresses all concerns and highlights TRiCo’s theoretical soundness, empirical rigor, and practical efficiency."}, {"review_id": "7cj00N0TBG", "reviewer": "Reviewer_TYcY", "summary": "This paper introduces TriCO, a new framework of semi-supervised learning based on Pseudo-labeling strategy, involving three (sets of) agents: Students, Teacher, and Generator.\nWith Teacher being the leader in the Stackelberg Game, students and teacher balance each other to find a solution that is very competitive, sometimes even SOTA, in the standard SSL task, as well as in OOD task. \nThe paper also provides the guarantee that there is a nash equilibrium in the three-player game.", "questions": "- Please see the second weakness section;  I wonder that Stackelberg Equilbrium is difficult in this context because the Generator is non-parametric?  Can this problem be resolved if we can impose some structural assumptions here? I believe that the result of Stackelberg Equilibrium would be beneficial to the community even if it requires additional (even possibly, a little unrealistic) assumption.\n\n\n- Also, I was a little worried that mentions were not made so much regarding the stability of the learning in the main manuscript, other than at the theorem section (Fig 5 in the Appendix and Table 15 are acknowledged).   As an devil's advocate, I would also like to know if there are any implicit hyperparameters (or the parameters of the training) that one must be particularly careful about in training TriCO,  because \"training multi-agents system\" has always been a challenge---this information would definitely benefit the researchers that would follow this study.", "limitations": "As claimed \"Limitations are implicitly discussed in Section 4\", regarding limited supervision and drastic distributional shift. I believe this should be explicitly stated, and this will not hurt the work's credibility.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "## Strength\n\n This paper is clearly strong in two aspects : first, it brings in “game theory” into  the challenging problem of interacting systems in the setting of SSL, and backs it up with a formal guarantee. “Three” player game is something that I have never encountered before in this genre, and strikes as a fundamental novelty.  \nSecond, the efficacy of the method is demonstrated on a strong basis. Ablations are also conducted thoroughly.  While I am committed not to evaluate the research based merely on “the depth of the experiments”, I believe that this thoroughness solidly supports the significance of the proposition. \n\n\n## Weaknesses \n\n- Several parts are difficult to read and some fix would greatly help conveying the idea. \nFor example, several definitions seems missing.  From the context, $p_i$ in (1) seems like the ensemble average, but this has never been explicitly stated in the main manuscript.  $H(p(y))$ is also confusing, since $p(y)$ is a scaler ;  I believe this is $p_Y$.    It also seems that $L_sup$ stands for $E_{i, X, Y}[ CE (f_i (v_i(X), Y) ]$, but I do not see it explicitly defined. \nThe same goes for $Accuracy_{val}(f_1, f_2)$.   Theorem 1, I believe, is supposed to be followed by “for all $f_i, G, \\pi_T$.  \n\n- Lastly, although the existence of Nash Equilibrium is wonderful and inspiring, I believe this problem is a Stackelberg game, and I believe Stackelberg Equilbrium and Nash Equilbrium are not always the same?   If I am not missing some lines, I believe some explanation in this aspect is warranted. I believe that if the paper can provide the existence of  Stackelberg Equilbrium, this work is irrefutable.\nI'm very much open to increasing the score when these subtlties and questions below are resolved to a degree.", "was_revised": true, "final_justification": "The authors have provided convincing rebuttal, providing the existence of Stackelberg-Nash equlibrium, making the theory in stronger alignment to the methods and experiments. The authors also conducted additional experiments to investigate the sensitivity of the hyperparameters, which is one of the most fearful factor in the systems involving multiple agents.  