When Forgotten Facts Return: Relearning-Aware Unlearning of Knowledge Graph Embeddings via Tangent Kernels
Organizations: Imperial College London · University of Oxford
Abstract
Approximate unlearning answers a deletion request on a knowledge-graph embedding by editing entity embeddings instead of retraining from scratch. These embeddings keep being updated as new facts arrive. Under this ordinary continued training, forgotten facts recover more than in the exact-unlearning baseline, a model retrained without them and given the same continued training. We trace this relearning to entity embeddings shared by forgotten and new facts. To our knowledge, this is the first study of relearning in knowledge-graph embeddings. Relearning-robust methods developed for deep networks do not account for the graph structure underlying this recovery. We propose Kernel-Anticipated Unlearning (KAU), which uses the tangent kernel to model how continued training changes forgotten facts' scores through these shared parameters. KAU expresses this first-order response in closed form and differentiates through it inside the forgetting loss. The objective steers edits toward directions that continued training reverses less. We compare KAU with state-of-the-art knowledge-graph unlearners and relearning-robust methods under identical continued training, measuring leakage beyond the exact-unlearning baseline. Across three backbones, KAU reduces the strongest knowledge-graph unlearner's leakage by 46-63% on FB15k-237 and 50-82% on CoDEx-L, at higher retained utility.
Figures & tables
| TransE | TransH | RotatE | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Method | ExRec | Leak | RER | ExRec | Leak | RER | ExRec | Leak | RER | ||||
| FB15k-237 | GraphDPO | 101.4 | 0.044 | 76.1 | 0.26 | 73.7 | 0.039 | 134.2 | 0.24 | 78.3 | 0.064 | 62.7 | 0.25 |
| ZeroFisher | 140.0 | 0.074 | 129.0 | 0.23 | 124.3 | 0.069 | 234.0 | 0.22 | 124.3 | 0.086 | 83.8 | 0.23 | |
| KAU | 48.5 | 0.022 | 37.5 | 0.28 | 6.6 | 0.021 | 72.0 | 0.26 | 23.8 | 0.024 | 23.4 | 0.33 | |
| Reduction (%) | 52.1 | 50.8 | 50.8 | – | 91.1 | 46.4 | 46.4 | – | 69.6 | 62.7 | 62.7 | – | |
| CoDEx-L | GraphDPO | 77.3 | 0.052 | 31.3 | 0.22 | 75.5 | 0.062 | 39.4 | 0.21 | 108.7 | 0.100 | 53.7 | 0.20 |
| TransE | TransH | RotatE | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Method | ExRec | Leak | RER | ExRec | Leak | RER | ExRec | Leak | RER | |||
| GraphDPO | 98.6 | 0.047 | 81.8 | 0.26 | 78.3 | 0.044 | 149.4 | 0.24 | 76.8 | 0.065 | 63.1 | 0.25 |
| GraphDPO+NTK | 95.7 | 0.046 | 79.7 | 0.26 | 74.8 | 0.042 | 144.7 | 0.24 | 69.7 | 0.067 | 65.2 | 0.25 |
| GraphDPO+SAM | 101.5 | 0.042 | 72.8 | 0.26 | 80.1 | 0.040 | 136.3 | 0.24 | 76.6 | 0.067 | 65.7 | 0.25 |
| GraphDPO+TAR | 91.0 | 0.043 | 74.6 | 0.26 | 79.1 | 0.040 | 136.9 | 0.24 | 75.4 | 0.066 | 64.6 | 0.25 |
| KAU | 46.3 | 0.022 | 37.7 | 0.28 | 6.5 | 0.023 | 77.0 | 0.26 | 26.2 | 0.026 | 25.3 | 0.32 |
Appendix figures & tables21 assets
Supplementary material from the paper’s appendix.
