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Revisiting Semi-Supervised Learning with Graph Embeddings

Zhilin YangWilliam W. CohenRuslan Salakhutdinov

article2016ICML2,603 citations

Proposes a graph embedding framework for semi-supervised learning that jointly predicts class labels and neighborhood context across both transductive and inductive settings, outperforming standard methods on text and entity classification benchmarks.

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The article addresses challenges in semi-supervised learning, where limited labeled data and graph structures between instances must be leveraged to improve classification accuracy in tasks such as text categorization and entity recognition. Traditional graph-based methods rely on regularization to enforce label consistency among connected nodes but often fail to fully exploit distributional patterns in the graph or produce reusable features. This matters now because many real-world applications, including knowledge-base completion and web-scale information extraction, face scarce labels yet abundant relational data.

The article set out to evaluate a framework called Planetoid that trains instance embeddings to jointly predict class labels and neighborhood context in the graph. It develops both a transductive version, which uses free embeddings for observed nodes, and an inductive version, which defines embeddings as a function of input features so predictions can be made on unseen instances.

The authors tested the approach through controlled experiments on five benchmark datasets spanning text classification, distantly supervised entity extraction, and entity classification in a knowledge base. They compared the new methods against established baselines including label propagation, manifold regularization, transductive support vector machines, semi-supervised embeddings, and unsupervised graph embeddings, using accuracy and recall metrics under varying label rates and consistent data splits.

The inductive variant outperformed the best prior inductive method by up to 18.7 percentage points and 7.8 points on average; the strongest of either variant beat the best competing method by up to 8.5 points and 4.1 on average. Joint training with label information improved embedding quality over purely unsupervised graph embedding, and predicting graph context proved more effective than Laplacian regularization alone. Performance gains were largest when label rates were low and when graph structure carried more signal than raw features.

These results indicate that embedding-based context prediction can deliver higher accuracy and better generalization in label-scarce settings, potentially reducing annotation costs and improving reliability for downstream systems that rely on node classification. The inductive formulation additionally enables deployment on streaming or large-scale data where new instances arrive after training.

The article recommends extending the framework to recurrent networks and testing cases where graphs are derived from feature distances rather than external relations. Further work is needed to assess sensitivity to hyper-parameters and to confirm gains on additional domains before broad adoption.

The findings rest on five datasets and fixed hyper-parameter choices; results may vary if feature vectors are uninformative or if graphs contain substantial noise. Confidence is moderate to high for the reported benchmarks but readers should verify behavior on their own data distributions before making deployment decisions.

Carnegie Mellon Universityclass labelsfeature vectorsgraph embeddingsPlanetoidGraph EmbeddingSemi‑Supervised Learning
arXiv: 1603.08861kimiyoung/planetoid
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Abstract

We present a semi-supervised learning framework based on graph embeddings. Given a graph between instances, we train an embedding for each instance to jointly predict the class label and the neighborhood context in the graph. We develop both transductive and inductive variants of our method. In the transductive variant of our method, the class labels are determined by both the learned embeddings and input feature vectors, while in the inductive variant, the embeddings are defined as a parametric function of the feature vectors, so predictions can be made on instances not seen during training. On a large and diverse set of benchmark tasks, including text classification, distantly supervised entity extraction, and entity classification, we show improved performance over many of the existing models.

Topics

class labelsfeature vectorsgraph embeddingssemi-supervised learningsupervised machine learningtext classificationunsupervised learninginput featuresword embeddingclassification models

Keywords

PlanetoidGraph EmbeddingSemi‑Supervised LearningTransductive LearningInductive LearningGraph Laplacian RegularizationNegative SamplingRandom WalkLabel PropagationDeepWalk

Table of Contents

  • 1 Introduction
  • 2 Related Work
  • 2.1 Semi-Supervised Learning
  • 2.2 Graph-Based Semi-Supervised Learning
  • 2.3 Learning Embeddings
  • 2.4 Comparison
  • 3 Semi-Supervised Learning with Graph Embeddings
  • 3.1 Sampling Context
  • 3.2 Transductive Formulation
  • 3.3 Inductive Formulation
  • 3.4 Training
  • 4 Experiments
  • 4.1 Text Classification
  • 4.2 Distantly-Supervised Entity Extraction
  • 4.3 Entity Classification
  • 5 Conclusion
  • References