I believe that this work contributes an important aspect of the semi-supervised learning, and I raised my score to 5.", "rebuttal": "### Q1: Clarification of Notation and Mathematical Precision\n\nWe thank the reviewer for highlighting the importance of notation clarity and agree that the current manuscript can be improved in this regard. We will revise the final version to explicitly define key symbols and clarify equations.\n\nIn Equation (1), we compute mutual information (MI) between model predictions $\\hat{y}$ and model posterior induced by dropout. Specifically, we use:\n\n$\\mathcal{I}[\\hat{y}; \\omega] = \\mathbb{H}\\left[\\frac{1}{K} \\sum_{k=1}^{K} p_{\\theta}(y|x;\\omega_k)\\right] - \\frac{1}{K} \\sum_{k=1}^{K} \\mathbb{H}\\left[p_{\\theta}(y|x;\\omega_k)\\right]$\n\nwhere $\\omega_k$ denotes the $k$-th Monte Carlo dropout mask. We will clarify that $\\bar{p}(y|x) := \\frac{1}{K} \\sum_{k=1}^{K} p_{\\theta}(y|x;\\omega_k)$ is the ensemble distribution, and $\\mathbb{H}[p] = -\\sum_{c=1}^{C} p_c \\log p_c$ is the entropy of a categorical distribution. Additionally, we will specify that $\\tau_{\\mathrm{MI}}$ is a scalar threshold used to select pseudo-labels when $\\mathcal{I}[y;\\omega] \\geq \\tau_{\\mathrm{MI}}$, and that its range lies in $[0, \\log C]$, where $C$ is the number of classes.\n\nTheorem 1 will also be revised to include the appropriate quantifiers. The corrected statement reads:  \n**Theorem 1.** Under assumptions (A1)–(A3), for all teacher strategies $\\pi_T \\in \\Pi$, there exists a Stackelberg equilibrium $(f_1^\\star, f_2^\\star, G^\\star)$ such that $(f_1^\\star, f_2^\\star, G^\\star)$ form a joint best-response to $\\pi_T$, and $\\pi_T$ minimizes validation loss given these optimal responses.\n\nMoreover, while the generator $G$ is non-parametric (via 1-step PGD), its updates are deterministic given student parameters, and can be treated as an implicit, structured response in the Stackelberg formulation. We will update Section 4.4 to clarify these points and explicitly distinguish Nash and Stackelberg equilibrium.\n\n\n---\n\n## Q2. Clarification on Equilibrium Type and Theoretical Guarantees\n\nWe sincerely thank the reviewer for their insightful question regarding the distinction between Nash and Stackelberg equilibria in our triadic game formulation. You are absolutely correct—our framework is designed as a *Stackelberg game*, where the teacher $\\pi_T$ acts as the leader, while the student classifiers $f_1$, $f_2$, and the generator $G$ act as followers responding to the teacher’s strategy. \n\nWhile our original statement focused on the existence of a Nash equilibrium, we emphasize that what we actually prove is a Stackelberg-Nash equilibrium. That is, the equilibrium point is derived from a Stackelberg setting, but satisfies the equilibrium conditions of all agents, as clarified below.\n\nIn particular, our proof (Appendix A, Theorem 1) proceeds in two stages:\n\t**In Equations (9–18)**, we establish that the strategy spaces of the teacher, students, and generator are all compact subsets of Euclidean space, and the payoff functions are jointly continuous. Therefore, by Glicksberg’s Theorem, a pure-strategy Nash equilibrium exists. Notably, Glicksberg’s theorem is agnostic to role asymmetry and is applicable to Stackelberg games as long as continuity and compactness hold.\n\t**In Equations (19–32)**, we construct the solution explicitly under the Stackelberg game formulation, where the teacher optimizes a meta-objective based on the best responses of the students and generator. This construction satisfies the Stackelberg equilibrium condition by solving the bilevel optimization problem where the followers’ reactions are uniquely defined.\n\nAlthough the generator $G$ is non-parametric, it is defined via a deterministic one-step projected gradient ascent (PGD) update in the embedding space, which depends on the current student parameters $\\theta_S$. This structure allows us to treat $G$ as an implicit function $ G(\\theta_S)$, which is continuous with respect to $\\theta_S$.