Appendix
| Models | |
|---|---|
| Backbones | TransE, RotatE, TransH at latent coordinates, Appendix B.8 |
| Training | margin ranking with Adam ( Kingma and Ba, 2015 ) , loss with the mean score of random corruptions of and margin (Section 3 ) |
| Original and retrained models | one training run each per graph and backbone with that recipe, the original model on , with the never-trained set of Section 5 , and the retrained model on alone, never visible to the unlearner. Table 9 adds two further retrained models on FB15k-237, at retraining seeds 2 and 3 |
| Update class assumed by the unlearner | |
| Known channel | each retained edge at with probability |
| Unseen channel | stand-ins per entity, best of four by plausibility under , refreshed every epochs, each entering with probability , |
| Method | Time to release (s) |
|---|---|
| Retraining (exact) | 240.4 [240.2, 240.5] |
| GraphDPO | 72.8 [72.7, 72.8] |
| KAU | 13.4 [13.4, 13.4] |
| TransE | TransH | RotatE | |||||||
| Method | |||||||||
| Original | 0.2733 | 0.2800 | 0.2137 | 0.2550 | 0.2596 | 0.2038 | 0.3075 | 0.3176 | 0.2214 |
| Gold | 0.2211 | 0.2932 | 0.2143 | 0.2097 | 0.2739 | 0.2051 | 0.2127 | 0.3301 | 0.2203 |
| GraphDPO | 0.2301 | 0.2628 | 0.2041 | 0.2121 | 0.2401 | 0.1927 | 0.2298 | 0.2531 | 0.1808 |
| ZeroFisher | 0.2249 | 0.2322 | 0.1857 | 0.2123 | 0.2164 | 0.1818 | 0.2180 | 0.2261 | 0.1693 |
| KAU | 0.2272 | 0.2839 | 0.2137 | 0.2105 | 0.2614 | 0.2041 | 0.2112 | 0.3254 | 0.2213 |
| TransE | TransH | RotatE | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Method | |||||||||
| GraphDPO | 0.0009 | 0.0012 | 0.0008 | 0.0008 | 0.0013 | 0.0005 | 0.0011 | 0.0013 | 0.0013 |
| ZeroFisher | 0.0327 | 0.0320 | 0.0181 | 0.0340 | 0.0346 | 0.0212 | 0.0440 | 0.0445 | 0.0306 |
| KAU | 0.0006 | 0.0008 | 0.0004 | 0.0004 | 0.0005 | 0.0005 | 0.0004 | 0.0007 | 0.0002 |
| Backbone | Method | ExRec (%) | Leak | RER (%) | |
|---|---|---|---|---|---|
| TransE | KAU | 48.91 4.96 | 0.0215 0.0021 | 37.45 3.66 | 0.2839 0.0008 |
| GraphDPO | 102.13 10.07 | 0.0437 0.0057 | 76.12 9.95 | 0.2628 0.0013 | |
| ZeroFisher | 231.11 222.71 | 0.0741 0.0470 | 129.00 81.77 | 0.2322 0.0346 | |
| TransH | KAU | 6.44 2.05 | 0.0211 0.0019 | 71.99 6.48 | 0.2614 0.0005 |
| GraphDPO | 72.95 23.00 | 0.0394 0.0079 | 134.22 26.94 | 0.2401 0.0014 | |
| ZeroFisher | 343.97 447.50 | 0.0687 0.0421 | 234.02 143.34 | 0.2164 0.0374 |
| Backbone | Comparison | Leak | Welch | |||
|---|---|---|---|---|---|---|
| TransE | KAU vs. GraphDPO | 0 | 0.0006 | |||
| KAU vs. ZeroFisher | 0.025 | 7 | 0.026 | |||
| TransH | KAU vs. GraphDPO | 0.0007 | 0 | 0.0006 | ||
| KAU vs. ZeroFisher | 0.025 | 7 | 0.026 | |||
| RotatE | KAU vs. GraphDPO | 0 | 0.0006 | |||
| KAU vs. ZeroFisher | 0.0018 | 2 | 0.0023 |
| Backbone | Method | Oracle rise (seeds 1/2/3) | Leak vs. seeds 1/2/3 | Method rise |
| TransE | GraphDPO | 0.0351/0.0342/0.0417 | 0.0437/0.0447/0.0372 | 0.0789 0.0024 |
