| Design choice | Description | Exper- iment | OLMOE-1B-7B |
| Active params | # active parameters per input token | §4.1.1 | 1.3B active |
| Total params | Total # of parameters in the model | §4.1.1 | 6.9B total |
| Expert granularity | Using fine-grained small experts vs. a few large experts [39] | §4.1.2 | 64 small experts with 8 activated |
| Expert sharing | Whether or not to include a shared expert [39] §4.1.3 | | No shared expert |
| Routing algorithm | How inputs are assigned to experts, e.g., as- signment on a per token basis (e.g., 2 experts per token) or per expert basis (e.g., 2 tokens per expert), and whether or not all tokens get assigned or some get dropped [58, 219] | §4.1.4 | Dropless [58] MoE with token choice |
| Sparse upcycling Load balancing | Whether to start from a dense model [85, 211] | §4.1.5 | Not used |
| loss | Auxiliary loss to penalize unequal assignment §4.1.6 to experts that may harm performance [154] | | Used with weight 0.01 |
| Router z-loss | Auxiliary loss to penalize large logits in the router that may cause instabilities [221] | §4.1.7 | Used with weight 0.001 |
+
+Table 2: Composition of the pretraining data for OLMOE-1B-7B. StarCoder, peS2o, and Wikipedia parts come from Dolma 1.7 [163]. Links to our data are in Appendix A.
+
+| Expert IDInput token IDs Predicted output token IDs |
| 27 O(100%)I(100%)3(100%)i(100%)I'(100%) § (100%)0(100%) j (100%)ijT(100%)Q(100%)D(100%)b(100%)(100%)O(100%)D(100%)F(100%)E(100%)WH(100%)a(100%) b3(100%):(100%)2T(100%) |
| E(100%)WH(100%)a(100%) |
| 58 | ("(100%)("(100%) (94%),(92%) | such(100%)486(100%)see(95%)which(91%)driving(91%)UK(90%)who(88%)including(88%)normal(88%) |
| “(92%)((92%)"(90%),(89%)"(88%)$(87%)[(87%)£(86%) |
| 7 | Him(100%)inde(100%)Jesus(98%) | rella(100%)Him(94%)sin(90%) |
| God(90%)pray(81%)Holy(80%) | prince(80%)glory(72%)Jesus(69%)Lord(68%)Christ(65%)Spirit(55%)Holy(53%)God(50%)Prayer(50%) |
| Quran(80%)God(77%)Lord(76%) |
| glory(75%)Spirit(66%)Christ(65%) |
| 37 | Sunday (100%) Tuesday (100%) | days(91%)anniversary(90%)month |
| Thursday (100%) Olympic (100%) | (88%)week(84%)mpi(83%)semester |
| Christmas (100%) rugby (100%) | (81%)mand(80%)Olympics(78%)cent |
| Championship(100%)weekends(100%) | (76%)season(76%)perm(75%) |
| 43 | Armenian(100%)ijan(100%)enia(96%) | enia(90%)invasion(80%)Arabia(76%) |
| | Iraq(95%)Iranian(92%)Iran | (92%) | irregular(66%)regions(64%)border |
| Saudi(90%)northern(90%)Lebanon | (63%)Kong(61%)ians(61%)bases |
| (90%)Singapore(88%)Turkey(88%) | (60%) Republic(59%)Ireland(58%)Korea(58%)War(55%)Carolina(52%) |
| Asia(87%)Egypt(86%)western(86%) |
| 4 | sq(89%)Main(70%)reversal(69%) | YR(90%) | Character(88%)sq(77%) |
| YR(63%)GC(56%)Overall(50%)79 | Os(76%) | GHz(71%)fluence(60%) |
| | | | | | |
| | | | |
| (50%)main(50%)RE(46%)PCR(46%) | amycin(60%)pixels(56%)=(53%)arc |
|
| tomb(45%)normal(43%)intensity | (52%)Story(52%)=(51%)anth(50%) |
| (41%)Overall(41%)median(41%) | GHz(50%)cm(46%) |
| 0 | ESM(100%)icillin(100%)agra(98%) | *,(100%)sil(96%)pills(91%)vi |
| aust(96%)asa(93%)pills(92%)mg | | (87%)pharmacy(87%)gener |
| (85%)uk(82%)login(82%)doc(81%) | (85%)aust(82%)mg(75%)Content |
| generic(81%)cd(81%)Essay(81%) | (75%)uk(73%)THAT(73%)dispens |
| password(81%)Content(80%) | (68%)icillin(68%)generic(66%) |
| 3 | grandmother(92%)brother(91%)Daisy | | mother(35%)inde(31%) |
| | | | | | |
| | | | |
| (83%) daughter (78%)mum(75%) | | girl(28%)married(27%) |
| | | father(72%)wife(70%)husband(70%) |
|
| lady(63%)dad(62%)boy(61%) | | |
| 48 | compared(42%)!)(41%)Then(41%), | except(60%)tennis(41%)Marks(40%)Dunn(33%)tears(30%)Arizona(30%) |
| (40%)),(35%)",(35%)instead(33%) |
| 23 | …(58%)Therefore(55%)So(46%)!!! | | Republican (50%)Jack |
| (46%)And(44%)According(41%)."(41%)!!(40%)?"(38%)But(38%) | (46%)And(44%)According(41%)." |
| according(39%)So(38%)Step(33%) |
+
+Expert 43 shows specialization on geographic terms in both input and output tokens. Experts 48 and 23 both focus on connector words, such as Then and Therefore . This is likely because they commonly process tokens together with a high co-activation of $60 \%$ in Figure 21 (middle). Based on our findings in $\ S 5 . 3$ that for GitHub and arXiv often the same experts in layer 7 activate, we display one such expert (expert ID 4) in Table 8. It seems to specialize in measurements, such as sq , YR (year), and $\mathrm { G H z }$ . These are common terms in scientific papers corresponding to the arXiv domain and likely also in GitHub code for computations related to measurements. They are less likely to appear in books, which explains the low activation of expert $\mathrm { I D 4 }$ in layer 7 for book data in Figure 22. Expert 3 is among the three most active experts of layer 7 for book data in Figure 22 (fourth yellow bar for layer 7). This resonates when looking at its specialization on family terms in Table 8, which are far more common in books than scientific papers or code. Overall, domain specialization and vocabulary specialization are closely linked to one another, as domains are usually characterized by their distinct word distribution. In Appendix G (Figure 32), we link them more closely by comparing the extent of vocabulary specialization across domains and expert IDs. In Appendix G (Figure 30, Figure 31) we also find that OLMOE-1B-7B exhibits stronger vocabulary specialization than Mixtral- ${ } ^ { 8 \mathrm { x 7 B } }$ .