\n\nThis design ensures that the generator’s response satisfies the continuity and stability conditions required for the teacher's optimization in the Stackelberg game to be well-posed. Consequently, we do **NOT** require any additional structural assumptions on $G$ beyond those already assumed in our setting. Moreover, as shown in Equations(29)-(31) in Appendix A, the joint strategy of the students and generator admits a well-defined best-response mapping under a fixed teacher strategy $\\pi_T$. By applying a fixed-point theorem over this composite response, we construct the Stackelberg equilibrium even in the presence of a non-parametric generator. This validates the existence of a triadic Stackelberg equilibrium under the assumptions already established.\n\nTo avoid ambiguity, we will revise the manuscript to explicitly state that our framework admits a Stackelberg equilibrium, and we will introduce this result formally as **Theorem 2** in the revised version. We believe this clarification strengthens the theoretical foundation and removes any residual confusion about equilibrium definitions.\n\n**Theorem 2 (Existence of Stackelberg Equilibrium).**\nIn the Stackelberg formulation, we assume that one party (the leader) commits to a strategy first, and the remaining parties (the followers) best-respond. In our case, the teacher $\\pi_T \\in \\Pi_T$ is the leader, while the students $(f_1, f_2) \\in \\Pi_S$ and generator $G \\in \\Pi_G$ are the followers. We use the same assumptions on compactness and continuity as in Theorem 1.\n\nGiven a fixed teacher strategy $\\pi_T \\in \\Pi_T$, the followers play a simultaneous game. Their equilibrium is defined by the following best-response conditions:\n\n$$\\forall f_i \\in \\Pi_S: \\quad f_i^* (\\pi_T) = \\arg\\min_{f_i \\in \\Pi_S} R_S(f_i, \\pi_T, G^* (\\pi_T)), \\quad i = 1,2,$$\n\n$$G^* (\\pi_T) = \\arg\\max_{G \\in \\Pi_G} R_G(G, \\pi_T, f_i^* (\\pi_T)).$$\n\nThe teacher then selects her strategy to maximize her own payoff given the best responses of the followers:\n$$\n\\pi_T^* = \\arg\\max_{\\pi_T \\in \\Pi_T} R_T\\left(\\pi_T, f_1^* (\\pi_T), f_2^* (\\pi_T), G^* (\\pi_T)\\right).\n$$\n\nThen, the tuple $\\left(\\pi_T^* , f_1^* (\\pi_T^* ), f_2^* (\\pi_T^* ), G^* (\\pi_T^* )\\right)$ is a Stackelberg equilibrium if the following conditions hold:\n\n**(Leader’s optimality)**\n$$\nR_T\\left(\\pi_T^* , f_1^* (\\pi_T^* ), f_2^* (\\pi_T^* ), G^* (\\pi_T^* )\\right)\n\\geq R_T\\left(\\pi_T, f_1^* (\\pi_T), f_2^* (\\pi_T), G^* (\\pi_T)\\right), \\quad \\forall \\pi_T \\in \\Pi_T,\n$$\n\n**(Followers’ best responses)**\n$$\n\\forall f_i \\in \\Pi_S:\\quad R_S\\left(f_i^* (\\pi_T^* ), \\pi_T^* , G^* (\\pi_T^* )\\right)\n\\leq R_S\\left(f_i, \\pi_T^* , G^* (\\pi_T^* )\\right), \\quad i=1,2,\n$$\n$$\n\\forall G \\in \\Pi_G:\\quad R_G\\left(G^* (\\pi_T^* ), \\pi_T^* , f_i^* (\\pi_T^* )\\right)\n\\geq R_G\\left(G, \\pi_T^* , f_i^* (\\pi_T^* )\\right).\n$$\n\n**Proof Sketch.**\nWe reuse the compactness and continuity assumptions established in Theorem 1 (Eqs. 9–18 in Appendix A). For each fixed $\\pi_T$, the follower subgame among $f_1, f_2, G$ admits a Nash equilibrium. The teacher’s objective is continuous in $\\pi_T$ given continuous best-response mappings. Therefore, the leader’s optimization has a maximizer $\\pi_T^* \\in \\Pi_T$, which implies the existence of a Stackelberg equilibrium. The specific equlibria is provided in (Eqs. 19–32 in Appendix A)\n\nWe greatly appreciate the reviewer’s suggestion and are confident that this update will improve both precision and clarity for readers and practitioners.\n\n---\n\n\n\n## Q3. Stability of Multi-Agent Training and Sensitivity to Hyperparameters\n\nWe thank the reviewer for raising the important question of stability and sensitivity in training multi-agent systems. Despite TRiCo’s triadic design, our training remains highly stable in practice, as evidenced by the low standard deviation across all benchmarks (see Tables 1, 2, and Appendix C). This is due to three core design choices:\n\n- **Structured Modularity**  \n  The roles of the teacher, students, and generator are cleanly separated via frozen encoders and cross-view pseudo-labeling, avoiding entangled feedback loops that commonly destabilize multi-agent learning.