| Type | Methods | Training Scheme | Training Data | Acc. | Base Model |
| KD | CoT-KD | Distillation (Full FT) | CoT data | GSM8K: 21.99% (↑ 13.88%) | T5 XXL |
| KD | MD | Mixed distillation (Freeze FT) | CoT and PoT data | GSM8K: 41.50% (↑ 28.20%) | LLaMA2-7B |
| KD | Mix | Mixed distillation (Full FT & LoRA) | Long and short CoT data | GSM8K: 79.20% (↑ 1.70%) | LLaMA3.2-3B |
| KD | NAT | Mixed distillation (LoRA) | Positive and negative data | GSM8K: 41.24% (↑ 23.73%) | LLaMA-7B |
| KD | CD | Counterfactual distillation (Full FT) | Original and counterfactual data | - | - |
| KD | FDD | Feedback-driven distillation (Full FT) | Progressively add generated data | GSM8K: 49.43% (↑ 42.53%) | FlanT5-Large |
| KD | DLCoT | Distillation (Full FT) | High-quality data | GSM8K: 93.60% (↑ 9.10%) | LLaMA3.1-8B |
| KD | SKIntern | Distillation (LoRA) | Progressively simplify data | GSM8K: 33.90% (↑ 30.80%) | LLaMA2-7B |
| RL | Open-RS | GRPO (Full FT) | Blended1 | AIME: 46.70% (↑ 17.80%) | DeepSeek-R1-Distill-Qwen-1.5B |
| RL | DeepSacreR | GRPO (Full FT) | Blended2 | AIME: 43.10% (↑ 14.20%) | DeepSeek-R1-Distill-Qwen-1.5B |
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+ "text": "SLM-Foresee (Srivastava et al., 2025) conducted a systematic study on the reasoning abilities of diverse small language models, demonstrating that small language models can exhibit strong reasoning potential. Certain models, such as the Qwen2.5 series (Yang et al., 2024a), even achieve performance comparable to or surpassing some LLMs. Open-RS (Dang & Ngo, 2025) enhanced the reasoning capability of small language models using RL with the GRPO algorithm (Guo et al., 2025) and curated a high-quality mathematical reasoning dataset derived from the s1 dataset (Muennighoff et al., 2025) and DeepScaleR dataset (Luo et al., 2025b). They further develop a cosine reward to control response length effectively. Their 1.5B model, trained on 7K samples within 24 hours on $4 \\times \\mathrm{A}40$ GPUs, achieved performance on benchmarks (e.g., AIME 24, MATH-500) that matches or surpasses models like o1-preview (AI., 2024). SimpleRL-Zoo (Zeng et al., 2025a) systematically evaluated the generality of ZeroRL (i.e., an RL paradigm that enables LMs to learn long-chain reasoning with only simple rule-based rewards and no additional supervision). The study proposed several key design strategies for successful ZeroRL training: using simple correctness-based rewards, aligning data difficulty with model capacity, and employing stable RL algorithms like GRPO. Remarkably, verification behavior was observed for the first time in small language models outside the Qwen2.5 series $^{2}$ , further validating the reasoning potential of small language models. Additionally, DeepScaleR $^{3}$ (Luo et al., 2025b) leverages iterative scaling of GRPO to extend thinking length (i.e., $8\\mathrm{K} \\rightarrow 16\\mathrm{K} \\rightarrow 24\\mathrm{K}$ ), significantly improving performance on math reasoning benchmarks. The 1.5B model, DeepScaleR-1.5B-Preview, surpasses o1-Preview and achieves $43.1\\%$ Pass@1 on AIME.",
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| KD | CoT-KD | Distillation (Full FT) | CoT data | GSM8K: 21.99% (↑ 13.88%) | T5 XXL |
| KD | MD | Mixed distillation (Freeze FT) | CoT and PoT data | GSM8K: 41.50% (↑ 28.20%) | LLaMA2-7B |
| KD | Mix | Mixed distillation (Full FT & LoRA) | Long and short CoT data | GSM8K: 79.20% (↑ 1.70%) | LLaMA3.2-3B |
| KD | NAT | Mixed distillation (LoRA) | Positive and negative data | GSM8K: 41.24% (↑ 23.73%) | LLaMA-7B |
| KD | CD | Counterfactual distillation (Full FT) | Original and counterfactual data | - | - |
| KD | FDD | Feedback-driven distillation (Full FT) | Progressively add generated data | GSM8K: 49.43% (↑ 42.53%) | FlanT5-Large |
| KD | DLCoT | Distillation (Full FT) | High-quality data | GSM8K: 93.60% (↑ 9.10%) | LLaMA3.1-8B |
| KD | SKIntern | Distillation (LoRA) | Progressively simplify data | GSM8K: 33.90% (↑ 30.80%) | LLaMA2-7B |
| RL | Open-RS | GRPO (Full FT) | Blended1 | AIME: 46.70% (↑ 17.80%) | DeepSeek-R1-Distill-Qwen-1.5B |
| RL | DeepSacreR | GRPO (Full FT) | Blended2 | AIME: 43.10% (↑ 14.20%) | DeepSeek-R1-Distill-Qwen-1.5B |
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+ "content": "In the previous sections, we discussed two main directions for improving reasoning efficiency. However, this section covers strategies to accelerate reasoning during the decoding stage. It begins with techniques to reduce computational overhead during TTS (see Section 3.3.1), followed by an overview of other methods for making reasoning faster, with details provided in Section 3.3.2. These methods are summarized in Table 4, showing that most methods achieve notable efficiency gains and further improve model performance without additional training."
+ },
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+ "content": "2Most existing works focus exclusively on Qwen2.5 models, whose strong instruction following and self-reflection abilities may skew results."
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+ "content": "| Type | Methods | Training Scheme | Training Data | Acc. | Base Model |
| KD | CoT-KD | Distillation (Full FT) | CoT data | GSM8K: 21.99% (↑ 13.88%) | T5 XXL |
| KD | MD | Mixed distillation (Freeze FT) | CoT and PoT data | GSM8K: 41.50% (↑ 28.20%) | LLaMA2-7B |
| KD | Mix | Mixed distillation (Full FT & LoRA) | Long and short CoT data | GSM8K: 79.20% (↑ 1.70%) | LLaMA3.2-3B |
| KD | NAT | Mixed distillation (LoRA) | Positive and negative data | GSM8K: 41.24% (↑ 23.73%) | LLaMA-7B |
| KD | CD | Counterfactual distillation (Full FT) | Original and counterfactual data | - | - |
| KD | FDD | Feedback-driven distillation (Full FT) | Progressively add generated data | GSM8K: 49.43% (↑ 42.53%) | FlanT5-Large |
| KD | DLCoT | Distillation (Full FT) | High-quality data | GSM8K: 93.60% (↑ 9.10%) | LLaMA3.1-8B |
| KD | SKIntern | Distillation (LoRA) | Progressively simplify data | GSM8K: 33.90% (↑ 30.80%) | LLaMA2-7B |
| RL | Open-RS | GRPO (Full FT) | Blended1 | AIME: 46.70% (↑ 17.80%) | DeepSeek-R1-Distill-Qwen-1.5B |
| RL | DeepSacreR | GRPO (Full FT) | Blended2 | AIME: 43.10% (↑ 14.20%) | DeepSeek-R1-Distill-Qwen-1.5B |
+
+on Full FT, with a few adopting PEFT techniques. Notably, methods that progressively incorporate refined or synthesized data (e.g., FDD and SKIntern) tend to achieve greater performance improvements.
+
+Apart from model compression and RL, some studies explore the reasoning ability of small language models from alternative perspectives. For example, Liu et al. (2025d) shows that small language models can match or even surpass the reasoning performance of much larger LLMs with carefully designed TTS strategies. However, the effectiveness of TTS strategies varies with model architecture, reward design, and task complexity. While small language models show potential in reasoning, their limitations in instruction following and self-reflection highlight the need for further adaptation to align with human intent.
+
+# 3.2.1 Distillation Transfers Reasoning Ability to Small Language Model
+
+CoT-KD (Magister et al., 2022) first demonstrated that distillation can transfer reasoning ability from LLMs to small language models. However, due to limited capacity, small language models struggle to learn complex reasoning (Li et al., 2025e), motivating the development of more advanced strategies. Based on the optimization target, existing methods can be grouped into two directions: (1) data-focused, which improves the quality or composition of training data, and (2) model-focused, which concentrates on the distilled model itself or its generation strategy.
+
+Data-focused. MD (Li et al., 2023a) adopts mix distillation by combining data generated with different prompting strategies (CoT and PoT) as training data, and Mix (Li et al., 2025e) applies a similar strategy using a mix of long and short CoT samples. CD (Feng et al., 2024c) enhances training diversity by mixing original data with counterfactual samples derived from it, while NAT (Li et al., 2024a) leverages negative data. DLCoT (Luo et al., 2025c) improves training data quality by segmenting and simplifying long reasoning paths. SCORE (Zhang et al., 2024) enables self-correction by allowing the model to generate, identify, and refine its reasoning, using the corrected outputs for further distillation. Distill2-to-1 (Yu et al., 2024) only retrans (input, answer) pairs as training data. The above methods rely on standard SFT, but some adopt progressive SFT. FDD (Zhu et al., 2024b) progressively adjusts data difficulty based on the small language model's performance on LLM-generated data, while SKIntern (Liao et al., 2025b) proposes a progressive process that removes symbolic knowledge and examples step by step, encouraging the model to internalize reasoning ability.
+
+Model-focused. PRR (Zhao et al., 2024) distills two separate models: a probing model for retrieving relevant knowledge and a reasoning model for generating answers based on the question and retrieved content. Thinking slow, fast (Paliotta et al., 2025) explores distilling reasoning ability from transformer-based models into Mamba or Mamba-Transformer architectures to reduce inference cost. Similarly, M1 (Wang et al., 2025b) builds on Mamba (Gu & Dao, 2024) to develop a hybrid linear RNN reasoning model that alleviates latency and memory overhead from long reasoning chains, further enhanced through RL after distillation. Additionally, works such as NSA (Yuan et al., 2025) and MoBA (Lu et al., 2025), which focus on lightweight architectures for general efficiency, can also be extended to improve reasoning efficiency. Additionally, ATM (Chen et al., 2024b) designs an adaptive mechanism that enables the student model to
+
+dynamically choose between pre-thinking (i.e., thinking before answering) and post-thinking (i.e., answering before thinking) based on question complexity.
+
+# 3.2.2 Pruning or Quantization Retain Reasoning Ability
+
+Recent work (Srivastava et al., 2025) systematically explores the impact of compression techniques like pruning and quantization on the reasoning capabilities of small language models, which shows that while quantization methods (Frantar et al., 2023b) have minimal impact on reasoning performance, pruning approaches (Li et al., 2023b) significantly degrade reasoning abilities. Similarly, When Reasoning Meets Compression (Zhang et al., 2025b) presents a comprehensive benchmark of compressed LRMs across various reasoning tasks. It also finds that quantized models retain strong reasoning performance and sometimes even surpass the original model, while aggressive pruning causes performance collapse at moderate sparsity. Furthermore, Quantization Hurts Reasoning? (Liu et al., 2025c) systematically evaluates the impact of quantization on reasoning models. It finds that high-bit (e.g., 8-bit) quantization is nearly lossless, while low-bit settings (e.g., 4-bit) significantly degrade performance, especially on complex tasks. Interestingly, the output length of CoT reasoning remains largely unchanged, except under aggressive quantization or when using small models. Notably, the results show that on certain large models, quantization can reduce GPU memory usage by over $75\%$ while retaining nearly $100\%$ of the original performance. Meanwhile, quantized versions of large models are often more effective than standalone small models, offering advantages in both memory efficiency and performance.
+
+# 3.2.3 Reinforcement Learning Helps Build Small Language Model
+
+SLM-Foresee (Srivastava et al., 2025) conducted a systematic study on the reasoning abilities of diverse small language models, demonstrating that small language models can exhibit strong reasoning potential. Certain models, such as the Qwen2.5 series (Yang et al., 2024a), even achieve performance comparable to or surpassing some LLMs. Open-RS (Dang & Ngo, 2025) enhanced the reasoning capability of small language models using RL with the GRPO algorithm (Guo et al., 2025) and curated a high-quality mathematical reasoning dataset derived from the s1 dataset (Muennighoff et al., 2025) and DeepScaleR dataset (Luo et al., 2025b). They further develop a cosine reward to control response length effectively. Their 1.5B model, trained on 7K samples within 24 hours on $4 \times \mathrm{A}40$ GPUs, achieved performance on benchmarks (e.g., AIME 24, MATH-500) that matches or surpasses models like o1-preview (AI., 2024). SimpleRL-Zoo (Zeng et al., 2025a) systematically evaluated the generality of ZeroRL (i.e., an RL paradigm that enables LMs to learn long-chain reasoning with only simple rule-based rewards and no additional supervision). The study proposed several key design strategies for successful ZeroRL training: using simple correctness-based rewards, aligning data difficulty with model capacity, and employing stable RL algorithms like GRPO. Remarkably, verification behavior was observed for the first time in small language models outside the Qwen2.5 series $^{2}$ , further validating the reasoning potential of small language models. Additionally, DeepScaleR $^{3}$ (Luo et al., 2025b) leverages iterative scaling of GRPO to extend thinking length (i.e., $8\mathrm{K} \rightarrow 16\mathrm{K} \rightarrow 24\mathrm{K}$ ), significantly improving performance on math reasoning benchmarks. The 1.5B model, DeepScaleR-1.5B-Preview, surpasses o1-Preview and achieves $43.1\%$ Pass@1 on AIME.
+
+# 3.3 Let Decoding More Efficient
+
+In the previous sections, we discussed two main directions for improving reasoning efficiency. However, this section covers strategies to accelerate reasoning during the decoding stage. It begins with techniques to reduce computational overhead during TTS (see Section 3.3.1), followed by an overview of other methods for making reasoning faster, with details provided in Section 3.3.2. These methods are summarized in Table 4, showing that most methods achieve notable efficiency gains and further improve model performance without additional training.
+
+Table 4: Overview of efficient reasoning methods in Section 3.3. The efficiency-up ratio is computed by comparing either the sampling count (S.), costs (C.), latency (L.), the correct trajectory count (T.), or FLOPs (F.). $C_1$ represents the consistency probability of the majority candidate. $C_2$ means the answer consistency within the sampling window. $C_3$ is the internal consistency via Chain-of-Embedding. $C_4$ is the probability of reaching the correct answer.
+
+| Type | Methods | Training Scheme | Training Data | Acc. | Base Model |
| KD | CoT-KD | Distillation (Full FT) | CoT data | GSM8K: 21.99% (↑ 13.88%) | T5 XXL |
| KD | MD | Mixed distillation (Freeze FT) | CoT and PoT data | GSM8K: 41.50% (↑ 28.20%) | LLaMA2-7B |
| KD | Mix | Mixed distillation (Full FT & LoRA) | Long and short CoT data | GSM8K: 79.20% (↑ 1.70%) | LLaMA3.2-3B |
| KD | NAT | Mixed distillation (LoRA) | Positive and negative data | GSM8K: 41.24% (↑ 23.73%) | LLaMA-7B |
| KD | CD | Counterfactual distillation (Full FT) | Original and counterfactual data | - | - |
| KD | FDD | Feedback-driven distillation (Full FT) | Progressively add generated data | GSM8K: 49.43% (↑ 42.53%) | FlanT5-Large |
| KD | DLCoT | Distillation (Full FT) | High-quality data | GSM8K: 93.60% (↑ 9.10%) | LLaMA3.1-8B |
| KD | SKIntern | Distillation (LoRA) | Progressively simplify data | GSM8K: 33.90% (↑ 30.80%) | LLaMA2-7B |
| RL | Open-RS | GRPO (Full FT) | Blended1 | AIME: 46.70% (↑ 17.80%) | DeepSeek-R1-Distill-Qwen-1.5B |
| RL | DeepSacreR | GRPO (Full FT) | Blended2 | AIME: 43.10% (↑ 14.20%) | DeepSeek-R1-Distill-Qwen-1.5B |
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+ "content": "| Notations | Annotation |
| X, xi, Xn, yn | X is the input data, which contains P tokens. xi is the i-th token of X. Xn is the n-th input data with yn as the corresponding label. |
| Ψ | Ψ = {{a(l)}Pl=1, WO, WV, WK, WQ} denotes the set of all the model parameters. a(l) ∈ Rm and WO ∈ Rm×ma are the weights in the MLP layer. WV ∈ Rma×d, WK, WQ ∈ Rmb×d are weights in the self-attention layer. |
| Ψ(0), ΨT*, ΔΨT | Ψ(0) is the pre-trained model. ΨT* is the fine-tuned model on a given task T. ΔΨT is the task vector of the task T, which is computed as ΔΨT = ΨT* - Ψ(0). |
| μT, vj | μT is the discriminative pattern of the task T. vj is the j-th task-irrelevant pattern, j ∈ [M]. |
| δ*, δ# | δ* is the average fraction of label-relevant pattern in the input data. δ# is the average fraction of confusion pattern in the input data. |
| q1(t),ζ1,t, pn(t) | q1(t) = μ1T W(t) μ1 denotes the value of the product, where the patterns on both sides of W(t) are the same.ζ1,t denotes the modified value embedding of μ1 at the t-th iteration. pn(t) refers to the summation of attention weights where the key and the query are the same discriminative pattern. |
| Wn,l,Un,l | Wn,l and Un,l respectively represent of sets of positive or negative neurons so that the Relu activation is activated with xln as the query. |
| Bb | Bb is the SGD batch at the b-th iteration. |
| O(), Ω(), Θ() | We follow the convention that f(x) = O(g(x)) (or Ω(g(x)), Θ(g(x))) means that f(x) increases at most, at least, or in the order of g(x), respectively. |
| a | a = |a(l)i| = 1/√m for i ∈ [m]. |
| ≥, ≤ | f(x) ≥ g(x) (or f(x) ≤ g(x)) means that f(x) ≥ Ω(g(x)) (or f(x) ≤ O(g(x))). |
",
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+ "type": "text",
+ "text": "Define $\\mathcal{W}_{n,l},\\mathcal{U}_{n,l}$ as the sets of lucky neurons such that",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {W} _ {n, l} = \\left\\{i: \\boldsymbol {V} _ {(i, \\cdot)} ^ {\\top} \\boldsymbol {R} _ {n, l} (0) > 0, l \\in \\mathcal {S} _ {1} ^ {n}, a _ {i} > 0 \\right\\}, \\tag {13}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\mathcal {U} _ {n, l} = \\left\\{i: \\boldsymbol {V} _ {(i, \\cdot)} ^ {\\top} \\boldsymbol {R} _ {n, l} (0) > 0, l \\in \\mathcal {S} _ {2} ^ {n}, a _ {i} < 0 \\right\\}. \\tag {14}\n$$\n",
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+ "type": "text",
+ "text": "Definition 5 ((Vershynin, 2010)). We say $X$ is a sub-Gaussian random variable with sub-Gaussian norm $K > 0$ , if $(\\mathbb{E}|X|^p)^{\\frac{1}{p}} \\leq K\\sqrt{p}$ for all $p \\geq 1$ . In addition, the sub-Gaussian norm of $X$ , denoted $\\| X\\|_{\\psi_2}$ , is defined as $\\| X\\|_{\\psi_2} = \\sup_{p \\geq 1} p^{-\\frac{1}{2}}(\\mathbb{E}|X|^p)^{\\frac{1}{p}}$ .",
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+ "type": "text",
+ "text": "Lemma 2 (Vershynin (2010) Proposition 5.1, Hoeffding's inequality). Let $X_{1}, X_{2}, \\dots, X_{N}$ be independent centered sub-gaussian random variables, and let $K = \\max_{i} \\|X_{i}\\|_{\\psi_{2}}$ . Then for every $\\mathbf{a} = (a_{1}, \\dots, a_{N}) \\in \\mathbb{R}^{N}$ and every $t \\geq 0$ , we have",
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+ "type": "equation",
+ "text": "\n$$\n\\Pr \\left(\\left| \\sum_ {i = 1} ^ {N} a _ {i} X _ {i} \\right| \\geq t\\right) \\leq e \\cdot \\exp \\left(- \\frac {c t ^ {2}}{K ^ {2} \\| \\boldsymbol {a} \\| ^ {2}}\\right), \\tag {15}\n$$\n",
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+ "type": "text",
+ "text": "where $c > 0$ is an absolute constant.",
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+ "type": "text",
+ "text": "Lemma 3. For task $\\mathcal{T}$ based on any $\\pmb{\\mu}_1$ , $0 \\leq t \\leq T$ , there exists $K(t) > 0$ , such that",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} = \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} + K (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 1} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\tag {16}\n$$\n",
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+ "type": "text",
+ "text": "where",
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+ "type": "equation",
+ "text": "\n$$\nK (t) \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\zeta_ {1, t} p _ {n} (t) \\phi_ {n} (t) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|), \\tag {17}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\iota_ {l} ^ {\\prime} \\leq K (t) \\cdot e ^ {- q _ {1} (t)}. \\tag {18}\n$$\n",
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+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
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+ "type": "page_number",
+ "text": "17",
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+ "type": "text",
+ "text": "For $k\\in [M]$",
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+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\lesssim \\sqrt {\\frac {\\log B}{B}} \\sum_ {b = 0} ^ {t} K (b), \\tag {19}\n$$\n",
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+ "type": "text",
+ "text": "and for $j\\neq k$ $j\\in [M]$",
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+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {v} _ {j} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\lesssim K (t) e ^ {- q _ {1} (t)}, \\tag {20}\n$$\n",
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+ "type": "text",
+ "text": "For any $\\pmb{\\mu}'$ such that $\\pmb{\\mu}_1^\\top \\pmb{\\mu}' = \\alpha$ and $\\pmb{\\mu}' \\perp \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M$ , we have",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} ^ {\\prime} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} ^ {\\prime} = \\alpha^ {2} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\cdot (1 \\pm \\Theta (\\epsilon)), \\tag {21}\n$$\n",
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+ "type": "text",
+ "text": "if $B \\geq \\epsilon^{-2} \\log M$ for some $\\epsilon < 1$ .",
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+ "type": "text",
+ "text": "Lemma 4. Given a task $\\mathcal{T}$ based on any $\\pmb{\\mu}_1$ , $0 \\leq t \\leq T$ . Then, for $i \\in \\mathcal{W}_{n,t}$ ,",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\gtrsim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {22}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {23}\n$$\n",
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+ "type": "text",
+ "text": "for $k\\in [M]$ .For $i\\in \\mathcal{U}_{n,l}$ , we similarly have",
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+ "type": "equation",
+ "text": "\n$$\n- \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\gtrsim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {24}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {25}\n$$\n",
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+ "type": "text",
+ "text": "for some $k\\in [M]$ . For $i\\notin \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}$ , we have that",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\lesssim \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {26}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k}, \\tag {27}\n$$\n",
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+ "type": "text",
+ "text": "where $k\\in [M],j\\in \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}$",
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+ "type": "text",
+ "text": "Lemma 5. (Full version of Lemma 1) Given a task $\\mathcal{T}$ defined in Definition 2 based on the discriminative pattern $\\pmb{\\mu}_{\\mathcal{T}}$ , we have that as long as conditions (i)-(iii) in Theorem 1 hold, then the returned model $\\Psi_{\\mathcal{T}}^{*}$ after $T$ iterations achieves a generalization error",
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+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\mathcal {T}}} \\left[ \\ell \\left(\\boldsymbol {X}, y; \\Psi_ {\\mathcal {T}} ^ {*}\\right) \\right] \\leq \\Theta (\\epsilon). \\tag {28}\n$$\n",
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+ "type": "text",
+ "text": "The required sample complexity is $N = BT$ , where $B$ is the batch size. We also have that",
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+ "type": "text",
+ "text": "1.",
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+ "type": "equation",
+ "text": "\n$$\np _ {n} (T) \\geq 1 - \\left(1 - \\delta_ {*}\\right) \\delta_ {*} ^ {- 1} T ^ {- C}, \\tag {29}\n$$\n",
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+ {
+ "type": "text",
+ "text": "for some constant $C > 1$ .",
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+ "type": "text",
+ "text": "2.",
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+ "type": "equation",
+ "text": "\n$$\n\\sum_ {k = 1} ^ {M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {v} _ {k} \\right\\| ^ {2} \\lesssim \\frac {1}{M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T}} \\right\\| ^ {2}, \\tag {30}\n$$\n",
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+ {
+ "type": "text",
+ "text": "for $i \\in \\mathcal{W}_{n,l}$ with $l \\in S_1^n$ and for $i \\in \\mathcal{U}_{n,l}$ with $l \\in S_2^n$ . We also have that (26) and (27) hold when $t = T$ .",
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+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
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+ "type": "page_number",
+ "text": "18",
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+ {
+ "type": "text",
+ "text": "D PROOF OF MAIN THEOREMS AND COROLLARIES",
+ "text_level": 1,
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+ "page_idx": 18
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+ {
+ "type": "text",
+ "text": "D.1 PROOF OF THEOREM 1 AND 2",
+ "text_level": 1,
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+ "type": "text",
+ "text": "Proof. Since the model is initialized close to zero, then $\\Delta \\Psi$ is close to $\\Psi$ . Denote $\\Psi_{1} = \\{\\{a_{(l,1)}^{P}\\}_{l=1}, V_{1}, W_{1}\\}$ and $\\Psi_{2} = \\{\\{a_{(l,2)}^{P}\\}_{l=1}, V_{2}, W_{2}\\}$ . We consider three cases of this learning problem.",
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+ "type": "text",
+ "text": "(1) Consider $\\alpha = 0$ . By (21) in Lemma 3, we know that",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} \\left(1 + \\lambda \\alpha^ {2} (1 \\pm \\Theta (\\epsilon))\\right) = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}}, \\tag {31}\n$$\n",
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+ {
+ "type": "equation",
+ "text": "\n$$\n- \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = - \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}}, \\tag {32}\n$$\n",
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+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = \\lambda \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}, \\tag {33}\n$$\n",
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+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n- \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = - \\lambda \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}. \\tag {34}\n$$\n",
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+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "Then, for any $l \\in [M]$ and for task $\\mathcal{T}_1$ ,",
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+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {s \\in S _ {1} ^ {n}} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}, \\tag {35}\n$$\n",
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+ {
+ "type": "text",
+ "text": "for task $\\mathcal{T}_2$",
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+ "type": "equation",
+ "text": "\n$$\n\\sum_ {s \\in \\mathcal {S} _ {1} ^ {n}} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq \\frac {\\delta_ {*} T ^ {\\lambda C}}{\\delta_ {*} T ^ {\\lambda C} + (1 - \\delta_ {*})} \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C}. \\tag {36}\n$$\n",
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+ {
+ "type": "text",
+ "text": "Since that $\\pmb{\\mu}_{\\mathcal{T}_2} \\perp \\{\\pmb{\\mu}_{\\mathcal{T}_1}, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\}$ and $\\pmb{\\mu}_{\\mathcal{T}_1} \\perp \\{\\pmb{\\mu}_{\\mathcal{T}_2}, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\}$ , we have",
+ "bbox": [
+ 169,
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = 0, \\tag {37}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "for $V\\in \\Psi_{1}$ , and",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = 0, \\tag {38}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "for $V \\in \\Psi_2$ . Then, for data with the label $y = 1$ , the network output for $\\Psi_1 + \\lambda \\Psi_2$ is almost the same as that for $\\Psi_1$ on task $\\mathcal{T}_1$ when $|\\lambda|$ is not too large. To see this, for $X$ from $\\mathcal{T}_1$ , we have",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} 1 - \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in [ m ]} \\frac {1}{a} \\operatorname {R e l u} \\left(\\left(\\boldsymbol {V} _ {1 (i, \\cdot)} ^ {(T)} + \\lambda \\boldsymbol {V} _ {2 (i, \\cdot)} ^ {(T)}\\right) \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ \\leq | \\lambda | \\cdot \\Theta \\left(\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}\\right) \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} + | \\lambda | \\cdot \\Theta \\left(\\sqrt {M \\frac {\\log B}{B}}\\right) \\tag {39} \\\\ \\leq | \\lambda | \\cdot \\Theta \\left(1 - \\delta_ {*}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\\\ = | \\lambda | \\beta , \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "where the second to last step is by (26) and (27) and $B \\gtrsim \\epsilon^2 \\log M$ . Therefore, a larger $|\\lambda|$ leads to a performance drop in task $\\mathcal{T}_1$ . For data of $\\mathcal{T}_1$ with the label $y = -1$ , we can choose $\\lambda$ to be greater than around 1 to make the network output smaller than $-1$ . Meanwhile, for $\\mathbf{X}$ from $\\mathcal{T}_2$ , we have",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f (\\boldsymbol {X} ^ {n}, \\Psi) \\\\ \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\lambda}\\right) \\cdot \\lambda - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right), \\tag {40} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "where we need $\\lambda \\geq 1 + \\beta$ so that $f(\\pmb{X}^n, \\Psi) \\geq 1 - \\Theta(\\epsilon)$ .",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "If $\\lambda \\leq 0$ , the attention map tends to be uniform. Then, for $X^n$ in task $\\mathcal{T}_2$ , we have",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}; \\Psi_ {1} + \\lambda \\Psi_ {2}\\right) \\lesssim - \\frac {1}{P}, \\tag {41}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "which leads to",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {42}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "page_number",
+ "text": "19",
+ "bbox": [
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+ ],
+ "page_idx": 18
+ },
+ {
+ "type": "text",
+ "text": "(2) Consider $\\alpha > 0$ . We first have",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} \\left(1 + \\lambda \\alpha^ {2}\\right), \\tag {43}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = (\\lambda + \\alpha^ {2}) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}. \\tag {44}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "Then, for $y^n = 1$ in task $\\widetilde{T}_1$ , we have that when $\\lambda > 0$ ,",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf (\\boldsymbol {X} ^ {n}, \\Psi)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\gtrsim (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {| \\mathcal {S} _ {1} ^ {n} |}{a P M}) \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C} \\\\ - | \\lambda | \\cdot \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {45} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\geq 1 + \\Theta (\\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right) \\\\ = 1 + \\Theta (\\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\frac {1 - \\delta_ {*}}{\\delta_ {*}}) \\cdot \\mathrm {p o l y} (\\eta \\delta_ {*}) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}), \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 245,
+ 276,
+ 651,
+ 339
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "and for $y^n = 1$ in task $\\mathcal{T}_2$ , we have that when $\\lambda \\geq 0$ ,",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) \\cdot (\\lambda + \\alpha) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {46} \\\\ - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "Therefore, when $\\lambda \\geq 1 - \\alpha +\\beta$ , we have that for task $\\mathcal{T}_1$",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq 1 - | \\lambda | \\beta - \\Theta (\\epsilon), \\tag {47}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "and for task $\\mathcal{T}_2$",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq (1 - \\Theta (\\epsilon)) (\\lambda + \\alpha) - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\cdot \\mathbf {p o l y} (\\eta \\delta_ {*}) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {48} \\\\ \\geq (1 - \\Theta (\\epsilon)) (\\lambda + \\alpha) - \\beta \\\\ \\geq 1 - \\Theta (\\epsilon). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "We can obtain corresponding conclusions for $y^n = -1$ . Hence,",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon) + | \\lambda | \\beta , \\tag {49}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 367,
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+ 575
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon). \\tag {50}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 392,
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "Meanwhile, for $y^n = 1$ in task $\\mathcal{T}_1$ , we have that when $\\lambda < 0$ ,",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)\\right) \\cdot (1 + \\lambda \\alpha) \\\\ - (| \\lambda | + 1) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\tag {51} \\\\ \\geq 1 + \\lambda \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})}\\right) - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) \\\\ - \\left(| \\lambda | + 1\\right) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right), \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 200,