Knowls

  1. Knowl 1 — Planetoid Transductive Semi-Supervised Learning Formulation (Planetoid-T)

    model/method

    In the transductive formulation of the Planetoid framework (Planetoid-T), each instance i∈{1,…,L+U}i \in \{1, \dots, L + U\} in a dataset with LL labeled instances and UU unlabeled instances is assigned a free embedding vector ei∈Rde_i \in \mathbb{R}^d. Predictions are computed from both the instance feature vector xi∈RFx_i \in \mathbb{R}^F and its embedding eie_i.

    A feed-forward neural network maps the feature vector xx through kk layers to produce hidden representation hk(x)h^k(x), and maps the embedding ee through ll layers to produce hl(e)h^l(e). Layer activations are defined by: hm(v)=ReLU(Wmhm−1(v)+bm)h^m(v) = \text{ReLU}(W^m h^{m-1}(v) + b^m) where WmW^m and bmb^m are layer weight matrices and bias vectors, h0(v)=vh^0(v) = v, and ReLU(z)=max⁡(0,z)\text{ReLU}(z) = \max(0, z).

    The conditional probability distribution p(y∣x,e)p(y \mid x, e) over class labels yy is computed by concatenating hk(x)h^k(x) and hl(e)h^l(e) and feeding the concatenated representation into a softmax layer: p(y∣x,e)=exp⁡([hk(x)T,hl(e)T]wy)∑y′exp⁡([hk(x)T,hl(e)T]wy′)p(y \mid x, e) = \frac{\exp\left([h^k(x)^T, h^l(e)^T] w_y\right)}{\sum_{y'} \exp\left([h^k(x)^T, h^l(e)^T] w_{y'}\right)} where [uT,vT][u^T, v^T] denotes vector concatenation, and wyw_y is the model parameter vector for class yy.

    The overall transductive objective function minimizes a joint loss composed of the supervised classification loss on labeled instances and an unsupervised negative-sampling context loss on graph and label contexts: L=−1L∑i=1Llog⁡p(yi∣xi,ei)−λE(i,c,γ)[log⁡σ(γwcTei)]\mathcal{L} = -\frac{1}{L} \sum_{i=1}^L \log p(y_i \mid x_i, e_i) - \lambda \mathbb{E}_{(i, c, \gamma)} \left[\log \sigma(\gamma w_c^T e_i)\right] where λ\lambda is a weighting factor, σ(z)=11+e−z\sigma(z) = \frac{1}{1 + e^{-z}} is the sigmoid function, wcw_c is the context parameter vector for context node cc, and (i,c,γ)(i, c, \gamma) is a context triplet sampled from graph and label distributions with binary indicator γ∈{+1,−1}\gamma \in \{+1, -1\}.

  2. Knowl 2 — Planetoid Inductive Semi-Supervised Learning Formulation (Planetoid-I)

    model/method

    In the inductive formulation of the Planetoid framework (Planetoid-I), instance embeddings are parameterized as functions of input feature vectors xx rather than maintained as free lookup parameters. This enables the model to generalize predictions to instances not observed in the graph at training time.

    For an instance with feature vector x∈RFx \in \mathbb{R}^F, the embedding e(x)e(x) is defined by applying l1l_1 feed-forward layers to xx: e(x)=hl1(x)e(x) = h^{l_1}(x) where layer activations are defined by hm(v)=ReLU(Wmhm−1(v)+bm)h^m(v) = \text{ReLU}(W^m h^{m-1}(v) + b^m) with h0(x)=xh^0(x) = x and ReLU(z)=max⁡(0,z)\text{ReLU}(z) = \max(0, z). A subsequent set of l2l_2 layers is applied to e(x)e(x), yielding hl(x)=hl2(hl1(x))h^l(x) = h^{l_2}(h^{l_1}(x)), where l=l1+l2l = l_1 + l_2. In parallel, a separate feed-forward branch applies kk layers directly to xx to yield hk(x)h^k(x).