\n\n- **Smooth Meta-Scheduling**  \n  The teacher $\\pi_T$ is warm-started and trained with conservative updates (first-order meta-gradients) using stop-gradient feedback, which helps avoid oscillation in threshold/loss scheduling during early training.\n\n- **Implicit Generator**  \n  The generator performs a single-step PGD perturbation in embedding space, without trainable parameters or gradients, keeping its behavior interpretable and consistent.\n\nIn addition to stability, we conducted a systematic sensitivity analysis of TRiCo’s key hyperparameters—MI threshold $\\tau_{\\mathrm{MI}}$, unsupervised loss weight $\\lambda_u$, and perturbation budget $\\epsilon$. As shown below, TRiCo exhibits strong robustness across a wide range of values:\n\n**Table: Sensitivity of TRiCo on CIFAR-10 w.r.t. hyperparameters** $\\tau_{\\mathrm{MI}}$, $\\lambda_u$, and $\\epsilon$. Other settings fixed as default.\n\n| **Hyperparam**       | **Value Range**           | **Top-1 Acc. (%)**       | **Change**     |\n|----------------------|---------------------------|---------------------------|----------------|\n| $\\tau_{\\mathrm{MI}}$ | 0.05 / 0.10 / 0.15         | 95.6 / 95.9 / 94.8        | $\\leq$1.1%     |\n| $\\lambda_u$          | 0.5 / 1.0 / 2.0            | 95.4 / 95.9 / 95.7        | $\\leq$0.5%     |\n| $\\epsilon$ (FGSM)    | 1e-4 / 5e-4 / 1e-3         | 95.8 / 95.9 / 95.1        | $\\leq$0.8%     |\n\n\n---\n\n### Q4: Explicit Limitations\nWe thank the reviewer and will clarify TRiCo’s limitations in Section 6. TRiCo relies on a small labeled validation set for meta-optimization, which may limit use in fully unsupervised settings—future work may explore self-supervised proxies. The non-parametric generator lacks memory and could benefit from learnable dynamics (e.g., conditional SSMs). TRiCo does not yet address extreme shifts like open-set scenarios, which could be tackled via OOD-aware extensions. These are trade-offs that offer clear paths for future work."}, {"review_id": "uFLodbiLrJ", "reviewer": "Reviewer_C9DT", "summary": "The paper introduces a semi-supervised learning (SSL) method that is based on a game-theoretic interaction among models. This paper observes how existing methods select pseduo-labels based on the models' confidence, which is susceptible to calibration error and can be detrimental. The proposed solution uses mutual information as an alternative to pseudo-label selection. It is shown that across a wide range of standard SSL benchmarks, the proposed method outperforms not only existing SSL methods but also fully-supervised methods despite using only 25% of the dataset.", "questions": "Is there any part of the method that can be removed or combined with others and yet retain the high performance?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n* The proposed method is tested on many image classification benchmarks. This is convincing, as many SSL methods are tested on fewer datasets due to the higher compute requirements. \n\n* This paper demonstrates various benefits to the proposed method, including feature clustering and CAM visualizations. \n\n**Weaknesses**\n* Complexity: This is a limitation that exists not only in the proposed method, but also in existing semi-supervised learning methods. The proposed method uses two encoders that are trained using different methods, several additional models, and adversarially-robust training, in addition to the many details. Such multi-stage training methods and those that combine various modules are hard to apply to new problems due to development costs and potentially requiring a whole new set of hyperparameters to tune.