| ZeroFisher | 0.0192/0.0165/0.0235 | 0.0741/0.0768/0.0698 | 0.0933 0.0499 | |
| KAU | 0.0419/0.0385/0.0471 | 0.0215/0.0249/0.0163 | 0.0634 0.0049 | |
| Oracle at release 0.2932/0.2899/0.2906 | ||||
| Leakage reduction of KAU over GraphDPO 50.8/44.4/56.0% | ||||
| TransH | GraphDPO | 0.0303/0.0365/0.0344 | 0.0394/0.0332/0.0353 | 0.0697 0.0137 |
| Backbone | Method | |||
|---|---|---|---|---|
| TransE | Original | 0.2354 | 0.2462 | 0.1501 |
| Gold | 0.1487 | 0.2655 | 0.1492 | |
| GraphDPO | 0.1630 0.0004 | 0.2154 0.0013 | 0.1389 0.0006 | |
| ZeroFisher | 0.1655 0.0655 | 0.1734 0.0683 | 0.1081 0.0404 | |
| KAU | 0.1504 0.0002 | 0.2474 0.0005 | 0.1466 0.0002 | |
| TransH | Original | 0.2412 | 0.2535 | 0.1505 |
| Backbone | Method | ExRec (%) | Leak | RER (%) | |
|---|---|---|---|---|---|
| TransE | GraphDPO | 77.35 3.60 | 0.0524 0.0031 | 31.30 1.83 | 0.2154 0.0013 |
| ZeroFisher | 218.34 122.10 | 0.1080 0.0844 | 64.49 50.37 | 0.1734 0.0683 | |
| KAU | 26.94 1.60 | 0.0214 0.0012 | 12.80 0.71 | 0.2474 0.0005 | |
| Leak, KAU vs. GraphDPO: Welch , , and , | |||||
| Leak, KAU vs. ZeroFisher: Welch , , and , | |||||
| TransH | GraphDPO | 75.61 4.80 | 0.0621 0.0034 | 39.41 2.15 | 0.2091 0.0008 |
| Axis | Value | GraphDPO | ZeroFisher | KAU | KAU /GraphDPO |
|---|---|---|---|---|---|
| drawn | 1,000 | 0.0467 | 0.0469 | 0.0248 | 0.53 |
| 3,000 | 0.0462 | 0.0475 | 0.0247 | 0.53 | |
| 6,000 | 0.0470 | 0.0440 | 0.0249 | 0.53 | |
| neighbour fraction | 0.15 | 0.0467 | 0.0471 | 0.0249 | 0.53 |
| 0.25 | 0.0456 | 0.0421 | 0.0215 | 0.47 | |
| 0.35 | 0.0470 | 0.0475 | 0.0248 | 0.53 |
| GraphDPO | ZeroFisher | KAU | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Update | ExRec | Leak | RER | ExRec | Leak | RER | ExRec | Leak | RER |
| standard | 95.26 | 0.0402 | 69.93 | 132.11 | 0.0697 | 121.35 | 48.53 | 0.0215 | 37.45 |
| never-trained edges only | 61.12 | 0.0274 | 47.75 | 104.15 | 0.0553 | 96.30 | 19.36 | 0.0089 | 15.55 |
| Model | Released | Every parameter | Involved rows | Relation rows | Other entity rows |
|---|---|---|---|---|---|
| KAU | 0.2265 | ||||
| GraphDPO | 0.2292 | ||||
| ZeroFisher | 0.2527 | ||||
| retrained oracle | 0.2240 |
| Response | Pearson | Spearman | Mean predicted | Mean realized |
| joint solve, exact evaluation ( KAU ) | 0.526 [0.294, 0.674] | 0.207 [0.145, 0.266] | ||
| per-entity block solve | 0.303 [0.167, 0.422] | 0.197 [0.135, 0.259] | ||
| first-order evaluation | 0.605 [0.333, 0.746] | 0.125 [0.061, 0.188] | ||
| Realized recovery by quintile of prediction (joint solve): | ||||
| Variant | ExRec | Leak | RER |
|---|---|---|---|
| KAU (full) | 31.91 | 0.0111 | 19.38 |
| no anticipated response ( ) | 48.60 | 0.0250 | 43.53 |
| first-order evaluation | 37.40 | 0.0167 | 29.07 |
| per-entity block solve | 42.24 | 0.0175 | 30.50 |
| signed residual | 30.78 | 0.0138 | 24.05 |
| no unseen channel ( ) | 37.55 | 0.0151 | 26.29 |
| Variant | ExRec | Leak | RER | |
|---|---|---|---|---|
| KAU (full) | 31.91 | 0.0111 | 19.38 | 0.2860 |