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+ 823,
+ 737
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "and for $y^n = 1$ in task $\\mathcal{T}_2$ , we have that when $\\lambda < 0$ ,",
+ "bbox": [
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+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) \\cdot (\\lambda + \\alpha) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\\\ \\geq \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)\\right) \\cdot (\\lambda + \\alpha) \\\\ - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\tag {52} \\\\ \\geq \\lambda + \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) - \\lambda \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) \\\\ - \\Theta (\\sqrt {\\frac {M \\log B}{B}}) - \\Theta (\\frac {1 - \\delta_ {*}}{\\delta_ {*}}) \\cdot \\mathrm {p o l y} (\\eta \\delta_ {*}). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 181,
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+ 823,
+ 926
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "page_number",
+ "text": "20",
+ "bbox": [
+ 488,
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+ 509,
+ 959
+ ],
+ "page_idx": 19
+ },
+ {
+ "type": "text",
+ "text": "Then, for task $\\mathcal{T}_1$ , when $0 > \\lambda \\geq -\\Theta (1 / \\alpha^2)$",
+ "bbox": [
+ 171,
+ 102,
+ 473,
+ 119
+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi) \\\\ = \\min \\left\\{\\Theta \\left(- \\lambda \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)}\\right) + \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) + \\epsilon \\right. \\right. \\\\ + (| \\lambda | + 1) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}), \\Theta (1) \\} \\tag {53} \\\\ \\geq \\min \\left\\{\\Theta (- \\lambda \\alpha + (| \\lambda | + 1) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M})) , \\Theta (1) \\right\\} \\\\ = \\min \\left\\{\\Theta (- \\lambda \\alpha + | \\lambda | \\beta + \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right)), \\Theta (1) \\right\\}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "Hence,",
+ "bbox": [
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+ 270
+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (- \\lambda \\alpha + (1 + | \\lambda |) \\beta), \\Theta (1) \\right\\}. \\tag {54}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "When $\\lambda < -\\Theta (1 / \\alpha^2)$",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\mathcal {T} _ {1}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 426,
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n= \\Theta \\left(1 - \\frac {1}{M} \\cdot \\frac {1}{M} \\cdot M\\right) \\tag {55}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\geq \\Theta (1).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "For task $\\mathcal{T}_2$ , when $0 > \\lambda \\geq \\Theta(1) - \\alpha^2$",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi) \\\\ = \\min \\left\\{\\Theta \\left(1 - \\lambda - \\alpha + \\alpha \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} + \\lambda \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) + \\epsilon \\right. \\right. \\\\ + \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) + \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right), \\Theta (1) \\} \\tag {56} \\\\ \\geq \\min \\{\\Theta (1 + \\eta^ {C} - \\lambda - \\alpha + \\Theta (\\operatorname {p o l y} (\\eta \\delta_ {*}) + \\epsilon \\sqrt {M})), \\Theta (1) \\} \\\\ = \\min \\left\\{\\Theta \\left(1 + \\eta^ {C} - \\lambda - \\alpha + \\beta\\right), \\Theta (1) \\right\\}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "where the second step is by $\\lambda +\\alpha \\geq \\Theta (1) + \\alpha -\\alpha^{2}\\geq \\Theta (1)$ . When $\\lambda < \\Theta (1) - \\alpha^2 < 0$",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {57}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "(3) Consider $\\alpha < 0$ . When $\\lambda \\in (-\\Theta (1 / \\alpha^2),0)$ , we have that for task $\\mathcal{T}_1$",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f (\\boldsymbol {X} ^ {n}, \\Psi) \\\\ \\gtrsim \\big (\\frac {1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})}}{1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}} - \\Theta (\\epsilon) \\big) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {| S _ {1} ^ {n} |}{a P M}) \\\\ \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C} - | \\lambda | \\cdot \\Theta (\\sqrt {\\frac {M \\log B}{B}}) \\\\ \\geq (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\tag {58} \\\\ - \\frac {\\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\left(T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - T ^ {- C}\\right)}{1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}} (1 + \\lambda \\alpha) \\\\ \\geq (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right) \\\\ - \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\lambda \\alpha^ {2} (- \\log \\eta \\delta_ {*}) (1 + \\lambda \\alpha), \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "Hence, if $\\lambda \\leq \\mathrm{poly}(\\eta \\delta_{*})\\alpha$ , we have",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq 1 - | \\lambda | \\beta - \\Theta (\\epsilon). \\tag {59}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon) + | \\lambda | \\beta . \\tag {60}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
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+ 47
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+ "page_idx": 20
+ },
+ {
+ "type": "page_number",
+ "text": "21",
+ "bbox": [
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+ "page_idx": 20
+ },
+ {
+ "type": "text",
+ "text": "If $\\lambda >\\frac{\\beta}{\\alpha - \\beta}$ , we have",
+ "bbox": [
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+ ],
+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (1), \\Theta (- \\lambda \\alpha + (| \\lambda | + 1) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right)) \\right\\}. \\tag {61}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "If $\\lambda \\leq -\\Theta (1 / \\alpha^2)$ , we have",
+ "bbox": [
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+ ],
+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {62}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "For task $\\mathcal{T}_2$ , we have that when $\\lambda \\geq 1 + \\eta^C - \\alpha + \\beta$ ,",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim (1 - \\eta^ {C}) (\\lambda + \\alpha) - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\geq 1, \\tag {63}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon). \\tag {64}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "When $\\lambda \\leq 1 + \\eta^C -\\alpha +\\Theta (\\mathrm{poly}(\\eta \\delta_*) + \\epsilon \\sqrt{M})$",
+ "bbox": [
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+ ],
+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} \\tau_ {2}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (1), 1 + \\eta^ {C} - \\lambda - \\alpha + \\beta \\right\\}. \\tag {65}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "One can easily find that there is no region of $\\lambda$ such that $\\Psi$ performs well on both $\\mathcal{T}_1$ and $\\mathcal{T}_2$ . However, when $-\\Theta (1 / \\alpha^2) < \\lambda < \\mathrm{poly}(\\eta \\delta_*)\\alpha < 1 + \\eta^c -\\alpha +\\beta$ , we can unlearn $\\mathcal{T}_2$ and retain the performance of $\\mathcal{T}_1$ .",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "image",
+ "img_path": "images/ea1cd5581af1d4f55f85c6d4da16411fb09ae3aa0fb76816a4c4ce49bfc3ef7f.jpg",
+ "image_caption": [],
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+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "D.2 PROOF OF THEOREM 3",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Proof. By Lemma 1, we know that",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} \\\\ = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} ^ {\\top} \\left(\\sum_ {j = 1} \\lambda_ {j} \\boldsymbol {W} _ {j} ^ {(T)}\\right) \\sum_ {k \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {k} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {k}} \\tag {66} \\\\ \\gtrsim \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} ^ {\\top} \\cdot \\lambda_ {i} \\boldsymbol {W} _ {i} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "For positive neurons, we also have",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\boldsymbol {V} _ {\\mathcal {T} _ {i}} ^ {(T)} \\sum_ {i \\in \\mathcal {V} ^ {\\prime}} \\gamma_ {i} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\boldsymbol {V} _ {\\mathcal {T} _ {i}} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} \\tag {67}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Then, we need",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\geq 1 + c, \\tag {68}\n$$\n",
+ "text_format": "latex",
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} ^ {2} \\geq 1 + c, \\tag {69}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left| \\lambda_ {i} \\right| \\left(\\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + \\epsilon \\sqrt {M}\\right)\\right) = \\left| \\lambda_ {i} \\right| \\beta \\leq c, \\text {f o r s o m e} c > 0 \\text {a n d a l l} i \\in \\mathcal {V} _ {\\Psi}, \\tag {70}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "to hold simultaneously.",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Then, when $\\gamma_{i} = k$ does not hold for all $i\\in \\mathcal{V}_{\\Psi}$ and for some fixed $k < 0$ , we can find $\\lambda_{i}$ in the middle of the normalized $\\gamma_{i}$ and $\\gamma_{i}^{2}$ to satisfy (68) and (69), i.e.,",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\lambda_ {i} \\propto \\frac {\\gamma_ {i}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\frac {\\gamma_ {i} ^ {2}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}. \\tag {71}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "By Cauchy-Schwarz inequality, we have",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n- \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} < \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3} < \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}. \\tag {72}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 21
+ },
+ {
+ "type": "page_number",
+ "text": "22",
+ "bbox": [
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+ "page_idx": 21
+ },
+ {
+ "type": "text",
+ "text": "Hence,",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\propto \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} + \\frac {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}} = \\frac {\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}} > 0, (7 3)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} ^ {2} \\propto \\frac {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} = \\frac {\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} > 0. \\tag {74}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Therefore, by letting",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\lambda_ {i} = C _ {\\gamma} \\cdot \\left(\\frac {\\gamma_ {i}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\frac {\\gamma_ {i} ^ {2}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}\\right), \\tag {75}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nC _ {\\gamma} = \\frac {(1 + c) \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}, \\tag {76}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "we can obtain (68) and (69) hold if $C_{\\gamma} \\lesssim \\beta^{-1}$ .",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "When $\\gamma_{i} = k$ hold for all $i\\in \\mathcal{V}_{\\Psi}$ and for some fixed $k < 0$ with $|\\mathcal{V}_{\\Psi}| > 0$ , we cannot find $\\lambda_{i}$ such that both (68) and (69) hold.",
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+ "page_idx": 22
+ },
+ {
+ "type": "image",
+ "img_path": "images/9e9206517ede8bd3c53e325cea1bc145788bb698c263f6d401cc17020e802cf8.jpg",
+ "image_caption": [],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "D.3 PROOF OF COROLLARY 1",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Proof. Let $\\{\\pmb{\\mu}_1, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\} \\cup \\{\\pmb{u}_1, \\pmb{u}_2, \\dots, \\pmb{u}_{d - M + 1}\\}$ form a set of orthonormal vectors, which is denoted by",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {U} = \\left(\\boldsymbol {\\mu} _ {1}, \\boldsymbol {v} _ {1}, \\boldsymbol {v} _ {2}, \\dots , \\boldsymbol {v} _ {M}, \\boldsymbol {u} _ {1}, \\boldsymbol {u} _ {2}, \\dots , \\boldsymbol {u} _ {d - M + 1}\\right). \\tag {77}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Note that for any $\\pmb{a},\\pmb{b}\\in \\{\\pmb{\\mu}_1,\\pmb{v}_1,\\pmb{v}_2,\\dots ,\\pmb{v}_M\\} \\cup \\{\\pmb{u}_1,\\pmb{u}_2,\\dots ,\\pmb{u}_{d - M + 1}\\}$",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {a} ^ {\\top} \\boldsymbol {W} ^ {(0)} \\boldsymbol {b} = \\sum_ {1 \\leq i, j \\leq d} a _ {i} b _ {j} W _ {i, j} ^ {(0)} \\sim \\mathcal {N} (0, \\sum_ {1 \\leq i, j \\leq d} | a _ {i} b _ {j} | \\xi^ {2}), \\tag {78}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "where the last step comes from that each entry of $\\mathbf{W}^{(0)} \\sim \\mathcal{N}(0, \\xi^2)$ . Given that $\\| \\mathbf{a} \\| = \\| \\mathbf{b} \\| = 1$ , we have",
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {1 \\leq i, j \\leq d} | a _ {i} b _ {j} | = \\left(| a _ {1} |, \\dots , | a _ {d} |\\right) ^ {\\top} \\left(| b _ {1} |, \\dots , | b _ {d} |\\right) \\leq 1. \\tag {79}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "By (90), we know that for $\\pmb{a} \\in \\{\\pmb{u}_1, \\pmb{u}_2, \\dots, \\pmb{u}_{d - M + 1}\\}$ and any $t = 0, 1, \\dots, T - 1$ ,",
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+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {a} = 0, \\tag {80}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 388,
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {a} ^ {\\top} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} = 0. \\tag {81}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 383,
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Then, we have that for some $C > 1$",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left[ \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} \\right] _ {i, j} = \\left\\{ \\begin{array}{l l} \\Theta (\\log T), & i = j = 1, \\\\ O \\left(\\epsilon \\cdot \\frac {1}{e ^ {\\Theta (\\log T)} \\cdot \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)}\\right) = O \\left(\\epsilon \\cdot T ^ {- C}\\right), & j = 1, 1 \\leq i \\leq M - 1, \\\\ O \\left(\\epsilon \\cdot \\log T\\right), & j \\in [ 2, M - 1 ], i \\in [ 1, M - 1 ], \\\\ O (\\xi), & \\text {e l s e .} \\end{array} \\right. \\tag {82}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
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+ 47
+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "page_number",
+ "text": "23",
+ "bbox": [
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+ ],
+ "page_idx": 22
+ },
+ {
+ "type": "text",
+ "text": "Let $E_{i,j}$ be the matrix that only the $(i,j)$ entry equals 1, while all other entries are 0. Therefore,",
+ "bbox": [
+ 169,
+ 103,
+ 803,
+ 119
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\left\\| \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\right\\| _ {F} ^ {2} \\\\ \\leq (\\epsilon \\cdot T ^ {- C}) ^ {2} \\cdot (M - 1) + (\\epsilon \\cdot \\log T) ^ {2} \\cdot (M - 1) (M - 2) + \\xi^ {2} (d ^ {2} - M ^ {2}) \\\\ \\leq \\epsilon^ {2} \\log^ {2} T \\cdot M ^ {2} + d ^ {2} / m \\tag {83} \\\\ \\lesssim \\epsilon^ {2} \\cdot M ^ {2} + \\frac {1}{\\log M}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 254,
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+ 227
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "where the last step comes from that $m \\gtrsim M^2 \\log M$ and $M = \\Theta(d)$ . Then,",
+ "bbox": [
+ 171,
+ 243,
+ 666,
+ 258
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\left\\| \\boldsymbol {W} ^ {(T)} - \\boldsymbol {U} \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\cdot \\boldsymbol {U} ^ {\\top} \\right\\| _ {F} \\\\ \\leq \\left\\| \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {U} \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\right\\| _ {F} \\cdot \\left\\| \\boldsymbol {U} ^ {\\top} \\right\\| \\tag {84} \\\\ \\leq \\| \\boldsymbol {U} \\| \\cdot \\| \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\| _ {F} \\\\ \\leq \\epsilon M + 1 / \\log M. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 352,
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Likewise, by (132), we know that neurons of $\\mathbf{V}^{(T)}$ with a non-trivial magnitude are in the direction of the iterative summation of $\\left(\\sum_{s=1}^{P} \\boldsymbol{x}_s^n \\operatorname{softmax}_l(\\boldsymbol{x}_s^{n\\top} \\boldsymbol{W}\\boldsymbol{x}_l^n)\\right)$ . Hence, there exists $\\hat{\\boldsymbol{v}}_1 \\in \\mathbb{R}^m$ and $\\hat{\\boldsymbol{v}}_2 \\in \\mathbb{R}^d$ such that",
+ "bbox": [
+ 169,
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+ 823,
+ 424
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {V} ^ {(T)} - \\hat {\\boldsymbol {v}} _ {1} \\hat {\\boldsymbol {v}} _ {2} ^ {\\top} \\right\\| _ {F} \\leq \\Theta (1) \\cdot \\sqrt {m} \\cdot \\sqrt {\\frac {\\log B}{B}} \\cdot \\delta_ {*} ^ {- 2} \\cdot \\delta_ {*} \\cdot \\frac {1}{\\sqrt {m}} \\leq \\delta_ {*} ^ {- 1} \\epsilon \\tag {85}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 474
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Then, for $n$ such that $y^{n} = +1$ , we have that the low-rank trained model, where $\\boldsymbol{W}_{LR}^{(T)} = \\boldsymbol{U}\\boldsymbol{E}_{1,1} \\cdot \\Theta (\\log T) \\cdot \\boldsymbol{U}^{\\top}$ , satisfies",
+ "bbox": [
+ 169,
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+ 525
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi_ {L R}\\right) \\geq 1 \\cdot \\left(1 - \\delta_ {*} \\epsilon\\right) \\cdot \\left(1 - \\Theta \\left(\\epsilon \\log T\\right)\\right) = 1 - \\Theta \\left(\\left(\\log T + \\delta_ {*}\\right) \\epsilon\\right), \\tag {86}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 556
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "which leads to",
+ "bbox": [
+ 171,
+ 571,
+ 272,
+ 585
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\ell \\left(\\boldsymbol {X} ^ {n}, y ^ {n}; \\Psi_ {L R}\\right) \\leq \\Theta \\left(\\epsilon_ {L R}\\right), \\text {w h e r e} \\epsilon_ {L R} = (\\log T + \\delta_ {*}) \\epsilon . \\tag {87}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 303,
+ 597,
+ 823,
+ 614
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "D.4 PROOF OF COROLLARY 2",
+ "text_level": 1,
+ "bbox": [
+ 171,
+ 671,
+ 393,
+ 684
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Proof. We know that from Lemma 1, there is a number of $\\Omega(m)$ lucky neurons with large weights. We can denote the set of lucky neurons as $\\mathcal{L} \\subset [m]$ . By combining (148) and (163), we have that for any lucky neuron $u_i$ ,",
+ "bbox": [
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+ 744
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {u} _ {i} \\right\\| \\geq \\eta \\eta^ {- 1} \\delta_ {*} ^ {- 1} \\cdot \\delta_ {*} \\cdot \\frac {1}{\\sqrt {m}} = m ^ {- 1 / 2}. \\tag {88}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 367,
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "For any unlucky neurons, by (149), we have",
+ "bbox": [
+ 171,
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+ 815
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {u} _ {i} \\right\\| \\leq m ^ {- 1 / 2} \\sqrt {\\frac {\\log B}{B}}. \\tag {89}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 415,
+ 832,
+ 823,
+ 864
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "Since that $B \\geq \\epsilon^{-2} \\log M$ by Lemma 1, we have that if we remove neurons from $m \\backslash \\mathcal{L}$ , the output in (158) and (159) will only be affected by a factor of $\\epsilon$ . Therefore, Lemma 1 still holds, so that Theorems 1-3 all hold.",
+ "bbox": [
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+ 825,
+ 922
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "page_number",
+ "text": "24",
+ "bbox": [
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+ ],
+ "page_idx": 23
+ },
+ {
+ "type": "text",
+ "text": "E PROOF OF KEY LEMMAS",
+ "text_level": 1,
+ "bbox": [
+ 171,
+ 102,
+ 413,
+ 118
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "E.1 PROOF OF LEMMA 3",
+ "text_level": 1,
+ "bbox": [
+ 171,
+ 133,
+ 359,
+ 148
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "For ease of presentation, we sometimes use $\\mu_{2}$ to represent $-\\mu_{1}$ in the proof. We first investigate the gradient of $W$ , i.e.,",
+ "bbox": [
+ 169,
+ 160,
+ 823,
+ 189
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi)}{\\partial \\boldsymbol {W}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi)}{\\partial f (\\boldsymbol {X} ^ {n} ; \\Psi)} \\frac {f (\\boldsymbol {X} ^ {n} ; \\Psi)}{\\partial \\boldsymbol {W}} \\\\ = \\eta \\frac {1}{B} \\sum_ {\\substack {n \\in \\mathcal {B} _ {b} \\\\ P}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] \\tag{90} \\\\ \\cdot \\left(\\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\left(\\boldsymbol {x} _ {s} ^ {n} - \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top}\\right) \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \\mathbb {1} \\left[ V _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] \\\\ \\cdot \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top}\\right) \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 210,
+ 196,
+ 823,
+ 455
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "For $j,l\\in S_1^n$ , we have",
+ "bbox": [
+ 171,
+ 462,
+ 323,
+ 478
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n ^ {\\top}} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\gtrsim \\frac {e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|}}{\\left| \\mathcal {S} _ {1} ^ {n} \\right| e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + \\left(P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|\\right)} \\tag {91}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 316,
+ 484,
+ 825,
+ 521
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "For $j \\notin S_1^n$ and $l \\in S_1^n$ , we have",
+ "bbox": [
+ 171,
+ 527,
+ 390,
+ 544
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\frac {1}{\\left| \\mathcal {S} _ {1} ^ {n} \\right| e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + \\left(P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|\\right)}, \\tag {92}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 315,
+ 551,
+ 823,
+ 584
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "where $\\| \\pmb{q}_1(0)\\| = 0$ . For $l\\notin S_1^n\\cup S_2^n$ , $j\\in [P]$ , we have",
+ "bbox": [
+ 171,
+ 590,
+ 542,
+ 607
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(0)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\frac {1}{P}. \\tag {93}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 393,
+ 614,
+ 825,
+ 643
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "Therefore, for $s,r,l\\in S_1^n$ , let",
+ "bbox": [
+ 171,
+ 648,
+ 372,
+ 665
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n} := \\beta_ {1} ^ {n} (t) \\boldsymbol {\\mu} _ {1} + \\beta_ {2} ^ {n} (t), \\tag {94}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 300,
+ 672,
+ 825,
+ 714
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 720,
+ 217,
+ 734
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\beta_ {1} ^ {n} (t) \\gtrsim \\frac {P - | \\mathcal {S} _ {1} ^ {n} |}{| \\mathcal {S} _ {1} ^ {n} | e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + P - | \\mathcal {S} _ {1} ^ {n} |} := \\phi_ {n} (t) (P - | \\mathcal {S} _ {1} ^ {n} |). \\tag {95}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 303,
+ 732,
+ 825,
+ 767
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\beta_ {2} ^ {n} (t) = \\sum_ {l = 2} ^ {M _ {1}} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\tag {96}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 437,
+ 772,
+ 825,
+ 813
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 816,
+ 217,
+ 830
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left| \\iota_ {l} ^ {\\prime} \\right| \\leq \\beta_ {1} ^ {n} (t) \\frac {\\left| \\mathcal {S} _ {l} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {97}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 421,
+ 828,
+ 825,
+ 863
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "Note that $|l_{l}^{\\prime}| = 0$ if $P = |\\mathcal{S}_1^n|, l \\geq 2$ .",
+ "bbox": [
+ 171,
+ 864,
+ 419,
+ 881
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "If $s \\in S_1^n$ , we have",
+ "bbox": [
+ 171,
+ 881,
+ 300,
+ 896
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq \\zeta_ {i, 1, t} \\cdot \\frac {p _ {n} (t)}{\\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {98}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 343,
+ 895,
+ 825,
+ 929
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "page_number",
+ "text": "25",
+ "bbox": [
+ 488,
+ 946,
+ 506,
+ 959
+ ],
+ "page_idx": 24
+ },
+ {
+ "type": "text",
+ "text": "If $s \\in S_2^n$ and $j \\in S_1^n$ , we have",
+ "bbox": [
+ 169,
+ 103,
+ 382,
+ 119
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {j} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\phi_ {n} (t) \\cdot \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{p _ {n} (t)}. \\tag {99}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 205,
+ 121,
+ 825,
+ 154
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "If $s \\notin (S_1^n \\cup S_2^n)$ and $j \\in S_1^n$ ,",
+ "bbox": [
+ 169,
+ 155,
+ 374,
+ 172
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {j} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n \\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\phi_ {n} (t) \\cdot \\frac {\\left| S _ {1} ^ {n} \\right|}{\\sqrt {B} p _ {n} (t)}. \\tag {100}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 187,
+ 174,
+ 825,
+ 208
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "Then, by combining (94) to (100), we have that for $l \\in S_1^n$ , $i \\in \\mathcal{W}_{n,l}$ ,",
+ "bbox": [
+ 169,
+ 209,
+ 629,
+ 226
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {101}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 210,
+ 228,
+ 825,
+ 268
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\gtrsim \\zeta_ {i, 1, t} \\cdot p _ {n} (t) \\phi_ {n} (t) (P - | S _ {1} ^ {n} |).\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 199,
+ 271,
+ 411,
+ 287
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "For $l \\in S_1^n$ , $i \\in \\mathcal{W}_{n,l}$ , we have that for $k \\neq 1,2$",
+ "bbox": [
+ 169,
+ 290,
+ 491,
+ 306
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {102} \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 187,
+ 309,
+ 823,
+ 393
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "For $l \\in S_1^n$ , $i \\in \\mathcal{W}_{n,l}$ , we have that for $k \\in [M]$",
+ "bbox": [
+ 169,
+ 395,
+ 491,
+ 412
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {103} \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|} \\cdot \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| \\phi_ {n} (t)}{p _ {n} (t)}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 199,
+ 415,
+ 825,
+ 535
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "For $i\\in \\mathcal{U}_{n,l}$ , by the definition of $\\mathcal{U}_{n,l}$ in Definition 4, we have",
+ "bbox": [
+ 169,
+ 537,
+ 581,
+ 551
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 0. \\tag {104}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 351,
+ 555,
+ 825,
+ 574
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "For $i \\notin \\mathcal{W}_{n,l} \\cup \\mathcal{U}_{n,l}$ , we have that for $j \\in \\mathcal{W}_{n,l}, k \\in [M]$",
+ "bbox": [
+ 169,
+ 575,
+ 553,
+ 592
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {105} \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 199,
+ 595,
+ 825,
+ 717
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1} (106) \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}. \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (107) \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)} \\cdot \\frac {| \\mathcal {R} _ {k} ^ {n} |}{P - | \\mathcal {S} _ {1} ^ {n} |}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 187,
+ 719,
+ 825,
+ 929
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "page_number",
+ "text": "26",
+ "bbox": [
+ 488,
+ 946,
+ 509,
+ 960
+ ],
+ "page_idx": 25
+ },
+ {
+ "type": "text",
+ "text": "When $l \\notin S_1^n$ , we have that $\\pmb{x}_l^{n^\\top} \\pmb{\\mu}_1 = 0$ . If $l \\in S_2^n$ , we can obtain that",
+ "bbox": [
+ 171,
+ 102,
+ 635,
+ 119
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\tag {108} \\\\ \\gtrsim \\zeta_ {i, 1, t} \\cdot \\frac {p _ {n} (t) | \\mathcal {S} _ {2} ^ {n} |}{| \\mathcal {S} _ {1} ^ {n} |} \\phi_ {n} (t) (P - | \\mathcal {S} _ {1} ^ {n} |), \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 200,
+ 125,
+ 823,
+ 199
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} (109) \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2}, \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} (110) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {2} ^ {n} \\right|} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| \\phi_ {n} (t)}{p _ {n} (t)}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 189,
+ 204,
+ 823,
+ 409
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "where $k\\in [M],i\\in \\mathcal{U}_{n,l}$ . If $i\\in \\mathcal{W}_{n,l}$",
+ "bbox": [
+ 171,
+ 410,
+ 423,
+ 426
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 0. \\tag {111}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 352,
+ 430,
+ 823,
+ 448
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "If $i \\notin \\mathcal{W}_{n,l} \\cup \\mathcal{U}_{n,l}$ , we have that for $j \\in \\mathcal{U}_{n,l}$ , $k \\in [M]$",
+ "bbox": [
+ 171,
+ 450,
+ 535,
+ 468
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} \\tag {112} \\\\ \\cdot \\phi_ {n} (t) \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{\\sqrt {B} p _ {n} (t)}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 200,
+ 472,
+ 823,
+ 592
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} (113) \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}. \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} (114) \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)} \\cdot \\frac {| \\mathcal {R} _ {k} ^ {n} |}{P - | \\mathcal {S} _ {1} ^ {n} |}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 189,
+ 597,
+ 823,
+ 804
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "text",
+ "text": "If $l \\in \\mathcal{R}_k^n$ , $k \\in [M]$ , we have that for $j \\in \\mathcal{W}_{n,l}$ , if $V_{(j,\\cdot)} \\sum_{s=1}^{P} \\pmb{x}_s^n \\mathrm{softmax}_l(\\pmb{x}_s^{n\\top} \\pmb{W} \\pmb{x}_l^n) > 0$ , $l' \\in S_1^n$ ,",
+ "bbox": [
+ 169,
+ 806,
+ 823,
+ 839
+ ],
+ "page_idx": 26
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k} \\tag {115} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
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+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
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+ "page_idx": 26
+ },
+ {
+ "type": "page_number",
+ "text": "27",
+ "bbox": [
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k} (116) \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k}, \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (117) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
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+ "type": "text",
+ "text": "Likewise, if $l \\in \\mathcal{R}_k^n$ , $k \\in [M]$ , $\\pmb{V}_{(j,\\cdot)}\\sum_{s=1}^{P}\\pmb{x}_s^n\\mathrm{softmax}_l(\\pmb{x}_s^{n^\\top}\\pmb{W}\\pmb{x}_l^n) > 0$ , $j \\in \\mathcal{U}_{n,l}$ , $l' \\in S_1^n$ , $l'' \\in S_2^n$ ,",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}, \\tag {118} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}, (118) \\\\ \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {2}, (119) \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (120) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "Therefore, by the update rule, we know",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} - \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {\\mu} _ {1} \\tag {121} \\\\ = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} + K (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 2} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nK (t) \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\zeta_ {1, t} p _ {n} (t) \\phi_ {n} (t) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|), \\tag {122}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\iota_ {l} ^ {\\prime} \\leq K (t) \\cdot \\max _ {n} \\left\\{\\frac {| S _ {1} ^ {n} | \\phi_ {n} (t)}{p _ {n} (t)} \\right\\} \\leq K (t) \\cdot e ^ {- q _ {1} (t)}. \\tag {123}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "We know that",
+ "bbox": [
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+ ],
+ "page_idx": 27
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {W} ^ {(0)} \\boldsymbol {\\mu} _ {1} \\approx 0. \\tag {124}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 27
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 27
+ },
+ {
+ "type": "page_number",
+ "text": "28",
+ "bbox": [
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+ "page_idx": 27
+ },
+ {
+ "type": "text",
+ "text": "Then,",
+ "bbox": [