+
+# 6 Related Work
+
+Advances in MoEs Current LMs still largely follow the transformer architecture [185] with only few architectural changes that have been widely adopted, such as decoder-only training [137], SwiGLU activations [153, 41], RoPE [166], MQA/GQA [152, 3] and RMSNorm [208]. Model sparsity via Mixture-of-Experts is one modification still under active exploration with some early adoption but most LMs, including Llama 3 [50], still rely on a dense architecture. There has been a lot of progress in improving the sparsely-gated MoE layer since its introduction [154]: New routing techniques [89, 146, 222, 66, 77, 49, 215, 195, 124], fine-grained expert segmentation [39, 69], stability [221] and efficiency [88, 141, 48, 218, 91, 168, 129, 145] improvements. In this work, we perform many experiments to provide insights into training Mixture-of-Experts LMs. Subsequently, we train OLMOE-1B-7B for 5T tokens. No prior MoE has been overtrained [57] to this extent to our knowledge making OLMOE-1B-7B the best testbed to research performance saturation of MoEs vs. dense models. With OLMOE we hope to facilitate such and other research to help the field uncover whether MoEs should make it into all future LMs and with what precise configuration.
+
+Open LMs A variety of model families have been proposed under varying degrees of openness commonly categorized based on whether model weights are available. Closed-weight models include GPT [24, 128], Gemini [174, 175], PaLM [30, 9], Reka [181], and open-weight ones include Llama [182, 183, 50], Mistral [78, 79], Gemma [176, 177], Falcon [8, 132], MPT [179], Qwen [13, 201], GLM [61], Yi [2], DeepSeek [42, 43, 39], Nemotron [130, 126], Zamba [62], InternLM [26], Baichuan [200], Phi [68, 94, 1], StableLM [16], OPT [212]. However, besides model weights, training data and code are key to enabling scientific research of these models [105, 106] and distributing their benefits broadly [23]. There have been few releases also including data and code in addition to model weights which we refer to as “fully open-source”: BLOOM [193, 151, 123, 203], GPT-NeoX [21, 22, 186], StarCoder [92, 109, 5, 120, 220], Pythia [18], OLMo [65], LLM360 [103], Cerebras-GPT [46], DCLM [90], MAP-Neo [209], RWKV [133, 134], and SmolLM [6]. For Mixture-of-Experts only OpenMoE [199] aims to be fully open-source, however, its poor performance limits its usefulness. We release OLMOE-1B-7B as the first state-of-the-art Mixture-of-Experts LM that is fully open-source: model weights, data, code, and logs.
+
+# 7 Conclusion
+
+We open-source OLMOE-1B-7B and OLMOE-1B-7B-INSTRUCT including model, data, code, and logs. At 1B active and 7B total parameters, our models yield state-of-the-art performance among models with a similar amount of active parameters even outperforming larger models including DeepSeekMoE-16B and Llama2-13B-Chat. We share various training experiments and define and analyze router saturation, expert co-activation, domain and vocabulary specialization of our model. Through our fully open release, we seek to help the field build better MoEs. We are excited about more iterations of OLMOE to close the gap between frontier models and fully open models.
+
+# Author Contributions
+
+Niklas Muennighoff proposed and led the project. He ran the pretraining experiments, pretrained the model, helped run adaptation and analysis, and wrote most of the paper.
+
+Luca Soldaini created the pretraining dataset and advised on pretraining.
+
+Dirk Groeneveld advised on pretraining, especially stability and throughput improvements.
+
+Kyle Lo helped with pretraining dataset creation, analyzed data experiments, and advised on data and framing, and helped edit the paper.
+
+Jacob Morrison co-created the adaptation dataset, ran most adaptation experiments, and helped edit the paper.
+
+Sewon Min analyzed router saturation, expert correlation, and vocabulary specialization, and helped frame and edit the paper.
+
+Weijia Shi analyzed domain and vocabulary specialization, advised at various project stages, and helped edit the paper.
+
+Pete Walsh advised on pretraining, especially stability and throughput improvements.
+
+Oyvind Tafjord ran OLMES evaluations.
+
+Nathan Lambert co-created the adaptation dataset, advised on adaptation, and helped edit the paper.
+
+uling Gu ran OLMES evaluations and helped edit the paper.
+
+ane Arora uploaded the models, helped with code review and framework integration.
+
+Akshita Bhagia supported stability investigations and helped with DCLM evaluations.
+
+Dustin Schwenk supported stability investigations.
+
+David Wadden ran DCLM evaluations and helped with Weights & Biases reports.
+
+Alexander Wettig advised on pretraining, analyzed load balancing, routing, and domain specializa tion, and helped edit the paper.
+
+Binyuan Hui advised on pretraining and helped with plotting and framework integration.
+
+Tim Dettmers advised on analysis and inference experiments.
+
+Douwe Kiela advised on framing.
+
+Ali Farhadi advised on pretraining and framing.
+
+Noah A. Smith advised on pretraining, and helped frame and edit the paper.
+
+Pang Wei Koh advised on analysis, and helped frame and edit the paper.
+
+Amanpreet Singh advised on pretraining, framing and helped edit the paper.
+
+Hannaneh Hajishirzi was responsible for direction and advising of the overall effort and helped frame and edit the paper.
+
+# Acknowledgements
+
+OLMOE would not be possible without the support of many individuals and institutions. We thank our teammates at the Allen Institute for AI, Contextual AI, and the University of Washington for their support, especially Aditya Kusupati, Ananya Harsh Jha, Caitlin Wittlif, Carissa Schoenick, Costa Huang, Crystal Nam, David Atkinson, Emma Strubell, Faeze Brahman, Hamish Ivison, Karel D’Oosterlinck, Matt Latzke, Ian Magnusson, Jack Merullo, Jay Chen, Jennifer Dumas, Jiacheng Liu, Johann Dahm, Luke Zettlemoyer, Michael Schmitz, Michael Wilson, Pradeep Dasigi, Sahil Verma, Sam Skjonsberg, Sophie Lebrecht, Stas Bekman, Taira Anderson, Valentina Pyatkin, Yanai Elazar, Yizhong Wang, and Yoganand Chandrasekhar. We also thank Armen Aghajanyan, Akshat Shrivastava, Colin Raffel, Haokun Liu, Ludwig Schmidt, Mengzhou Xia, Shayne Longpre, Sheng Shen, and Zexuan Zhong. PWK is supported by the Singapore National Research Foundation and the National AI Group in the Singapore Ministry of Digital Development and Innovation under the AI Visiting Professorship Programme (award number AIVP-2024-001).
+
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+
+# A Artifacts
+
+Table 9: All artifacts released and used in this work. We point from the name used for a given artifact in this work (e.g. Figure 1) to the URL where it can be obtained.