\n\n* It's unclear what the role of each stage plays. Can some of the components be removed yet achieve similar performance? While Section 2 motivates each component, it's unclear that the intended effects are the cause for enhanced performance. For example, describing the teacher-student training as a Stackelberg game doesn't seem to use anything from game theory. Existence of a Nash equilibrium often doesn't say much about the quality of the equilibrium or convergence to equilibrium. While the teacher-student training is described as a bi-level (Stackelberg) game, the students/teacher are updated iteratively instead of being solved at different time scales. \n\nOverall, the practical benefits seem to outweigh the limitations as an off-the-shelf method. My main concern is that the method seems to be a bit ad-hoc, and therefore may have limited applicability outside of the considered benchmarks. \n\n\n**Minor Comments**\n\n* The figures should have thicker curves; they are hard to read, for example Figure 1.a.\n\n* It seems unnecessary to put the results figure in page 1. I recommend moving this to the experiments/results sections.", "was_revised": true, "final_justification": "The authors addressed my technical concerns regarding computational overhead and the Stackelberg formulation. The method is new and the authors demonstrate moderate gains on standard benchmarks. The contribution is solid, but does not substantially advance the state-of-the-art or open up new applications, which prevents me from giving a stronger recommendation. Therefore, I hold my rating as weak accept.", "rebuttal": "### Q1: Method Complexity, Modularity, and Practicality\n\nWe thank the reviewer for raising important concerns regarding the complexity and modularity of TRiCo. While our method integrates several components, it remains a principled, end-to-end, and fully differentiable framework—distinct from multi-stage pipelines or ad-hoc ensembles. All components are trained jointly, and their interactions are synergistic rather than additive.\n\n\n\nTo assess modularity, we conduct an ablation study to examine the contribution of each component. Results (Table: Ablation on TRiCo Components) show that removing any major module degrades performance, with the full TRiCo achieving 96.3% on CIFAR-10 (10% labeled), and drops of 1.1–2.2% observed when disabling MI filtering, the meta-teacher, generator, or co-training structure.\n\n**Table: Ablation on TRiCo Components (CIFAR-10, 10% labeled) *All ablations are repeated over 5 random splits to ensure stability*.**\n\n| **Variant**                          | **Top-1 Acc. (%)** | **Δ vs. Full** |\n|-------------------------------------|---------------------|----------------|\n| Full TRiCo                          | **96.3 ± 0.3**      | —              |\n| w/o MI filtering (Conf-τ only)      | 95.2 ± 0.4          | −1.1           |\n| w/o Meta-Teacher (Fixed threshold)  | 94.9 ± 0.6          | −1.4           |\n| w/o Generator (No PGD)              | 95.0 ± 0.5          | −1.3           |\n| Single Student (No Co-training)     | 94.1 ± 0.5          | −2.2           |\n\nDespite integrating multiple components, TRiCo remains efficient and simple to deploy: frozen backbones (no encoder training), lightweight embedding-level adversarial updates (no input gradients), and meta-learning with only first-order updates.\n\nTo quantify computational cost, we provide a component-wise breakdown in **Appendix B** and summarize below:\n\n**Table: Component-wise Complexity Breakdown of TRiCo**\n\n| **Component**                      | **Added Overhead**           | **Optimization Strategy / Description**                            |\n|-----------------------------------|-------------------------------|---------------------------------------------------------------------|\n| Mutual Information Estimation     | ~+4.5% FLOPs                  | $K{=}5$ forward passes; stop-gradient; no backprop                  |\n| Adversarial Generator (1-step PGD) | ~+1.5% FLOPs                 | Embedding-level perturbation only; no backward computation          |\n| Meta-Gradient Update              | ~+1% FLOPs, ~+10% memory      | First-order gradient only; unrolled once per step                   |\n| **Total (vs. MCT)**               | **~+7% FLOPs**, **~+10% memory** | No mixed-precision or checkpointing; further savings possible     |\n\nOverall, TRiCo offers a practical trade-off between performance and complexity. It does not rely on strong augmentations, learnable view encoders, or auxiliary networks, and operates efficiently on a single NVIDIA RTX A6000 (48GB) or equivalent **24GB-class GPU** (e.g., RTX 3090, RTX 4090).