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+ 118
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} q _ {1} (t + 1) = \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} \\\\ = \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} + K (t) \\\\ = q _ {1} (t) + K (t) \\tag {125} \\\\ = \\sum_ {b = 0} ^ {t} K (b). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "Similarly,",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {2} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {2} - \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {\\mu} _ {2} \\tag {126}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n= \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {2} + K (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l \\neq 2} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}.\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 619,
+ 311
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {2} = \\sum_ {b = 0} ^ {t} K (b). \\tag {127}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "For $k\\in [M]$",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {v} _ {k} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} + J _ {1} (t) \\boldsymbol {\\mu} _ {1} + J _ {2} (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l = 1} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {v} _ {l}. \\tag {128}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "By Hoeffding's inequality (15), with high probability,",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {v} _ {k} \\right\\| \\leq \\Theta (1) \\cdot \\sqrt {\\frac {\\log B}{B}} \\sum_ {b = 0} ^ {t} K (b) \\lesssim \\epsilon \\cdot \\sum_ {b = 0} ^ {t} K (b), \\tag {129}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "where the second step holds if $B \\geq \\epsilon^{-2} \\log M$ . And for $j \\neq k$ , $j \\in [M]$",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left\\| \\boldsymbol {v} _ {j} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\leq K (t) e ^ {- q _ {1} (t)}. \\tag {130}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "For any $\\pmb{\\mu}'$ such that $\\pmb{\\mu}_1^\\top \\pmb{\\mu}' = \\alpha$ and $\\pmb{\\mu}' \\perp \\{v_1, v_2, \\dots, v_M\\}$ , we can write $\\pmb{\\mu}'$ as $\\alpha \\pmb{\\mu}_1 \\pm \\sqrt{1 - \\alpha^2} \\pmb{\\mu}_\\perp$ for some $\\pmb{\\mu}_\\perp \\perp \\{\\pmb{\\mu}_1, v_1, v_2, \\dots, v_M\\}$ . Therefore,",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\boldsymbol {\\mu} ^ {\\prime} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} ^ {\\prime} = \\left(\\alpha \\boldsymbol {\\mu} _ {1} \\pm \\sqrt {1 - \\alpha^ {2}} \\boldsymbol {\\mu} _ {\\perp}\\right) ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\left(\\alpha \\boldsymbol {\\mu} _ {1} \\pm \\sqrt {1 - \\alpha^ {2}} \\boldsymbol {\\mu} _ {\\perp}\\right) \\tag {131} \\\\ = \\alpha^ {2} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} \\pm \\Theta (\\epsilon) \\cdot \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "E.2 PROOF OF LEMMA 4",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "For ease of presentation, we sometimes use $\\pmb{\\mu}_{2}$ to represent $-\\pmb{\\mu}_{1}$ in the proof.",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)} \\frac {f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\tag {132} \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
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+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)} \\frac {f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\tag {132} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} a _ {(l) _ {i}} \\mathbb {1} [ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\geq 0 ] \\\\ \\cdot \\left(\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "For $n$ such that $y^{n} = +1$ and $i\\in \\mathcal{W}_{n,l}$ , we have that",
+ "bbox": [
+ 171,
+ 882,
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+ 897
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 1, \\tag {133}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
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+ 47
+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "page_number",
+ "text": "29",
+ "bbox": [
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+ 508,
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+ ],
+ "page_idx": 28
+ },
+ {
+ "type": "text",
+ "text": "and for $l\\in S_1^n$",
+ "bbox": [
+ 171,
+ 103,
+ 277,
+ 119
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 1} ^ {M _ {2}} \\iota_ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + \\iota_ {M _ {2} + 1} ^ {\\prime} \\boldsymbol {\\mu} _ {2}, \\tag {134}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 285,
+ 127,
+ 825,
+ 169
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 176,
+ 217,
+ 188
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\iota_ {l} ^ {\\prime} \\leq (1 - p _ {n} (t)) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {135}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 400,
+ 186,
+ 825,
+ 220
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "If $l\\in \\mathcal{S}_2^n$ , we have",
+ "bbox": [
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+ 300,
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l = 1} ^ {M _ {2}} \\kappa_ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + \\kappa_ {M _ {2} + 1} ^ {\\prime} \\boldsymbol {\\mu} _ {2}, \\tag {136}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 281,
+ 246,
+ 825,
+ 287
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 296,
+ 217,
+ 308
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\np _ {n} ^ {\\prime} (t) \\leq p _ {n} (t), \\tag {137}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 444,
+ 308,
+ 825,
+ 325
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\kappa_ {l} ^ {\\prime} \\leq (1 - p _ {n} (t)) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {2} ^ {n} \\right|}. \\tag {138}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 398,
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+ 364
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "If $l\\in \\mathcal{R}_k^n$ $k\\in [M]$ , we have",
+ "bbox": [
+ 171,
+ 366,
+ 366,
+ 383
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {1} + p _ {n} ^ {\\prime \\prime} (t) \\boldsymbol {\\mu} _ {2} + o _ {n} (t) \\boldsymbol {v} _ {k} + \\sum_ {l \\neq k} u _ {l} ^ {\\prime} \\boldsymbol {v} _ {l}, \\tag {139}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 250,
+ 391,
+ 825,
+ 433
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 441,
+ 217,
+ 453
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\np _ {n} ^ {\\prime} (t) \\leq \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot p _ {n} (t), \\tag {140}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 423,
+ 450,
+ 825,
+ 481
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\np _ {n} ^ {\\prime \\prime} (t) \\leq \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{P} \\cdot p _ {n} (t), \\tag {141}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 424,
+ 484,
+ 825,
+ 515
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\no _ {n} (t) \\leq \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P} \\cdot p _ {n} (t) \\tag {142}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 424,
+ 518,
+ 825,
+ 549
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nu _ {l} ^ {\\prime} \\leq \\left(1 - \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| + \\left| \\mathcal {S} _ {2} ^ {n} \\right| + \\left| \\mathcal {R} _ {k} ^ {n} \\right|}{\\left| \\mathcal {S} _ {1} ^ {n} \\right|} \\cdot p _ {n} (t)\\right) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right| - \\left| \\mathcal {S} _ {2} ^ {n} \\right| - \\left| \\mathcal {R} _ {k} ^ {n} \\right|}. \\tag {143}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Therefore, we have",
+ "bbox": [
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+ 303,
+ 603
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n- \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V}} = \\sum_ {l = 1} ^ {M} u _ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + q _ {n} (t) \\boldsymbol {\\mu} _ {1} + q _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {2}, \\tag {144}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 284,
+ 611,
+ 825,
+ 654
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "where",
+ "bbox": [
+ 171,
+ 661,
+ 217,
+ 674
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nq _ {n} (t) ^ {\\prime} \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (t), \\tag {145}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 390,
+ 672,
+ 825,
+ 709
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left| q _ {n} ^ {\\prime} (t) \\right| \\lesssim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (t), \\tag {146}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 388,
+ 714,
+ 825,
+ 752
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\left| u _ {k} ^ {\\prime} \\right| \\lesssim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{a P} \\cdot (1 - p _ {n} (t)) \\frac {1}{M}. \\tag {147}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 364,
+ 756,
+ 825,
+ 795
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "Then,",
+ "bbox": [
+ 171,
+ 797,
+ 215,
+ 811
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\geq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| S _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {148}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 369,
+ 811,
+ 825,
+ 853
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2} = - \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {149}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 423,
+ 858,
+ 825,
+ 881
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {150}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 390,
+ 886,
+ 825,
+ 926
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "page_number",
+ "text": "30",
+ "bbox": [
+ 488,
+ 946,
+ 509,
+ 959
+ ],
+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "for $k\\in [M]$ . For $i\\in \\mathcal{U}_{n,l}$ , we similarly have",
+ "bbox": [
+ 169,
+ 103,
+ 468,
+ 119
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2} \\geq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| S _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {151}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 369,
+ 123,
+ 825,
+ 164
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} = - \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2}, \\tag {152}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 424,
+ 167,
+ 825,
+ 191
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {153}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 390,
+ 193,
+ 823,
+ 234
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "for some $k\\in [M]$ . For $i\\notin \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}$ , we have that",
+ "bbox": [
+ 169,
+ 236,
+ 524,
+ 252
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k}, \\tag {154}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 401,
+ 255,
+ 825,
+ 287
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\leq \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {155}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 401,
+ 292,
+ 825,
+ 325
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "where $k\\in [M],j\\in \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}$",
+ "bbox": [
+ 169,
+ 325,
+ 393,
+ 340
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "E.3 PROOF OF LEMMA 1",
+ "text_level": 1,
+ "bbox": [
+ 171,
+ 356,
+ 357,
+ 369
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We know that by Lemma 3 and 4 in (Li et al., 2023a), for $i \\in \\mathcal{W}_{n,l}(0)$ and $l \\in S_1^n$ , we have that",
+ "bbox": [
+ 169,
+ 382,
+ 800,
+ 398
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {R} _ {l} ^ {n} (t) \\right] = 1, \\tag {156}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 431,
+ 402,
+ 823,
+ 424
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "and for $i\\in \\mathcal{U}_{n,l}(0)$ and $l\\in S_2^n$ , we have that",
+ "bbox": [
+ 169,
+ 426,
+ 468,
+ 443
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {R} _ {l} ^ {n} (t) \\right] = 1. \\tag {157}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 431,
+ 446,
+ 823,
+ 468
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We also have that the size of $\\mathcal{W}_{n,l}$ and $\\mathcal{V}_{n,l}$ are larger than $\\Omega(m)$ . Therefore, for $y^n = +1$ , by Lemma 4 and 3, we have",
+ "bbox": [
+ 169,
+ 470,
+ 823,
+ 500
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}; \\Psi\\right) = \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in \\mathcal {W} _ {l, n} (0)} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ + \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\notin \\mathcal {W} _ {l, n} (0), a _ {(l) _ {i}} > 0} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\tag {158} \\\\ - \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i: a _ {(l) _ {i}} < 0} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 220,
+ 503,
+ 825,
+ 643
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We know that",
+ "bbox": [
+ 171,
+ 646,
+ 266,
+ 659
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in \\mathcal {W} _ {l, n} (0)} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ \\gtrsim \\frac {\\left| S _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a} \\cdot \\zeta_ {T} \\cdot p _ {n} (T) \\tag {159} \\\\ \\gtrsim \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a ^ {2}} \\cdot \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{P} p _ {h} (b) \\cdot p _ {n} (T). \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 297,
+ 657,
+ 825,
+ 779
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We can derive that",
+ "bbox": [
+ 171,
+ 780,
+ 297,
+ 792
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\begin{array}{l} q _ {1} (T) = \\sum_ {b = 0} ^ {T - 1} K (b) \\\\ \\geq \\sum_ {b = 0} ^ {T - 1} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} p _ {n} (b) \\phi_ {n} (b) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|) \\eta \\sum_ {c = 0} ^ {b - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {c}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{a P} p _ {h} (c) \\tag {160} \\\\ \\gtrsim \\delta_ {*} ^ {4} \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{e ^ {q _ {1} (b)}}. \\\\ \\end{array}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 222,
+ 796,
+ 825,
+ 926
+ ],
+ "page_idx": 30
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "page_number",
+ "text": "31",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Therefore, we have that when $q_{1}(T) \\leq O(1)$ or $q_{1}(T) \\geq \\Theta(T^{c})$ for $c = \\Theta(1)$ , (160) does not hold. When $q_{1}(T) = \\Theta(\\log T)$ , we have that (160) holds. In this case,",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\np _ {n} (T) \\geq \\frac {\\delta_ {*} T ^ {C}}{\\delta_ {*} T ^ {C} + 1 - \\delta_ {*}} \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}, \\tag {161}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "where $C > 1$ . Meanwhile, for $l \\in \\mathcal{R}_k^n$ , $k \\in [M]$ , and any $s \\in [P]$",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) = \\Theta \\left(\\frac {1}{P}\\right). \\tag {162}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "We can then derive that as long as",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nT \\gtrsim \\eta^ {- 1} \\delta_ {*} ^ {- 2}, \\tag {163}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "we have",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a ^ {2}} \\cdot \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{P} p _ {h} (b) \\cdot p _ {n} (T) \\geq 1. \\tag {164}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "Then,",
+ "bbox": [
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+ 340
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf \\left(\\boldsymbol {X} ^ {n}; \\Psi\\right) \\geq 1, \\ell \\left(\\boldsymbol {X} ^ {n}, y ^ {n}; \\Psi\\right) = 0. \\tag {165}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "With (163), we can also derive that",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\sum_ {k = 1} ^ {M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {v} _ {k} \\right\\| ^ {2} \\lesssim \\frac {1}{M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {1} \\right\\| ^ {2}, \\tag {166}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "which means that for $i \\in \\mathcal{W}_{n,l}$ with $l \\in S_1^n$ , $V_{(i,\\cdot)}^{(T)}$ is mainly in the direction of $\\pmb{\\mu}_1$ . This verifies condition (B) of Lemma 1. Therefore, by Hoeffding's inequality (15), for any $W' \\in \\Psi$ ,",
+ "bbox": [
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+ 464
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\Pr \\left( \\right.\\left\\| \\frac {1}{| \\mathcal {B} _ {b} |} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} - \\mathbb {E} \\left[ \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} \\right]\\right\\| \\geq \\left| \\right. \\mathbb {E} \\left[ \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} \\right] \\epsilon\\left. \\right) \\tag {167}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\leq e ^ {- B \\epsilon^ {2}} \\leq M ^ {- C},\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 194,
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+ 316,
+ 532
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "as long as",
+ "bbox": [
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+ 241,
+ 553
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nB \\gtrsim \\epsilon^ {- 2} \\log M. \\tag {168}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
+ 442,
+ 550,
+ 825,
+ 566
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "Then,",
+ "bbox": [
+ 171,
+ 571,
+ 215,
+ 584
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\epsilon . \\tag {169}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "F EXTENSION TO MULTI-CLASSIFICATION",
+ "text_level": 1,
+ "bbox": [
+ 171,
+ 619,
+ 537,
+ 633
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "Define that a $2^{c}$ -classification is achieved by $c$ times of binary classification with the orthonormal set $\\{\\pmb{\\mu}_{\\mathcal{T}}^{(1)}, \\dots, \\pmb{\\mu}_{\\mathcal{T}}^{(c)}\\}$ as the discriminative patterns for the task $\\mathcal{T}$ . We have $\\pmb{\\mu}_{\\mathcal{T}}^{(i)} \\perp \\pmb{v}_m$ , $m \\in [M]$ , $i \\in [c]$ . The label $\\pmb{y}$ is $c$ -dimensional with each entry chosen from $\\{+1, -1\\}$ . Specifically, each $(X \\in \\mathbb{R}^{d \\times P}, y \\in \\mathbb{R}^c) \\sim \\mathcal{D}_{\\mathcal{T}}$ is generated as follows:",
+ "bbox": [
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+ "page_idx": 31
+ },
+ {
+ "type": "list",
+ "sub_type": "text",
+ "list_items": [
+ "- Randomly generate the $k$ -th entry $y_{k}, k \\in [c]$ of the label $\\mathbf{y}$ from $\\{+1, -1\\}$ with an equal probability.",
+ "- Each token is randomly chosen from $\\{\\pmb{\\mu}_{\\mathcal{T}}^{(i)}, - \\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\}_{i = 1}^{c}\\cup \\{\\pmb{v}_1,\\dots ,\\pmb{v}_M\\}$ . If $y_{k} = 1$ (or $-1$ ), the number of tokens corresponding to $\\pmb{\\mu}_{\\mathcal{T}_k}$ (or $-\\pmb{\\mu}_{\\mathcal{T}_k}$ ) is larger than that of $-\\pmb{\\mu}_{\\mathcal{T}_k}$ (or $\\pmb{\\mu}_{\\mathcal{T}_k}$ ). $\\pmb{\\mu}_{\\mathcal{T}}^{(i)}$ and $-\\pmb{\\mu}_{\\mathcal{T}}^{(i)}$ (or “ $-\\pmb{\\mu}_{\\mathcal{T}}^{(i)}$ and $\\pmb{\\mu}_{\\mathcal{T}}^{(i),}$ ” are referred to label-relevant and confusion patterns for $y_{k} = 1$ (or $y_{k} = -1$ ), respectively. The average fractions of label-relevant and confusion tokens of $\\pmb{\\mu}_{\\mathcal{T}}^{(i)}$ are $\\delta_{*}^{(i)}$ and $\\delta_{\\#}^{(i)}$ , respectively."
+ ],
+ "bbox": [
+ 215,
+ 720,
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "We then need $c$ sets of our binary model (4) to generate the output for $2^{c}$ -classification, i.e.,",
+ "bbox": [
+ 171,
+ 845,
+ 771,
+ 861
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf (\\boldsymbol {X}; \\Psi) = \\left(f _ {1} (\\boldsymbol {X}; \\Psi), f _ {2} (\\boldsymbol {X}; \\Psi), \\dots , f _ {c} (\\boldsymbol {X}; \\Psi)\\right)\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ 864,
+ 581,
+ 883
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nf _ {i} (\\boldsymbol {X}; \\Psi) = \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\boldsymbol {a} _ {(l) _ {i}} ^ {\\top} \\operatorname {R e l u} \\left(\\boldsymbol {W} _ {O _ {i}} \\sum_ {s = 1} ^ {P} \\boldsymbol {W} _ {V _ {i}} \\boldsymbol {x} _ {s} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {\\top} \\boldsymbol {W} _ {K _ {i}} ^ {\\top} \\boldsymbol {W} _ {Q _ {i}} \\boldsymbol {x} _ {l}\\right)\\right), \\tag {170}\n$$\n",
+ "text_format": "latex",
+ "bbox": [
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+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
+ 32,
+ 478,
+ 47
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "page_number",
+ "text": "32",
+ "bbox": [
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+ 509,
+ 960
+ ],
+ "page_idx": 31
+ },
+ {
+ "type": "text",
+ "text": "with $\\Psi = \\{\\{a_{(l)i}\\}_{l=1}^{P}, W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}\\}_{i=1}^{c}$ . The dimensions of $W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}$ , $i \\in [c]$ follow Section 3.2.",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "text",
+ "text": "The learning process is then $c$ independent and parallel binary classification problems for each entry of the $c$ -dimensional output. After fine-tuning, the trained model of each output entry has a similar property to Lemma 1 for single binary classification. $\\delta_{*}^{(i)}$ , the fraction of label-relevant pattern $\\mu_{\\mathcal{T}}^{(i)}$ , $i \\in [c]$ , may decrease by $c$ times in average from the binary classification scenario. Therefore, by condition (iii) of Theorem 1, the number of iterations and samples increases by $c^2$ times, which is a polynomial of log scale of the number of classes $2^c$ . Then, for the disriminative patterns $\\{\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\}_{i=1}^c$ of task $\\mathcal{T}_1$ and $\\{\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}\\}_{i=1}^c$ and $\\mathcal{T}_2$ of task $\\mathcal{T}_2$ , if for any $\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}$ , there exists a unique $\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}$ close to be orthogonal to $\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}$ , then $\\mathcal{T}_1$ and $\\mathcal{T}_2$ are irrelevant. If for any $\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}$ , there exists a unique $\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}$ with a small angle to (or almost opposite to) $\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}$ , then $\\mathcal{T}_1$ and $\\mathcal{T}_2$ are aligned (or contradictory). We can then derive similar conclusions as our Theorems 1 and 2 by combining the results of all the output entries.",
+ "bbox": [
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+ ],
+ "page_idx": 32
+ },
+ {
+ "type": "header",
+ "text": "Published as a conference paper at ICLR 2025",
+ "bbox": [
+ 171,
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+ "page_idx": 32
+ },
+ {
+ "type": "page_number",
+ "text": "33",
+ "bbox": [
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+ "page_idx": 32
+ }
+]
\ No newline at end of file
diff --git a/data/2025/2504_10xxx/2504.10957/aee32c72-0906-4851-a50f-6b02b7f21eea_model.json b/data/2025/2504_10xxx/2504.10957/aee32c72-0906-4851-a50f-6b02b7f21eea_model.json
new file mode 100644
index 0000000000000000000000000000000000000000..65554f5ae003634ef675ea70666ba40018ffc469
--- /dev/null
+++ b/data/2025/2504_10xxx/2504.10957/aee32c72-0906-4851-a50f-6b02b7f21eea_model.json
@@ -0,0 +1,7064 @@
+[
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.1,
+ 0.825,
+ 0.175
+ ],
+ "angle": 0,
+ "content": "WHEN IS TASK VECTOR Provably EFFECTIVE FOR MODEL EDITING? A GENERALIZATION ANALYSIS OF NONLINEAR TRANSFORMERS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.18,
+ 0.195,
+ 0.798,
+ 0.24
+ ],
+ "angle": 0,
+ "content": "Hongkang Li\\(^{1}\\), Yihua Zhang\\(^{2}\\), Shuai Zhang\\(^{3}\\), Pin-Yu Chen\\(^{4}\\), Sijia Liu\\(^{2,4}\\), Meng Wang\\(^{1,*}\\) \n\\(^{1}\\)Rensselaer Polytechnic Institute, \\(^{2}\\)Michigan State University, \\(^{3}\\)New Jersey Institute of Technology, \\(^{4}\\)IBM Research"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.451,
+ 0.276,
+ 0.547,
+ 0.291
+ ],
+ "angle": 0,
+ "content": "ABSTRACT"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.23,
+ 0.309,
+ 0.77,
+ 0.575
+ ],
+ "angle": 0,
+ "content": "Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors, each of which is the weight update from the pre-trained model to fine-tuned models for certain tasks. This approach recently gained attention as a computationally efficient inference method for model editing, e.g., multi-task learning, forgetting, and out-of-domain generalization capabilities. However, the theoretical understanding of why task vectors can execute various conceptual operations remains limited, due to the highly non-convexity of training Transformer-based models. To the best of our knowledge, this paper provides the first theoretical characterization of the generalization guarantees of task vector methods on nonlinear Transformers. We consider a conceptual learning setting, where each task is a binary classification problem based on a discriminative pattern. We theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or aligned tasks, as well as the success of task negation in unlearning one task from irrelevant or contradictory tasks. Moreover, we prove the proper selection of linear coefficients for task arithmetic to achieve guaranteed generalization to out-of-domain tasks. All of our theoretical results hold for both dense-weight parameters and their low-rank approximations. Although established in a conceptual setting, our theoretical findings were validated on a practical machine unlearning task using the large language model Phi-1.5 (1.3B)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.173,
+ 0.601,
+ 0.338,
+ 0.617
+ ],
+ "angle": 0,
+ "content": "1 INTRODUCTION"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.627,
+ 0.827,
+ 0.74
+ ],
+ "angle": 0,
+ "content": "Large pre-trained models (Chowdhery et al., 2022; Touvron et al., 2023; Achiam et al., 2023) have recently served as a foundational module in deep learning systems. Under the pre-training-and-fine-tuning paradigm, although the traditional and straightforward full-parameter fine-tuning can demonstrate superior performance in downstream tasks, its immense computational and memory costs have become a serious practical issue. Consequently, many Parameter-Efficient Fine-Tuning (PEFT) methods (Li & Liang, 2021; Hu et al., 2022; Jia et al., 2022; Wei et al., 2022b;a) have been proposed to address this concern. Among them, the recent task vector approach receives increasing attention (Ilharco et al., 2022a; Ortiz-Jimenez et al., 2023; Hendel et al., 2023; Todd et al., 2024)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.745,
+ 0.827,
+ 0.9
+ ],
+ "angle": 0,
+ "content": "The task vector approach first fine-tunes a pre-trained model on several simpler tasks to obtain task vectors, which represent the weight differences between the fine-tuned models and the pre-trained model. To handle more complex tasks, a proper model can be edited by adding a linear combination of these task vectors to the pre-trained model. Since this approach only requires determining the appropriate arithmetic hyperparameters, with no need for further fine-tuning on complicated tasks, the task vector method offers a significant efficiency advantage and is particularly effective when adapting to a wide range of downstream tasks. Empirical evidence shows that adding multiple task vectors can improve the model's performance on corresponding tasks, while subtracting certain task vectors allows the model to forget associated tasks. A proper linear combination of task vectors can even enable the model to generalize on an out-of-domain task that has an analogous relationship with the given task vectors, without needing labeled data. Additionally, it has been found that using low-"
+ },
+ {
+ "type": "page_footnote",
+ "bbox": [
+ 0.191,
+ 0.91,
+ 0.492,
+ 0.925
+ ],
+ "angle": 0,
+ "content": "*Corresponding author. Email: wangm7@rpi.edu."
+ },
+ {
+ "type": "aside_text",
+ "bbox": [
+ 0.023,
+ 0.256,
+ 0.061,
+ 0.709
+ ],
+ "angle": 270,
+ "content": "arXiv:2504.10957v3 [cs.LG] 25 May 2025"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.495,
+ 0.949,
+ 0.505,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "1"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.825,
+ 0.133
+ ],
+ "angle": 0,
+ "content": "rank and/or sparse task vectors can further improve efficiency while maintaining the performance (Yadav et al., 2023; Chitale et al., 2023; Yu et al., 2024; He et al., 2025)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.14,
+ 0.825,
+ 0.169
+ ],
+ "angle": 0,
+ "content": "Despite empirical successes, theoretical analysis of task vectors is less investigated. In particular, we ask the following question:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.179,
+ 0.174,
+ 0.82,
+ 0.203
+ ],
+ "angle": 0,
+ "content": "When and why can the task vector approach perform well in multi-task learning, unlearning, and out-of-domain generalization successfully and efficiently?"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.209,
+ 0.827,
+ 0.378
+ ],
+ "angle": 0,
+ "content": "Some related theoretical works focus on analyzing the performance of machine unlearning from a purely optimization perspective (Ginart et al., 2019; Neel et al., 2021; Guo et al., 2020; Mu & Klabjan, 2024). However, these analyses do not apply to Transformer-based neural networks, which are key components of large pre-trained models. Moreover, these works cannot be extended to study multi-task learning or out-of-domain generalization to new tasks. Frankle et al. (2020) proposes the concept of linear mode connectivity, suggesting that there exists a small-loss connected region in the loss landscape of the model, thereby demonstrating that linear interpolation between models can yield good performance. The most relevant work to this paper is (Ortiz-Jimenez et al., 2023), which uses the Neural Tangent Kernel (NTK) framework (Jacot et al., 2018) to study neural networks as linearized models under specific assumptions, to justify the use of linear arithmetic on task vectors for targeted model editing. However, this work does not have generalization guarantees and cannot explain the success of task vectors in nonlinear models without NTK assumptions."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.173,
+ 0.39,
+ 0.389,
+ 0.403
+ ],
+ "angle": 0,
+ "content": "1.1 MAJOR CONTRIBUTIONS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.41,
+ 0.825,
+ 0.495
+ ],
+ "angle": 0,
+ "content": "To the best of our knowledge, this work is the first theoretical generalization analysis of task arithmetic on a nonlinear Transformer model for multi-task learning, unlearning, and out-of-domain generalization. Focusing on binary classification tasks, we provide a quantitative analysis of the dependence of the task arithmetic effect on arithmetic hyperparameters. Although our analysis is centered on a simplified single-head and one-layer nonlinear Transformer, our theoretical insights are validated on practical architectures. Our major contributions include:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.5,
+ 0.825,
+ 0.641
+ ],
+ "angle": 0,
+ "content": "1. A fine-grained feature-learning analysis of the effectiveness of task addition and negation. We consider a data model in which binary labels are determined by the majority of discriminative tokens, rather than their opposing discriminative counterparts, while other tokens do not affect the labels. We begin by analyzing the learning dynamics of fine-tuning a Transformer and characterize the properties of the resulting task vectors. Next, we provide sufficient conditions on the arithmetic hyperparameters for the task vector approach to be successful. We prove that task addition is effective for multi-task learning when the tasks are either irrelevant or aligned. Aligned tasks are those where solving one task contributes positively to solving the other. In contrast, task negation is provably successful for unlearning tasks that are either irrelevant or contradictory. Contradictory tasks are defined as those where improving performance on one task harms the performance of the other."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.646,
+ 0.825,
+ 0.717
+ ],
+ "angle": 0,
+ "content": "2. The first provable out-of-domain generalization guarantees through task arithmetic. Focusing on task vectors representing a set of irrelevant tasks, we prove a linear combination of these task vectors can generalize to a wide range of new tasks by properly selecting the arithmetic coefficients. Additionally, we characterize the range of suitable arithmetic coefficients sufficient for successful generalization. This is the first theoretical justification of task vectors' ability to adapt to new tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "3. Theoretical justification of low-rank approximation and magnitude-based pruning for task vectors. We construct low-rank and sparse approximations to task vectors and prove that the generalization guarantees are minimally affected by these approximations. This provides the first theoretical support for the practice of using low-rank and sparse approximations to task vectors in order to reduce computational complexity."