+
+| Artifact | Public link |
| OLMOE-1B-7B | https://hf.co/allenai/OLMoE-1B-7B-0924 |
| OLMOE-1B-7B-INSTRUCT | https://hf.co/allenai/OLMoE-1B-7B-0924-Instruct |
| OLMoE-1B-7B-SFT | https://hf.co/allenai/OLMoE-1B-7B-0924-SFT |
| OLMoE-Mix | https://hf.co/datasets/allenai/OLMoE-mix-0924 |
| SFT data | https://hf.co/datasets/allenai/ |
| tulu-v3.1-mix-preview-4096-OLMoE https://hf.co/datasets/allenai/ |
| KTO/DPO data | ultrafeedback_binarized_cleaned |
| Code | https://github.com/allenai/OLMoE |
| Logs | https://wandb.ai/ai2-llm/olmoe/reports/ |
| BLOOM-7B DeepSeekMoE-3B-16B | OLMoE-1B-7B-0924--Vmlldzo4OTcyMjU3 https://hf.co/bigscience/bloom-7b1 |
| https://hf.co/deepseek-ai/deepseek-moe-16b-base |
| DeepSeekMoE-3B-16B+chat | |
| DeepSeekV2-2B-16B | https://hf.co/deepseek-ai/deepseek-moe-16b-chat |
| https://hf.co/deepseek-ai/DeepSeek-V2-Lite |
| DCLM-1B | https://hf.co/TRI-ML/DCLM-1B |
| DCLM-7B | https://hf.co/TRI-ML/DCLM-7B |
| Falcon-7B | https://hf.co/tiiuae/falcon-7b |
| Gemma2-3B | |
| Gemma2-9B | https://hf.co/google/gemma-2-2b |
| JetMoE-2B-9B | https://hf.co/google/gemma-2-9b |
| JetMoE-2B-9B+SFT | https://hf.co/jetmoe/jetmoe-8b |
| JetMoE-2B-9B+Chat | https://hf.co/jetmoe/jetmoe-8b-sft |
| https://hf.co/jetmoe/jetmoe-8b-chat |
| Llama-7B | https://hf.co/huggyllama/1lama-7b |
| Llama2-7B | https://hf.co/meta-1lama/Llama-2-7b-hf |
| Llama3.1-8B | https://hf.co/meta-1lama/Meta-Llama-3.1-8B |
| MPT-7B | https://hf.co/mosaicml/mpt-7b |
| Mistral-7B | https://hf.co/mistralai/Mistral-7B-v0.1 |
| Mixtral-8x7B | https://hf.co/mistralai/Mixtral-8x7B-v0.1 |
| OLMo-1B (0724) | https://hf.co/allenai/OLMo-1B-0724-hf |
| OLMo-7B (0724) | https://hf.co/allenai/OLMo-7B-0724-hf |
| OpenMoE-3B-9B | https://hf.co/OrionZheng/openmoe-8b |
| Pythia-7B | https://hf.co/EleutherAI/pythia-6.9b |
| Qwen1.5-3B-14B | https://hf.co/Qwen/Qwen1.5-MoE-A2.7B |
| Qwen1.5-3B-14B+Chat | https://hf.co/Qwen/Qwen1.5-MoE-A2.7B-Chat |
| StableLM2-2B | https://hf.co/stabilityai/stablelm-2-1_6b |
| TinyLlama-1B | https://hf.co/TinyLlama/TinyLlama_v1.1 |
+
+# B Training Configuration
+
+Pretraining We display the pretraining hyperparameter configuration of OLMOE-1B-7B in Appendix B comparing with other relevant models. We follow Groeneveld et al. [65] using the AdamW optimizer [107] with ZeRO [142] via PyTorch FSDP [213] and mixed-precision training [116]. Our main model settings differing from Groeneveld et al. [65] are: (1) MoE-related changes: OLMOE-1B-7B is a sparsely activated decoder-only transformer [185] using dropless Mixtureof-Experts [58]. Unlike most prior MoEs, we use a high granularity [39, 86] with 64 small experts with an FFN dimension of just 1,024 rather than a few large experts. We further use two auxiliary losses: router z-loss [221] and load balancing loss [154]. (2) Stability improvements: (a) We use a truncated normal initialization with a standard deviation of 0.02 and a minimum (maximum) cut-off of -0.06 (0.06) corresponding to three standard deviations. (b) We use QK normalization [173, 113, 44]. (c) We use RMSNorm [208] instead of the non-parametric LayerNorm used in Groeneveld et al. [65]. (3) Performance improvements: Besides some of the stability improvements which also impact performance, we also reduce the AdamW epsilon to 1.0E-08 from the 1.0E-05 used in Groeneveld et al. [65] to speed up convergence. Finally, we train OLMOE-1B-7B for significantly longer than all prior OLMo models amounting to 5T tokens and thus more than one epoch (1.3) following Muennighoff et al. [121]. We shuffle the pretraining dataset before starting the second epoch. For the final 100B tokens, we decay the learning rate linearly from 5.0E-04 to 0. We experiment with many of these settings in $\ S 4$ .
+
+Adaptation For finetuning we use Open Instruct [188, 76].8 We filter all SFT samples to a length of fewer than 4096 tokens to match the sequence length of the model. Following Muennighoff et al. [122], we aggregate loss at the token level during SFT to improve performance on long generative tasks, such as AlpacaEval. We finetune in BF16 with a global batch size of 128 (4 H100 nodes with 8 GPUs each, a per device batch size of 2, and 2 gradient accumulation steps). We train for 2 epochs with a constant learning rate of 2.0E-5. For DPO [138], we reduce the global batch size to 32 (4 H100 nodes with 8 GPUs each and a per device batch size of 1). We train for 3 epochs with a learning rate of 5.0E-7 and a DPO beta of 0.1. Our adapted models are built on top of our annealed checkpoint, and we include the load balancing loss during both SFT and DPO based on our experiments in $\ S 4 . 3$ . Our preference tuning recipe is heavily optimized for DPO based on extensive experiments by Ivison et al. [76], thus for KTO [54] we experiment with a few settings in Appendix F. Our final KTO adaptation uses the same hyperparameters as DPO, except that we use the RMSProp optimizer instead of Adam, which we use for SFT and DPO, and that we reduce the training duration to 1.3 epochs (5,000 steps) for KTO instead of the 3 epochs used for DPO.
+
+Hardware We pretrain OLMOE-1B-7B on 256 H100 GPUs for approximately 10 days with NVlink interconnect across GPUs and InfiniBand interconnect across nodes. We also use H100 GPUs for all our experiments but some use a cluster with GCP TCPx interconnect across nodes instead. For adaptation, we use 32 H100 GPUs for 33 hours to instruction tune and for another 14 hours to preference tune via DPO. For KTO adaptation we use 8 H100 GPUs for 30 hours instead.
+
+Table 10: Pretraining hyperparameters of OLMOE-1B-7B and comparable models trained from scratch. We highlight rows where OLMOE-1B-7B differs from OLMo-1B. Active params include vocab params. “?” $=$ undisclosed settings, $\mathrm { F F N = }$ feed-forward network, Attn $=$ Attention, $\mathrm { L R } =$ learning rate, ${ \bf W } { \bf S } { \bf D } = { \bf \Phi }$ Weight-Stable-Decay [74], LBL $=$ load balancing loss, Inv Sq Root $=$ Inverse Square Root decay [155], trunc $=$ truncation, std $=$ standard deviation, “varies” $=$ stds that are layer or weight-dependent.