\n\n---\n\n### Q2: Theoretical Justification and Stackelberg Formulation\n\nWe sincerely thank the reviewer for their insightful question regarding the distinction between Nash and Stackelberg equilibria in our triadic game formulation. You are absolutely correct—our framework is designed as a *Stackelberg game*, where the teacher $\\pi_T$ acts as the leader, while the student classifiers $f_1$, $f_2$, and the generator $G$ act as followers responding to the teacher’s strategy. \n\nWhile our original statement focused on the existence of a Nash equilibrium, we emphasize that what we actually prove is a Stackelberg-Nash equilibrium. That is, the equilibrium point is derived from a Stackelberg setting, but satisfies the equilibrium conditions of all agents, as clarified below.\n\nIn particular, our proof (Appendix A, Theorem 1) proceeds in two stages:\n\t**In Equations (9–18)**, we establish that the strategy spaces of the teacher, students, and generator are all compact subsets of Euclidean space, and the payoff functions are jointly continuous. Therefore, by Glicksberg’s Theorem, a pure-strategy Nash equilibrium exists. Notably, Glicksberg’s theorem is agnostic to role asymmetry and is applicable to Stackelberg games as long as continuity and compactness hold.\n\t**In Equations (19–32)**, we construct the solution explicitly under the Stackelberg game formulation, where the teacher optimizes a meta-objective based on the best responses of the students and generator. This construction satisfies the Stackelberg equilibrium condition by solving the bilevel optimization problem where the followers’ reactions are uniquely defined.\n\nAlthough the generator $G$ is non-parametric, it is defined via a deterministic one-step projected gradient ascent (PGD) update in the embedding space, which depends on the current student parameters $\\theta_S$. This structure allows us to treat $G$ as an implicit function $ G(\\theta_S)$, which is continuous with respect to $\\theta_S$.\n\nThis design ensures that the generator’s response satisfies the continuity and stability conditions required for the teacher's optimization in the Stackelberg game to be well-posed. Consequently, we do **NOT** require any additional structural assumptions on $G$ beyond those already assumed in our setting. Moreover, as shown in Equations(29)-(31) in Appendix A, the joint strategy of the students and generator admits a well-defined best-response mapping under a fixed teacher strategy $\\pi_T$. By applying a fixed-point theorem over this composite response, we construct the Stackelberg equilibrium even in the presence of a non-parametric generator. This validates the existence of a triadic Stackelberg equilibrium under the assumptions already established.\n\nTo avoid ambiguity, we will revise the manuscript to explicitly state that our framework admits a Stackelberg equilibrium, and we will introduce this result formally as **Theorem 2** in the revised version. We believe this clarification strengthens the theoretical foundation and removes any residual confusion about equilibrium definitions.\n\n**Theorem 2 (Existence of Stackelberg Equilibrium).**\nIn the Stackelberg formulation, we assume that one party (the leader) commits to a strategy first, and the remaining parties (the followers) best-respond. In our case, the teacher $\\pi_T \\in \\Pi_T$ is the leader, while the students $(f_1, f_2) \\in \\Pi_S$ and generator $G \\in \\Pi_G$ are the followers. We use the same assumptions on compactness and continuity as in Theorem 1.\n\nGiven a fixed teacher strategy $\\pi_T \\in \\Pi_T$, the followers play a simultaneous game. Their equilibrium is defined by the following best-response conditions:\n\n$$\\forall f_i \\in \\Pi_S: \\quad f_i^* (\\pi_T) = \\arg\\min_{f_i \\in \\Pi_S} R_S(f_i, \\pi_T, G^* (\\pi_T)), \\quad i = 1,2,$$\n\n$$G^* (\\pi_T) = \\arg\\max_{G \\in \\Pi_G} R_G(G, \\pi_T, f_i^* (\\pi_T)).$$\n\nThe teacher then selects her strategy to maximize her own payoff given the best responses of the followers:\n$$\n\\pi_T^* = \\arg\\max_{\\pi_T \\in \\Pi_T} R_T\\left(\\pi_T, f_1^* (\\pi_T), f_2^* (\\pi_T), G^* (\\pi_T)\\right).