+ },
+ {
+ "type": "list",
+ "bbox": [
+ 0.171,
+ 0.5,
+ 0.825,
+ 0.793
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "1.2 RELATED WORKS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.827,
+ 0.827,
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+ ],
+ "angle": 0,
+ "content": "Weight interpolation technique. Weight interpolation or model merging (Matena & Raffel, 2022; Ilharco et al., 2022b; Yadav et al., 2023; Yu et al., 2024; He et al., 2025) refers to the practice of linearly interpolating weights of multiple models, where these models may be fine-tuned from different downstream tasks or using different hyperparameters (model soups (Wortsman et al., 2022a)). Weight interpolation is empirically observed to be able to guide the model towards wider optima (Izmailov et al., 2018; Frankle et al., 2020) and better generalization in both single-task performance and multi-task abilities, even surpassing fine-tuning methods in some cases (Rame et al.,"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "2"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.825,
+ 0.133
+ ],
+ "angle": 0,
+ "content": "2022; Wortsman et al., 2022b; Ramé et al., 2023). Task arithmetic can be viewed as a special type of weight interpolation, where linear operations are performed on task vectors."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.14,
+ 0.827,
+ 0.252
+ ],
+ "angle": 0,
+ "content": "Feature learning analysis for Transformers. Several recent works study the optimization and generalization analysis of Transformers following the feature learning framework, which describes how neural networks gradually focus on important features while discarding unimportant features during training. Jelassi et al. (2022); Li et al. (2023e); Oymak et al. (2023); Ildiz et al. (2024); Nichani et al. (2024); Chen et al. (2024); Li et al. (2023a; 2024c; 2023b); Huang et al. (2024); Luo et al. (2024) study the generalization of one-layer Transformers on different data models such as spatial association, semantic/contextual structure, causal structure/Markov Chain of data, and the majority voting of tokens in the data. However, no discussion was provided for merged models."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.258,
+ 0.828,
+ 0.399
+ ],
+ "angle": 0,
+ "content": "Theoretical study of PEFT methods. These are recent theoretical analyses on other PEFT methods. For example, in-context learning is analyzed from the perspective of expressive power (Bai et al., 2023; Akyurek et al., 2023; Von Oswald et al., 2023), the training dynamics or generalization (Xie et al., 2021; Zhang et al., 2023a; Li et al., 2023c; 2024a;b; Huang et al., 2023). Some other works focus on prompt engineering with a tunable prompt (Wei et al., 2021; Oymak et al., 2023; Zhang et al., 2024). Another line of work theoretically investigates the low-rank adaptation in terms of the implicit bias of the optimization process (Damian et al., 2022; Abbe et al., 2022; 2023; Boix-Adsera et al., 2023; Jang et al., 2024; Li et al., 2024d) or model pruning with generalization analysis (Zhang et al., 2021; Yang & Wang, 2023; Yang et al., 2023; Zhang et al., 2023b; Li et al., 2024a). However, none of these works involve the task vector method or related approaches."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2 TASK VECTOR: DEFINITION AND OBSERVATIONS"
+ },
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+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "2.1 PRELIMINARIES"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
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+ "angle": 0,
+ "content": "Let \\(f:\\mathcal{X}\\times \\Theta \\to \\mathcal{Y}\\) be a neural network that maps inputs \\(\\pmb {X}\\in \\mathcal{X}\\) to labels \\(\\pmb {y}\\in \\mathcal{V}\\) with \\(\\Psi \\in \\Theta\\) as the model parameters. Denote \\(\\Psi^{(0)}\\) as the pre-trained model and \\(\\Psi_T^*\\) as the fine-tuned model on a given task \\(\\mathcal{T}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Definition 1. (Task Vector) The task vector \\(\\Delta \\Psi_{\\mathcal{T}}\\) for the task \\(\\mathcal{T}\\) is computed as the element-wise difference between the pre-trained and fine-tuned weights, i.e., \\(\\Delta \\Psi_{\\mathcal{T}} = \\Psi_{\\mathcal{T}}^{*} - \\Psi^{(0)}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Task Arithmetic and Generalization. Given the pre-trained model \\(\\Psi^{(0)}\\) and a set of task vectors \\(\\{\\Delta \\Psi_{\\mathcal{T}_i}\\}_{i\\in \\mathcal{V}}\\) on tasks \\(\\{\\mathcal{T}_i\\}_{i\\in \\mathcal{V}}\\), one can construct a merged model \\(\\Psi = \\Psi^{(0)} + \\sum_{i\\in \\mathcal{V}}\\lambda_i\\Delta \\Psi_{\\mathcal{T}_i}\\) for inference on downstream tasks, where \\(\\lambda_{i}\\in \\mathbb{R}\\) are arithmetic hyperparameters. Denote \\(\\ell (X,y;\\Psi)\\) as the loss function for the input \\(X\\in \\mathcal{X}\\), output \\(y\\in \\mathcal{Y}\\), and the model \\(\\Psi \\in \\Theta\\). Hence, the generalization error on the task \\(\\mathcal{T}'\\) with data \\((X,y)\\sim \\mathcal{D}_{\\mathcal{T}'}\\) is defined as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau^ {\\prime}}} \\ell (\\boldsymbol {X}, y; \\Psi). \\tag {1}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.647,
+ 0.825,
+ 0.704
+ ],
+ "angle": 0,
+ "content": "Existing works (Ilharco et al., 2022a; Ortiz-Jimenez et al., 2023) conclude that by controlling \\(\\lambda_{i}\\), the merged model \\(\\Psi\\) can generalize across different tasks. Specifically, adding several \\(\\Delta \\Psi_{\\mathcal{T}_i}\\) via making \\(\\lambda_{i} > 0\\), \\(i \\in \\mathcal{V}_{A} \\subset \\mathcal{V}\\), leads to a model that exhibits desired performance on multiple tasks from \\(\\mathcal{V}_{A}\\). Such a successful multi-task learning result can be mathematically represented as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.357,
+ 0.708,
+ 0.825,
+ 0.726
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {i}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon), \\forall i \\in \\mathcal {V} _ {A}. \\tag {2}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.736,
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+ "angle": 0,
+ "content": "Meanwhile, negating \\(\\Delta \\Psi_{\\mathcal{T}_i}\\) with \\(\\lambda_i < 0\\), \\(i \\in \\mathcal{V}_N \\subset \\mathcal{V}\\), results in a machine unlearning model that performs poorly on \\(\\mathcal{V}_N\\) but roughly retains the accuracy on \\(\\mathcal{V} \\backslash \\mathcal{V}_N\\), i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.195,
+ 0.769,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {i}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1), \\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {j}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon), \\forall i \\in \\mathcal {V} _ {N}, \\forall j \\in \\mathcal {V} \\backslash \\mathcal {V} _ {N}. \\tag {3}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.81,
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+ 0.856
+ ],
+ "angle": 0,
+ "content": "Moreover, task arithmetic is empirically (Ilharco et al., 2022a) shown to produce a model \\(\\Psi = \\Psi^{(0)} + \\lambda \\cdot \\Delta \\Psi_{\\mathcal{T}'}\\) that performs well on task analogy, in the form that \"the target out-of-domain task \\(\\mathcal{T}'(\\notin \\mathcal{V})\\) is to \\(\\mathcal{T}_A\\) as \\(\\mathcal{T}_B\\) is to \\(\\mathcal{T}_C\\),\" by constructing a task vector \\(\\Delta \\Psi_{\\mathcal{T}'} = \\Delta \\Psi_{\\mathcal{T}_A} + (\\Delta \\Psi_{\\mathcal{T}_B} - \\Delta \\Psi_{\\mathcal{T}_C})\\)."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "2.2 EMPIRICAL OBSERVATIONS"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
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+ "angle": 0,
+ "content": "Note that experiments in (Ilharco et al., 2022a) only summarize the empirical findings when tasks are almost \"orthogonal\" to each other, while non-orthogonal cases are less explored. Therefore, in Table 1, we further construct binary classification tasks on the parity of digits of Colored-MNIST"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "3"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.104,
+ 0.827,
+ 0.148
+ ],
+ "angle": 0,
+ "content": "(Arjovsky et al., 2019; Chapel et al., 2020). We control the colors of digits to generate a pair of two datasets so that the parity classification tasks on different pairs of datasets are conceptually \"irrelevant,\" \"aligned,\" or \"contradictory\" to each other, respectively."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.153,
+ 0.825,
+ 0.227
+ ],
+ "angle": 0,
+ "content": "For irrelevant tasks, odd and even digits are highly correlated with red and green colors in one dataset but independent of colors in the other. In aligned tasks, the odd and even digits are correlated with red and green colors in both datasets. In contradictory tasks, the color-parity correspondence is the opposite in the two datasets. Let \\(\\mathcal{T}_1\\) and \\(\\mathcal{T}_2\\) denote the parity classification task on two different datasets. \\(\\Psi = \\Psi^{(0)} + \\Delta \\Psi_{\\mathcal{T}_1} + \\lambda \\Delta \\Psi_{\\mathcal{T}_2}\\) is used to evaluate the performance of \\(\\mathcal{T}_1\\) and \\(\\mathcal{T}_2\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.231,
+ 0.825,
+ 0.303
+ ],
+ "angle": 0,
+ "content": "A key finding from Table 1 is that the task vector method performs quite differently with different task correlations. To be concrete, given \\(\\Delta \\Psi_{\\mathcal{T}_1}\\) and \\(\\Delta \\Psi_{\\mathcal{T}_2}\\) for aligned tasks, the merged model \\(\\Psi\\) can acquire strong multi-task learning abilities but have poor unlearning capabilities. The conclusion is exactly opposite for contradictory tasks. For irrelevant tasks, using task arithmetic can result in good performance in both unlearning and multi-task learning. A question arises, i.e.,"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.191,
+ 0.31,
+ 0.806,
+ 0.34
+ ],
+ "angle": 0,
+ "content": "(Q1) How does task correlation quantitatively affect the performance of task arithmetic in multi-task learning and unlearning?"
+ },
+ {
+ "type": "table",
+ "bbox": [
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+ "angle": 0,
+ "content": "| Notations | Annotation |
| X, xi, Xn, yn | X is the input data, which contains P tokens. xi is the i-th token of X. Xn is the n-th input data with yn as the corresponding label. |
| Ψ | Ψ = {{a(l)}Pl=1, WO, WV, WK, WQ} denotes the set of all the model parameters. a(l) ∈ Rm and WO ∈ Rm×ma are the weights in the MLP layer. WV ∈ Rma×d, WK, WQ ∈ Rmb×d are weights in the self-attention layer. |
| Ψ(0), ΨT*, ΔΨT | Ψ(0) is the pre-trained model. ΨT* is the fine-tuned model on a given task T. ΔΨT is the task vector of the task T, which is computed as ΔΨT = ΨT* - Ψ(0). |
| μT, vj | μT is the discriminative pattern of the task T. vj is the j-th task-irrelevant pattern, j ∈ [M]. |
| δ*, δ# | δ* is the average fraction of label-relevant pattern in the input data. δ# is the average fraction of confusion pattern in the input data. |
| q1(t),ζ1,t, pn(t) | q1(t) = μ1T W(t) μ1 denotes the value of the product, where the patterns on both sides of W(t) are the same.ζ1,t denotes the modified value embedding of μ1 at the t-th iteration. pn(t) refers to the summation of attention weights where the key and the query are the same discriminative pattern. |
| Wn,l,Un,l | Wn,l and Un,l respectively represent of sets of positive or negative neurons so that the Relu activation is activated with xln as the query. |
| Bb | Bb is the SGD batch at the b-th iteration. |
| O(), Ω(), Θ() | We follow the convention that f(x) = O(g(x)) (or Ω(g(x)), Θ(g(x))) means that f(x) increases at most, at least, or in the order of g(x), respectively. |
| a | a = |a(l)i| = 1/√m for i ∈ [m]. |
| ≥, ≤ | f(x) ≥ g(x) (or f(x) ≤ g(x)) means that f(x) ≥ Ω(g(x)) (or f(x) ≤ O(g(x))). |
"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.533,
+ 0.538,
+ 0.548
+ ],
+ "angle": 0,
+ "content": "Define \\(\\mathcal{W}_{n,l},\\mathcal{U}_{n,l}\\) as the sets of lucky neurons such that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.337,
+ 0.556,
+ 0.825,
+ 0.575
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {W} _ {n, l} = \\left\\{i: \\boldsymbol {V} _ {(i, \\cdot)} ^ {\\top} \\boldsymbol {R} _ {n, l} (0) > 0, l \\in \\mathcal {S} _ {1} ^ {n}, a _ {i} > 0 \\right\\}, \\tag {13}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.34,
+ 0.584,
+ 0.825,
+ 0.603
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathcal {U} _ {n, l} = \\left\\{i: \\boldsymbol {V} _ {(i, \\cdot)} ^ {\\top} \\boldsymbol {R} _ {n, l} (0) > 0, l \\in \\mathcal {S} _ {2} ^ {n}, a _ {i} < 0 \\right\\}. \\tag {14}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.606,
+ 0.825,
+ 0.659
+ ],
+ "angle": 0,
+ "content": "Definition 5 ((Vershynin, 2010)). We say \\( X \\) is a sub-Gaussian random variable with sub-Gaussian norm \\( K > 0 \\), if \\( (\\mathbb{E}|X|^p)^{\\frac{1}{p}} \\leq K\\sqrt{p} \\) for all \\( p \\geq 1 \\). In addition, the sub-Gaussian norm of \\( X \\), denoted \\( \\| X\\|_{\\psi_2} \\), is defined as \\( \\| X\\|_{\\psi_2} = \\sup_{p \\geq 1} p^{-\\frac{1}{2}}(\\mathbb{E}|X|^p)^{\\frac{1}{p}} \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.661,
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+ 0.707
+ ],
+ "angle": 0,
+ "content": "Lemma 2 (Vershynin (2010) Proposition 5.1, Hoeffding's inequality). Let \\(X_{1}, X_{2}, \\dots, X_{N}\\) be independent centered sub-gaussian random variables, and let \\(K = \\max_{i} \\|X_{i}\\|_{\\psi_{2}}\\). Then for every \\(\\mathbf{a} = (a_{1}, \\dots, a_{N}) \\in \\mathbb{R}^{N}\\) and every \\(t \\geq 0\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.338,
+ 0.714,
+ 0.826,
+ 0.755
+ ],
+ "angle": 0,
+ "content": "\\[\n\\Pr \\left(\\left| \\sum_ {i = 1} ^ {N} a _ {i} X _ {i} \\right| \\geq t\\right) \\leq e \\cdot \\exp \\left(- \\frac {c t ^ {2}}{K ^ {2} \\| \\boldsymbol {a} \\| ^ {2}}\\right), \\tag {15}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.763,
+ 0.414,
+ 0.776
+ ],
+ "angle": 0,
+ "content": "where \\( c > 0 \\) is an absolute constant."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Lemma 3. For task \\(\\mathcal{T}\\) based on any \\(\\pmb{\\mu}_1\\), \\(0 \\leq t \\leq T\\), there exists \\(K(t) > 0\\), such that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.338,
+ 0.805,
+ 0.825,
+ 0.845
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} = \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} + K (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 1} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\tag {16}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ 0.864,
+ 0.825,
+ 0.903
+ ],
+ "angle": 0,
+ "content": "\\[\nK (t) \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\zeta_ {1, t} p _ {n} (t) \\phi_ {n} (t) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|), \\tag {17}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.432,
+ 0.907,
+ 0.825,
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\iota_ {l} ^ {\\prime} \\leq K (t) \\cdot e ^ {- q _ {1} (t)}. \\tag {18}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "17"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.104,
+ 0.264,
+ 0.12
+ ],
+ "angle": 0,
+ "content": "For \\(k\\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.377,
+ 0.122,
+ 0.826,
+ 0.162
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\lesssim \\sqrt {\\frac {\\log B}{B}} \\sum_ {b = 0} ^ {t} K (b), \\tag {19}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.168,
+ 0.334,
+ 0.184
+ ],
+ "angle": 0,
+ "content": "and for \\(j\\neq k\\) \\(j\\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.185,
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+ 0.204
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {v} _ {j} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\lesssim K (t) e ^ {- q _ {1} (t)}, \\tag {20}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.21,
+ 0.623,
+ 0.227
+ ],
+ "angle": 0,
+ "content": "For any \\(\\pmb{\\mu}'\\) such that \\(\\pmb{\\mu}_1^\\top \\pmb{\\mu}' = \\alpha\\) and \\(\\pmb{\\mu}' \\perp \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.352,
+ 0.236,
+ 0.826,
+ 0.256
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} ^ {\\prime} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} ^ {\\prime} = \\alpha^ {2} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\cdot (1 \\pm \\Theta (\\epsilon)), \\tag {21}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.263,
+ 0.398,
+ 0.279
+ ],
+ "angle": 0,
+ "content": "if \\(B \\geq \\epsilon^{-2} \\log M\\) for some \\(\\epsilon < 1\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ 0.3
+ ],
+ "angle": 0,
+ "content": "Lemma 4. Given a task \\(\\mathcal{T}\\) based on any \\(\\pmb{\\mu}_1\\), \\(0 \\leq t \\leq T\\). Then, for \\(i \\in \\mathcal{W}_{n,t}\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\gtrsim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {22}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.363,
+ 0.826,
+ 0.405
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {23}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.169,
+ 0.411,
+ 0.469,
+ 0.428
+ ],
+ "angle": 0,
+ "content": "for \\(k\\in [M]\\) .For \\(i\\in \\mathcal{U}_{n,l}\\) , we similarly have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.363,
+ 0.438,
+ 0.826,
+ 0.479
+ ],
+ "angle": 0,
+ "content": "\\[\n- \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\gtrsim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {24}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.492,
+ 0.826,
+ 0.533
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {25}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.169,
+ 0.54,
+ 0.526,
+ 0.556
+ ],
+ "angle": 0,
+ "content": "for some \\(k\\in [M]\\). For \\(i\\notin \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}\\), we have that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.401,
+ 0.566,
+ 0.826,
+ 0.598
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\lesssim \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {26}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.402,
+ 0.611,
+ 0.826,
+ 0.644
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\lesssim \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k}, \\tag {27}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.649,
+ 0.395,
+ 0.666
+ ],
+ "angle": 0,
+ "content": "where \\(k\\in [M],j\\in \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}\\)"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.17,
+ 0.669,
+ 0.827,
+ 0.713
+ ],
+ "angle": 0,
+ "content": "Lemma 5. (Full version of Lemma 1) Given a task \\(\\mathcal{T}\\) defined in Definition 2 based on the discriminative pattern \\(\\pmb{\\mu}_{\\mathcal{T}}\\), we have that as long as conditions (i)-(iii) in Theorem 1 hold, then the returned model \\(\\Psi_{\\mathcal{T}}^{*}\\) after \\(T\\) iterations achieves a generalization error"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.382,
+ 0.72,
+ 0.826,
+ 0.739
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\mathcal {T}}} \\left[ \\ell \\left(\\boldsymbol {X}, y; \\Psi_ {\\mathcal {T}} ^ {*}\\right) \\right] \\leq \\Theta (\\epsilon). \\tag {28}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.746,
+ 0.763,
+ 0.761
+ ],
+ "angle": 0,
+ "content": "The required sample complexity is \\( N = BT \\), where \\( B \\) is the batch size. We also have that"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.212,
+ 0.774,
+ 0.227,
+ 0.785
+ ],
+ "angle": 0,
+ "content": "1."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.419,
+ 0.787,
+ 0.826,
+ 0.806
+ ],
+ "angle": 0,
+ "content": "\\[\np _ {n} (T) \\geq 1 - \\left(1 - \\delta_ {*}\\right) \\delta_ {*} ^ {- 1} T ^ {- C}, \\tag {29}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.228,
+ 0.811,
+ 0.402,
+ 0.826
+ ],
+ "angle": 0,
+ "content": "for some constant \\(C > 1\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.211,
+ 0.836,
+ 0.227,
+ 0.848
+ ],
+ "angle": 0,
+ "content": "2."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.408,
+ 0.85,
+ 0.826,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {k = 1} ^ {M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {v} _ {k} \\right\\| ^ {2} \\lesssim \\frac {1}{M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T}} \\right\\| ^ {2}, \\tag {30}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.228,
+ 0.896,
+ 0.825,
+ 0.924
+ ],
+ "angle": 0,
+ "content": "for \\( i \\in \\mathcal{W}_{n,l} \\) with \\( l \\in S_1^n \\) and for \\( i \\in \\mathcal{U}_{n,l} \\) with \\( l \\in S_2^n \\). We also have that (26) and (27) hold when \\( t = T \\)."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.491,
+ 0.949,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "18"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.103,
+ 0.616,
+ 0.119
+ ],
+ "angle": 0,
+ "content": "D PROOF OF MAIN THEOREMS AND COROLLARIES"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.134,
+ 0.427,
+ 0.147
+ ],
+ "angle": 0,
+ "content": "D.1 PROOF OF THEOREM 1 AND 2"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.16,
+ 0.825,
+ 0.205
+ ],
+ "angle": 0,
+ "content": "Proof. Since the model is initialized close to zero, then \\(\\Delta \\Psi\\) is close to \\(\\Psi\\). Denote \\(\\Psi_{1} = \\{\\{a_{(l,1)}^{P}\\}_{l=1}, V_{1}, W_{1}\\}\\) and \\(\\Psi_{2} = \\{\\{a_{(l,2)}^{P}\\}_{l=1}, V_{2}, W_{2}\\}\\). We consider three cases of this learning problem."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.205,
+ 0.544,
+ 0.219
+ ],
+ "angle": 0,
+ "content": "(1) Consider \\(\\alpha = 0\\). By (21) in Lemma 3, we know that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.206,
+ 0.224,
+ 0.826,
+ 0.247
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} \\left(1 + \\lambda \\alpha^ {2} (1 \\pm \\Theta (\\epsilon))\\right) = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}}, \\tag {31}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.338,
+ 0.251,
+ 0.825,
+ 0.272
+ ],
+ "angle": 0,
+ "content": "\\[\n- \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = - \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}}, \\tag {32}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.345,
+ 0.274,
+ 0.825,
+ 0.295
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = \\lambda \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}, \\tag {33}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.332,
+ 0.297,
+ 0.825,
+ 0.319
+ ],
+ "angle": 0,
+ "content": "\\[\n- \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = - \\lambda \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}. \\tag {34}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.32,
+ 0.427,
+ 0.335
+ ],
+ "angle": 0,
+ "content": "Then, for any \\( l \\in [M] \\) and for task \\( \\mathcal{T}_1 \\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.331,
+ 0.341,
+ 0.826,
+ 0.379
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {s \\in S _ {1} ^ {n}} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}, \\tag {35}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.385,
+ 0.251,
+ 0.4
+ ],
+ "angle": 0,
+ "content": "for task \\(\\mathcal{T}_2\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.255,
+ 0.405,
+ 0.825,
+ 0.445
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {s \\in \\mathcal {S} _ {1} ^ {n}} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq \\frac {\\delta_ {*} T ^ {\\lambda C}}{\\delta_ {*} T ^ {\\lambda C} + (1 - \\delta_ {*})} \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C}. \\tag {36}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.452,
+ 0.741,
+ 0.469
+ ],
+ "angle": 0,
+ "content": "Since that \\(\\pmb{\\mu}_{\\mathcal{T}_2} \\perp \\{\\pmb{\\mu}_{\\mathcal{T}_1}, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\}\\) and \\(\\pmb{\\mu}_{\\mathcal{T}_1} \\perp \\{\\pmb{\\mu}_{\\mathcal{T}_2}, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\}\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.449,
+ 0.474,
+ 0.825,
+ 0.497
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = 0, \\tag {37}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.501,
+ 0.289,
+ 0.517
+ ],
+ "angle": 0,
+ "content": "for \\(V\\in \\Psi_{1}\\) , and"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.449,
+ 0.515,
+ 0.825,
+ 0.537
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = 0, \\tag {38}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.539,
+ 0.825,
+ 0.569
+ ],
+ "angle": 0,
+ "content": "for \\( V \\in \\Psi_2 \\). Then, for data with the label \\( y = 1 \\), the network output for \\( \\Psi_1 + \\lambda \\Psi_2 \\) is almost the same as that for \\( \\Psi_1 \\) on task \\( \\mathcal{T}_1 \\) when \\( |\\lambda| \\) is not too large. To see this, for \\( X \\) from \\( \\mathcal{T}_1 \\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.199,
+ 0.575,
+ 0.826,
+ 0.704
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} 1 - \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in [ m ]} \\frac {1}{a} \\operatorname {R e l u} \\left(\\left(\\boldsymbol {V} _ {1 (i, \\cdot)} ^ {(T)} + \\lambda \\boldsymbol {V} _ {2 (i, \\cdot)} ^ {(T)}\\right) \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ \\leq | \\lambda | \\cdot \\Theta \\left(\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}\\right) \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} + | \\lambda | \\cdot \\Theta \\left(\\sqrt {M \\frac {\\log B}{B}}\\right) \\tag {39} \\\\ \\leq | \\lambda | \\cdot \\Theta \\left(1 - \\delta_ {*}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\\\ = | \\lambda | \\beta , \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.712,
+ 0.825,
+ 0.755
+ ],
+ "angle": 0,
+ "content": "where the second to last step is by (26) and (27) and \\(B \\gtrsim \\epsilon^2 \\log M\\). Therefore, a larger \\(|\\lambda|\\) leads to a performance drop in task \\(\\mathcal{T}_1\\). For data of \\(\\mathcal{T}_1\\) with the label \\(y = -1\\), we can choose \\(\\lambda\\) to be greater than around 1 to make the network output smaller than \\(-1\\). Meanwhile, for \\(\\mathbf{X}\\) from \\(\\mathcal{T}_2\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.268,
+ 0.761,
+ 0.825,
+ 0.814
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f (\\boldsymbol {X} ^ {n}, \\Psi) \\\\ \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\lambda}\\right) \\cdot \\lambda - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right), \\tag {40} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.82,
+ 0.554,
+ 0.836
+ ],
+ "angle": 0,
+ "content": "where we need \\(\\lambda \\geq 1 + \\beta\\) so that \\(f(\\pmb{X}^n, \\Psi) \\geq 1 - \\Theta(\\epsilon)\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.841,
+ 0.713,
+ 0.857
+ ],
+ "angle": 0,
+ "content": "If \\(\\lambda \\leq 0\\), the attention map tends to be uniform. Then, for \\(X^n\\) in task \\(\\mathcal{T}_2\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.406,
+ 0.862,
+ 0.825,
+ 0.891
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}; \\Psi_ {1} + \\lambda \\Psi_ {2}\\right) \\lesssim - \\frac {1}{P}, \\tag {41}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.897,
+ 0.273,
+ 0.91
+ ],
+ "angle": 0,
+ "content": "which leads to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.909,
+ 0.825,
+ 0.929
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {42}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.491,
+ 0.948,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "19"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.104,
+ 0.4,
+ 0.118
+ ],
+ "angle": 0,
+ "content": "(2) Consider \\(\\alpha > 0\\). We first have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.317,
+ 0.12,
+ 0.825,
+ 0.141
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} = \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} ^ {\\top} \\boldsymbol {W} _ {1} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {1}} \\left(1 + \\lambda \\alpha^ {2}\\right), \\tag {43}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.32,
+ 0.142,
+ 0.825,
+ 0.162
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\left(\\boldsymbol {W} _ {1} ^ {(T)} + \\lambda \\boldsymbol {W} _ {2} ^ {(T)}\\right) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} = (\\lambda + \\alpha^ {2}) \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}} ^ {\\top} \\boldsymbol {W} _ {2} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {2}}. \\tag {44}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.16,
+ 0.534,
+ 0.175
+ ],
+ "angle": 0,
+ "content": "Then, for \\(y^n = 1\\) in task \\(\\widetilde{T}_1\\), we have that when \\(\\lambda > 0\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.258,
+ 0.176,
+ 0.329,
+ 0.191
+ ],
+ "angle": 0,
+ "content": "\\[\nf (\\boldsymbol {X} ^ {n}, \\Psi)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.246,
+ 0.196,
+ 0.824,
+ 0.275
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\gtrsim (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {| \\mathcal {S} _ {1} ^ {n} |}{a P M}) \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C} \\\\ - | \\lambda | \\cdot \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {45} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.246,
+ 0.277,
+ 0.652,
+ 0.34
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\geq 1 + \\Theta (\\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right) \\\\ = 1 + \\Theta (\\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\frac {1 - \\delta_ {*}}{\\delta_ {*}}) \\cdot \\mathrm {p o l y} (\\eta \\delta_ {*}) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}), \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.341,
+ 0.52,
+ 0.356
+ ],
+ "angle": 0,
+ "content": "and for \\(y^n = 1\\) in task \\(\\mathcal{T}_2\\), we have that when \\(\\lambda \\geq 0\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.281,
+ 0.358,
+ 0.824,
+ 0.422
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) \\cdot (\\lambda + \\alpha) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {46} \\\\ - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.424,
+ 0.557,
+ 0.438
+ ],
+ "angle": 0,
+ "content": "Therefore, when \\(\\lambda \\geq 1 - \\alpha +\\beta\\) , we have that for task \\(\\mathcal{T}_1\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.396,
+ 0.44,
+ 0.825,
+ 0.456
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq 1 - | \\lambda | \\beta - \\Theta (\\epsilon), \\tag {47}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.457,
+ 0.279,
+ 0.471
+ ],
+ "angle": 0,
+ "content": "and for task \\(\\mathcal{T}_2\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.255,
+ 0.473,
+ 0.825,
+ 0.541
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq (1 - \\Theta (\\epsilon)) (\\lambda + \\alpha) - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\cdot \\mathbf {p o l y} (\\eta \\delta_ {*}) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\tag {48} \\\\ \\geq (1 - \\Theta (\\epsilon)) (\\lambda + \\alpha) - \\beta \\\\ \\geq 1 - \\Theta (\\epsilon). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.543,
+ 0.591,
+ 0.558
+ ],
+ "angle": 0,
+ "content": "We can obtain corresponding conclusions for \\(y^n = -1\\). Hence,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.368,
+ 0.559,
+ 0.825,
+ 0.576
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon) + | \\lambda | \\beta , \\tag {49}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.393,
+ 0.578,
+ 0.825,
+ 0.595
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon). \\tag {50}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.595,
+ 0.573,
+ 0.609
+ ],
+ "angle": 0,
+ "content": "Meanwhile, for \\(y^n = 1\\) in task \\(\\mathcal{T}_1\\), we have that when \\(\\lambda < 0\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.201,
+ 0.61,
+ 0.824,
+ 0.738
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)\\right) \\cdot (1 + \\lambda \\alpha) \\\\ - (| \\lambda | + 1) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\tag {51} \\\\ \\geq 1 + \\lambda \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})}\\right) - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) \\\\ - \\left(| \\lambda | + 1\\right) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right), \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.74,
+ 0.52,
+ 0.754
+ ],
+ "angle": 0,
+ "content": "and for \\(y^n = 1\\) in task \\(\\mathcal{T}_2\\), we have that when \\(\\lambda < 0\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.182,
+ 0.757,
+ 0.825,
+ 0.928
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) \\cdot (\\lambda + \\alpha) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\\\ \\geq \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C} - \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)\\right) \\cdot (\\lambda + \\alpha) \\\\ - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) - \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\tag {52} \\\\ \\geq \\lambda + \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)}\\right) - \\lambda \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) \\\\ - \\Theta (\\sqrt {\\frac {M \\log B}{B}}) - \\Theta (\\frac {1 - \\delta_ {*}}{\\delta_ {*}}) \\cdot \\mathrm {p o l y} (\\eta \\delta_ {*}). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.51,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "20"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.103,
+ 0.474,
+ 0.12
+ ],
+ "angle": 0,
+ "content": "Then, for task \\(\\mathcal{T}_1\\), when \\(0 > \\lambda \\geq -\\Theta (1 / \\alpha^2)\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.206,
+ 0.129,
+ 0.824,
+ 0.25
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi) \\\\ = \\min \\left\\{\\Theta \\left(- \\lambda \\alpha \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)}\\right) + \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) + \\epsilon \\right. \\right. \\\\ + (| \\lambda | + 1) \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}), \\Theta (1) \\} \\tag {53} \\\\ \\geq \\min \\left\\{\\Theta (- \\lambda \\alpha + (| \\lambda | + 1) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M})) , \\Theta (1) \\right\\} \\\\ = \\min \\left\\{\\Theta (- \\lambda \\alpha + | \\lambda | \\beta + \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right)), \\Theta (1) \\right\\}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.257,
+ 0.223,
+ 0.271
+ ],
+ "angle": 0,
+ "content": "Hence,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.293,
+ 0.271,
+ 0.824,
+ 0.29
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (- \\lambda \\alpha + (1 + | \\lambda |) \\beta), \\Theta (1) \\right\\}. \\tag {54}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.294,
+ 0.327,
+ 0.311
+ ],
+ "angle": 0,
+ "content": "When \\(\\lambda < -\\Theta (1 / \\alpha^2)\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.428,
+ 0.31,
+ 0.582,
+ 0.328
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\mathcal {T} _ {1}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.416,
+ 0.33,
+ 0.824,
+ 0.358
+ ],
+ "angle": 0,
+ "content": "\\[\n= \\Theta \\left(1 - \\frac {1}{M} \\cdot \\frac {1}{M} \\cdot M\\right) \\tag {55}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.418,
+ 0.361,
+ 0.468,
+ 0.375
+ ],
+ "angle": 0,
+ "content": "\\[\n\\geq \\Theta (1).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.381,
+ 0.434,
+ 0.397
+ ],
+ "angle": 0,
+ "content": "For task \\(\\mathcal{T}_2\\), when \\(0 > \\lambda \\geq \\Theta(1) - \\alpha^2\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.195,
+ 0.406,
+ 0.824,
+ 0.535
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\mathbb {E} _ {(\\boldsymbol {X}, \\boldsymbol {y}) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, \\boldsymbol {y}; \\Psi) \\\\ = \\min \\left\\{\\Theta \\left(1 - \\lambda - \\alpha + \\alpha \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} + \\lambda \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C \\left(\\lambda + \\alpha^ {2}\\right)} - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right) + \\epsilon \\right. \\right. \\\\ + \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) + \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right), \\Theta (1) \\} \\tag {56} \\\\ \\geq \\min \\{\\Theta (1 + \\eta^ {C} - \\lambda - \\alpha + \\Theta (\\operatorname {p o l y} (\\eta \\delta_ {*}) + \\epsilon \\sqrt {M})), \\Theta (1) \\} \\\\ = \\min \\left\\{\\Theta \\left(1 + \\eta^ {C} - \\lambda - \\alpha + \\beta\\right), \\Theta (1) \\right\\}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.543,
+ 0.767,
+ 0.56
+ ],
+ "angle": 0,
+ "content": "where the second step is by \\(\\lambda +\\alpha \\geq \\Theta (1) + \\alpha -\\alpha^{2}\\geq \\Theta (1)\\). When \\(\\lambda < \\Theta (1) - \\alpha^2 < 0\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.392,
+ 0.567,
+ 0.824,
+ 0.586
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {57}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.601,
+ 0.654,
+ 0.617
+ ],
+ "angle": 0,
+ "content": "(3) Consider \\(\\alpha < 0\\). When \\(\\lambda \\in (-\\Theta (1 / \\alpha^2),0)\\), we have that for task \\(\\mathcal{T}_1\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.214,
+ 0.625,
+ 0.824,
+ 0.854
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f (\\boldsymbol {X} ^ {n}, \\Psi) \\\\ \\gtrsim \\big (\\frac {1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C (1 + \\lambda \\alpha^ {2})}}{1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}} - \\Theta (\\epsilon) \\big) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta (\\eta \\sum_ {b = 0} ^ {T - 1} \\sum_ {i \\in [ m ]} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {| S _ {1} ^ {n} |}{a P M}) \\\\ \\cdot \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- \\lambda C} - | \\lambda | \\cdot \\Theta (\\sqrt {\\frac {M \\log B}{B}}) \\\\ \\geq (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta (\\epsilon \\sqrt {M}) \\tag {58} \\\\ - \\frac {\\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\left(T ^ {- C \\left(1 + \\lambda \\alpha^ {2}\\right)} - T ^ {- C}\\right)}{1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}} (1 + \\lambda \\alpha) \\\\ \\geq (1 - \\Theta (\\epsilon)) \\cdot (1 + \\lambda \\alpha) - | \\lambda | \\cdot \\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}}\\right) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right) \\\\ - \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) \\lambda \\alpha^ {2} (- \\log \\eta \\delta_ {*}) (1 + \\lambda \\alpha), \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.862,
+ 0.411,
+ 0.877
+ ],
+ "angle": 0,
+ "content": "Hence, if \\(\\lambda \\leq \\mathrm{poly}(\\eta \\delta_{*})\\alpha\\) , we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.397,
+ 0.886,
+ 0.824,
+ 0.901
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\geq 1 - | \\lambda | \\beta - \\Theta (\\epsilon). \\tag {59}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.369,
+ 0.91,
+ 0.824,
+ 0.928
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon) + | \\lambda | \\beta . \\tag {60}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.508,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "21"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.102,
+ 0.317,
+ 0.123
+ ],
+ "angle": 0,
+ "content": "If \\(\\lambda >\\frac{\\beta}{\\alpha - \\beta}\\) , we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.187,
+ 0.131,
+ 0.825,
+ 0.151
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (1), \\Theta (- \\lambda \\alpha + (| \\lambda | + 1) \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + | \\lambda | \\cdot \\Theta \\left(\\epsilon \\sqrt {M}\\right)) \\right\\}. \\tag {61}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.159,
+ 0.359,
+ 0.175
+ ],
+ "angle": 0,
+ "content": "If \\(\\lambda \\leq -\\Theta (1 / \\alpha^2)\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.174,
+ 0.825,
+ 0.192
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} _ {\\tau_ {1}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\Theta (1). \\tag {62}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.197,
+ 0.529,
+ 0.214
+ ],
+ "angle": 0,
+ "content": "For task \\(\\mathcal{T}_2\\), we have that when \\(\\lambda \\geq 1 + \\eta^C - \\alpha + \\beta\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.243,
+ 0.221,
+ 0.826,
+ 0.255
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi\\right) \\gtrsim (1 - \\eta^ {C}) (\\lambda + \\alpha) - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\cdot \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) - \\Theta \\left(\\sqrt {\\frac {M \\log B}{B}}\\right) \\geq 1, \\tag {63}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.263,
+ 0.826,
+ 0.282
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau_ {2}}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\Theta (\\epsilon). \\tag {64}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.287,
+ 0.504,
+ 0.305
+ ],
+ "angle": 0,
+ "content": "When \\(\\lambda \\leq 1 + \\eta^C -\\alpha +\\Theta (\\mathrm{poly}(\\eta \\delta_*) + \\epsilon \\sqrt{M})\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.295,
+ 0.311,
+ 0.825,
+ 0.331
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {\\left(\\boldsymbol {X}, y\\right) \\sim \\mathcal {D} \\tau_ {2}} \\ell (\\boldsymbol {X}, y; \\Psi) \\geq \\min \\left\\{\\Theta (1), 1 + \\eta^ {C} - \\lambda - \\alpha + \\beta \\right\\}. \\tag {65}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.338,
+ 0.825,
+ 0.381
+ ],
+ "angle": 0,
+ "content": "One can easily find that there is no region of \\(\\lambda\\) such that \\(\\Psi\\) performs well on both \\(\\mathcal{T}_1\\) and \\(\\mathcal{T}_2\\). However, when \\(-\\Theta (1 / \\alpha^2) < \\lambda < \\mathrm{poly}(\\eta \\delta_*)\\alpha < 1 + \\eta^c -\\alpha +\\beta\\), we can unlearn \\(\\mathcal{T}_2\\) and retain the performance of \\(\\mathcal{T}_1\\)."