    The predicted class probability distribution depends solely on feature vector xx: p(y∣x)=exp⁡([hk(x)T,hl(x)T]wy)∑y′exp⁡([hk(x)T,hl(x)T]wy′)p(y \mid x) = \frac{\exp\left([h^k(x)^T, h^l(x)^T] w_y\right)}{\sum_{y'} \exp\left([h^k(x)^T, h^l(x)^T] w_{y'}\right)} where [uT,vT][u^T, v^T] denotes vector concatenation and wyw_y is the parameter vector for class yy.

    The inductive loss function jointly optimizes classification loss on labeled instances and context prediction loss: L=−1L∑i=1Llog⁡p(yi∣xi)−λE(i,c,γ)[log⁡σ(γwcThl1(xi))]\mathcal{L} = -\frac{1}{L} \sum_{i=1}^L \log p(y_i \mid x_i) - \lambda \mathbb{E}_{(i, c, \gamma)} \left[\log \sigma\left(\gamma w_c^T h^{l_1}(x_i)\right)\right] where LL is the number of labeled instances, λ\lambda is a weighting constant, σ(z)=11+e−z\sigma(z) = \frac{1}{1 + e^{-z}}, wcw_c is the context parameter vector, and (i,c,γ)(i, c, \gamma) is a context triplet sampled from graph random walks and label sets with γ∈{+1,−1}\gamma \in \{+1, -1\}.

  3. Knowl 3 — Context Sampling Distribution for Graph and Label Embeddings in Planetoid

    algorithm

    Planetoid formulates an unsupervised loss Lu=−E(i,c,γ)[log⁡σ(γwcTei)]\mathcal{L}_u = -\mathbb{E}_{(i, c, \gamma)} [\log \sigma(\gamma w_c^T e_i)] based on predicting graph context and class label context using negative sampling, where eie_i is the embedding of instance ii, wcw_c is the parameter vector for context node cc, γ∈{+1,−1}\gamma \in \{+1, -1\} is the positive/negative sample indicator, and σ(z)=(1+e−z)−1\sigma(z) = (1 + e^{-z})^{-1}.

    The context distribution p(i,c,γ)p(i, c, \gamma) is sampled using positive sample ratio r1∈(0,1)r_1 \in (0, 1), graph context ratio r2∈(0,1)r_2 \in (0, 1), random walk length qq, and context window size dd. With probability r2r_2, context is sampled from graph adjacency matrix AA via random walk; with probability (1−r2)(1 - r_2), context is sampled from labeled instance pairs to inject class supervision directly into the embedding space.

    Input: Graph adjacency matrix AA of size (L+U)×(L+U)(L+U) \times (L+U), labeled instance set with labels y1:Ly_{1:L}, positive sample ratio r1∈(0,1)r_1 \in (0,1), graph context ratio r2∈(0,1)r_2 \in (0,1), random walk length qq, window size dd
    Output: Context triplet (i,c,γ)(i, c, \gamma)
    Sample uniform random number u1∼Uniform(0,1)u_1 \sim \text{Uniform}(0, 1)
    if u1<r1u_1 < r_1 then
        γ←+1\gamma \leftarrow +1
    else
        γ←−1\gamma \leftarrow -1
    Sample uniform random number u2∼Uniform(0,1)u_2 \sim \text{Uniform}(0, 1)
    if u2<r2u_2 < r_2 then
        Uniformly sample initial node S1∈{1,…,L+U}S_1 \in \{1, \dots, L + U\}
        for t←2t \leftarrow 2 to qq do
            Sample next node St=jS_t = j with probability ASt−1,j/∑j′=1L+UASt−1,j′A_{S_{t-1}, j} / \sum_{j'=1}^{L+U} A_{S_{t-1}, j'}
        Uniformly sample a pair (Sj,Sk)(S_j, S_k) from walk SS such that ∣j−k∣<d|j - k| < d
        i←Sji \leftarrow S_j
        c←Skc \leftarrow S_k
        if γ=−1\gamma = -1 then
            Uniformly sample context node cc from {1,…,L+U}\{1, \dots, L + U\}
    else
        if γ=+1\gamma = +1 then
            Uniformly sample instance pair (i,c)(i, c) from labeled set such that yi=ycy_i = y_c
        else
            Uniformly sample instance pair (i,c)(i, c) from labeled set such that yi≠ycy_i \neq y_c
    return (i,c,γ)(i, c, \gamma)

    Standard hyperparameter settings used in experiments are r1=5/6r_1 = 5/6, q=10q = 10, and d=3d = 3.