\n$$\n\nThen, the tuple $\\left(\\pi_T^* , f_1^* (\\pi_T^* ), f_2^* (\\pi_T^* ), G^* (\\pi_T^* )\\right)$ is a Stackelberg equilibrium if the following conditions hold:\n\n**(Leader’s optimality)**\n$$\nR_T\\left(\\pi_T^* , f_1^* (\\pi_T^* ), f_2^* (\\pi_T^* ), G^* (\\pi_T^* )\\right)\n\\geq R_T\\left(\\pi_T, f_1^* (\\pi_T), f_2^* (\\pi_T), G^* (\\pi_T)\\right), \\quad \\forall \\pi_T \\in \\Pi_T,\n$$\n\n**(Followers’ best responses)**\n$$\n\\forall f_i \\in \\Pi_S:\\quad R_S\\left(f_i^* (\\pi_T^* ), \\pi_T^* , G^* (\\pi_T^* )\\right)\n\\leq R_S\\left(f_i, \\pi_T^* , G^* (\\pi_T^* )\\right), \\quad i=1,2,\n$$\n$$\n\\forall G \\in \\Pi_G:\\quad R_G\\left(G^* (\\pi_T^* ), \\pi_T^* , f_i^* (\\pi_T^* )\\right)\n\\geq R_G\\left(G, \\pi_T^* , f_i^* (\\pi_T^* )\\right).\n$$\n\n**Proof Sketch.**\nWe reuse the compactness and continuity assumptions established in Theorem 1 (Eqs. 9–18 in Appendix A). For each fixed $\\pi_T$, the follower subgame among $f_1, f_2, G$ admits a Nash equilibrium. The teacher’s objective is continuous in $\\pi_T$ given continuous best-response mappings. Therefore, the leader’s optimization has a maximizer $\\pi_T^* \\in \\Pi_T$, which implies the existence of a Stackelberg equilibrium. The specific equlibria is provided in (Eqs. 19–32 in Appendix A)\n\nWe greatly appreciate the reviewer’s suggestion and are confident that this update will improve both precision and clarity for readers and practitioners.\n\n---\n\n### Q3: Clarity and Presentation Improvements\nWe thank the reviewer for the suggestions on figure quality. We will increase curve thickness and relocate Figure 1 to the experiments section in the final version to improve clarity.\nRather than a loose combination of techniques, TRiCo’s design is guided by a unified game-theoretic principle: the teacher regulates pseudo-label quality and loss dynamics to shape stable, complementary student learning trajectories.\n\n---\n\n## Summary\n\nWe clarify that TRiCo is a principled, end-to-end differentiable framework—not a multi-stage ensemble—where all components are co-trained for synergy. Through ablations, we show that each module (e.g., MI filtering, generator, meta-teacher) contributes significantly to performance. Despite its triadic structure, TRiCo incurs only modest computational overhead and runs efficiently on a 24GB-class GPU.\n\nTheoretically, we explicitly formalize TRiCo as a triadic Stackelberg game, not just a heuristic co-training strategy. We prove the existence of a Stackelberg equilibrium using compactness and continuity assumptions, even with a non-parametric generator. This resolves the ambiguity around equilibrium type and strengthens the theoretical foundation.\n\nWe also appreciate the reviewer’s suggestions on figure clarity and will revise visuals accordingly. Overall, TRiCo is both theoretically grounded and practically viable for semi-supervised learning."}]

Author rebuttal:

Meta-review: (a) Summary of claims and findings:
The paper proposes TRiCo, a triadic game-theoretic semi-supervised learning framework with three interacting roles: two student classifiers on complementary frozen representations, a meta-learned teacher that adaptively controls pseudo-label filtering and loss weights via validation feedback, and a non-parametric adversarial generator that perturbs embeddings to probe decision boundaries. Pseudo-label selection is based on mutual information (from MC-dropout) rather than confidence. The interaction is framed as a Stackelberg game with the teacher as leader. Experiments on CIFAR-10/100, SVHN, STL-10, ImageNet and class-imbalanced variants show gains over strong baselines in low-label regimes. During rebuttal, the authors added: (i) class imbalance results (CIFAR-10-LT, ImageNet-LT), (ii) variance reporting across runs, (iii) results with weaker encoders (MAE, SwAV), (iv) ablations indicating all components contribute, (v) computational overhead analysis (~7% FLOPs over Meta Co-Training), and (vi) a theoretical clarification proving existence of a Stackelberg equilibrium (Stackelberg–Nash) with a sketch and plan to add a new theorem.