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.808,
+ 0.388,
+ 0.824,
+ 0.4
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.419,
+ 0.377,
+ 0.433
+ ],
+ "angle": 0,
+ "content": "D.2 PROOF OF THEOREM 3"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.446,
+ 0.408,
+ 0.461
+ ],
+ "angle": 0,
+ "content": "Proof. By Lemma 1, we know that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.364,
+ 0.468,
+ 0.825,
+ 0.558
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} \\\\ = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} ^ {\\top} \\left(\\sum_ {j = 1} \\lambda_ {j} \\boldsymbol {W} _ {j} ^ {(T)}\\right) \\sum_ {k \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {k} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {k}} \\tag {66} \\\\ \\gtrsim \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} ^ {\\top} \\cdot \\lambda_ {i} \\boldsymbol {W} _ {i} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.567,
+ 0.405,
+ 0.58
+ ],
+ "angle": 0,
+ "content": "For positive neurons, we also have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.294,
+ 0.589,
+ 0.826,
+ 0.623
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} ^ {\\prime}} = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\boldsymbol {V} _ {\\mathcal {T} _ {i}} ^ {(T)} \\sum_ {i \\in \\mathcal {V} ^ {\\prime}} \\gamma_ {i} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} = \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\boldsymbol {V} _ {\\mathcal {T} _ {i}} ^ {(T)} \\boldsymbol {\\mu} _ {\\mathcal {T} _ {i}} \\tag {67}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.63,
+ 0.274,
+ 0.643
+ ],
+ "angle": 0,
+ "content": "Then, we need"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.434,
+ 0.644,
+ 0.826,
+ 0.677
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\geq 1 + c, \\tag {68}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.434,
+ 0.683,
+ 0.826,
+ 0.715
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} ^ {2} \\geq 1 + c, \\tag {69}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.232,
+ 0.72,
+ 0.825,
+ 0.752
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left| \\lambda_ {i} \\right| \\left(\\Theta \\left(\\frac {1 - \\delta_ {*}}{\\delta_ {*}} \\operatorname {p o l y} \\left(\\eta \\delta_ {*}\\right) + \\epsilon \\sqrt {M}\\right)\\right) = \\left| \\lambda_ {i} \\right| \\beta \\leq c, \\text {f o r s o m e} c > 0 \\text {a n d a l l} i \\in \\mathcal {V} _ {\\Psi}, \\tag {70}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.756,
+ 0.329,
+ 0.771
+ ],
+ "angle": 0,
+ "content": "to hold simultaneously."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.777,
+ 0.825,
+ 0.806
+ ],
+ "angle": 0,
+ "content": "Then, when \\(\\gamma_{i} = k\\) does not hold for all \\(i\\in \\mathcal{V}_{\\Psi}\\) and for some fixed \\(k < 0\\), we can find \\(\\lambda_{i}\\) in the middle of the normalized \\(\\gamma_{i}\\) and \\(\\gamma_{i}^{2}\\) to satisfy (68) and (69), i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.376,
+ 0.813,
+ 0.825,
+ 0.858
+ ],
+ "angle": 0,
+ "content": "\\[\n\\lambda_ {i} \\propto \\frac {\\gamma_ {i}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\frac {\\gamma_ {i} ^ {2}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}. \\tag {71}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.865,
+ 0.446,
+ 0.88
+ ],
+ "angle": 0,
+ "content": "By Cauchy-Schwarz inequality, we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.296,
+ 0.887,
+ 0.825,
+ 0.929
+ ],
+ "angle": 0,
+ "content": "\\[\n- \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} < \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3} < \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}. \\tag {72}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.51,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "22"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.105,
+ 0.224,
+ 0.119
+ ],
+ "angle": 0,
+ "content": "Hence,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.189,
+ 0.128,
+ 0.826,
+ 0.181
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} \\propto \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} + \\frac {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}} = \\frac {\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}} > 0, (7 3)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.188,
+ 0.19,
+ 0.826,
+ 0.245
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\lambda_ {i} \\gamma_ {i} ^ {2} \\propto \\frac {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} = \\frac {\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} > 0. \\tag {74}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.249,
+ 0.313,
+ 0.264
+ ],
+ "angle": 0,
+ "content": "Therefore, by letting"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.345,
+ 0.264,
+ 0.826,
+ 0.315
+ ],
+ "angle": 0,
+ "content": "\\[\n\\lambda_ {i} = C _ {\\gamma} \\cdot \\left(\\frac {\\gamma_ {i}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}}} + \\frac {\\gamma_ {i} ^ {2}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}\\right), \\tag {75}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.32,
+ 0.218,
+ 0.332
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.336,
+ 0.333,
+ 0.826,
+ 0.386
+ ],
+ "angle": 0,
+ "content": "\\[\nC _ {\\gamma} = \\frac {(1 + c) \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}}}{\\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {2}} \\cdot \\sqrt {\\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {4}} + \\sum_ {i \\in \\mathcal {V} _ {\\Psi}} \\gamma_ {i} ^ {3}}, \\tag {76}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.392,
+ 0.482,
+ 0.407
+ ],
+ "angle": 0,
+ "content": "we can obtain (68) and (69) hold if \\(C_{\\gamma} \\lesssim \\beta^{-1}\\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.406,
+ 0.825,
+ 0.434
+ ],
+ "angle": 0,
+ "content": "When \\(\\gamma_{i} = k\\) hold for all \\(i\\in \\mathcal{V}_{\\Psi}\\) and for some fixed \\(k < 0\\) with \\(|\\mathcal{V}_{\\Psi}| > 0\\), we cannot find \\(\\lambda_{i}\\) such that both (68) and (69) hold."
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.808,
+ 0.441,
+ 0.825,
+ 0.454
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.474,
+ 0.394,
+ 0.488
+ ],
+ "angle": 0,
+ "content": "D.3 PROOF OF COROLLARY 1"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.501,
+ 0.825,
+ 0.53
+ ],
+ "angle": 0,
+ "content": "Proof. Let \\(\\{\\pmb{\\mu}_1, \\pmb{v}_1, \\pmb{v}_2, \\dots, \\pmb{v}_M\\} \\cup \\{\\pmb{u}_1, \\pmb{u}_2, \\dots, \\pmb{u}_{d - M + 1}\\}\\) form a set of orthonormal vectors, which is denoted by"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.329,
+ 0.539,
+ 0.826,
+ 0.555
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {U} = \\left(\\boldsymbol {\\mu} _ {1}, \\boldsymbol {v} _ {1}, \\boldsymbol {v} _ {2}, \\dots , \\boldsymbol {v} _ {M}, \\boldsymbol {u} _ {1}, \\boldsymbol {u} _ {2}, \\dots , \\boldsymbol {u} _ {d - M + 1}\\right). \\tag {77}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.563,
+ 0.664,
+ 0.58
+ ],
+ "angle": 0,
+ "content": "Note that for any \\(\\pmb{a},\\pmb{b}\\in \\{\\pmb{\\mu}_1,\\pmb{v}_1,\\pmb{v}_2,\\dots ,\\pmb{v}_M\\} \\cup \\{\\pmb{u}_1,\\pmb{u}_2,\\dots ,\\pmb{u}_{d - M + 1}\\}\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.307,
+ 0.587,
+ 0.826,
+ 0.621
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {a} ^ {\\top} \\boldsymbol {W} ^ {(0)} \\boldsymbol {b} = \\sum_ {1 \\leq i, j \\leq d} a _ {i} b _ {j} W _ {i, j} ^ {(0)} \\sim \\mathcal {N} (0, \\sum_ {1 \\leq i, j \\leq d} | a _ {i} b _ {j} | \\xi^ {2}), \\tag {78}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.631,
+ 0.825,
+ 0.658
+ ],
+ "angle": 0,
+ "content": "where the last step comes from that each entry of \\( \\mathbf{W}^{(0)} \\sim \\mathcal{N}(0, \\xi^2) \\). Given that \\( \\| \\mathbf{a} \\| = \\| \\mathbf{b} \\| = 1 \\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.318,
+ 0.659,
+ 0.826,
+ 0.692
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {1 \\leq i, j \\leq d} | a _ {i} b _ {j} | = \\left(| a _ {1} |, \\dots , | a _ {d} |\\right) ^ {\\top} \\left(| b _ {1} |, \\dots , | b _ {d} |\\right) \\leq 1. \\tag {79}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.698,
+ 0.734,
+ 0.714
+ ],
+ "angle": 0,
+ "content": "By (90), we know that for \\( \\pmb{a} \\in \\{\\pmb{u}_1, \\pmb{u}_2, \\dots, \\pmb{u}_{d - M + 1}\\} \\) and any \\( t = 0, 1, \\dots, T - 1 \\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.722,
+ 0.826,
+ 0.76
+ ],
+ "angle": 0,
+ "content": "\\[\n\\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {a} = 0, \\tag {80}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.384,
+ 0.771,
+ 0.826,
+ 0.809
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {a} ^ {\\top} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} = 0. \\tag {81}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.815,
+ 0.414,
+ 0.83
+ ],
+ "angle": 0,
+ "content": "Then, we have that for some \\(C > 1\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.171,
+ 0.839,
+ 0.829,
+ 0.926
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left[ \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} \\right] _ {i, j} = \\left\\{ \\begin{array}{l l} \\Theta (\\log T), & i = j = 1, \\\\ O \\left(\\epsilon \\cdot \\frac {1}{e ^ {\\Theta (\\log T)} \\cdot \\left(1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}\\right)}\\right) = O \\left(\\epsilon \\cdot T ^ {- C}\\right), & j = 1, 1 \\leq i \\leq M - 1, \\\\ O \\left(\\epsilon \\cdot \\log T\\right), & j \\in [ 2, M - 1 ], i \\in [ 1, M - 1 ], \\\\ O (\\xi), & \\text {e l s e .} \\end{array} \\right. \\tag {82}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "23"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.804,
+ 0.12
+ ],
+ "angle": 0,
+ "content": "Let \\(E_{i,j}\\) be the matrix that only the \\((i,j)\\) entry equals 1, while all other entries are 0. Therefore,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.256,
+ 0.136,
+ 0.824,
+ 0.228
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\left\\| \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\right\\| _ {F} ^ {2} \\\\ \\leq (\\epsilon \\cdot T ^ {- C}) ^ {2} \\cdot (M - 1) + (\\epsilon \\cdot \\log T) ^ {2} \\cdot (M - 1) (M - 2) + \\xi^ {2} (d ^ {2} - M ^ {2}) \\\\ \\leq \\epsilon^ {2} \\log^ {2} T \\cdot M ^ {2} + d ^ {2} / m \\tag {83} \\\\ \\lesssim \\epsilon^ {2} \\cdot M ^ {2} + \\frac {1}{\\log M}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.244,
+ 0.668,
+ 0.26
+ ],
+ "angle": 0,
+ "content": "where the last step comes from that \\(m \\gtrsim M^2 \\log M\\) and \\(M = \\Theta(d)\\). Then,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.354,
+ 0.276,
+ 0.824,
+ 0.353
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\left\\| \\boldsymbol {W} ^ {(T)} - \\boldsymbol {U} \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\cdot \\boldsymbol {U} ^ {\\top} \\right\\| _ {F} \\\\ \\leq \\left\\| \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {U} \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\right\\| _ {F} \\cdot \\left\\| \\boldsymbol {U} ^ {\\top} \\right\\| \\tag {84} \\\\ \\leq \\| \\boldsymbol {U} \\| \\cdot \\| \\boldsymbol {U} ^ {\\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {U} - \\boldsymbol {E} _ {1, 1} \\cdot \\Theta (\\log T) \\| _ {F} \\\\ \\leq \\epsilon M + 1 / \\log M. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.371,
+ 0.825,
+ 0.425
+ ],
+ "angle": 0,
+ "content": "Likewise, by (132), we know that neurons of \\( \\mathbf{V}^{(T)} \\) with a non-trivial magnitude are in the direction of the iterative summation of \\( \\left(\\sum_{s=1}^{P} \\boldsymbol{x}_s^n \\operatorname{softmax}_l(\\boldsymbol{x}_s^{n\\top} \\boldsymbol{W}\\boldsymbol{x}_l^n)\\right) \\). Hence, there exists \\( \\hat{\\boldsymbol{v}}_1 \\in \\mathbb{R}^m \\) and \\( \\hat{\\boldsymbol{v}}_2 \\in \\mathbb{R}^d \\) such that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.27,
+ 0.441,
+ 0.825,
+ 0.476
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {V} ^ {(T)} - \\hat {\\boldsymbol {v}} _ {1} \\hat {\\boldsymbol {v}} _ {2} ^ {\\top} \\right\\| _ {F} \\leq \\Theta (1) \\cdot \\sqrt {m} \\cdot \\sqrt {\\frac {\\log B}{B}} \\cdot \\delta_ {*} ^ {- 2} \\cdot \\delta_ {*} \\cdot \\frac {1}{\\sqrt {m}} \\leq \\delta_ {*} ^ {- 1} \\epsilon \\tag {85}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.493,
+ 0.825,
+ 0.526
+ ],
+ "angle": 0,
+ "content": "Then, for \\(n\\) such that \\(y^{n} = +1\\), we have that the low-rank trained model, where \\(\\boldsymbol{W}_{LR}^{(T)} = \\boldsymbol{U}\\boldsymbol{E}_{1,1} \\cdot \\Theta (\\log T) \\cdot \\boldsymbol{U}^{\\top}\\), satisfies"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.254,
+ 0.541,
+ 0.825,
+ 0.558
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}, \\Psi_ {L R}\\right) \\geq 1 \\cdot \\left(1 - \\delta_ {*} \\epsilon\\right) \\cdot \\left(1 - \\Theta \\left(\\epsilon \\log T\\right)\\right) = 1 - \\Theta \\left(\\left(\\log T + \\delta_ {*}\\right) \\epsilon\\right), \\tag {86}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.573,
+ 0.273,
+ 0.586
+ ],
+ "angle": 0,
+ "content": "which leads to"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.305,
+ 0.598,
+ 0.825,
+ 0.615
+ ],
+ "angle": 0,
+ "content": "\\[\n\\ell \\left(\\boldsymbol {X} ^ {n}, y ^ {n}; \\Psi_ {L R}\\right) \\leq \\Theta \\left(\\epsilon_ {L R}\\right), \\text {w h e r e} \\epsilon_ {L R} = (\\log T + \\delta_ {*}) \\epsilon . \\tag {87}\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.672,
+ 0.395,
+ 0.685
+ ],
+ "angle": 0,
+ "content": "D.4 PROOF OF COROLLARY 2"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.702,
+ 0.825,
+ 0.745
+ ],
+ "angle": 0,
+ "content": "Proof. We know that from Lemma 1, there is a number of \\(\\Omega(m)\\) lucky neurons with large weights. We can denote the set of lucky neurons as \\(\\mathcal{L} \\subset [m]\\). By combining (148) and (163), we have that for any lucky neuron \\(u_i\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.368,
+ 0.755,
+ 0.825,
+ 0.786
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {u} _ {i} \\right\\| \\geq \\eta \\eta^ {- 1} \\delta_ {*} ^ {- 1} \\cdot \\delta_ {*} \\cdot \\frac {1}{\\sqrt {m}} = m ^ {- 1 / 2}. \\tag {88}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.802,
+ 0.466,
+ 0.816
+ ],
+ "angle": 0,
+ "content": "For any unlucky neurons, by (149), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.416,
+ 0.833,
+ 0.825,
+ 0.866
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {u} _ {i} \\right\\| \\leq m ^ {- 1 / 2} \\sqrt {\\frac {\\log B}{B}}. \\tag {89}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.882,
+ 0.826,
+ 0.924
+ ],
+ "angle": 0,
+ "content": "Since that \\( B \\geq \\epsilon^{-2} \\log M \\) by Lemma 1, we have that if we remove neurons from \\( m \\backslash \\mathcal{L} \\), the output in (158) and (159) will only be affected by a factor of \\( \\epsilon \\). Therefore, Lemma 1 still holds, so that Theorems 1-3 all hold."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "24"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.103,
+ 0.414,
+ 0.119
+ ],
+ "angle": 0,
+ "content": "E PROOF OF KEY LEMMAS"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.135,
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+ ],
+ "angle": 0,
+ "content": "E.1 PROOF OF LEMMA 3"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.161,
+ 0.825,
+ 0.19
+ ],
+ "angle": 0,
+ "content": "For ease of presentation, we sometimes use \\(\\mu_{2}\\) to represent \\(-\\mu_{1}\\) in the proof. We first investigate the gradient of \\(W\\), i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.211,
+ 0.198,
+ 0.825,
+ 0.456
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi)}{\\partial \\boldsymbol {W}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi)}{\\partial f (\\boldsymbol {X} ^ {n} ; \\Psi)} \\frac {f (\\boldsymbol {X} ^ {n} ; \\Psi)}{\\partial \\boldsymbol {W}} \\\\ = \\eta \\frac {1}{B} \\sum_ {\\substack {n \\in \\mathcal {B} _ {b} \\\\ P}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] \\tag{90} \\\\ \\cdot \\left(\\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\left(\\boldsymbol {x} _ {s} ^ {n} - \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top}\\right) \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \\mathbb {1} \\left[ V _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] \\\\ \\cdot \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top}\\right) \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.463,
+ 0.325,
+ 0.479
+ ],
+ "angle": 0,
+ "content": "For \\(j,l\\in S_1^n\\) , we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.318,
+ 0.486,
+ 0.826,
+ 0.522
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n ^ {\\top}} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\gtrsim \\frac {e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|}}{\\left| \\mathcal {S} _ {1} ^ {n} \\right| e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + \\left(P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|\\right)} \\tag {91}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.528,
+ 0.391,
+ 0.545
+ ],
+ "angle": 0,
+ "content": "For \\( j \\notin S_1^n \\) and \\( l \\in S_1^n \\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.316,
+ 0.552,
+ 0.825,
+ 0.585
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\frac {1}{\\left| \\mathcal {S} _ {1} ^ {n} \\right| e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + \\left(P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|\\right)}, \\tag {92}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.591,
+ 0.543,
+ 0.608
+ ],
+ "angle": 0,
+ "content": "where \\(\\| \\pmb{q}_1(0)\\| = 0\\). For \\(l\\notin S_1^n\\cup S_2^n\\), \\(j\\in [P]\\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.395,
+ 0.615,
+ 0.826,
+ 0.645
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(0)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\frac {1}{P}. \\tag {93}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.65,
+ 0.373,
+ 0.666
+ ],
+ "angle": 0,
+ "content": "Therefore, for \\(s,r,l\\in S_1^n\\) , let"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.301,
+ 0.674,
+ 0.826,
+ 0.715
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n} := \\beta_ {1} ^ {n} (t) \\boldsymbol {\\mu} _ {1} + \\beta_ {2} ^ {n} (t), \\tag {94}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.722,
+ 0.218,
+ 0.735
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.305,
+ 0.733,
+ 0.826,
+ 0.768
+ ],
+ "angle": 0,
+ "content": "\\[\n\\beta_ {1} ^ {n} (t) \\gtrsim \\frac {P - | \\mathcal {S} _ {1} ^ {n} |}{| \\mathcal {S} _ {1} ^ {n} | e ^ {\\left\\| \\boldsymbol {q} _ {1} (t) \\right\\|} + P - | \\mathcal {S} _ {1} ^ {n} |} := \\phi_ {n} (t) (P - | \\mathcal {S} _ {1} ^ {n} |). \\tag {95}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.438,
+ 0.773,
+ 0.826,
+ 0.814
+ ],
+ "angle": 0,
+ "content": "\\[\n\\beta_ {2} ^ {n} (t) = \\sum_ {l = 2} ^ {M _ {1}} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\tag {96}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.818,
+ 0.218,
+ 0.831
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.422,
+ 0.829,
+ 0.826,
+ 0.864
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left| \\iota_ {l} ^ {\\prime} \\right| \\leq \\beta_ {1} ^ {n} (t) \\frac {\\left| \\mathcal {S} _ {l} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {97}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.866,
+ 0.42,
+ 0.882
+ ],
+ "angle": 0,
+ "content": "Note that \\( |l_{l}^{\\prime}| = 0 \\) if \\( P = |\\mathcal{S}_1^n|, l \\geq 2 \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.882,
+ 0.302,
+ 0.897
+ ],
+ "angle": 0,
+ "content": "If \\( s \\in S_1^n \\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.344,
+ 0.896,
+ 0.826,
+ 0.93
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq \\zeta_ {i, 1, t} \\cdot \\frac {p _ {n} (t)}{\\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {98}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.508,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "25"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.383,
+ 0.12
+ ],
+ "angle": 0,
+ "content": "If \\( s \\in S_2^n \\) and \\( j \\in S_1^n \\), we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.206,
+ 0.122,
+ 0.826,
+ 0.155
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {j} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\phi_ {n} (t) \\cdot \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{p _ {n} (t)}. \\tag {99}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.156,
+ 0.375,
+ 0.173
+ ],
+ "angle": 0,
+ "content": "If \\( s \\notin (S_1^n \\cup S_2^n) \\) and \\( j \\in S_1^n \\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.189,
+ 0.175,
+ 0.826,
+ 0.209
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\lesssim \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {x} _ {j} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {j} ^ {n \\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {x} _ {l} ^ {n}\\right) \\phi_ {n} (t) \\cdot \\frac {\\left| S _ {1} ^ {n} \\right|}{\\sqrt {B} p _ {n} (t)}. \\tag {100}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.21,
+ 0.63,
+ 0.227
+ ],
+ "angle": 0,
+ "content": "Then, by combining (94) to (100), we have that for \\(l \\in S_1^n\\), \\(i \\in \\mathcal{W}_{n,l}\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.212,
+ 0.229,
+ 0.826,
+ 0.27
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {101}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.2,
+ 0.272,
+ 0.412,
+ 0.289
+ ],
+ "angle": 0,
+ "content": "\\[\n\\gtrsim \\zeta_ {i, 1, t} \\cdot p _ {n} (t) \\phi_ {n} (t) (P - | S _ {1} ^ {n} |).\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.291,
+ 0.492,
+ 0.307
+ ],
+ "angle": 0,
+ "content": "For \\( l \\in S_1^n \\), \\( i \\in \\mathcal{W}_{n,l} \\), we have that for \\( k \\neq 1,2 \\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.189,
+ 0.31,
+ 0.825,
+ 0.394
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {102} \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.396,
+ 0.492,
+ 0.413
+ ],
+ "angle": 0,
+ "content": "For \\( l \\in S_1^n \\), \\( i \\in \\mathcal{W}_{n,l} \\), we have that for \\( k \\in [M] \\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.2,
+ 0.416,
+ 0.826,
+ 0.536
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {103} \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|} \\cdot \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| \\phi_ {n} (t)}{p _ {n} (t)}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.538,
+ 0.583,
+ 0.553
+ ],
+ "angle": 0,
+ "content": "For \\(i\\in \\mathcal{U}_{n,l}\\), by the definition of \\(\\mathcal{U}_{n,l}\\) in Definition 4, we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.352,
+ 0.556,
+ 0.826,
+ 0.575
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 0. \\tag {104}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.577,
+ 0.554,
+ 0.593
+ ],
+ "angle": 0,
+ "content": "For \\(i \\notin \\mathcal{W}_{n,l} \\cup \\mathcal{U}_{n,l}\\), we have that for \\(j \\in \\mathcal{W}_{n,l}, k \\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.2,
+ 0.596,
+ 0.826,
+ 0.718
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\tag {105} \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.189,
+ 0.72,
+ 0.826,
+ 0.93
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1} (106) \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}. \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (107) \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)} \\cdot \\frac {| \\mathcal {R} _ {k} ^ {n} |}{P - | \\mathcal {S} _ {1} ^ {n} |}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.51,
+ 0.961
+ ],
+ "angle": 0,
+ "content": "26"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.103,
+ 0.637,
+ 0.121
+ ],
+ "angle": 0,
+ "content": "When \\(l \\notin S_1^n\\), we have that \\(\\pmb{x}_l^{n^\\top} \\pmb{\\mu}_1 = 0\\). If \\(l \\in S_2^n\\), we can obtain that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.202,
+ 0.125,
+ 0.824,
+ 0.2
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\tag {108} \\\\ \\gtrsim \\zeta_ {i, 1, t} \\cdot \\frac {p _ {n} (t) | \\mathcal {S} _ {2} ^ {n} |}{| \\mathcal {S} _ {1} ^ {n} |} \\phi_ {n} (t) (P - | \\mathcal {S} _ {1} ^ {n} |), \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.19,
+ 0.205,
+ 0.824,
+ 0.41
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} (109) \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2}, \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} (110) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {2} ^ {n} \\right|} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| \\phi_ {n} (t)}{p _ {n} (t)}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.411,
+ 0.424,
+ 0.427
+ ],
+ "angle": 0,
+ "content": "where \\(k\\in [M],i\\in \\mathcal{U}_{n,l}\\) . If \\(i\\in \\mathcal{W}_{n,l}\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.354,
+ 0.431,
+ 0.824,
+ 0.449
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 0. \\tag {111}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.452,
+ 0.536,
+ 0.469
+ ],
+ "angle": 0,
+ "content": "If \\(i \\notin \\mathcal{W}_{n,l} \\cup \\mathcal{U}_{n,l}\\), we have that for \\(j \\in \\mathcal{U}_{n,l}\\), \\(k \\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.202,
+ 0.473,
+ 0.824,
+ 0.593
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} \\tag {112} \\\\ \\cdot \\phi_ {n} (t) \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{\\sqrt {B} p _ {n} (t)}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.19,
+ 0.598,
+ 0.824,
+ 0.805
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {\\mu} _ {2} (113) \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}. \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(i, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2} (114) \\\\ \\cdot \\phi_ {n} (t) \\frac {| \\mathcal {S} _ {1} ^ {n} |}{\\sqrt {B} p _ {n} (t)} \\cdot \\frac {| \\mathcal {R} _ {k} ^ {n} |}{P - | \\mathcal {S} _ {1} ^ {n} |}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.807,
+ 0.824,
+ 0.84
+ ],
+ "angle": 0,
+ "content": "If \\( l \\in \\mathcal{R}_k^n \\), \\( k \\in [M] \\), we have that for \\( j \\in \\mathcal{W}_{n,l} \\), if \\( V_{(j,\\cdot)} \\sum_{s=1}^{P} \\pmb{x}_s^n \\mathrm{softmax}_l(\\pmb{x}_s^{n\\top} \\pmb{W} \\pmb{x}_l^n) > 0 \\), \\( l' \\in S_1^n \\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.187,
+ 0.844,
+ 0.824,
+ 0.927
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\mathbf {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k} \\tag {115} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {1}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "27"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.189,
+ 0.101,
+ 0.824,
+ 0.316
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} _ {2} ^ {\\top} \\mathbf {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k} (116) \\\\ = - \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\cdot (\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\boldsymbol {x} _ {r} ^ {n}) \\boldsymbol {x} _ {l} ^ {n} ^ {\\top} \\boldsymbol {v} _ {k}, \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (117) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.321,