+
+| OLMoE-1B-7B checkpoint (→) | step 1,200,000 | step 1,220,000 | annealed | OLMo-1B | OLMo-7B |
| AGI Eval LSAT-AR* | 24.3 | 26.5 | 28.7 | 28.3 | 28.3 |
| AGI Eval LSAT-LR | 40.2 | 38.6 | 37.3 | 30.2 | 42.9 |
| AGI Eval LSAT-RC | 47.4 | 43.7 | 46.6 | 23.5 | 61.6 |
| AGI Eval SAT-En | 55.3 | 54.9 | 52.9 | 28.2 | 73.8 |
| AGI Eval SAT-Math CoT | 5.5 | 4.1 | 6.4 | 1.8 | 6.8 |
| AQuA CoT | 2.4 | 2.9 | 2.0 | 2.9 | 6.1 |
| ARC Challenge* | 53.3 | 53.4 | 53.8 | 34.6 | 48.1 |
| ARC Easy* | 77.1 | 78.5 | 77.7 | 64.4 | 75.9 |
| BBQ | 49.8 | 48.3 | 50.6 | 45.8 | 67.2 |
| BigBench CS Algorithms* | 47.1 | 50.2 | 47.2 | 47.5 | 53.6 |
| BigBench Conceptual Combinations | 51.5 | 50.5 | 56.3 | 31.1 | 68.0 |
| BigBench Conlang Translation | 3.7 | 6.1 | 7.3 | 4.3 | 7.3 |
| BigBench Dyck Languages* | 19.3 | 15.9 | 21.5 | 26.6 | 22.2 |
| BigBench Elementary Math QA | 26.2 | 27.0 | 26.9 | 26.2 | 30.4 |
| BigBench Language Identification* | 31.9 | 34.0 | 31.0 | 27.0 | 39.1 |
| BigBench Logical Deduction | 26.6 | 25.3 | 24.6 | 23.6 | 27.3 |
| BigBench Misconceptions | 59.8 | 55.3 | 62.6 | 55.7 | 58.0 |
| BigBench Novel Concepts | 62.5 | 62.5 | 65.6 | 43.8 | 53.1 |
| BigBench Operators* | 36.2 | 34.3 | 33.8 | 23.8 | 45.2 |
| BigBench QA Wikidata* | 68.2 | 68.8 | 69.2 | 67.0 | 69.9 |
| BigBench Repeat Copy Logic* | 15.6 | 15.6 | 18.8 | 3.1 | 9.4 |
| BigBench Strange Stories | 66.7 | 68.4 | 69.5 | 53.4 | 66.1 |
| BigBench Strategy QA BigBench Understanding Fables | 56.2 | 58.1 | 57.0 | 51.5 | 68.6 |
| BoolQ* | 47.1 | 44.4 | 47.6 | 28.0 | 61.4 |
| COPA* | 73.3 | 72.8 | 73.2 | 63.7 | 83.9 |
| 81.0 | 80.0 | 78.0 | 75.0 | 77.0 |
| CoQA* | 43.7 | 44.4 | 43.7 | 3.4 | 45.4 |
| CommonsenseQA* | 67.2 | 67.0 | 69.3 | 19.6 | 86.0 |
| Enterprise PII Classification | 52.3 | 53.7 | 52.2 | 57.3 | 50.6 |
| GPQA Diamond | 22.2 | 21.2 | 19.7 | 19.7 | 20.2 |
| GPQA Main | 24.8 | 22.3 | 22.5 | 20.3 | 23.0 |
| GSM8K CoT | 6.4 | 7.4 | 7.4 | 4.9 | 30.6 |
| HellaSwag 0-shot* | 76.0 | 76.0 | 77.0 | 65.8 | 76.7 |
| HellaSwag 10-shot* | 77.6 | 77.5 | 78.6 | 66.3 | 78.9 |
| Jeopardy* | 48.8 | 48.7 | 50.3 | 22.6 | 46.5 |
| LAMBADA* | 72.7 | 72.2 | 73.3 | 61.1 | 71.8 |
| LogiQA | 34.9 | 34.3 | 34.6 | 28.7 | 31.0 |
| MMLU Few-shot | 52.2 | 51.9 | 53.3 | 28.4 | 55.1 |
| MMLU Zero-shot | 41.6 | 42.7 | 43.3 | 26.2 | 50.0 |
| Math QA | 26.4 | 27.1 | 27.5 | 24.1 | 29.8 |
| OpenBookQA* | 41.4 | 44.0 | 44.8 | 36.6 | 43.4 |
| PIQA* | 81.3 | 81.2 | 82.0 | 76.4 | 81.7 |
| PubMedQA SQQuAD* | 56.1 | 46.6 | 57.9 | 0.2 | 57.9 |
| SVAMP CoT | 52.9 | 52.4 | 52.4 | 0.0 | 65.5 |
| Simple Arithmetic, no spaces | 30.0 | 28.0 | 33.0 | 14.3 | 44.7 |
| 17.6 | 18.1 | 20.1 | 1.2 | 15.3 |
| Simple Arithmetic, with spaces | 19.5 | 20.6 | 22.1 | 1.8 | 16.0 |
| Social IQA | 71.5 | 70.7 | 69.3 | 69.5 | 84.4 |
| Trivia QA | 54.2 | 53.0 | 55.9 | 25.1 | 51.8 |
| Winogender Female | 50.0 | 46.7 | 50.0 | 41.7 | 58.3 |
| Winogender Male | 55.0 | 58.3 | 60.0 | 63.3 | 58.3 |
| Winogd* | 82.8 | 83.2 | 84.6 | 79.9 | 83.2 |
| Winogrande* | 68.0 | 68.5 | 69.0 | 61.8 | 67.6 |
| Core | 46.3 | 46.5 | 47.2 | 30.2 | 49.8 |
| Extended | 31.3 | 30.9 | 32.5 | 16.9 | 37.0 |
+
+Table 13: DCLM evaluation metrics on the Core and Extended task subsets [90]. =Core tasks. “annealed” is the final pretraining checkpoint we use for OLMOE-1B-7B and was annealed from the checkpoint at step 1,200,000. We left the non-annealing pretraining run train a little longer resulting in the 1,220,000 checkpoint.
+
+
+Figure 26: Adding Reddit or FLAN to OLMOE-MIX. More results, logs, and configurations: https://wandb.ai/ai2-llm/olmoe/reports/ Plot-Adding-Reddit-FLAN--Vmlldzo4OTg1NTg4
+
+Adding Reddit or FLAN to OLMOE-MIX In Figure 26 we benchmark adding the Reddit or FLAN [190] subsets of Dolma 1.7 [163] to our pretraining data mix (§2). Overall, we do not find either one to lead to consistent gains, thus we do not use them in our final data mix.
+
+Load balancing precision Fedus et al. [56] selectively perform operations related to routing in full precision (FP32) to improve stability. In Figure 27, we test whether computing the load balancing loss in full precision improves stability, but do not find it to reduce spikes. Thus, we stick with bfloat16 (BF16).
+
+Noise upcycling For the creation of Qwen2-MoE [201, 180, 13], the authors add $50 \%$ of gaussian noise to feedforward networks before continuing training in an upcycled setup [85]. Komatsuzaki et al. [85] also report that they experimented with adding noise but did not find it beneficial. In Figure 28, we experiment with regular upcycling versus adding noise by randomly replacing $50 \%$ of each MLP with numbers drawn from a normal distribution with a standard deviation of 0.02 following. We find that after 700 billion tokens, the no noise variant still performs slightly better but both appear to converge to the same performance. If training further, it is possible that the noise variant eventually outperforms the no noise variant, but at that point, it may make more sense to just train the MoE from scratch (§4.1.5).