| response damped (relative ridge ) | 47.13 | 0.0238 | 41.45 | 0.2869 |
| no pessimistic composition | 32.60 | 0.0110 | 19.18 | 0.2854 |
| per-relation gate | 34.18 | 0.0110 | 19.12 | 0.2859 |
| no best-of- stand-in choice ( ) | 37.90 | 0.0147 | 25.52 | 0.2857 |
| hinge of the expected draw | 31.98 | 0.0148 | 25.81 | 0.2857 |
| Forgetting objective | ExRec | Leak | RER |
|---|---|---|---|
| Plain forgetting loss ( ) | 48.60 | 0.0250 | 43.53 |
| Retained-span projection | 33.32 | 0.0158 | 27.51 |
| SAM-RS | 40.15 | 0.0154 | 26.85 |
| SAM-GP | 42.51 | 0.0150 | 26.09 |
| SAM-CR | 36.58 | 0.0152 | 26.55 |
| SAM-WA | 43.00 | 0.0151 | 26.30 |
| Study | Domain | Update model | Object reasoned about | Coupling | Data effect | Reference | Matched | Cost |
|---|---|---|---|---|---|---|---|---|
| Relearning-robust unlearning methods | ||||||||
| TAR ( Tamirisa et al., 2025 ) | LLM | forget-set, on forget, retain and mixed corpora, adaptive only by adding fine-tunes that broke earlier versions | weights and residual-stream activations, entropy of one fixed forget batch along a simulated multi-step trajectory | no, retain loss at the weights before the simulated fine-tune | partial, simulated, one fixed batch, straight-through gradient | no | none | a simulated fine-tune per outer step, multi-GPU |
| Meta-unlearning ( Gao et al., 2025 ) | diffusion | forget-set, on forget images, benign set as test only, not adaptive | weights, two whole-set gradients, a few unrolled inner steps | partial, one inner product of whole-set gradients | partial, forget-data fine-tune, unrolled | no | none | second-order autodiff through the inner steps |
| ResAlign ( Li et al., 2025 ) | diffusion | forget-set, sampled fine-tunes on a benign and harmful pool, benign at test, two adaptive attacks | weights, harmful loss at a simulated end point, implicit proximal Jacobian | no, retain as distillation at the current weights | partial, simulated, mean harmful loss at the end point | no | none | three simulated fine-tunes per outer step, about one GPU hour in total |
| MUDMAN ( Sondej et al., 2025 ) | LLM | forget-set, on the forget set, fixed rate, not adaptive | per-parameter gradients, retain momentum, a forked copy | partial, per-coordinate sign agreement | partial, first-order, interleaved fine-tune on the forget set | no, unedited model after the same relearning | none | several passes per loop, hours of search |
| CIR ( Sondej and Yang, 2025 ) | LLM | forget-set, long fine-tune on most of the forget set, not adaptive | principal components of forget-set activations and gradients | no, retain step optional | no, a hypothesis that broken representations are repaired | no | none | seconds per fact |
Machine Unlearning
Machine unlearning removes the influence of selected data or concepts from trained models while preserving useful behavior and avoiding full retraining.
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