+ 0.825,
+ 0.355
+ ],
+ "angle": 0,
+ "content": "Likewise, if \\(l \\in \\mathcal{R}_k^n\\), \\(k \\in [M]\\), \\(\\pmb{V}_{(j,\\cdot)}\\sum_{s=1}^{P}\\pmb{x}_s^n\\mathrm{softmax}_l(\\pmb{x}_s^{n^\\top}\\pmb{W}\\pmb{x}_l^n) > 0\\), \\(j \\in \\mathcal{U}_{n,l}\\), \\(l' \\in S_1^n\\), \\(l'' \\in S_2^n\\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.192,
+ 0.363,
+ 0.824,
+ 0.461
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}, \\tag {118} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.192,
+ 0.363,
+ 0.824,
+ 0.68
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} 0 \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n} ^ {\\top} \\boldsymbol {\\mu} _ {2}, (118) \\\\ \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ = - \\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime \\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime \\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {2}, (119) \\\\ \\boldsymbol {v} _ {k} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l} ^ {n \\top} \\boldsymbol {v} _ {k} \\\\ \\leq \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {V} _ {(j, \\cdot)} \\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\cdot \\left(\\boldsymbol {x} _ {s} ^ {n} - \\sum_ {r = 1} ^ {P} \\operatorname {s o f t m a x} _ {l ^ {\\prime}} \\left(\\boldsymbol {x} _ {r} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n}\\right) \\boldsymbol {x} _ {r} ^ {n}\\right) \\boldsymbol {x} _ {l ^ {\\prime}} ^ {n \\top} \\boldsymbol {\\mu} _ {1} (120) \\\\ \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.684,
+ 0.434,
+ 0.699
+ ],
+ "angle": 0,
+ "content": "Therefore, by the update rule, we know"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.321,
+ 0.707,
+ 0.825,
+ 0.79
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} - \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {\\mu} _ {1} \\tag {121} \\\\ = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} + K (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 2} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}, \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.798,
+ 0.218,
+ 0.811
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.326,
+ 0.81,
+ 0.826,
+ 0.849
+ ],
+ "angle": 0,
+ "content": "\\[\nK (t) \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\zeta_ {1, t} p _ {n} (t) \\phi_ {n} (t) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|), \\tag {122}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.334,
+ 0.856,
+ 0.825,
+ 0.89
+ ],
+ "angle": 0,
+ "content": "\\[\n\\iota_ {l} ^ {\\prime} \\leq K (t) \\cdot \\max _ {n} \\left\\{\\frac {| S _ {1} ^ {n} | \\phi_ {n} (t)}{p _ {n} (t)} \\right\\} \\leq K (t) \\cdot e ^ {- q _ {1} (t)}. \\tag {123}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.895,
+ 0.268,
+ 0.908
+ ],
+ "angle": 0,
+ "content": "We know that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.452,
+ 0.907,
+ 0.825,
+ 0.926
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {W} ^ {(0)} \\boldsymbol {\\mu} _ {1} \\approx 0. \\tag {124}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.509,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "28"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.105,
+ 0.217,
+ 0.119
+ ],
+ "angle": 0,
+ "content": "Then,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.401,
+ 0.119,
+ 0.824,
+ 0.219
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} q _ {1} (t + 1) = \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} \\\\ = \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {1} + K (t) \\\\ = q _ {1} (t) + K (t) \\tag {125} \\\\ = \\sum_ {b = 0} ^ {t} K (b). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.224,
+ 0.241,
+ 0.24
+ ],
+ "angle": 0,
+ "content": "Similarly,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.323,
+ 0.239,
+ 0.825,
+ 0.282
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {2} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {2} - \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {W} ^ {(t)}} \\boldsymbol {\\mu} _ {2} \\tag {126}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.4,
+ 0.283,
+ 0.62,
+ 0.313
+ ],
+ "angle": 0,
+ "content": "\\[\n= \\boldsymbol {W} ^ {(t)} \\boldsymbol {\\mu} _ {2} + K (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l \\neq 2} \\iota_ {l} ^ {\\prime} \\boldsymbol {\\mu} _ {l}.\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.406,
+ 0.322,
+ 0.824,
+ 0.362
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {\\mu} _ {2} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {2} = \\sum_ {b = 0} ^ {t} K (b). \\tag {127}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.367,
+ 0.264,
+ 0.384
+ ],
+ "angle": 0,
+ "content": "For \\(k\\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.312,
+ 0.384,
+ 0.825,
+ 0.425
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {v} _ {k} = \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} + J _ {1} (t) \\boldsymbol {\\mu} _ {1} + J _ {2} (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l = 1} ^ {M} \\iota_ {l} ^ {\\prime} \\boldsymbol {v} _ {l}. \\tag {128}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.43,
+ 0.528,
+ 0.446
+ ],
+ "angle": 0,
+ "content": "By Hoeffding's inequality (15), with high probability,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.295,
+ 0.455,
+ 0.825,
+ 0.495
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {v} _ {k} \\right\\| \\leq \\Theta (1) \\cdot \\sqrt {\\frac {\\log B}{B}} \\sum_ {b = 0} ^ {t} K (b) \\lesssim \\epsilon \\cdot \\sum_ {b = 0} ^ {t} K (b), \\tag {129}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.504,
+ 0.651,
+ 0.522
+ ],
+ "angle": 0,
+ "content": "where the second step holds if \\(B \\geq \\epsilon^{-2} \\log M\\). And for \\(j \\neq k\\), \\(j \\in [M]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.398,
+ 0.528,
+ 0.825,
+ 0.548
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left\\| \\boldsymbol {v} _ {j} ^ {\\top} \\boldsymbol {W} ^ {(t)} \\boldsymbol {v} _ {k} \\right\\| \\leq K (t) e ^ {- q _ {1} (t)}. \\tag {130}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.557,
+ 0.823,
+ 0.588
+ ],
+ "angle": 0,
+ "content": "For any \\(\\pmb{\\mu}'\\) such that \\(\\pmb{\\mu}_1^\\top \\pmb{\\mu}' = \\alpha\\) and \\(\\pmb{\\mu}' \\perp \\{v_1, v_2, \\dots, v_M\\}\\), we can write \\(\\pmb{\\mu}'\\) as \\(\\alpha \\pmb{\\mu}_1 \\pm \\sqrt{1 - \\alpha^2} \\pmb{\\mu}_\\perp\\) for some \\(\\pmb{\\mu}_\\perp \\perp \\{\\pmb{\\mu}_1, v_1, v_2, \\dots, v_M\\}\\). Therefore,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.26,
+ 0.597,
+ 0.825,
+ 0.64
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\boldsymbol {\\mu} ^ {\\prime} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} ^ {\\prime} = \\left(\\alpha \\boldsymbol {\\mu} _ {1} \\pm \\sqrt {1 - \\alpha^ {2}} \\boldsymbol {\\mu} _ {\\perp}\\right) ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\left(\\alpha \\boldsymbol {\\mu} _ {1} \\pm \\sqrt {1 - \\alpha^ {2}} \\boldsymbol {\\mu} _ {\\perp}\\right) \\tag {131} \\\\ = \\alpha^ {2} \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1} \\pm \\Theta (\\epsilon) \\cdot \\boldsymbol {\\mu} _ {1} ^ {\\top} \\boldsymbol {W} ^ {(t + 1)} \\boldsymbol {\\mu} _ {1}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.657,
+ 0.361,
+ 0.671
+ ],
+ "angle": 0,
+ "content": "E.2 PROOF OF LEMMA 4"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.684,
+ 0.683,
+ 0.699
+ ],
+ "angle": 0,
+ "content": "For ease of presentation, we sometimes use \\(\\pmb{\\mu}_{2}\\) to represent \\(-\\pmb{\\mu}_{1}\\) in the proof."
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.279,
+ 0.706,
+ 0.825,
+ 0.797
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)} \\frac {f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\tag {132} \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.279,
+ 0.706,
+ 0.825,
+ 0.875
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)} \\frac {f \\left(\\boldsymbol {X} ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V} _ {(i , .)}} \\tag {132} \\\\ = \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} (- y ^ {n}) \\frac {1}{P} \\sum_ {l = 1} ^ {P} a _ {(l) _ {i}} \\mathbb {1} [ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} (\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}) \\geq 0 ] \\\\ \\cdot \\left(\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.883,
+ 0.522,
+ 0.898
+ ],
+ "angle": 0,
+ "content": "For \\(n\\) such that \\(y^{n} = +1\\) and \\(i\\in \\mathcal{W}_{n,l}\\), we have that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.359,
+ 0.907,
+ 0.825,
+ 0.927
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) \\geq 0 \\right] = 1, \\tag {133}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.509,
+ 0.961
+ ],
+ "angle": 0,
+ "content": "29"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.104,
+ 0.279,
+ 0.121
+ ],
+ "angle": 0,
+ "content": "and for \\(l\\in S_1^n\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.286,
+ 0.128,
+ 0.826,
+ 0.17
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} (t) \\boldsymbol {\\mu} _ {1} + \\sum_ {l = 1} ^ {M _ {2}} \\iota_ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + \\iota_ {M _ {2} + 1} ^ {\\prime} \\boldsymbol {\\mu} _ {2}, \\tag {134}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.177,
+ 0.218,
+ 0.189
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.401,
+ 0.187,
+ 0.826,
+ 0.222
+ ],
+ "angle": 0,
+ "content": "\\[\n\\iota_ {l} ^ {\\prime} \\leq (1 - p _ {n} (t)) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|}. \\tag {135}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.224,
+ 0.301,
+ 0.24
+ ],
+ "angle": 0,
+ "content": "If \\(l\\in \\mathcal{S}_2^n\\) , we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.282,
+ 0.247,
+ 0.826,
+ 0.289
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {2} + \\sum_ {l = 1} ^ {M _ {2}} \\kappa_ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + \\kappa_ {M _ {2} + 1} ^ {\\prime} \\boldsymbol {\\mu} _ {2}, \\tag {136}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.297,
+ 0.218,
+ 0.309
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.446,
+ 0.309,
+ 0.826,
+ 0.327
+ ],
+ "angle": 0,
+ "content": "\\[\np _ {n} ^ {\\prime} (t) \\leq p _ {n} (t), \\tag {137}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.399,
+ 0.33,
+ 0.826,
+ 0.365
+ ],
+ "angle": 0,
+ "content": "\\[\n\\kappa_ {l} ^ {\\prime} \\leq (1 - p _ {n} (t)) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {2} ^ {n} \\right|}. \\tag {138}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.367,
+ 0.367,
+ 0.384
+ ],
+ "angle": 0,
+ "content": "If \\(l\\in \\mathcal{R}_k^n\\) \\(k\\in [M]\\) , we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.251,
+ 0.392,
+ 0.826,
+ 0.434
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {s = 1} ^ {P} \\boldsymbol {x} _ {s} ^ {n} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right) = p _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {1} + p _ {n} ^ {\\prime \\prime} (t) \\boldsymbol {\\mu} _ {2} + o _ {n} (t) \\boldsymbol {v} _ {k} + \\sum_ {l \\neq k} u _ {l} ^ {\\prime} \\boldsymbol {v} _ {l}, \\tag {139}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.442,
+ 0.218,
+ 0.454
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.424,
+ 0.452,
+ 0.826,
+ 0.482
+ ],
+ "angle": 0,
+ "content": "\\[\np _ {n} ^ {\\prime} (t) \\leq \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot p _ {n} (t), \\tag {140}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.425,
+ 0.485,
+ 0.826,
+ 0.516
+ ],
+ "angle": 0,
+ "content": "\\[\np _ {n} ^ {\\prime \\prime} (t) \\leq \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{P} \\cdot p _ {n} (t), \\tag {141}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.426,
+ 0.519,
+ 0.826,
+ 0.55
+ ],
+ "angle": 0,
+ "content": "\\[\no _ {n} (t) \\leq \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{P} \\cdot p _ {n} (t) \\tag {142}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.276,
+ 0.552,
+ 0.826,
+ 0.588
+ ],
+ "angle": 0,
+ "content": "\\[\nu _ {l} ^ {\\prime} \\leq \\left(1 - \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right| + \\left| \\mathcal {S} _ {2} ^ {n} \\right| + \\left| \\mathcal {R} _ {k} ^ {n} \\right|}{\\left| \\mathcal {S} _ {1} ^ {n} \\right|} \\cdot p _ {n} (t)\\right) \\cdot \\frac {\\left| \\mathcal {R} _ {k} ^ {l} \\right|}{P - \\left| \\mathcal {S} _ {1} ^ {n} \\right| - \\left| \\mathcal {S} _ {2} ^ {n} \\right| - \\left| \\mathcal {R} _ {k} ^ {n} \\right|}. \\tag {143}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.59,
+ 0.304,
+ 0.604
+ ],
+ "angle": 0,
+ "content": "Therefore, we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.285,
+ 0.612,
+ 0.826,
+ 0.655
+ ],
+ "angle": 0,
+ "content": "\\[\n- \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell \\left(\\boldsymbol {X} ^ {n} , y ^ {n} ; \\Psi\\right)}{\\partial \\boldsymbol {V}} = \\sum_ {l = 1} ^ {M} u _ {l} ^ {\\prime} \\boldsymbol {v} _ {l} + q _ {n} (t) \\boldsymbol {\\mu} _ {1} + q _ {n} ^ {\\prime} (t) \\boldsymbol {\\mu} _ {2}, \\tag {144}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.662,
+ 0.218,
+ 0.675
+ ],
+ "angle": 0,
+ "content": "where"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.391,
+ 0.673,
+ 0.826,
+ 0.71
+ ],
+ "angle": 0,
+ "content": "\\[\nq _ {n} (t) ^ {\\prime} \\gtrsim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (t), \\tag {145}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.39,
+ 0.715,
+ 0.826,
+ 0.753
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left| q _ {n} ^ {\\prime} (t) \\right| \\lesssim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (t), \\tag {146}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.366,
+ 0.757,
+ 0.826,
+ 0.796
+ ],
+ "angle": 0,
+ "content": "\\[\n\\left| u _ {k} ^ {\\prime} \\right| \\lesssim \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {R} _ {k} ^ {n} \\right|}{a P} \\cdot (1 - p _ {n} (t)) \\frac {1}{M}. \\tag {147}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.173,
+ 0.799,
+ 0.217,
+ 0.813
+ ],
+ "angle": 0,
+ "content": "Then,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.37,
+ 0.813,
+ 0.826,
+ 0.854
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\geq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| S _ {1} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {148}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.424,
+ 0.859,
+ 0.826,
+ 0.882
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2} = - \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {149}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.391,
+ 0.887,
+ 0.826,
+ 0.927
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {150}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.948,
+ 0.51,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "30"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.47,
+ 0.12
+ ],
+ "angle": 0,
+ "content": "for \\(k\\in [M]\\) . For \\(i\\in \\mathcal{U}_{n,l}\\) , we similarly have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.37,
+ 0.124,
+ 0.826,
+ 0.165
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2} \\geq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| S _ {2} ^ {n} \\right|}{a P} \\cdot p _ {n} (b), \\tag {151}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.425,
+ 0.169,
+ 0.826,
+ 0.192
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} = - \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {2}, \\tag {152}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.391,
+ 0.194,
+ 0.825,
+ 0.235
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\eta \\sum_ {b = 0} ^ {t - 1} \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P M}, \\tag {153}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.237,
+ 0.525,
+ 0.253
+ ],
+ "angle": 0,
+ "content": "for some \\(k\\in [M]\\) . For \\(i\\notin \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}\\) , we have that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.402,
+ 0.256,
+ 0.826,
+ 0.289
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k} \\leq \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {v} _ {k}, \\tag {154}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.402,
+ 0.293,
+ 0.826,
+ 0.326
+ ],
+ "angle": 0,
+ "content": "\\[\n\\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1} \\leq \\sqrt {\\frac {\\log B}{B}} \\boldsymbol {V} _ {(j, \\cdot)} ^ {(t)} \\boldsymbol {\\mu} _ {1}, \\tag {155}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.326,
+ 0.395,
+ 0.342
+ ],
+ "angle": 0,
+ "content": "where \\(k\\in [M],j\\in \\mathcal{W}_{n,l}\\cup \\mathcal{U}_{n,l}\\)"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.357,
+ 0.358,
+ 0.371
+ ],
+ "angle": 0,
+ "content": "E.3 PROOF OF LEMMA 1"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.383,
+ 0.801,
+ 0.399
+ ],
+ "angle": 0,
+ "content": "We know that by Lemma 3 and 4 in (Li et al., 2023a), for \\( i \\in \\mathcal{W}_{n,l}(0) \\) and \\( l \\in S_1^n \\), we have that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.432,
+ 0.403,
+ 0.825,
+ 0.425
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {R} _ {l} ^ {n} (t) \\right] = 1, \\tag {156}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.428,
+ 0.47,
+ 0.444
+ ],
+ "angle": 0,
+ "content": "and for \\(i\\in \\mathcal{U}_{n,l}(0)\\) and \\(l\\in S_2^n\\) , we have that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.432,
+ 0.447,
+ 0.825,
+ 0.469
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {1} \\left[ \\boldsymbol {V} _ {(i, \\cdot)} ^ {(t)} \\boldsymbol {R} _ {l} ^ {n} (t) \\right] = 1. \\tag {157}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.472,
+ 0.825,
+ 0.5
+ ],
+ "angle": 0,
+ "content": "We also have that the size of \\(\\mathcal{W}_{n,l}\\) and \\(\\mathcal{V}_{n,l}\\) are larger than \\(\\Omega(m)\\). Therefore, for \\(y^n = +1\\), by Lemma 4 and 3, we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.221,
+ 0.504,
+ 0.826,
+ 0.644
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} f \\left(\\boldsymbol {X} ^ {n}; \\Psi\\right) = \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in \\mathcal {W} _ {l, n} (0)} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ + \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\notin \\mathcal {W} _ {l, n} (0), a _ {(l) _ {i}} > 0} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\tag {158} \\\\ - \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i: a _ {(l) _ {i}} < 0} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n} ^ {\\top} \\boldsymbol {W} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.647,
+ 0.267,
+ 0.66
+ ],
+ "angle": 0,
+ "content": "We know that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.299,
+ 0.658,
+ 0.826,
+ 0.78
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\sum_ {i \\in \\mathcal {W} _ {l, n} (0)} \\frac {1}{a} \\operatorname {R e l u} \\left(\\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {X} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {X} ^ {n ^ {\\top}} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right)\\right) \\\\ \\gtrsim \\frac {\\left| S _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a} \\cdot \\zeta_ {T} \\cdot p _ {n} (T) \\tag {159} \\\\ \\gtrsim \\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a ^ {2}} \\cdot \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{P} p _ {h} (b) \\cdot p _ {n} (T). \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.781,
+ 0.299,
+ 0.794
+ ],
+ "angle": 0,
+ "content": "We can derive that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.223,
+ 0.797,
+ 0.826,
+ 0.928
+ ],
+ "angle": 0,
+ "content": "\\[\n\\begin{array}{l} q _ {1} (T) = \\sum_ {b = 0} ^ {T - 1} K (b) \\\\ \\geq \\sum_ {b = 0} ^ {T - 1} \\eta \\frac {1}{B} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {m \\left| \\mathcal {S} _ {1} ^ {n} \\right|}{a P} p _ {n} (b) \\phi_ {n} (b) (P - \\left| \\mathcal {S} _ {1} ^ {n} \\right|) \\eta \\sum_ {c = 0} ^ {b - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {c}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{a P} p _ {h} (c) \\tag {160} \\\\ \\gtrsim \\delta_ {*} ^ {4} \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{e ^ {q _ {1} (b)}}. \\\\ \\end{array}\n\\]"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.949,
+ 0.508,
+ 0.96
+ ],
+ "angle": 0,
+ "content": "31"
+ }
+ ],
+ [
+ {
+ "type": "header",
+ "bbox": [
+ 0.173,
+ 0.033,
+ 0.48,
+ 0.049
+ ],
+ "angle": 0,
+ "content": "Published as a conference paper at ICLR 2025"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.104,
+ 0.825,
+ 0.134
+ ],
+ "angle": 0,
+ "content": "Therefore, we have that when \\( q_{1}(T) \\leq O(1) \\) or \\( q_{1}(T) \\geq \\Theta(T^{c}) \\) for \\( c = \\Theta(1) \\), (160) does not hold. When \\( q_{1}(T) = \\Theta(\\log T) \\), we have that (160) holds. In this case,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.343,
+ 0.139,
+ 0.826,
+ 0.171
+ ],
+ "angle": 0,
+ "content": "\\[\np _ {n} (T) \\geq \\frac {\\delta_ {*} T ^ {C}}{\\delta_ {*} T ^ {C} + 1 - \\delta_ {*}} \\geq 1 - \\frac {1 - \\delta_ {*}}{\\delta_ {*}} T ^ {- C}, \\tag {161}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.177,
+ 0.608,
+ 0.195
+ ],
+ "angle": 0,
+ "content": "where \\(C > 1\\). Meanwhile, for \\(l \\in \\mathcal{R}_k^n\\), \\(k \\in [M]\\), and any \\(s \\in [P]\\)"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.38,
+ 0.199,
+ 0.826,
+ 0.228
+ ],
+ "angle": 0,
+ "content": "\\[\n\\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {n \\top} \\boldsymbol {W} ^ {(T)} \\boldsymbol {x} _ {l} ^ {n}\\right) = \\Theta \\left(\\frac {1}{P}\\right). \\tag {162}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.24,
+ 0.4,
+ 0.255
+ ],
+ "angle": 0,
+ "content": "We can then derive that as long as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.451,
+ 0.251,
+ 0.826,
+ 0.27
+ ],
+ "angle": 0,
+ "content": "\\[\nT \\gtrsim \\eta^ {- 1} \\delta_ {*} ^ {- 2}, \\tag {163}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.273,
+ 0.232,
+ 0.285
+ ],
+ "angle": 0,
+ "content": "we have"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.334,
+ 0.283,
+ 0.826,
+ 0.325
+ ],
+ "angle": 0,
+ "content": "\\[\n\\frac {\\left| \\mathcal {S} _ {1} ^ {n} \\right|}{P} \\cdot \\frac {m}{a ^ {2}} \\cdot \\eta \\sum_ {b = 0} ^ {T - 1} \\frac {1}{B} \\sum_ {h \\in \\mathcal {B} _ {b}} \\frac {\\left| \\mathcal {S} _ {1} ^ {h} \\right|}{P} p _ {h} (b) \\cdot p _ {n} (T) \\geq 1. \\tag {164}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.328,
+ 0.216,
+ 0.342
+ ],
+ "angle": 0,
+ "content": "Then,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.381,
+ 0.34,
+ 0.826,
+ 0.358
+ ],
+ "angle": 0,
+ "content": "\\[\nf \\left(\\boldsymbol {X} ^ {n}; \\Psi\\right) \\geq 1, \\ell \\left(\\boldsymbol {X} ^ {n}, y ^ {n}; \\Psi\\right) = 0. \\tag {165}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.36,
+ 0.408,
+ 0.374
+ ],
+ "angle": 0,
+ "content": "With (163), we can also derive that"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.381,
+ 0.381,
+ 0.826,
+ 0.422
+ ],
+ "angle": 0,
+ "content": "\\[\n\\sum_ {k = 1} ^ {M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {v} _ {k} \\right\\| ^ {2} \\lesssim \\frac {1}{M} \\left\\| \\boldsymbol {V} _ {(i, \\cdot)} ^ {(T)} \\boldsymbol {\\mu} _ {1} \\right\\| ^ {2}, \\tag {166}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.43,
+ 0.825,
+ 0.465
+ ],
+ "angle": 0,
+ "content": "which means that for \\( i \\in \\mathcal{W}_{n,l} \\) with \\( l \\in S_1^n \\), \\( V_{(i,\\cdot)}^{(T)} \\) is mainly in the direction of \\( \\pmb{\\mu}_1 \\). This verifies condition (B) of Lemma 1. Therefore, by Hoeffding's inequality (15), for any \\( W' \\in \\Psi \\),"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.208,
+ 0.47,
+ 0.826,
+ 0.511
+ ],
+ "angle": 0,
+ "content": "\\[\n\\Pr \\left( \\right.\\left\\| \\frac {1}{| \\mathcal {B} _ {b} |} \\sum_ {n \\in \\mathcal {B} _ {b}} \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} - \\mathbb {E} \\left[ \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} \\right]\\right\\| \\geq \\left| \\right. \\mathbb {E} \\left[ \\frac {\\partial \\ell (\\Psi ; \\boldsymbol {P} ^ {n} , z ^ {n})}{\\partial \\boldsymbol {W} ^ {\\prime}} \\right] \\epsilon\\left. \\right) \\tag {167}\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.196,
+ 0.513,
+ 0.318,
+ 0.534
+ ],
+ "angle": 0,
+ "content": "\\[\n\\leq e ^ {- B \\epsilon^ {2}} \\leq M ^ {- C},\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.54,
+ 0.243,
+ 0.554
+ ],
+ "angle": 0,
+ "content": "as long as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.443,
+ 0.551,
+ 0.826,
+ 0.568
+ ],
+ "angle": 0,
+ "content": "\\[\nB \\gtrsim \\epsilon^ {- 2} \\log M. \\tag {168}\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.172,
+ 0.572,
+ 0.216,
+ 0.585
+ ],
+ "angle": 0,
+ "content": "Then,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.405,
+ 0.584,
+ 0.826,
+ 0.602
+ ],
+ "angle": 0,
+ "content": "\\[\n\\mathbb {E} _ {(\\boldsymbol {X}, y) \\sim \\mathcal {D} _ {\\tau}} \\ell (\\boldsymbol {X}, y; \\Psi) \\leq \\epsilon . \\tag {169}\n\\]"
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.172,
+ 0.62,
+ 0.538,
+ 0.635
+ ],
+ "angle": 0,
+ "content": "F EXTENSION TO MULTI-CLASSIFICATION"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.171,
+ 0.651,
+ 0.825,
+ 0.711
+ ],
+ "angle": 0,
+ "content": "Define that a \\(2^{c}\\)-classification is achieved by \\(c\\) times of binary classification with the orthonormal set \\(\\{\\pmb{\\mu}_{\\mathcal{T}}^{(1)}, \\dots, \\pmb{\\mu}_{\\mathcal{T}}^{(c)}\\}\\) as the discriminative patterns for the task \\(\\mathcal{T}\\). We have \\(\\pmb{\\mu}_{\\mathcal{T}}^{(i)} \\perp \\pmb{v}_m\\), \\(m \\in [M]\\), \\(i \\in [c]\\). The label \\(\\pmb{y}\\) is \\(c\\)-dimensional with each entry chosen from \\(\\{+1, -1\\}\\). Specifically, each \\((X \\in \\mathbb{R}^{d \\times P}, y \\in \\mathbb{R}^c) \\sim \\mathcal{D}_{\\mathcal{T}}\\) is generated as follows:"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.216,
+ 0.721,
+ 0.825,
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+ ],
+ "angle": 0,
+ "content": "- Randomly generate the \\(k\\)-th entry \\(y_{k}, k \\in [c]\\) of the label \\(\\mathbf{y}\\) from \\(\\{+1, -1\\}\\) with an equal probability."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "- Each token is randomly chosen from \\(\\{\\pmb{\\mu}_{\\mathcal{T}}^{(i)}, - \\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\}_{i = 1}^{c}\\cup \\{\\pmb{v}_1,\\dots ,\\pmb{v}_M\\}\\). If \\(y_{k} = 1\\) (or \\(-1\\)), the number of tokens corresponding to \\(\\pmb{\\mu}_{\\mathcal{T}_k}\\) (or \\(-\\pmb{\\mu}_{\\mathcal{T}_k}\\)) is larger than that of \\(-\\pmb{\\mu}_{\\mathcal{T}_k}\\) (or \\(\\pmb{\\mu}_{\\mathcal{T}_k}\\)). \\(\\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\) and \\(-\\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\) (or “\\(-\\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\) and \\(\\pmb{\\mu}_{\\mathcal{T}}^{(i),}\\)” are referred to label-relevant and confusion patterns for \\(y_{k} = 1\\) (or \\(y_{k} = -1\\)), respectively. The average fractions of label-relevant and confusion tokens of \\(\\pmb{\\mu}_{\\mathcal{T}}^{(i)}\\) are \\(\\delta_{*}^{(i)}\\) and \\(\\delta_{\\#}^{(i)}\\), respectively."