+
+
+Figure 27: Load balancing precision. More results, logs, and configurations: https://wandb. ai/ai2-llm/olmoe/reports/Plot-FP32-LBL--Vmlldzo4NDMxNDA4
+
+
+Figure 28: Adding noise to the upcycled checkpoint. More results, logs, and configurations: https://wandb.ai/ai2-llm/olmoe/reports/ Plot-Noise-upcycle---Vmlldzo4NDA3MzI2
+
+
+Figure 29: Sharing the same MoE across layers versus a regular dense LM. The number of experts in the MoE is equivalent to its number of layers. Thus, because the MoE is shared across layers, it has the same number of total and active parameters as the dense model. More results, logs, and configurations: https://wandb.ai/ai2-llm/olmoe/reports/ Plot-Shared-vs-Dense--Vmlldzo4NDI0MTc5
+
+Shared Layer Some work has investigated Mixture-of-Experts with weights shared across layers in the context of Universal Transformers [171, 37, 45]. We test whether layer-shared Mixture-of-Experts can beat non-shared dense models in Figure 29. The layer-shared MoE uses a load balancing loss that is applied at the model level rather than at the layer level. This gives the model more flexibility by allowing it to completely deactivate certain experts for some layers and even emulate a dense model by always activating one separate expert for each layer. This makes it a generalization of the dense model which motivated our hypothesis that it may perform better than the dense model. However, in practice, we find that both perform similarly with the regular dense models even maintaining a small advantage on validation loss and HellaSwag. One possible advantage of layer-shared MoEs is that they can allow for better load balancing at inference. If prompts come in continuously, then newly incoming prompts can be batched with previous prompts that have already passed through several layers and sent through the MoE module together, as the MoE module is the same regardless of whether it is the first or last layer. Sharing also reduces throughput by around $20 \%$ during training, which further motivates our decision not to use it for OLMOE-1B-7B.
+
+KTO experiments In Table 14 we experiment with the number of steps (5,000 vs. 10,000) and the optimizer (Adam [83] vs. RMS) used for KTO [54]. Based on these experiments we use the RMS optimizer and the checkpoint at 5,000 steps in $\ S 4 . 3$ .
+
+| Method | Hyperparamters | #Params (M) |
| VPT-Shallow | Prompt Number:: [5, 10, 50, 100, 200] | 0.0003~0.153 |
| VPT-Deep | Prompt Number: [5, 10, 50, 1100] | 0.046 ∼ 0.921 |
| BitFit | N/A | 0.102 |
| DiffFit | N/A | 0.140 |
| LayerNorm | N/A | 0.038 |
| SSF | N/A | 0.205 |
| Pfeif. Adapter | Adapter Scale Factor:[0.01, 0.1, 1, 10]Adapter Bottleneck: [4, 8, 16, 32] | 0.082 ~0.599 |
| Houl. Adapter | Adapter Scale Factor: [0.01, 0.1, 1, 10]Adapter Bottleneck: [4, 8, 16, 32] | 0.165 ~1.198 |
| AdaptFormer | Adapter Scale Factor:[0.05, 0.1, 0.2]Adapter Bottleneck:[4, 16, 32] | 0.082 ~0.599 |
| RepAdapter | RepAdapter Scale Factor:[0.1, 0.5, 1, 5, 10]RepAdapter Bottleneck: [8, 16, 32] | 0.239 ~0.903 |
| Convpass | Convpass Scale Factor: [0.01, 0.1, 1, 10, 100]Convpass Bottleneck: [8, 16]Convpass Xavier Init:[True, False] | 0.327 ~0.664 |
| LoRA | LoRA Bottleneck: [1, 8, 16, 32] | 0.036 ~1.179 |
| FacT_TT | FacT Scale Factor: [0.01, 0.1, 1, 10, 100]FacT Bottleneck: [8, 16, 32] | 0.021 ~0.196 |
| FacT_TK | FacT Bottleneck: [16, 32, 64]FacT Scale Factor: [0.01, 0.1, 1, 10, 100] | 0.030 ~0.369 |
+
+$$
+{ \pmb Z } _ { 0 } = \left[ { \pmb x } _ { 0 } ^ { ( \mathrm { C l a s s } ) } , { \pmb x } _ { 0 } ^ { ( 1 ) } , { \pmb x } _ { 0 } ^ { ( 2 ) } , \cdots , { \pmb x } _ { 0 } ^ { ( N ) } \right] + { \bf E } _ { \mathrm { p o s } }
+$$
+
+Table 4. We apply simple data augmentations (DA) (RandomResizedCrop, RandomVerticalFlip and RandomHorizontalFlip) on three datasets in each group. Data augmentation does not benefit most of VTAB-1K datasets and thus, most recent PEFT papers [42–44, 55, 63, 63, 107] skip it. Figure 8 shows examples of how some data augmentation transforms are harmful for a specific task. Therefore, to ensure that our results are directly comparable to existing papers, we don’t apply data augmentation.
+
+| Method | What | TunableParameters | HyperParameters | ModifiedType | InferenceEfficient |
| VPT-Deep | h1 = [h1, P] | P Rl×D | l: Number of prompts | Feature | × |
| AdaptFormer | h9 = h9 + Adapter(h7) | Wdown /up Rr×D/D×r in Adapter | s: Scale factor in Adapterr: Bottleneck dimension | Feature | × |
| Pfeif. Adapter | h9 = Adapter(h9) | Wown /up R×D/D in Adapter | s: Scale factor in Adapterr: Bottleneck dimension | Feature | × |
| Houl. Adapter | h5 = Adapter1(h5)h9 = Adapter2(h9) | Rr×D/D×r in Adapter2 | s: Scale factor in Adapterr: Bottleneck dimension | Feature | × |
| Convpass | h5 = Convpass1(h2) + h5h9 = Convpass2(h7) + h9 | W 2 Rr×D/D×r in Convpass2 | s: Scale factor in Convpassr: Bottleneck dimensionk: Kernel size of conv2d | Feature | × |
| RepAdpater | h2 = RepAdapter1(h2)h7 = RepAdapter2(h7) | bR in RepAdapter1ben RD× in RepAdapter2 | s: Scale factor in RepAdapterr: Bottleneck dimensionG: Number of groups | Feature | × |
| LayerNorm | h2 = LayerNorm1(h1)h7 = LayerNorm2(h6) | W 1(2), b1(2) RD in LayerNorm1(2) | N/A | Backbone | ✓ |
| BitFit | Fine-tune all bias termsin the network | b1(2) RD in LayerNorm1(2)bQ//V RD in Q/K/Vttn R in FCattnb1 R4D, in FC1, b2 RD in FC2 | N/A | Backbone | ✓ |
| DiffFit | LayerNorm + BitFith5 = γ1 · h5hg = γ2 · h9 | All tunable parametersin LayerNorm & BitFitγ1, γ2 RD | N/A | Backbone | √ |
| SSF | h2 = SSF2(h2), h3 = SSF3(h3)h5 = SSF5(h5), h7 = SSF7(h7)h8 = SSF7(h8), h9 = SSF9(h9) | W 2,5,7,9 RD , b2,5,7,9 RDW3 R3D , b3 R3DW 8 R4D, 68 R4D | N/A | Backbone | ✓ |
| LoRA | h3 = LoRA(h2) + h3 | WQ/K/V Rr×D/D× in LoRA | r: Bottleneck dimension | Backbone | ✓ |
| FacTTT(TK) | h3 = FacTTT(TK)(h2) + h3h5 = FacTTT(TK)(h4) + h5h8 = FacTTT(TK)(h7) + h8h9 = FacTTT(TK)(h8) + h9 | U RDr, V RDr,∑ R12L×r×r in FacTTT | s: Scale factor in FacTTT(TK)r: Bottleneck dimension | Backbone | V |
| U RD×r, V RD×r, |
+
+$$
+\begin{array} { r l } & { [ \tilde { P } _ { 1 } , \boldsymbol { Z } _ { 1 } ] = L _ { m } ( [ P _ { 0 } , \boldsymbol { Z } _ { 0 } ] ) } \\ & { [ \tilde { P _ { m } } , \boldsymbol { Z } _ { m } ] = L _ { m } ( [ \tilde { P } _ { m - 1 } , \boldsymbol { Z } _ { m - 1 } ] ) \quad m = 2 , 3 , \dots , M } \end{array}
+$$
+
+$$
+[ \ , Z _ { m } ] = { \cal L } _ { m } ( [ P _ { m - 1 } , Z _ { m - 1 } ] ) \quad m = 1 , 2 , 3 , \ldots , M
+$$
+
+Throughout the adaptation process, the pre-trained model is frozen and no additional weights are introduced to the model, thereby preserving the model’s original behaviour. During the forward pass, the output $Z _ { m }$ of layer $m$ is changed because of the interaction between $Z _ { m - 1 }$ and $P _ { m - 1 }$ (or $\tilde { P } _ { m - 1 } ^ { \mathrm { ~ ~ } }$ ) in the MSA block. Thus, the output feature is adapted to the downstream tasks by iteratively tuning the prompts through gradient descent.