(b) Strengths:

- Clear problem motivation; addresses pseudo-label reliability, stability, and hard-sample modeling in SSL.
- Principled formulation with a teacher-students-generator Stackelberg game; improved theoretical framing in rebuttal.
- MI-based filtering is better aligned with epistemic uncertainty; new head-to-head shows MI vs meta-learned confidence favors MI across datasets.
- Broad empirical coverage, including long-tailed settings and weaker backbones; consistent gains and reduced variance reported.
- Practicality: frozen encoders, single-step embedding PGD, first-order meta-updates; overhead quantified and modest relative to related meta co-training.

(c) Weaknesses / missing pieces:

- Complexity: multi-agent design with several moving parts; while ablations show contributions, the method remains heavier than many SSL baselines and could be perceived as ensemble-like.
- Novelty is moderate at the component level (co-training, meta-learning, MI filtering, adversarial perturbation); contribution is in the integration and game-theoretic framing.
- Theory focuses on existence; no guarantees on convergence rates or equilibrium quality; the original Nash vs Stackelberg mismatch required rebuttal to fix.
- MI estimation via MC-dropout adds stochasticity and can be noisy early; mitigations are described, but a comparison with alternative uncertainty estimators (e.g., Dirichlet) remains future work.
- Some reported gains on large-scale settings (e.g., ImageNet 10%) are modest; strongest advantages appear in lower-label or imbalanced regimes.

(d) Reasons for decision:

The rebuttal addressed major reviewer concerns: clarified theory with a Stackelberg-equilibrium existence result; provided ablations showing each component’s necessity; added class-imbalance results and variance reporting; compared MI vs meta-learned confidence; and quantified overhead (~+7% FLOPs, ~+10% memory vs Meta Co-Training). While the design is complex and component-level novelty is incremental, the integration is coherent and empirically effective, especially in challenging low-label/imbalanced scenarios. This justifies a poster acceptance. The contribution is more methodological/practical than theoretical; appropriate for poster rather than a higher track.

(e) Rebuttal impact and discussion:

- R1 (fairness, imbalance, variance, complexity): Added imbalance (CIFAR-10-LT, ImageNet-LT), std across runs, weaker backbones; ablations and cost breakdown; concerns alleviated; score raised.
- R2 (novelty vs engineering, overhead): Agreed overhead exists but quantified; maintained that triadic design plus MI filtering improves robustness; reviewer remained positive.
- R3 (MI vs confidence, clarity, lossless theory analogy not applicable here): Provided head-to-head MI vs meta-learned confidence favoring MI; stability mitigations added; score increased.
- R4 (Nash vs Stackelberg): Authors corrected framing and provided Stackelberg-equilibrium existence; reviewer satisfied and raised score.
- R5 (complexity, modularity): Ablations show every component matters; overhead modest; reviewer kept weak accept.

I read the author response and discussion. No external information beyond the reviews and rebuttal was used.
Decision: Accept (poster)