+ },
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+ "type": "list",
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+ "bbox": [
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+ "angle": 0,
+ "content": "We then need \\(c\\) sets of our binary model (4) to generate the output for \\(2^{c}\\)-classification, i.e.,"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.243,
+ 0.866,
+ 0.582,
+ 0.884
+ ],
+ "angle": 0,
+ "content": "\\[\nf (\\boldsymbol {X}; \\Psi) = \\left(f _ {1} (\\boldsymbol {X}; \\Psi), f _ {2} (\\boldsymbol {X}; \\Psi), \\dots , f _ {c} (\\boldsymbol {X}; \\Psi)\\right)\n\\]"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.243,
+ 0.887,
+ 0.826,
+ 0.929
+ ],
+ "angle": 0,
+ "content": "\\[\nf _ {i} (\\boldsymbol {X}; \\Psi) = \\frac {1}{P} \\sum_ {l = 1} ^ {P} \\boldsymbol {a} _ {(l) _ {i}} ^ {\\top} \\operatorname {R e l u} \\left(\\boldsymbol {W} _ {O _ {i}} \\sum_ {s = 1} ^ {P} \\boldsymbol {W} _ {V _ {i}} \\boldsymbol {x} _ {s} \\operatorname {s o f t m a x} _ {l} \\left(\\boldsymbol {x} _ {s} ^ {\\top} \\boldsymbol {W} _ {K _ {i}} ^ {\\top} \\boldsymbol {W} _ {Q _ {i}} \\boldsymbol {x} _ {l}\\right)\\right), \\tag {170}\n\\]"
+ },
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+ "type": "page_number",
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+ "content": "Published as a conference paper at ICLR 2025"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "with \\(\\Psi = \\{\\{a_{(l)i}\\}_{l=1}^{P}, W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}\\}_{i=1}^{c}\\). The dimensions of \\(W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}\\), \\(i \\in [c]\\) follow Section 3.2."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The learning process is then \\(c\\) independent and parallel binary classification problems for each entry of the \\(c\\)-dimensional output. After fine-tuning, the trained model of each output entry has a similar property to Lemma 1 for single binary classification. \\(\\delta_{*}^{(i)}\\), the fraction of label-relevant pattern \\(\\mu_{\\mathcal{T}}^{(i)}\\), \\(i \\in [c]\\), may decrease by \\(c\\) times in average from the binary classification scenario. Therefore, by condition (iii) of Theorem 1, the number of iterations and samples increases by \\(c^2\\) times, which is a polynomial of log scale of the number of classes \\(2^c\\). Then, for the disriminative patterns \\(\\{\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\}_{i=1}^c\\) of task \\(\\mathcal{T}_1\\) and \\(\\{\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}\\}_{i=1}^c\\) and \\(\\mathcal{T}_2\\) of task \\(\\mathcal{T}_2\\), if for any \\(\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\), there exists a unique \\(\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}\\) close to be orthogonal to \\(\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\), then \\(\\mathcal{T}_1\\) and \\(\\mathcal{T}_2\\) are irrelevant. If for any \\(\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\), there exists a unique \\(\\pmb{\\mu}_{\\mathcal{T}_2}^{(i)}\\) with a small angle to (or almost opposite to) \\(\\pmb{\\mu}_{\\mathcal{T}_1}^{(i)}\\), then \\(\\mathcal{T}_1\\) and \\(\\mathcal{T}_2\\) are aligned (or contradictory). We can then derive similar conclusions as our Theorems 1 and 2 by combining the results of all the output entries."
+ },
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+# WHEN IS TASK VECTOR Provably EFFECTIVE FOR MODEL EDITING? A GENERALIZATION ANALYSIS OF NONLINEAR TRANSFORMERS
+
+Hongkang Li $^{1}$ , Yihua Zhang $^{2}$ , Shuai Zhang $^{3}$ , Pin-Yu Chen $^{4}$ , Sijia Liu $^{2,4}$ , Meng Wang $^{1,*}$ $^{1}$ Rensselaer Polytechnic Institute, $^{2}$ Michigan State University, $^{3}$ New Jersey Institute of Technology, $^{4}$ IBM Research
+
+# ABSTRACT
+
+Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors, each of which is the weight update from the pre-trained model to fine-tuned models for certain tasks. This approach recently gained attention as a computationally efficient inference method for model editing, e.g., multi-task learning, forgetting, and out-of-domain generalization capabilities. However, the theoretical understanding of why task vectors can execute various conceptual operations remains limited, due to the highly non-convexity of training Transformer-based models. To the best of our knowledge, this paper provides the first theoretical characterization of the generalization guarantees of task vector methods on nonlinear Transformers. We consider a conceptual learning setting, where each task is a binary classification problem based on a discriminative pattern. We theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or aligned tasks, as well as the success of task negation in unlearning one task from irrelevant or contradictory tasks. Moreover, we prove the proper selection of linear coefficients for task arithmetic to achieve guaranteed generalization to out-of-domain tasks. All of our theoretical results hold for both dense-weight parameters and their low-rank approximations. Although established in a conceptual setting, our theoretical findings were validated on a practical machine unlearning task using the large language model Phi-1.5 (1.3B).
+
+# 1 INTRODUCTION
+
+Large pre-trained models (Chowdhery et al., 2022; Touvron et al., 2023; Achiam et al., 2023) have recently served as a foundational module in deep learning systems. Under the pre-training-and-fine-tuning paradigm, although the traditional and straightforward full-parameter fine-tuning can demonstrate superior performance in downstream tasks, its immense computational and memory costs have become a serious practical issue. Consequently, many Parameter-Efficient Fine-Tuning (PEFT) methods (Li & Liang, 2021; Hu et al., 2022; Jia et al., 2022; Wei et al., 2022b;a) have been proposed to address this concern. Among them, the recent task vector approach receives increasing attention (Ilharco et al., 2022a; Ortiz-Jimenez et al., 2023; Hendel et al., 2023; Todd et al., 2024).
+
+The task vector approach first fine-tunes a pre-trained model on several simpler tasks to obtain task vectors, which represent the weight differences between the fine-tuned models and the pre-trained model. To handle more complex tasks, a proper model can be edited by adding a linear combination of these task vectors to the pre-trained model. Since this approach only requires determining the appropriate arithmetic hyperparameters, with no need for further fine-tuning on complicated tasks, the task vector method offers a significant efficiency advantage and is particularly effective when adapting to a wide range of downstream tasks. Empirical evidence shows that adding multiple task vectors can improve the model's performance on corresponding tasks, while subtracting certain task vectors allows the model to forget associated tasks. A proper linear combination of task vectors can even enable the model to generalize on an out-of-domain task that has an analogous relationship with the given task vectors, without needing labeled data. Additionally, it has been found that using low-
+
+rank and/or sparse task vectors can further improve efficiency while maintaining the performance (Yadav et al., 2023; Chitale et al., 2023; Yu et al., 2024; He et al., 2025).
+
+Despite empirical successes, theoretical analysis of task vectors is less investigated. In particular, we ask the following question:
+
+When and why can the task vector approach perform well in multi-task learning, unlearning, and out-of-domain generalization successfully and efficiently?
+
+Some related theoretical works focus on analyzing the performance of machine unlearning from a purely optimization perspective (Ginart et al., 2019; Neel et al., 2021; Guo et al., 2020; Mu & Klabjan, 2024). However, these analyses do not apply to Transformer-based neural networks, which are key components of large pre-trained models. Moreover, these works cannot be extended to study multi-task learning or out-of-domain generalization to new tasks. Frankle et al. (2020) proposes the concept of linear mode connectivity, suggesting that there exists a small-loss connected region in the loss landscape of the model, thereby demonstrating that linear interpolation between models can yield good performance. The most relevant work to this paper is (Ortiz-Jimenez et al., 2023), which uses the Neural Tangent Kernel (NTK) framework (Jacot et al., 2018) to study neural networks as linearized models under specific assumptions, to justify the use of linear arithmetic on task vectors for targeted model editing. However, this work does not have generalization guarantees and cannot explain the success of task vectors in nonlinear models without NTK assumptions.
+
+# 1.1 MAJOR CONTRIBUTIONS
+
+To the best of our knowledge, this work is the first theoretical generalization analysis of task arithmetic on a nonlinear Transformer model for multi-task learning, unlearning, and out-of-domain generalization. Focusing on binary classification tasks, we provide a quantitative analysis of the dependence of the task arithmetic effect on arithmetic hyperparameters. Although our analysis is centered on a simplified single-head and one-layer nonlinear Transformer, our theoretical insights are validated on practical architectures. Our major contributions include:
+
+1. A fine-grained feature-learning analysis of the effectiveness of task addition and negation. We consider a data model in which binary labels are determined by the majority of discriminative tokens, rather than their opposing discriminative counterparts, while other tokens do not affect the labels. We begin by analyzing the learning dynamics of fine-tuning a Transformer and characterize the properties of the resulting task vectors. Next, we provide sufficient conditions on the arithmetic hyperparameters for the task vector approach to be successful. We prove that task addition is effective for multi-task learning when the tasks are either irrelevant or aligned. Aligned tasks are those where solving one task contributes positively to solving the other. In contrast, task negation is provably successful for unlearning tasks that are either irrelevant or contradictory. Contradictory tasks are defined as those where improving performance on one task harms the performance of the other.
+2. The first provable out-of-domain generalization guarantees through task arithmetic. Focusing on task vectors representing a set of irrelevant tasks, we prove a linear combination of these task vectors can generalize to a wide range of new tasks by properly selecting the arithmetic coefficients. Additionally, we characterize the range of suitable arithmetic coefficients sufficient for successful generalization. This is the first theoretical justification of task vectors' ability to adapt to new tasks.
+3. Theoretical justification of low-rank approximation and magnitude-based pruning for task vectors. We construct low-rank and sparse approximations to task vectors and prove that the generalization guarantees are minimally affected by these approximations. This provides the first theoretical support for the practice of using low-rank and sparse approximations to task vectors in order to reduce computational complexity.
+
+# 1.2 RELATED WORKS
+
+Weight interpolation technique. Weight interpolation or model merging (Matena & Raffel, 2022; Ilharco et al., 2022b; Yadav et al., 2023; Yu et al., 2024; He et al., 2025) refers to the practice of linearly interpolating weights of multiple models, where these models may be fine-tuned from different downstream tasks or using different hyperparameters (model soups (Wortsman et al., 2022a)). Weight interpolation is empirically observed to be able to guide the model towards wider optima (Izmailov et al., 2018; Frankle et al., 2020) and better generalization in both single-task performance and multi-task abilities, even surpassing fine-tuning methods in some cases (Rame et al.,
+
+2022; Wortsman et al., 2022b; Ramé et al., 2023). Task arithmetic can be viewed as a special type of weight interpolation, where linear operations are performed on task vectors.
+
+Feature learning analysis for Transformers. Several recent works study the optimization and generalization analysis of Transformers following the feature learning framework, which describes how neural networks gradually focus on important features while discarding unimportant features during training. Jelassi et al. (2022); Li et al. (2023e); Oymak et al. (2023); Ildiz et al. (2024); Nichani et al. (2024); Chen et al. (2024); Li et al. (2023a; 2024c; 2023b); Huang et al. (2024); Luo et al. (2024) study the generalization of one-layer Transformers on different data models such as spatial association, semantic/contextual structure, causal structure/Markov Chain of data, and the majority voting of tokens in the data. However, no discussion was provided for merged models.
+
+Theoretical study of PEFT methods. These are recent theoretical analyses on other PEFT methods. For example, in-context learning is analyzed from the perspective of expressive power (Bai et al., 2023; Akyurek et al., 2023; Von Oswald et al., 2023), the training dynamics or generalization (Xie et al., 2021; Zhang et al., 2023a; Li et al., 2023c; 2024a;b; Huang et al., 2023). Some other works focus on prompt engineering with a tunable prompt (Wei et al., 2021; Oymak et al., 2023; Zhang et al., 2024). Another line of work theoretically investigates the low-rank adaptation in terms of the implicit bias of the optimization process (Damian et al., 2022; Abbe et al., 2022; 2023; Boix-Adsera et al., 2023; Jang et al., 2024; Li et al., 2024d) or model pruning with generalization analysis (Zhang et al., 2021; Yang & Wang, 2023; Yang et al., 2023; Zhang et al., 2023b; Li et al., 2024a). However, none of these works involve the task vector method or related approaches.
+
+# 2 TASK VECTOR: DEFINITION AND OBSERVATIONS
+
+# 2.1 PRELIMINARIES
+
+Let $f:\mathcal{X}\times \Theta \to \mathcal{Y}$ be a neural network that maps inputs $\pmb {X}\in \mathcal{X}$ to labels $\pmb {y}\in \mathcal{V}$ with $\Psi \in \Theta$ as the model parameters. Denote $\Psi^{(0)}$ as the pre-trained model and $\Psi_T^*$ as the fine-tuned model on a given task $\mathcal{T}$ .
+
+Definition 1. (Task Vector) The task vector $\Delta \Psi_{\mathcal{T}}$ for the task $\mathcal{T}$ is computed as the element-wise difference between the pre-trained and fine-tuned weights, i.e., $\Delta \Psi_{\mathcal{T}} = \Psi_{\mathcal{T}}^{*} - \Psi^{(0)}$ .
+
+Task Arithmetic and Generalization. Given the pre-trained model $\Psi^{(0)}$ and a set of task vectors $\{\Delta \Psi_{\mathcal{T}_i}\}_{i\in \mathcal{V}}$ on tasks $\{\mathcal{T}_i\}_{i\in \mathcal{V}}$ , one can construct a merged model $\Psi = \Psi^{(0)} + \sum_{i\in \mathcal{V}}\lambda_i\Delta \Psi_{\mathcal{T}_i}$ for inference on downstream tasks, where $\lambda_{i}\in \mathbb{R}$ are arithmetic hyperparameters. Denote $\ell (X,y;\Psi)$ as the loss function for the input $X\in \mathcal{X}$ , output $y\in \mathcal{Y}$ , and the model $\Psi \in \Theta$ . Hence, the generalization error on the task $\mathcal{T}'$ with data $(X,y)\sim \mathcal{D}_{\mathcal{T}'}$ is defined as
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau^ {\prime}}} \ell (\boldsymbol {X}, y; \Psi). \tag {1}
+$$
+
+Existing works (Ilharco et al., 2022a; Ortiz-Jimenez et al., 2023) conclude that by controlling $\lambda_{i}$ , the merged model $\Psi$ can generalize across different tasks. Specifically, adding several $\Delta \Psi_{\mathcal{T}_i}$ via making $\lambda_{i} > 0$ , $i \in \mathcal{V}_{A} \subset \mathcal{V}$ , leads to a model that exhibits desired performance on multiple tasks from $\mathcal{V}_{A}$ . Such a successful multi-task learning result can be mathematically represented as
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\tau_ {i}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon), \forall i \in \mathcal {V} _ {A}. \tag {2}
+$$
+
+Meanwhile, negating $\Delta \Psi_{\mathcal{T}_i}$ with $\lambda_i < 0$ , $i \in \mathcal{V}_N \subset \mathcal{V}$ , results in a machine unlearning model that performs poorly on $\mathcal{V}_N$ but roughly retains the accuracy on $\mathcal{V} \backslash \mathcal{V}_N$ , i.e.,
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {i}}} \ell (\boldsymbol {X}, y; \Psi) \geq \Theta (1), \mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {j}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon), \forall i \in \mathcal {V} _ {N}, \forall j \in \mathcal {V} \backslash \mathcal {V} _ {N}. \tag {3}
+$$
+
+Moreover, task arithmetic is empirically (Ilharco et al., 2022a) shown to produce a model $\Psi = \Psi^{(0)} + \lambda \cdot \Delta \Psi_{\mathcal{T}'}$ that performs well on task analogy, in the form that "the target out-of-domain task $\mathcal{T}'(\notin \mathcal{V})$ is to $\mathcal{T}_A$ as $\mathcal{T}_B$ is to $\mathcal{T}_C$ ," by constructing a task vector $\Delta \Psi_{\mathcal{T}'} = \Delta \Psi_{\mathcal{T}_A} + (\Delta \Psi_{\mathcal{T}_B} - \Delta \Psi_{\mathcal{T}_C})$ .
+
+# 2.2 EMPIRICAL OBSERVATIONS
+
+Note that experiments in (Ilharco et al., 2022a) only summarize the empirical findings when tasks are almost "orthogonal" to each other, while non-orthogonal cases are less explored. Therefore, in Table 1, we further construct binary classification tasks on the parity of digits of Colored-MNIST
+
+(Arjovsky et al., 2019; Chapel et al., 2020). We control the colors of digits to generate a pair of two datasets so that the parity classification tasks on different pairs of datasets are conceptually "irrelevant," "aligned," or "contradictory" to each other, respectively.
+
+For irrelevant tasks, odd and even digits are highly correlated with red and green colors in one dataset but independent of colors in the other. In aligned tasks, the odd and even digits are correlated with red and green colors in both datasets. In contradictory tasks, the color-parity correspondence is the opposite in the two datasets. Let $\mathcal{T}_1$ and $\mathcal{T}_2$ denote the parity classification task on two different datasets. $\Psi = \Psi^{(0)} + \Delta \Psi_{\mathcal{T}_1} + \lambda \Delta \Psi_{\mathcal{T}_2}$ is used to evaluate the performance of $\mathcal{T}_1$ and $\mathcal{T}_2$ .
+
+A key finding from Table 1 is that the task vector method performs quite differently with different task correlations. To be concrete, given $\Delta \Psi_{\mathcal{T}_1}$ and $\Delta \Psi_{\mathcal{T}_2}$ for aligned tasks, the merged model $\Psi$ can acquire strong multi-task learning abilities but have poor unlearning capabilities. The conclusion is exactly opposite for contradictory tasks. For irrelevant tasks, using task arithmetic can result in good performance in both unlearning and multi-task learning. A question arises, i.e.,
+
+(Q1) How does task correlation quantitatively affect the performance of task arithmetic in multi-task learning and unlearning?
+
+| Notations | Annotation |
| X, xi, Xn, yn | X is the input data, which contains P tokens. xi is the i-th token of X. Xn is the n-th input data with yn as the corresponding label. |
| Ψ | Ψ = {{a(l)}Pl=1, WO, WV, WK, WQ} denotes the set of all the model parameters. a(l) ∈ Rm and WO ∈ Rm×ma are the weights in the MLP layer. WV ∈ Rma×d, WK, WQ ∈ Rmb×d are weights in the self-attention layer. |
| Ψ(0), ΨT*, ΔΨT | Ψ(0) is the pre-trained model. ΨT* is the fine-tuned model on a given task T. ΔΨT is the task vector of the task T, which is computed as ΔΨT = ΨT* - Ψ(0). |
| μT, vj | μT is the discriminative pattern of the task T. vj is the j-th task-irrelevant pattern, j ∈ [M]. |
| δ*, δ# | δ* is the average fraction of label-relevant pattern in the input data. δ# is the average fraction of confusion pattern in the input data. |
| q1(t),ζ1,t, pn(t) | q1(t) = μ1T W(t) μ1 denotes the value of the product, where the patterns on both sides of W(t) are the same.ζ1,t denotes the modified value embedding of μ1 at the t-th iteration. pn(t) refers to the summation of attention weights where the key and the query are the same discriminative pattern. |
| Wn,l,Un,l | Wn,l and Un,l respectively represent of sets of positive or negative neurons so that the Relu activation is activated with xln as the query. |
| Bb | Bb is the SGD batch at the b-th iteration. |
| O(), Ω(), Θ() | We follow the convention that f(x) = O(g(x)) (or Ω(g(x)), Θ(g(x))) means that f(x) increases at most, at least, or in the order of g(x), respectively. |
| a | a = |a(l)i| = 1/√m for i ∈ [m]. |
| ≥, ≤ | f(x) ≥ g(x) (or f(x) ≤ g(x)) means that f(x) ≥ Ω(g(x)) (or f(x) ≤ O(g(x))). |
+
+Define $\mathcal{W}_{n,l},\mathcal{U}_{n,l}$ as the sets of lucky neurons such that
+
+$$
+\mathcal {W} _ {n, l} = \left\{i: \boldsymbol {V} _ {(i, \cdot)} ^ {\top} \boldsymbol {R} _ {n, l} (0) > 0, l \in \mathcal {S} _ {1} ^ {n}, a _ {i} > 0 \right\}, \tag {13}
+$$
+
+$$
+\mathcal {U} _ {n, l} = \left\{i: \boldsymbol {V} _ {(i, \cdot)} ^ {\top} \boldsymbol {R} _ {n, l} (0) > 0, l \in \mathcal {S} _ {2} ^ {n}, a _ {i} < 0 \right\}. \tag {14}
+$$
+
+Definition 5 ((Vershynin, 2010)). We say $X$ is a sub-Gaussian random variable with sub-Gaussian norm $K > 0$ , if $(\mathbb{E}|X|^p)^{\frac{1}{p}} \leq K\sqrt{p}$ for all $p \geq 1$ . In addition, the sub-Gaussian norm of $X$ , denoted $\| X\|_{\psi_2}$ , is defined as $\| X\|_{\psi_2} = \sup_{p \geq 1} p^{-\frac{1}{2}}(\mathbb{E}|X|^p)^{\frac{1}{p}}$ .
+
+Lemma 2 (Vershynin (2010) Proposition 5.1, Hoeffding's inequality). Let $X_{1}, X_{2}, \dots, X_{N}$ be independent centered sub-gaussian random variables, and let $K = \max_{i} \|X_{i}\|_{\psi_{2}}$ . Then for every $\mathbf{a} = (a_{1}, \dots, a_{N}) \in \mathbb{R}^{N}$ and every $t \geq 0$ , we have
+
+$$
+\Pr \left(\left| \sum_ {i = 1} ^ {N} a _ {i} X _ {i} \right| \geq t\right) \leq e \cdot \exp \left(- \frac {c t ^ {2}}{K ^ {2} \| \boldsymbol {a} \| ^ {2}}\right), \tag {15}
+$$
+
+where $c > 0$ is an absolute constant.
+
+Lemma 3. For task $\mathcal{T}$ based on any $\pmb{\mu}_1$ , $0 \leq t \leq T$ , there exists $K(t) > 0$ , such that
+
+$$
+\boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1} = \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1} + K (t) \boldsymbol {\mu} _ {1} + \sum_ {l = 1} ^ {M} \iota_ {l} ^ {\prime} \boldsymbol {\mu} _ {l}, \tag {16}
+$$
+
+where
+
+$$
+K (t) \gtrsim \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {m \left| \mathcal {S} _ {1} ^ {n} \right|}{a P} \zeta_ {1, t} p _ {n} (t) \phi_ {n} (t) (P - \left| \mathcal {S} _ {1} ^ {n} \right|), \tag {17}
+$$
+
+$$
+\iota_ {l} ^ {\prime} \leq K (t) \cdot e ^ {- q _ {1} (t)}. \tag {18}
+$$
+
+For $k\in [M]$
+
+$$
+\left\| \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {v} _ {k} \right\| \lesssim \sqrt {\frac {\log B}{B}} \sum_ {b = 0} ^ {t} K (b), \tag {19}
+$$
+
+and for $j\neq k$ $j\in [M]$
+
+$$
+\left\| \boldsymbol {v} _ {j} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {v} _ {k} \right\| \lesssim K (t) e ^ {- q _ {1} (t)}, \tag {20}
+$$
+
+For any $\pmb{\mu}'$ such that $\pmb{\mu}_1^\top \pmb{\mu}' = \alpha$ and $\pmb{\mu}' \perp \pmb{v}_1, \pmb{v}_2, \dots, \pmb{v}_M$ , we have
+
+$$
+\boldsymbol {\mu} ^ {\prime} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} ^ {\prime} = \alpha^ {2} \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {1} \cdot (1 \pm \Theta (\epsilon)), \tag {21}
+$$
+
+if $B \geq \epsilon^{-2} \log M$ for some $\epsilon < 1$ .
+
+Lemma 4. Given a task $\mathcal{T}$ based on any $\pmb{\mu}_1$ , $0 \leq t \leq T$ . Then, for $i \in \mathcal{W}_{n,t}$ ,
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} \gtrsim \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P} \cdot p _ {n} (b), \tag {22}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \lesssim \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P M}, \tag {23}
+$$
+
+for $k\in [M]$ .For $i\in \mathcal{U}_{n,l}$ , we similarly have
+
+$$
+- \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} \gtrsim \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {2} ^ {n} \right|}{a P} \cdot p _ {n} (b), \tag {24}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \lesssim \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P M}, \tag {25}
+$$
+
+for some $k\in [M]$ . For $i\notin \mathcal{W}_{n,l}\cup \mathcal{U}_{n,l}$ , we have that
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} \lesssim \sqrt {\frac {\log B}{B}} \boldsymbol {V} _ {(j, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1}, \tag {26}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \lesssim \sqrt {\frac {\log B}{B}} \boldsymbol {V} _ {(j, \cdot)} ^ {(t)} \boldsymbol {v} _ {k}, \tag {27}
+$$
+
+where $k\in [M],j\in \mathcal{W}_{n,l}\cup \mathcal{U}_{n,l}$
+
+Lemma 5. (Full version of Lemma 1) Given a task $\mathcal{T}$ defined in Definition 2 based on the discriminative pattern $\pmb{\mu}_{\mathcal{T}}$ , we have that as long as conditions (i)-(iii) in Theorem 1 hold, then the returned model $\Psi_{\mathcal{T}}^{*}$ after $T$ iterations achieves a generalization error
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\mathcal {T}}} \left[ \ell \left(\boldsymbol {X}, y; \Psi_ {\mathcal {T}} ^ {*}\right) \right] \leq \Theta (\epsilon). \tag {28}
+$$
+
+The required sample complexity is $N = BT$ , where $B$ is the batch size. We also have that
+
+1.
+
+$$
+p _ {n} (T) \geq 1 - \left(1 - \delta_ {*}\right) \delta_ {*} ^ {- 1} T ^ {- C}, \tag {29}
+$$
+
+for some constant $C > 1$ .
+
+2.
+
+$$
+\sum_ {k = 1} ^ {M} \left\| \boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {v} _ {k} \right\| ^ {2} \lesssim \frac {1}{M} \left\| \boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T}} \right\| ^ {2}, \tag {30}
+$$
+
+for $i \in \mathcal{W}_{n,l}$ with $l \in S_1^n$ and for $i \in \mathcal{U}_{n,l}$ with $l \in S_2^n$ . We also have that (26) and (27) hold when $t = T$ .
+
+# D PROOF OF MAIN THEOREMS AND COROLLARIES
+
+# D.1 PROOF OF THEOREM 1 AND 2
+
+Proof. Since the model is initialized close to zero, then $\Delta \Psi$ is close to $\Psi$ . Denote $\Psi_{1} = \{\{a_{(l,1)}^{P}\}_{l=1}, V_{1}, W_{1}\}$ and $\Psi_{2} = \{\{a_{(l,2)}^{P}\}_{l=1}, V_{2}, W_{2}\}$ . We consider three cases of this learning problem.
+
+(1) Consider $\alpha = 0$ . By (21) in Lemma 3, we know that
+
+$$
+\boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {1}} = \boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \boldsymbol {W} _ {1} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {1}} \left(1 + \lambda \alpha^ {2} (1 \pm \Theta (\epsilon))\right) = \boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \boldsymbol {W} _ {1} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {1}}, \tag {31}
+$$
+
+$$
+- \boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {1}} = - \boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \boldsymbol {W} _ {1} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {1}}, \tag {32}
+$$
+
+$$
+\boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {2}} = \lambda \boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \boldsymbol {W} _ {2} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {2}}, \tag {33}
+$$
+
+$$
+- \boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {2}} = - \lambda \boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \boldsymbol {W} _ {2} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {2}}. \tag {34}
+$$
+
+Then, for any $l \in [M]$ and for task $\mathcal{T}_1$ ,
+
+$$
+\sum_ {s \in S _ {1} ^ {n}} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {x} _ {l} ^ {n}\right) \geq 1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}, \tag {35}
+$$
+
+for task $\mathcal{T}_2$
+
+$$
+\sum_ {s \in \mathcal {S} _ {1} ^ {n}} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {x} _ {l} ^ {n}\right) \geq \frac {\delta_ {*} T ^ {\lambda C}}{\delta_ {*} T ^ {\lambda C} + (1 - \delta_ {*})} \geq 1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- \lambda C}. \tag {36}
+$$
+
+Since that $\pmb{\mu}_{\mathcal{T}_2} \perp \{\pmb{\mu}_{\mathcal{T}_1}, \pmb{v}_1, \pmb{v}_2, \dots, \pmb{v}_M\}$ and $\pmb{\mu}_{\mathcal{T}_1} \perp \{\pmb{\mu}_{\mathcal{T}_2}, \pmb{v}_1, \pmb{v}_2, \dots, \pmb{v}_M\}$ , we have
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {2}} = 0, \tag {37}
+$$
+
+for $V\in \Psi_{1}$ , and
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {1}} = 0, \tag {38}
+$$
+
+for $V \in \Psi_2$ . Then, for data with the label $y = 1$ , the network output for $\Psi_1 + \lambda \Psi_2$ is almost the same as that for $\Psi_1$ on task $\mathcal{T}_1$ when $|\lambda|$ is not too large. To see this, for $X$ from $\mathcal{T}_1$ , we have
+
+$$
+\begin{array}{l} 1 - \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i \in [ m ]} \frac {1}{a} \operatorname {R e l u} \left(\left(\boldsymbol {V} _ {1 (i, \cdot)} ^ {(T)} + \lambda \boldsymbol {V} _ {2 (i, \cdot)} ^ {(T)}\right) \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n ^ {\top}} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {x} _ {l} ^ {n}\right)\right) \\ \leq | \lambda | \cdot \Theta \left(\eta \sum_ {b = 0} ^ {T - 1} \sum_ {i \in [ m ]} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P M}\right) \cdot \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C} + | \lambda | \cdot \Theta \left(\sqrt {M \frac {\log B}{B}}\right) \tag {39} \\ \leq | \lambda | \cdot \Theta \left(1 - \delta_ {*}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) + | \lambda | \cdot \Theta (\epsilon \sqrt {M}) \\ = | \lambda | \beta , \\ \end{array}
+$$
+
+where the second to last step is by (26) and (27) and $B \gtrsim \epsilon^2 \log M$ . Therefore, a larger $|\lambda|$ leads to a performance drop in task $\mathcal{T}_1$ . For data of $\mathcal{T}_1$ with the label $y = -1$ , we can choose $\lambda$ to be greater than around 1 to make the network output smaller than $-1$ . Meanwhile, for $\mathbf{X}$ from $\mathcal{T}_2$ , we have
+
+$$
+\begin{array}{l} f (\boldsymbol {X} ^ {n}, \Psi) \\ \gtrsim \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \lambda}\right) \cdot \lambda - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) - \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right), \tag {40} \\ \end{array}
+$$
+
+where we need $\lambda \geq 1 + \beta$ so that $f(\pmb{X}^n, \Psi) \geq 1 - \Theta(\epsilon)$ .