+
+
+Figure 9. An overview of a Transformer block in ViT. We adopt the notation $\{ h _ { 1 } , \ldots , h _ { 1 0 } \}$ to denote the intermediate features within a Transformer block to facilitate a clearer illustration of the PEFT methods discussed in subsection B.2.
+
+# B.2.2. Adapter-based Methods
+
+Adapter-based methods typically introduce additional trainable parameters into a frozen pre-trained model to facilitate learning of downstream tasks [54]. Initially developed for multi-domain adaptation [76, 77] and continual learning [66, 80], the idea of Adapters is subsequently embraced by Houlsby et al. [36] in the NLP domain to adapt Transformer-based networks for downstream tasks, and it also has garnered increasing interest in the computer vision field [102]. In this comparative analysis, we concentrate onfive popular Adapter-based methods, encompassing the original Adapter, along with variants focusing on adjusting the positions of Adapters [11, 74], introducing visual inductive biases [43], as well as employing re-parameterization to reduce the number of trainable parameters and inference latency [63].
+
+Houl. Adapter [36] inserts two lightweight bottleneckstructured modules into each Transformer layer: one after the MSA block and the other after the MLP block. As depicted in Figure 10a, the Adapter is composed of a down-projection layer with $W _ { \mathrm { d o w n } } \in \mathbb { R } ^ { r \times D }$ , a nonlinear activation function $\sigma$ , an up-projection layer with $W _ { \mathsf { u p } } \in \mathbb { R } ^ { D \times r }$ , a scaling factor $s$ and a skip-connection. To limit the number of trainable parameters, the bottleneck dimension is much smaller than the feature dimension $r \ll D$ . Formally, Houl. Adapter can be defined as:
+
+$$
+\begin{array} { r l } & { h _ { 5 } = \mathrm { A d a p t e r } _ { 1 } ( h _ { 5 } ) \quad h _ { 9 } = \mathrm { A d a p t e r } _ { 2 } ( h _ { 9 } ) } \\ & { \quad \mathrm { A d a p t e r } ( h ) = s \cdot W _ { \mathrm { u p } } \sigma ( W _ { \mathrm { d o w n } } h ) + h } \end{array}
+$$
+
+Pfeif. Adapter [74] is a more efficient variant that introduces the Adapter solely after the MLP block, a strategy that has demonstrated effectiveness in recent studies [37]. Pfeif. Adapter can be defined formally as $h _ { 9 } =$ Adapter $\left( h _ { 9 } \right)$ where Adapter follows Equation 14.
+
+AdaptFormer [11] proposed to insert the Adapter in parallel with the MLP block, which differs from the sequential design of Houl. and Pfeif. Adapter. The rationale behind this parallel design lies in the belief that the domain-specific features generated by the Adapter can complement the domain-agnostic features derived from the original MLP block, leading to an improved feature ensemble [85]. Formally, AdaptFormer can be defined as $h _ { 9 } = h _ { 9 } + \mathrm { A d a p t e r } ( h 7 )$ where Adapter follows Equation 14.
+
+ConvPass (Convolutional By-Passes) [43] addresses the concern that many existing Adapters lack visual inductive bias, potentially limiting their performance for downstream vision tasks with limited data. To this end, the authors introduce a convolutional bottleneck module, running in parallel with the MSA or(and) MLP block. This module encompasses a $1 \times 1$ convolution reducing the channel with $W _ { \mathrm { d o w n } } \ \in \mathbb { R } ^ { r \times D }$ , a $3 \times 3$ convolution with the same input and output channel, a $1 \times 1$ convolution expanding the channel $W _ { \mathsf { u p } } \in \mathbb { R } ^ { D \times r }$ , two nonlinear functions $\sigma$ and a scaling factor $s$ , as shown in Figure 10b. The authors argue that Convpass is more efficient at capturing visual information in low-data scenarios due to its hard-coded locality of convolutional layers. The formal definition of Convpass is shown in Equation 15.
+
+$$
+\begin{array} { r } { h _ { 5 } = \mathrm { C o n v p a s s } _ { 1 } ( h _ { 2 } ) + h _ { 5 } \quad h _ { 9 } = \mathrm { C o n v p a s s } _ { 2 } ( h _ { 7 } ) + h _ { 9 } } \\ { \mathrm { C o n v p a s s } ( h ) = s \cdot W _ { \mathrm { u p } } \sigma ( \mathrm { C o n v 2 d } ( \sigma ( W _ { \mathrm { d o w n } } h ) ) ) } \end{array}
+$$
+
+
+Figure 10. Comparison of three Adapter structures.
+
+RepAdapter [63] found that the removal of the nonlinear function in the Adapter does not result in performance degradation for vision tasks. In light of this finding, the authors propose a linear Adapter with group-wise transformation [62] and sequentially added two of these linear Adapters to both MSA and MLP blocks. Owing to the sequential placement of the RepAdapter and its inherent linearity, the additional parameters can be re-parameterized to the original MSA or MLP block after training, thereby incurring zero additional costs during inference. RepAdapter is illustrated in Figure 10c and formally defined in Equation 16.