  4. Knowl 4 — Mini-Batch SGD Training Procedure for Planetoid

    algorithm

    Planetoid trains neural network parameters θ\theta (and free embeddings ee in the transductive case) using alternating mini-batch stochastic gradient descent (SGD). Rather than explicitly fixing a weighting factor λ\lambda in the joint loss, the relative balance between the supervised loss Ls\mathcal{L}_s and unsupervised context loss Lu\mathcal{L}_u is controlled by alternating T1T_1 iterations of SGD on supervised mini-batches of size N1N_1 and T2T_2 iterations of SGD on unsupervised context mini-batches of size N2N_2.

    Before joint training, the model applies a sequence of unsupervised pretraining iterations that optimize Lu\mathcal{L}_u alone, initializing embeddings ee.

    Input: Graph adjacency matrix AA, feature vectors x1:L+Ux_{1:L+U}, labels y1:Ly_{1:L}, supervised batch iterations T1T_1, unsupervised batch iterations T2T_2, batch sizes N1N_1 and N2N_2
    Output: Model parameters θ\theta and embeddings ee
    Pretrain embeddings ee by optimizing unsupervised loss Lu\mathcal{L}_u alone for a designated number of iterations
    repeat
        for t←1t \leftarrow 1 to T1T_1 do
            Sample a mini-batch of labeled instances {i1,…,iN1}\{i_1, \dots, i_{N_1}\} from {1,…,L}\{1, \dots, L\}
            Compute supervised batch loss Ls=−1N1∑ilog⁡p(yi∣xi,ei)\mathcal{L}_s = -\frac{1}{N_1} \sum_{i} \log p(y_i \mid x_i, e_i)
            Update parameters θ\theta (and embeddings ee if transductive) via gradient descent step on Ls\mathcal{L}_s
        for t←1t \leftarrow 1 to T2T_2 do
            Sample a mini-batch of context triplets {(im,cm,γm)}m=1N2\{(i_m, c_m, \gamma_m)\}_{m=1}^{N_2} from context distribution p(i,c,γ)p(i, c, \gamma)
            Compute unsupervised batch loss Lu=−1N2∑m=1N2log⁡σ(γmwcmTeim)\mathcal{L}_u = -\frac{1}{N_2} \sum_{m=1}^{N_2} \log \sigma(\gamma_m w_{c_m}^T e_{i_m})
            Update context parameters ww, network weights θ\theta, and embeddings ee via gradient descent step on Lu\mathcal{L}_u
    until stopping criterion is met

    In the inductive formulation (Planetoid-I), eie_i is replaced by hl1(xi)h^{l_1}(x_i), p(yi∣xi,ei)p(y_i \mid x_i, e_i) is replaced by p(yi∣xi)p(y_i \mid x_i), and updates apply only to network parameters θ\theta. Standard batch sizes used in experiments are N1=200N_1 = 200 and N2=200N_2 = 200.

  5. Knowl 5 — Planetoid-G: Feature-Free Semi-Supervised Graph Embedding Variant

    model/method

    Planetoid-G is a variant of the Planetoid framework designed for graph-based semi-supervised learning when node feature vectors xx are unavailable or omitted.

    In Planetoid-G, each node ii is assigned a free embedding vector eie_i. A feed-forward network with ll layers maps eie_i to hidden representation hl(ei)h^l(e_i), defined recursively by hm(e)=ReLU(Wmhm−1(e)+bm)h^m(e) = \text{ReLU}(W^m h^{m-1}(e) + b^m) where h0(e)=eh^0(e) = e and ReLU(z)=max⁡(0,z)\text{ReLU}(z) = \max(0, z). Class predictions are computed via softmax directly from hl(ei)h^l(e_i): p(y∣ei)=exp⁡(hl(ei)Twy)∑y′exp⁡(hl(ei)Twy′)p(y \mid e_i) = \frac{\exp(h^l(e_i)^T w_y)}{\sum_{y'} \exp(h^l(e_i)^T w_{y'})} where wyw_y is the model parameter vector for class yy.