+
+$$
+\begin{array} { r l } & { h _ { 5 } = \mathrm { R e p A d a p t e r } _ { 1 } ( h _ { 2 } ) \quad h _ { 7 } = \mathrm { R e p A d a p t e r } _ { 2 } ( h _ { 7 } ) } \\ & { \qquad \mathrm { R e p A d a p t e r } ( h ) = s \cdot \phi _ { \mathrm { u p } } ( \phi _ { \mathrm { d o w n } } \left( h \right) ) + h } \\ & { \qquad \tilde { h } = \phi _ { \mathrm { d o w n } } \left( h \right) = W _ { \mathrm { d o w n } } h } \\ & { \qquad \phi _ { \mathrm { u p } } ( \tilde { h } ) = [ W _ { g 1 } \tilde { h } _ { g 1 } , \dots , W _ { g G } \tilde { h } _ { g G } ] } \end{array}
+$$
+
+where $W _ { \mathrm { d o w n } } \ \in \ \mathbb { R } ^ { r \times D }$ , $\tilde { h } _ { g ( 1 , \dots , G ) } \in \mathbb { R } ^ { \frac { r } { G } \times ( N + 1 ) }$ is the features splitted from $\tilde { h } \in \mathbb { R } ^ { r \times ( N + 1 ) }$ and $G$ is the number of groups in group-wise transformation [62]. $W _ { g ( 1 , \dots , G ) } \in$ $\textstyle \mathbb { R } ^ { \frac { D } { G } \times \frac { r } { G } }$ is the projection weight matrix.
+
+# B.2.3. Selective Parameter Tuning Methods
+
+The methods falling within this category aim to selectively update the parameters of a pre-trained model for downstream tasks. Within transfer learning, two prominent strategies, namely full fine-tuning and linear probing [48, 111], represent the two extremes of this category. Full fine-tuning updates all the model parameters end-to-end based on the new dataset while linear probing treats the pre-trained model as a feature extractor and only updates the prediction heads while keeping the backbone frozen. Although full fine-tuning generally exhibits superior performance compared to linear probing [104], it possesses certain limitations that may hinder its practicality in real-world production settings. Firstly, it requires running gradient descent for all parameters and necessitates storing a separate fine-tuned model for each task, incurring significant computational, memory, and storage overhead. These challenges become more salient with Transformer-based models whose parameters grow exponentially. Secondly, full fine-tuning may distort pre-trained features and underperform linear probing in out-of-distribution (OOD) scenarios [50].
+
+To cope with the above issues, a cohort of PEFT methods has emerged under this category. In addition to the two common approaches mentioned above, our investigation encompasses seven methods that can be further categorized into two groups: direct selective tuning [7, 97, 103], which involves the direct modification of selective weights, and efficient selective tuning [37, 44, 55], which approximates the weight updates with low-rank factors.
+
+Notably, an extra advantage of methods in this category is that they introduce no additional inference latency, making them particularly favourable when inference efficiency is a priority. Methods within the direct selective tuning group abstain from introducing any new modules, thus inherently avoiding extra inference latency. Meanwhile, for methods in the efficient selective tuning group, the added modules can often be seamlessly integrated into weights of the pre-trained models through the re-parameterization techniques [18, 41], thereby ensuring the absence of increased inference latency as well.
+
+# Direct Selective Tuning
+
+BitFit [103] is a simple yet effective method that only tunes the bias parts of the pre-trained model. For each Transformer layer in ViT, BitFit updates the bias terms in the QKV projections and the FC layer in the MSA block, two FC layers in the MLP block and two LN blocks. It also updates the bias in the projection for patch embedding. The original authors underscore BitFit’s capability to achieve performance comparable to full fine-tuning or even surpass it under low and medium-data scenarios in BERT models [46].
+
+LayerNorm [7] represents another simple but strong baseline that solely tunes the two LN blocks in each Transformer layer - one before the MSA block and another before the MLP block. Given that each LN block contains merely two trainable parameters $\{ W _ { L N } , b _ { L N } \} \in \mathbb { R } ^ { D }$ , LNtune stands out as an exceedingly light-weight approach compared to other PEFT methods. For instance, ViT-B/16 $\mathrm { \sim } 8 6 \mathrm { M }$ parameters) has only ${ \sim } 3 8 \mathrm { K }$ LN parameters, accounting for ${ \sim } 0 . 0 4 \%$ of the total parameters.
+
+DiffFit [97] is a recently proposed PEFT strategy designed for adapting large pre-trained diffusion models to the new domains. DiffFit exclusively fine-tunes the bias terms and the LN blocks within the network. Furthermore, it inserts learnable scale factors $\gamma$ to shift the features after the MSA and the MLP blocks, as shown in Equation 17. Consequently, DiffFit can be regarded as a combination of the BitFit and Ln-Tune, incorporating additional feature shift factors.
+
+$$
+\begin{array} { r } { h _ { 5 } = \gamma _ { 1 } \cdot h _ { 5 } } \\ { h _ { 9 } = \gamma _ { 2 } \cdot h _ { 9 } } \end{array}
+$$
+
+# Efficient Selective Tuning
+
+LoRA (Low-Rank Adaptation) [37] drew inspiration from recent investigations demonstrating that the learned over-parametrized models in fact reside on a low intrinsic dimension [1, 52]. Building upon this insight, the authors hypothesize that the change in weights during model adaptation also exhibits a low intrinsic rank and injects trainable low-rank decomposition matrices to approximate the weight updates. The LoRA update methodology is strategically applied to the Query/Value projection weights
+
+$W _ { Q / V } \in \mathbb { R } ^ { D \times D }$ within the MSA block. Concretely, the
+weight updates are approximated as $W _ { Q / V } + \Delta W _ { Q / V } =$
+$W _ { Q / V } + W _ { \mathrm { d o w n } } ^ { Q / V } W _ { \mathrm { u p } } ^ { Q / V }$ where W Q/V $W _ { \mathrm { d o w n / u p } } ^ { Q / V } \in \mathbb { R } ^ { D \times r / r \times D }$ and $r \ll D$ and zero for zero at the b so that g of tra $W _ { \mathrm { u p } } ^ { Q / V }$ $W _ { \mathrm { d o w n } } ^ { Q / V }$ $\Delta W _ { Q / V } =$
+$W _ { \mathrm { d o w n } } ^ { Q / V } W _ { \mathrm { u p } } ^ { Q / V }$
+formal definition of LoRA is articulated in Equation 18,
+utilizing the notations delineated in Figure 9.
+
+$$
+\begin{array} { c } { { h _ { 3 } = \mathrm { L o R A } ( h _ { 2 } ) + h _ { 3 } } } \\ { { h _ { 3 } = [ { \cal Q } , { \cal K } , { \cal V } ] } } \\ { { \mathrm { L o R A } ( h _ { 2 } ) = [ W _ { \mathrm { d o w n } } ^ { \cal Q } W _ { \mathrm { u p } } ^ { \cal Q } h _ { 2 } , 0 , W _ { \mathrm { d o w n } } ^ { \cal V } W _ { \mathrm { u p } } ^ { \cal V } h _ { 2 } ] } } \end{array}
+$$
+
+FacT (Factor Tuning) [44] is inspired by the recent advances in Transformer compression [92, 106]and exploited the low-rank update paradigm (e.g., LoRA) to the extreme. While LoRA posits that the update for an individual weight matrix manifests a low-rank characteristic during fine-tuning, FacT advances the proposition that the weight updates spanning different matrices can also be effectively approximated using low-rank decomposition matrices. Specifically, FacT encapsulates the four weight matrices $W _ { Q / K / V / O } \ \in \ \mathbb { R } ^ { D \times D }$ in the MSA block and the two weight matrices ${ \pmb W } _ { 1 } ~ \in ~ \mathbb { R } ^ { D \times 4 D }$ , $W _ { 2 } ~ \in ~ \mathbb { R } ^ { 4 D \times D }$ in the MLP block into a single $W _ { F a c T } \in \mathbb { R } ^ { 1 2 M \times D \times D }$ tensor where $M$ is the number of Transformer layer. The update of $W _ { F a c T }$ , $\Delta W _ { F a c T }$ , can be decomposed into several factors to promote parameter efficiency. To this end, the authors leverage the well-established Tensor-Train (TT) [72]and the Tucker (TK) [14] format to decompose $\Delta W _ { F a c T }$ . FacTTT and $\mathrm { F a c T } _ { \mathrm { T K } }$ are used to denote different decomposition formats for FacT and their formal definitions can be found in Equation 19.