    Planetoid-G jointly optimizes the supervised classification loss on labeled nodes and the unsupervised context loss on sampled graph and label context triplets: L=−1L∑i=1Llog⁡p(yi∣ei)−λE(i,c,γ)[log⁡σ(γwcTei)]\mathcal{L} = -\frac{1}{L} \sum_{i=1}^L \log p(y_i \mid e_i) - \lambda \mathbb{E}_{(i, c, \gamma)} \left[\log \sigma(\gamma w_c^T e_i)\right] Unlike unsupervised graph embedding algorithms (such as DeepWalk) that train node embeddings independently of classification labels, Planetoid-G jointly optimizes context prediction and class label classification while sampling positive and negative context pairs from class labels.

  6. Knowl 6 — Classification Accuracy on Citation Network Text Benchmarks

    data/table

    Evaluation on text classification across three citation networks (Citeseer, Cora, and Pubmed) compares inductive and transductive semi-supervised learning methods. Document bag-of-words represent feature vectors xx, citation links form the symmetric graph adjacency matrix AA (aij=aji=1a_{ij} = a_{ji} = 1 if document ii cites jj), 20 instances per class serve as labeled data, 1,000 instances serve as test data, and the remainder serve as unlabeled data.

    Method Citeseer Cora Pubmed
    Inductive methods
    Feat 0.572 0.574 0.698
    ManiReg 0.601 0.595 0.707
    SemiEmb 0.596 0.590 0.711
    Planetoid-I 0.647 0.612 0.772
    Transductive methods
    TSVM 0.640 0.575 0.622
    LP 0.453 0.680 0.630
    GraphEmb 0.432 0.672 0.653
    Planetoid-G 0.493 0.691 0.664
    Planetoid-T 0.629 0.757 0.757

    Planetoid-I achieves the highest classification accuracy among all inductive models on all three datasets, outperforming the second-best inductive baseline (SemiEmb) by up to 6.16.1 absolute percentage points on Pubmed. Planetoid-T achieves the highest performance on Cora (0.7570.757) and matches Planetoid-I on Pubmed (0.7570.757). Planetoid-G consistently outperforms unsupervised GraphEmb across all three datasets.

  7. Knowl 7 — Recall@k Performance on DIEL Distantly-Supervised Entity Extraction

    data/table

    Evaluation on the Distant Information Extraction using coordinate-term Lists (DIEL) benchmark measures extraction of medical entities from text given entity mention feature vectors and a bipartite graph connecting entity mentions to coordinate lists. The dataset contains 4,373,008 nodes and 4,464,261 edges across 4 entity classes. The evaluation metric is Recall@k with k=240,000k = 240,000 positive instances selected, evaluated against Freebase ground truth (with an empirical upper bound of 0.6170.617 due to unmentioned entities in text).

    Method Recall@k
    Inductive methods
    Feat 0.349
    ManiReg 0.477
    SemiEmb 0.486
    Planetoid-I 0.501
    Transductive methods
    DIEL 0.405
    LP 0.162
    GraphEmb 0.258
    Planetoid-G 0.394
    Planetoid-T 0.500
    Upper Bound 0.617

    Averaged over 10 runs, Planetoid-I (0.5010.501) and Planetoid-T (0.5000.500) outperform all comparison methods, each achieving the top recall in 5 out of 10 runs. Planetoid-G (0.3940.394) improves by 13.613.6 absolute percentage points over unsupervised GraphEmb (0.2580.258).

  8. Knowl 8 — Accuracy on NELL Entity Classification Across Labeling Rates

    data/table

    Evaluation on 210-class entity classification from the Never Ending Language Learning (NELL) knowledge base linked to ClueWeb09 text consists of 65,755 nodes and 266,144 edges. Document bag-of-words from ClueWeb09 provide entity feature vectors, while knowledge base relational triplets (e1,r,e2)(e_1, r, e_2) are represented as bipartite graph connections between entities e1,e2e_1, e_2 and split relation nodes r1,r2r_1, r_2. Accuracy is evaluated across labeling rates β∈{0.1,0.01,0.001}\beta \in \{0.1, 0.01, 0.001\}, where max⁡(βN,1)\max(\beta N, 1) labeled instances are provided for a class with NN entities.

    Method β=0.1\beta = 0.1 β=0.01\beta = 0.01 β=0.001\beta = 0.001
    Inductive methods
    Feat 0.621 0.404 0.217
    ManiReg 0.634 0.413 0.218
    SemiEmb 0.654 0.438 0.267
    Planetoid-I 0.702 0.598 0.454
    Transductive methods
    LP 0.714 0.448 0.265
    GraphEmb 0.795 0.725 0.581
    Planetoid-G / Planetoid-T 0.845 0.757 0.619

    Planetoid-I outperforms the strongest competing inductive baseline (SemiEmb) by 4.8%4.8\%, 16.0%16.0\%, and 18.7%18.7\% absolute percentage points at labeling rates 0.10.1, 0.010.01, and 0.0010.001 respectively. Transductive models achieve higher accuracy than inductive models because graph connectivity is substantially more informative than bag-of-words text features on this dataset. Planetoid-G/T outperforms unsupervised GraphEmb by 5.0%5.0\%, 3.2%3.2\%, and 3.8%3.8\% across the three labeling rates.

  9. Knowl 9 — Trade-offs and Optimization Constraints in Inductive vs Transductive Graph Embeddings

    empirical result

    The relative performance between inductive (Planetoid-I) and transductive (Planetoid-T) graph embedding models depends on the quality and informativeness of instance feature vectors relative to the graph structure:

    1. In the inductive formulation, defining embeddings as a parametric neural network function of input features (e=hl1(x)e = h^{l_1}(x)) imposes structural constraints on the embedding space. When feature representations align well with the target classification task (e.g., Citeseer where Planetoid-I achieves 0.6470.647 accuracy vs. 0.6290.629 for Planetoid-T, and Pubmed where Planetoid-I achieves 0.7720.772 vs. 0.7570.757), these constraints act as beneficial regularizers during non-convex optimization, leading to higher test accuracy than unconstrained transductive embeddings.
    2. Conversely, when graph structure is substantially more discriminative than input features (e.g., Cora where Planetoid-T achieves 0.7570.757 vs. 0.6120.612 for Planetoid-I, and NELL where Planetoid-T achieves 0.8450.845 vs. 0.7020.702 for Planetoid-I at labeling rate β=0.1\beta = 0.1), parameterizing embeddings as functions of features restricts the model away from optimal embedding configurations, causing inductive models to perform worse than free-embedding transductive formulations.

Coverage note — None. All major contributions—including the transductive formulation, inductive formulation, context sampling algorithm, training algorithm, feature-free variant, benchmark evaluations across three domains, and trade-off analysis—are covered.

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Citation

MLA
Yang, Z., et al. “Revisiting Semi-Supervised Learning with Graph Embeddings”. arXiv, 2016, https://doi.org/10.48550/arxiv.1603.08861.
APA
Yang, Z., Cohen, W. W., & Salakhutdinov, R. (2016). Revisiting Semi-Supervised Learning with Graph Embeddings. arXiv. https://doi.org/10.48550/arxiv.1603.08861
Chicago
Yang, Z., W. W. Cohen, and R. Salakhutdinov. 2016. “Revisiting Semi-Supervised Learning with Graph Embeddings”. Preprint, ArXiv. https://doi.org/10.48550/arxiv.1603.08861.
Harvard
Yang, Z., Cohen, W.W. and Salakhutdinov, R. (2016) “Revisiting Semi-Supervised Learning with Graph Embeddings”. arXiv. Available at: https://doi.org/10.48550/arxiv.1603.08861.
Vancouver
1. Yang Z, Cohen WW, Salakhutdinov R (2016) Revisiting Semi-Supervised Learning with Graph Embeddings. https://doi.org/10.48550/arxiv.1603.08861

BibTeX

@misc{https://doi.org/10.48550/arxiv.1603.08861,
  doi = {10.48550/ARXIV.1603.08861},
  url = {https://arxiv.org/abs/1603.08861},
  author = {Yang, Zhilin and Cohen, William W. and Salakhutdinov, Ruslan},
  keywords = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},
  title = {Revisiting Semi-Supervised Learning with Graph Embeddings},
  publisher = {arXiv},
  year = {2016},
  copyright = {arXiv.org perpetual, non-exclusive license}
}
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