+
+$$
+\begin{array} { r l } & { \mathrm { F a c T } _ { \mathrm { T T } } : \Delta W _ { F a c T } = s \cdot \Sigma \times _ { 2 } U ^ { \top } \times _ { 3 } V ^ { \top } } \\ & { \mathrm { F a c T } _ { \mathrm { T K } } : \Delta W _ { F a c T } = s \cdot A \times _ { 1 } B ^ { \top } \times _ { 2 } U ^ { \top } \times _ { 3 } V ^ { \top } } \end{array}
+$$
+
+where ${ \pmb U } \ \in \ \mathbb { R } ^ { D \times r } , { \pmb V } \ \in \ \mathbb { R } ^ { D \times r } , { \pmb \Sigma } \ \in \ \mathbb { R } ^ { 1 2 L \times r \times r } , { \pmb B } \ \in$ $\mathbb { R } ^ { 1 2 L \times r } A \in \mathbb { R } ^ { r \times r \times r }$ and the $\times _ { j }$ denotes mode- $j$ product and $s$ is the scaling factor.
+
+Since $\Delta W _ { F a c T }$ contains the updates for $W _ { Q / K / V / O } , W _ { 1 / 2 }$ , the modified forward pass inherently influences $h _ { 3 } , h _ { 5 } , h _ { 8 } , h _ { 9 }$ . Let’s consider $h _ { 5 }$ for elucidation. Once the weight update $\Delta W _ { F a c T }$ is calculated with $\mathrm { F a c T } _ { \mathrm { T T ( T K ) } }$ in Equation 19, the corresponding update for $W _ { O }$ , $\Delta W _ { O }$ , is extracted from $\Delta W _ { F a c T }$ . Similar to the modified forward pass of LoRA, $h _ { 5 } = h _ { 4 } \Delta { \cal W } _ { O } + h _ { 5 }$ .
+
+SSF (Scale & Shift deep Features) [55] employs linear transformations to adapt the intermediate features extracted by a pre-trained model. Motivated by the feature modulation methods [39, 73], SSF is designed to accommodate the distribution difference between the upstream and downstream datasets. Specifically, SSF modulates the features residing at $h _ { 2 } , h _ { 3 } , h _ { 5 } , h _ { 7 } , h _ { 8 } , h _ { 9 }$ by incorporating scale and shift factors. To demonstrate the mechanism of SSF, let’s consider $h _ { 5 } \in \mathbb { R } ^ { ( N + 1 ) \times D }$ as an illustrative example and other features can similarly undergo the same transformative process. Formally, the modulated $h _ { 5 }$ is formulated as follows.
+
+
+Figure 11. Performance gain for PEFT methods by turning droppath-rate on.
+
+$$
+h _ { 5 } = \mathrm { S S F } _ { 5 } ( h _ { 5 } ) = \pmb { w } ^ { 5 } \odot h _ { 5 } + \pmb { b } ^ { 5 }
+$$
+
+where ${ \pmb w } ^ { 5 } \in \mathbb { R } ^ { D }$ , $\pmb { b } ^ { 5 } \in \mathbb { R } ^ { D }$ are the scale and shift factors affiliated with the SSF module attributed to $h _ { 5 }$ , and $\odot$ is the dot product. It is noteworthy that each modulated feature has its own SSF module with corresponding scale and shift factors. The modification details for other features are summarized in Table 6.
+
+# C. More Detailed Results
+
+Drop-path-rate. Learning with low-shot data is prone to over-fitting. We find that if the drop path rate — which stochastically drops a transformer block per sample [38] — is set not as default (i.e., nonzero), all the methods can benefit from such a regularization. Figure 11 shows the performance gain by tuning the drop-path-rate on compared with the default 0.
+
+More results on prediction similarity analysis. Figure 13 shows the prediction analysis discussed in section 4 for all the datasets in VTAB-1K. It is expected that their predictions are similar for datasets with very high accuracy, such as Flowers102 $( \mathrm { a v g 9 9 . 1 \% } )$ and Caltech101 $( \mathrm { a v g 9 l . 4 \% } )$ . Beyond them, we find that most PEFT methods show diverse predictions in other datasets in VTAB-1K.
+
+Prediction similarity within the same PEFT group. To verify if methods within the same PEFT group share more prediction similarity, we plotted the prediction overlap for adapter-based methods, selective-tuning methods, and methods from different groups. As shown in Figure 12, methods within the same group share slightly more prediction similarity than those from different groups, but they still exhibit distinct predictions. Figure 3a in the main paper also supports this observation. Methods are grouped based on the categories defined in subsection 2.2. If methods within the same group had very high similarities, we would see bright squares, which are only slightly evident around BitFit, Diff-Fit, LayerNorm, and SSF.
+
+# WiSE PEFT results for all distribution shift datasets.
+
+We provide detailed WiSE PEFT performance for each distribution shift dataset in Figure 14. WiSE improves both the robustness and the in-distribution performance of PEFT methods. Interestingly, even though full fine-tuning is generally less robust than PEFT methods, applying WiSE allows it to achieve better performance in both target distribution and distribution shift data.
+
+Performance comparison between DINOv2 and IN21k. To compare the performance of DINOv2 and IN21k, we selected several PEFT methods and datasets from VTAB-1K. The results presented in Table 7 reveal several interesting findings:
+
+(1) Improved Linear Probing Performance: Linear probing generally shows improved results with DINOv2, indicating that its extracted features are more robust and discriminative than those from IN21k.
+
+(2) Deteriorated Full Fine-Tuning Performance: Conversely, full fine-tuning performance significantly worsens with DINOv2, suggesting that models fully fine-tuned on DINOv2 are more susceptible to overfitting.
+
+(3) Adapter-Based Methods Performance: Among the three adapter-based methods evaluated—Houl. Adapter, AdaptFormer, and Convpass—we observe performance enhancements in most datasets for AdaptFormer and Convpass. In contrast, Houl. Adapter exhibits significant degradation across all datasets. This disparity may be attributed to architectural differences: AdaptFormer and Convpass insert their adapter modules in parallel with existing modules such as Multi-Head Self-Attention (MSA) and/or Multi-Layer Perceptron (MLP), whereas Houl. Adapter inserts its adapter sequentially after the MSA and MLP layers. We hypothesize that the sequential design of Houl. Adapter leads to more substantial alterations of intermediate features compared to the parallel design, potentially explaining the observed decrease in performance.
+
+
+Figure 12. Prediction overlap for the 5K most confident samples. Although methods from the same group share slightly more prediction overlap than methods from other groups, they still have quite different predictions
+
+Table 7. Performance comparison between DINOv2 and IN21k.
+
+