+
+If $\lambda \leq 0$ , the attention map tends to be uniform. Then, for $X^n$ in task $\mathcal{T}_2$ , we have
+
+$$
+f \left(\boldsymbol {X} ^ {n}; \Psi_ {1} + \lambda \Psi_ {2}\right) \lesssim - \frac {1}{P}, \tag {41}
+$$
+
+which leads to
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\tau_ {2}}} \ell (\boldsymbol {X}, y; \Psi) \geq \Theta (1). \tag {42}
+$$
+
+(2) Consider $\alpha > 0$ . We first have
+
+$$
+\boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {1}} = \boldsymbol {\mu} _ {\mathcal {T} _ {1}} ^ {\top} \boldsymbol {W} _ {1} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {1}} \left(1 + \lambda \alpha^ {2}\right), \tag {43}
+$$
+
+$$
+\boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \left(\boldsymbol {W} _ {1} ^ {(T)} + \lambda \boldsymbol {W} _ {2} ^ {(T)}\right) \boldsymbol {\mu} _ {\mathcal {T} _ {2}} = (\lambda + \alpha^ {2}) \boldsymbol {\mu} _ {\mathcal {T} _ {2}} ^ {\top} \boldsymbol {W} _ {2} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {2}}. \tag {44}
+$$
+
+Then, for $y^n = 1$ in task $\widetilde{T}_1$ , we have that when $\lambda > 0$ ,
+
+$$
+f (\boldsymbol {X} ^ {n}, \Psi)
+$$
+
+$$
+\begin{array}{l} \gtrsim (1 - \Theta (\epsilon)) \cdot (1 + \lambda \alpha) - | \lambda | \cdot \Theta (\eta \sum_ {b = 0} ^ {T - 1} \sum_ {i \in [ m ]} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {| \mathcal {S} _ {1} ^ {n} |}{a P M}) \cdot \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- \lambda C} \\ - | \lambda | \cdot \Theta \left(\sqrt {\frac {M \log B}{B}}\right) \tag {45} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \geq 1 + \Theta (\lambda \alpha) - | \lambda | \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - | \lambda | \cdot \Theta \left(\epsilon \sqrt {M}\right) \\ = 1 + \Theta (\lambda \alpha) - | \lambda | \cdot \Theta (\frac {1 - \delta_ {*}}{\delta_ {*}}) \cdot \mathrm {p o l y} (\eta \delta_ {*}) - | \lambda | \cdot \Theta (\epsilon \sqrt {M}), \\ \end{array}
+$$
+
+and for $y^n = 1$ in task $\mathcal{T}_2$ , we have that when $\lambda \geq 0$ ,
+
+$$
+\begin{array}{l} f \left(\boldsymbol {X} ^ {n}, \Psi\right) \gtrsim \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)}\right) \cdot (\lambda + \alpha) - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) \tag {46} \\ - \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right). \\ \end{array}
+$$
+
+Therefore, when $\lambda \geq 1 - \alpha +\beta$ , we have that for task $\mathcal{T}_1$
+
+$$
+f \left(\boldsymbol {X} ^ {n}, \Psi\right) \geq 1 - | \lambda | \beta - \Theta (\epsilon), \tag {47}
+$$
+
+and for task $\mathcal{T}_2$
+
+$$
+\begin{array}{l} f \left(\boldsymbol {X} ^ {n}, \Psi\right) \geq (1 - \Theta (\epsilon)) (\lambda + \alpha) - \frac {1 - \delta_ {*}}{\delta_ {*}} \cdot \mathbf {p o l y} (\eta \delta_ {*}) - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) \tag {48} \\ \geq (1 - \Theta (\epsilon)) (\lambda + \alpha) - \beta \\ \geq 1 - \Theta (\epsilon). \\ \end{array}
+$$
+
+We can obtain corresponding conclusions for $y^n = -1$ . Hence,
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon) + | \lambda | \beta , \tag {49}
+$$
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {2}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon). \tag {50}
+$$
+
+Meanwhile, for $y^n = 1$ in task $\mathcal{T}_1$ , we have that when $\lambda < 0$ ,
+
+$$
+\begin{array}{l} f \left(\boldsymbol {X} ^ {n}, \Psi\right) \gtrsim \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C} - \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(1 + \lambda \alpha^ {2}\right)} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right)\right) \cdot (1 + \lambda \alpha) \\ - (| \lambda | + 1) \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - | \lambda | \cdot \Theta (\epsilon \sqrt {M}) \tag {51} \\ \geq 1 + \lambda \alpha \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C (1 + \lambda \alpha^ {2})}\right) - \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C (1 + \lambda \alpha^ {2})} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right) \\ - \left(| \lambda | + 1\right) \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - | \lambda | \cdot \Theta \left(\epsilon \sqrt {M}\right), \\ \end{array}
+$$
+
+and for $y^n = 1$ in task $\mathcal{T}_2$ , we have that when $\lambda < 0$ ,
+
+$$
+\begin{array}{l} f \left(\boldsymbol {X} ^ {n}, \Psi\right) \gtrsim \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)}\right) \cdot (\lambda + \alpha) - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) - \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) \\ \geq \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C} - \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right)\right) \cdot (\lambda + \alpha) \\ - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) - \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) \tag {52} \\ \geq \lambda + \alpha \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)}\right) - \lambda \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right) \\ - \Theta (\sqrt {\frac {M \log B}{B}}) - \Theta (\frac {1 - \delta_ {*}}{\delta_ {*}}) \cdot \mathrm {p o l y} (\eta \delta_ {*}). \\ \end{array}
+$$
+
+Then, for task $\mathcal{T}_1$ , when $0 > \lambda \geq -\Theta (1 / \alpha^2)$
+
+$$
+\begin{array}{l} \mathbb {E} _ {(\boldsymbol {X}, \boldsymbol {y}) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, \boldsymbol {y}; \Psi) \\ = \min \left\{\Theta \left(- \lambda \alpha \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(1 + \lambda \alpha^ {2}\right)}\right) + \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(1 + \lambda \alpha^ {2}\right)} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right) + \epsilon \right. \right. \\ + (| \lambda | + 1) \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) + | \lambda | \cdot \Theta (\epsilon \sqrt {M}), \Theta (1) \} \tag {53} \\ \geq \min \left\{\Theta (- \lambda \alpha + (| \lambda | + 1) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) + | \lambda | \cdot \Theta (\epsilon \sqrt {M})) , \Theta (1) \right\} \\ = \min \left\{\Theta (- \lambda \alpha + | \lambda | \beta + \operatorname {p o l y} \left(\eta \delta_ {*}\right)), \Theta (1) \right\}, \\ \end{array}
+$$
+
+Hence,
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, y; \Psi) \geq \min \left\{\Theta (- \lambda \alpha + (1 + | \lambda |) \beta), \Theta (1) \right\}. \tag {54}
+$$
+
+When $\lambda < -\Theta (1 / \alpha^2)$
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, \boldsymbol {y}) \sim \mathcal {D} _ {\mathcal {T} _ {1}}} \ell (\boldsymbol {X}, \boldsymbol {y}; \Psi)
+$$
+
+$$
+= \Theta \left(1 - \frac {1}{M} \cdot \frac {1}{M} \cdot M\right) \tag {55}
+$$
+
+$$
+\geq \Theta (1).
+$$
+
+For task $\mathcal{T}_2$ , when $0 > \lambda \geq \Theta(1) - \alpha^2$
+
+$$
+\begin{array}{l} \mathbb {E} _ {(\boldsymbol {X}, \boldsymbol {y}) \sim \mathcal {D} _ {\tau_ {2}}} \ell (\boldsymbol {X}, \boldsymbol {y}; \Psi) \\ = \min \left\{\Theta \left(1 - \lambda - \alpha + \alpha \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)} + \lambda \left(\frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C \left(\lambda + \alpha^ {2}\right)} - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right) + \epsilon \right. \right. \\ + \Theta \left(\sqrt {\frac {M \log B}{B}}\right) + \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right), \Theta (1) \} \tag {56} \\ \geq \min \{\Theta (1 + \eta^ {C} - \lambda - \alpha + \Theta (\operatorname {p o l y} (\eta \delta_ {*}) + \epsilon \sqrt {M})), \Theta (1) \} \\ = \min \left\{\Theta \left(1 + \eta^ {C} - \lambda - \alpha + \beta\right), \Theta (1) \right\}, \\ \end{array}
+$$
+
+where the second step is by $\lambda +\alpha \geq \Theta (1) + \alpha -\alpha^{2}\geq \Theta (1)$ . When $\lambda < \Theta (1) - \alpha^2 < 0$
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\tau_ {2}}} \ell (\boldsymbol {X}, y; \Psi) \geq \Theta (1). \tag {57}
+$$
+
+(3) Consider $\alpha < 0$ . When $\lambda \in (-\Theta (1 / \alpha^2),0)$ , we have that for task $\mathcal{T}_1$
+
+$$
+\begin{array}{l} f (\boldsymbol {X} ^ {n}, \Psi) \\ \gtrsim \big (\frac {1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C (1 + \lambda \alpha^ {2})}}{1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}} - \Theta (\epsilon) \big) \cdot (1 + \lambda \alpha) - | \lambda | \cdot \Theta (\eta \sum_ {b = 0} ^ {T - 1} \sum_ {i \in [ m ]} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {| S _ {1} ^ {n} |}{a P M}) \\ \cdot \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- \lambda C} - | \lambda | \cdot \Theta (\sqrt {\frac {M \log B}{B}}) \\ \geq (1 - \Theta (\epsilon)) \cdot (1 + \lambda \alpha) - | \lambda | \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - | \lambda | \cdot \Theta (\epsilon \sqrt {M}) \tag {58} \\ - \frac {\frac {1 - \delta_ {*}}{\delta_ {*}} \left(T ^ {- C \left(1 + \lambda \alpha^ {2}\right)} - T ^ {- C}\right)}{1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}} (1 + \lambda \alpha) \\ \geq (1 - \Theta (\epsilon)) \cdot (1 + \lambda \alpha) - | \lambda | \cdot \Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}}\right) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - | \lambda | \cdot \Theta \left(\epsilon \sqrt {M}\right) \\ - \operatorname {p o l y} \left(\eta \delta_ {*}\right) \lambda \alpha^ {2} (- \log \eta \delta_ {*}) (1 + \lambda \alpha), \\ \end{array}
+$$
+
+Hence, if $\lambda \leq \mathrm{poly}(\eta \delta_{*})\alpha$ , we have
+
+$$
+f \left(\boldsymbol {X} ^ {n}, \Psi\right) \geq 1 - | \lambda | \beta - \Theta (\epsilon). \tag {59}
+$$
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon) + | \lambda | \beta . \tag {60}
+$$
+
+If $\lambda >\frac{\beta}{\alpha - \beta}$ , we have
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, y; \Psi) \geq \min \left\{\Theta (1), \Theta (- \lambda \alpha + (| \lambda | + 1) \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) + | \lambda | \cdot \Theta \left(\epsilon \sqrt {M}\right)) \right\}. \tag {61}
+$$
+
+If $\lambda \leq -\Theta (1 / \alpha^2)$ , we have
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} _ {\tau_ {1}}} \ell (\boldsymbol {X}, y; \Psi) \geq \Theta (1). \tag {62}
+$$
+
+For task $\mathcal{T}_2$ , we have that when $\lambda \geq 1 + \eta^C - \alpha + \beta$ ,
+
+$$
+f \left(\boldsymbol {X} ^ {n}, \Psi\right) \gtrsim (1 - \eta^ {C}) (\lambda + \alpha) - \frac {1 - \delta_ {*}}{\delta_ {*}} \cdot \operatorname {p o l y} \left(\eta \delta_ {*}\right) - \Theta \left(\sqrt {\frac {M \log B}{B}}\right) \geq 1, \tag {63}
+$$
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau_ {2}}} \ell (\boldsymbol {X}, y; \Psi) \leq \Theta (\epsilon). \tag {64}
+$$
+
+When $\lambda \leq 1 + \eta^C -\alpha +\Theta (\mathrm{poly}(\eta \delta_*) + \epsilon \sqrt{M})$
+
+$$
+\mathbb {E} _ {\left(\boldsymbol {X}, y\right) \sim \mathcal {D} \tau_ {2}} \ell (\boldsymbol {X}, y; \Psi) \geq \min \left\{\Theta (1), 1 + \eta^ {C} - \lambda - \alpha + \beta \right\}. \tag {65}
+$$
+
+One can easily find that there is no region of $\lambda$ such that $\Psi$ performs well on both $\mathcal{T}_1$ and $\mathcal{T}_2$ . However, when $-\Theta (1 / \alpha^2) < \lambda < \mathrm{poly}(\eta \delta_*)\alpha < 1 + \eta^c -\alpha +\beta$ , we can unlearn $\mathcal{T}_2$ and retain the performance of $\mathcal{T}_1$ .
+
+
+
+# D.2 PROOF OF THEOREM 3
+
+Proof. By Lemma 1, we know that
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {\mathcal {T} ^ {\prime}} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} ^ {\prime}} \\ = \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} \boldsymbol {\mu} _ {\mathcal {T} _ {i}} ^ {\top} \left(\sum_ {j = 1} \lambda_ {j} \boldsymbol {W} _ {j} ^ {(T)}\right) \sum_ {k \in \mathcal {V} _ {\Psi}} \gamma_ {k} \boldsymbol {\mu} _ {\mathcal {T} _ {k}} \tag {66} \\ \gtrsim \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2} \boldsymbol {\mu} _ {\mathcal {T} _ {i}} ^ {\top} \cdot \lambda_ {i} \boldsymbol {W} _ {i} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {i}}. \\ \end{array}
+$$
+
+For positive neurons, we also have
+
+$$
+\boldsymbol {V} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} ^ {\prime}} = \sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \boldsymbol {V} _ {\mathcal {T} _ {i}} ^ {(T)} \sum_ {i \in \mathcal {V} ^ {\prime}} \gamma_ {i} \boldsymbol {\mu} _ {\mathcal {T} _ {i}} = \sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \gamma_ {i} \boldsymbol {V} _ {\mathcal {T} _ {i}} ^ {(T)} \boldsymbol {\mu} _ {\mathcal {T} _ {i}} \tag {67}
+$$
+
+Then, we need
+
+$$
+\sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \gamma_ {i} \geq 1 + c, \tag {68}
+$$
+
+$$
+\sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \gamma_ {i} ^ {2} \geq 1 + c, \tag {69}
+$$
+
+$$
+\left| \lambda_ {i} \right| \left(\Theta \left(\frac {1 - \delta_ {*}}{\delta_ {*}} \operatorname {p o l y} \left(\eta \delta_ {*}\right) + \epsilon \sqrt {M}\right)\right) = \left| \lambda_ {i} \right| \beta \leq c, \text {f o r s o m e} c > 0 \text {a n d a l l} i \in \mathcal {V} _ {\Psi}, \tag {70}
+$$
+
+to hold simultaneously.
+
+Then, when $\gamma_{i} = k$ does not hold for all $i\in \mathcal{V}_{\Psi}$ and for some fixed $k < 0$ , we can find $\lambda_{i}$ in the middle of the normalized $\gamma_{i}$ and $\gamma_{i}^{2}$ to satisfy (68) and (69), i.e.,
+
+$$
+\lambda_ {i} \propto \frac {\gamma_ {i}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}}} + \frac {\gamma_ {i} ^ {2}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}}. \tag {71}
+$$
+
+By Cauchy-Schwarz inequality, we have
+
+$$
+- \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} \cdot \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}} < \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3} < \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} \cdot \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}. \tag {72}
+$$
+
+Hence,
+
+$$
+\sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \gamma_ {i} \propto \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} + \frac {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}} = \frac {\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} \cdot \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}} + \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3}}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}} > 0, (7 3)
+$$
+
+$$
+\sum_ {i \in \mathcal {V} _ {\Psi}} \lambda_ {i} \gamma_ {i} ^ {2} \propto \frac {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}}} + \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}} = \frac {\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} \cdot \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}} + \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3}}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}}} > 0. \tag {74}
+$$
+
+Therefore, by letting
+
+$$
+\lambda_ {i} = C _ {\gamma} \cdot \left(\frac {\gamma_ {i}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}}} + \frac {\gamma_ {i} ^ {2}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}}\right), \tag {75}
+$$
+
+where
+
+$$
+C _ {\gamma} = \frac {(1 + c) \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}}}{\sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {2}} \cdot \sqrt {\sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {4}} + \sum_ {i \in \mathcal {V} _ {\Psi}} \gamma_ {i} ^ {3}}, \tag {76}
+$$
+
+we can obtain (68) and (69) hold if $C_{\gamma} \lesssim \beta^{-1}$ .
+
+When $\gamma_{i} = k$ hold for all $i\in \mathcal{V}_{\Psi}$ and for some fixed $k < 0$ with $|\mathcal{V}_{\Psi}| > 0$ , we cannot find $\lambda_{i}$ such that both (68) and (69) hold.
+
+
+
+# D.3 PROOF OF COROLLARY 1
+
+Proof. Let $\{\pmb{\mu}_1, \pmb{v}_1, \pmb{v}_2, \dots, \pmb{v}_M\} \cup \{\pmb{u}_1, \pmb{u}_2, \dots, \pmb{u}_{d - M + 1}\}$ form a set of orthonormal vectors, which is denoted by
+
+$$
+\boldsymbol {U} = \left(\boldsymbol {\mu} _ {1}, \boldsymbol {v} _ {1}, \boldsymbol {v} _ {2}, \dots , \boldsymbol {v} _ {M}, \boldsymbol {u} _ {1}, \boldsymbol {u} _ {2}, \dots , \boldsymbol {u} _ {d - M + 1}\right). \tag {77}
+$$
+
+Note that for any $\pmb{a},\pmb{b}\in \{\pmb{\mu}_1,\pmb{v}_1,\pmb{v}_2,\dots ,\pmb{v}_M\} \cup \{\pmb{u}_1,\pmb{u}_2,\dots ,\pmb{u}_{d - M + 1}\}$
+
+$$
+\boldsymbol {a} ^ {\top} \boldsymbol {W} ^ {(0)} \boldsymbol {b} = \sum_ {1 \leq i, j \leq d} a _ {i} b _ {j} W _ {i, j} ^ {(0)} \sim \mathcal {N} (0, \sum_ {1 \leq i, j \leq d} | a _ {i} b _ {j} | \xi^ {2}), \tag {78}
+$$
+
+where the last step comes from that each entry of $\mathbf{W}^{(0)} \sim \mathcal{N}(0, \xi^2)$ . Given that $\| \mathbf{a} \| = \| \mathbf{b} \| = 1$ , we have
+
+$$
+\sum_ {1 \leq i, j \leq d} | a _ {i} b _ {j} | = \left(| a _ {1} |, \dots , | a _ {d} |\right) ^ {\top} \left(| b _ {1} |, \dots , | b _ {d} |\right) \leq 1. \tag {79}
+$$
+
+By (90), we know that for $\pmb{a} \in \{\pmb{u}_1, \pmb{u}_2, \dots, \pmb{u}_{d - M + 1}\}$ and any $t = 0, 1, \dots, T - 1$ ,
+
+$$
+\eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {W} ^ {(t)}} \boldsymbol {a} = 0, \tag {80}
+$$
+
+$$
+\boldsymbol {a} ^ {\top} \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {W} ^ {(t)}} = 0. \tag {81}
+$$
+
+Then, we have that for some $C > 1$
+
+$$
+\left[ \boldsymbol {U} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {U} \right] _ {i, j} = \left\{ \begin{array}{l l} \Theta (\log T), & i = j = 1, \\ O \left(\epsilon \cdot \frac {1}{e ^ {\Theta (\log T)} \cdot \left(1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}\right)}\right) = O \left(\epsilon \cdot T ^ {- C}\right), & j = 1, 1 \leq i \leq M - 1, \\ O \left(\epsilon \cdot \log T\right), & j \in [ 2, M - 1 ], i \in [ 1, M - 1 ], \\ O (\xi), & \text {e l s e .} \end{array} \right. \tag {82}
+$$
+
+Let $E_{i,j}$ be the matrix that only the $(i,j)$ entry equals 1, while all other entries are 0. Therefore,
+
+$$
+\begin{array}{l} \left\| \boldsymbol {U} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {U} - \boldsymbol {E} _ {1, 1} \cdot \Theta (\log T) \right\| _ {F} ^ {2} \\ \leq (\epsilon \cdot T ^ {- C}) ^ {2} \cdot (M - 1) + (\epsilon \cdot \log T) ^ {2} \cdot (M - 1) (M - 2) + \xi^ {2} (d ^ {2} - M ^ {2}) \\ \leq \epsilon^ {2} \log^ {2} T \cdot M ^ {2} + d ^ {2} / m \tag {83} \\ \lesssim \epsilon^ {2} \cdot M ^ {2} + \frac {1}{\log M}, \\ \end{array}
+$$
+
+where the last step comes from that $m \gtrsim M^2 \log M$ and $M = \Theta(d)$ . Then,
+
+$$
+\begin{array}{l} \left\| \boldsymbol {W} ^ {(T)} - \boldsymbol {U} \boldsymbol {E} _ {1, 1} \cdot \Theta (\log T) \cdot \boldsymbol {U} ^ {\top} \right\| _ {F} \\ \leq \left\| \boldsymbol {W} ^ {(T)} \boldsymbol {U} - \boldsymbol {U} \boldsymbol {E} _ {1, 1} \cdot \Theta (\log T) \right\| _ {F} \cdot \left\| \boldsymbol {U} ^ {\top} \right\| \tag {84} \\ \leq \| \boldsymbol {U} \| \cdot \| \boldsymbol {U} ^ {\top} \boldsymbol {W} ^ {(T)} \boldsymbol {U} - \boldsymbol {E} _ {1, 1} \cdot \Theta (\log T) \| _ {F} \\ \leq \epsilon M + 1 / \log M. \\ \end{array}
+$$
+
+Likewise, by (132), we know that neurons of $\mathbf{V}^{(T)}$ with a non-trivial magnitude are in the direction of the iterative summation of $\left(\sum_{s=1}^{P} \boldsymbol{x}_s^n \operatorname{softmax}_l(\boldsymbol{x}_s^{n\top} \boldsymbol{W}\boldsymbol{x}_l^n)\right)$ . Hence, there exists $\hat{\boldsymbol{v}}_1 \in \mathbb{R}^m$ and $\hat{\boldsymbol{v}}_2 \in \mathbb{R}^d$ such that
+
+$$
+\left\| \boldsymbol {V} ^ {(T)} - \hat {\boldsymbol {v}} _ {1} \hat {\boldsymbol {v}} _ {2} ^ {\top} \right\| _ {F} \leq \Theta (1) \cdot \sqrt {m} \cdot \sqrt {\frac {\log B}{B}} \cdot \delta_ {*} ^ {- 2} \cdot \delta_ {*} \cdot \frac {1}{\sqrt {m}} \leq \delta_ {*} ^ {- 1} \epsilon \tag {85}
+$$
+
+Then, for $n$ such that $y^{n} = +1$ , we have that the low-rank trained model, where $\boldsymbol{W}_{LR}^{(T)} = \boldsymbol{U}\boldsymbol{E}_{1,1} \cdot \Theta (\log T) \cdot \boldsymbol{U}^{\top}$ , satisfies
+
+$$
+f \left(\boldsymbol {X} ^ {n}, \Psi_ {L R}\right) \geq 1 \cdot \left(1 - \delta_ {*} \epsilon\right) \cdot \left(1 - \Theta \left(\epsilon \log T\right)\right) = 1 - \Theta \left(\left(\log T + \delta_ {*}\right) \epsilon\right), \tag {86}
+$$
+
+which leads to
+
+$$
+\ell \left(\boldsymbol {X} ^ {n}, y ^ {n}; \Psi_ {L R}\right) \leq \Theta \left(\epsilon_ {L R}\right), \text {w h e r e} \epsilon_ {L R} = (\log T + \delta_ {*}) \epsilon . \tag {87}
+$$
+
+# D.4 PROOF OF COROLLARY 2
+
+Proof. We know that from Lemma 1, there is a number of $\Omega(m)$ lucky neurons with large weights. We can denote the set of lucky neurons as $\mathcal{L} \subset [m]$ . By combining (148) and (163), we have that for any lucky neuron $u_i$ ,
+
+$$
+\left\| \boldsymbol {u} _ {i} \right\| \geq \eta \eta^ {- 1} \delta_ {*} ^ {- 1} \cdot \delta_ {*} \cdot \frac {1}{\sqrt {m}} = m ^ {- 1 / 2}. \tag {88}
+$$
+
+For any unlucky neurons, by (149), we have
+
+$$
+\left\| \boldsymbol {u} _ {i} \right\| \leq m ^ {- 1 / 2} \sqrt {\frac {\log B}{B}}. \tag {89}
+$$
+
+Since that $B \geq \epsilon^{-2} \log M$ by Lemma 1, we have that if we remove neurons from $m \backslash \mathcal{L}$ , the output in (158) and (159) will only be affected by a factor of $\epsilon$ . Therefore, Lemma 1 still holds, so that Theorems 1-3 all hold.
+
+# E PROOF OF KEY LEMMAS
+
+# E.1 PROOF OF LEMMA 3
+
+For ease of presentation, we sometimes use $\mu_{2}$ to represent $-\mu_{1}$ in the proof. We first investigate the gradient of $W$ , i.e.,
+
+$$
+\begin{array}{l} \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell (\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi)}{\partial \boldsymbol {W}} \\ = \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell (\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi)}{\partial f (\boldsymbol {X} ^ {n} ; \Psi)} \frac {f (\boldsymbol {X} ^ {n} ; \Psi)}{\partial \boldsymbol {W}} \\ = \eta \frac {1}{B} \sum_ {\substack {n \in \mathcal {B} _ {b} \\ P}} (- y ^ {n}) \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq 0 \right] \tag{90} \\ \cdot \left(\mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \left(\boldsymbol {x} _ {s} ^ {n} - \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top}\right) \\ = \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} (- y ^ {n}) \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i = 1} ^ {m} a _ {(l) _ {i}} \mathbb {1} \left[ V _ {(i, \cdot)} \boldsymbol {X} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq 0 \right] \\ \cdot \left(\boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n} ^ {\top}\right) \\ \end{array}
+$$
+
+For $j,l\in S_1^n$ , we have
+
+$$
+\operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {j} ^ {n ^ {\top}} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \gtrsim \frac {e ^ {\left\| \boldsymbol {q} _ {1} (t) \right\|}}{\left| \mathcal {S} _ {1} ^ {n} \right| e ^ {\left\| \boldsymbol {q} _ {1} (t) \right\|} + \left(P - \left| \mathcal {S} _ {1} ^ {n} \right|\right)} \tag {91}
+$$
+
+For $j \notin S_1^n$ and $l \in S_1^n$ , we have
+
+$$
+\operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {j} ^ {n} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \lesssim \frac {1}{\left| \mathcal {S} _ {1} ^ {n} \right| e ^ {\left\| \boldsymbol {q} _ {1} (t) \right\|} + \left(P - \left| \mathcal {S} _ {1} ^ {n} \right|\right)}, \tag {92}
+$$
+
+where $\| \pmb{q}_1(0)\| = 0$ . For $l\notin S_1^n\cup S_2^n$ , $j\in [P]$ , we have
+
+$$
+\operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {j} ^ {n} ^ {\top} \boldsymbol {W} ^ {(0)} \boldsymbol {x} _ {l} ^ {n}\right) \lesssim \frac {1}{P}. \tag {93}
+$$
+
+Therefore, for $s,r,l\in S_1^n$ , let
+
+$$
+\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n} := \beta_ {1} ^ {n} (t) \boldsymbol {\mu} _ {1} + \beta_ {2} ^ {n} (t), \tag {94}
+$$
+
+where
+
+$$
+\beta_ {1} ^ {n} (t) \gtrsim \frac {P - | \mathcal {S} _ {1} ^ {n} |}{| \mathcal {S} _ {1} ^ {n} | e ^ {\left\| \boldsymbol {q} _ {1} (t) \right\|} + P - | \mathcal {S} _ {1} ^ {n} |} := \phi_ {n} (t) (P - | \mathcal {S} _ {1} ^ {n} |). \tag {95}
+$$
+
+$$
+\beta_ {2} ^ {n} (t) = \sum_ {l = 2} ^ {M _ {1}} \iota_ {l} ^ {\prime} \boldsymbol {\mu} _ {l}, \tag {96}
+$$
+
+where
+
+$$
+\left| \iota_ {l} ^ {\prime} \right| \leq \beta_ {1} ^ {n} (t) \frac {\left| \mathcal {S} _ {l} ^ {n} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right|}. \tag {97}
+$$
+
+Note that $|l_{l}^{\prime}| = 0$ if $P = |\mathcal{S}_1^n|, l \geq 2$ .
+
+If $s \in S_1^n$ , we have
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq \zeta_ {i, 1, t} \cdot \frac {p _ {n} (t)}{\left| \mathcal {S} _ {1} ^ {n} \right|}. \tag {98}
+$$
+
+If $s \in S_2^n$ and $j \in S_1^n$ , we have
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \lesssim \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {x} _ {j} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {j} ^ {n} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \phi_ {n} (t) \cdot \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{p _ {n} (t)}. \tag {99}
+$$
+
+If $s \notin (S_1^n \cup S_2^n)$ and $j \in S_1^n$ ,
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \lesssim \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {x} _ {j} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {j} ^ {n \top} \boldsymbol {W} ^ {(t)} \boldsymbol {x} _ {l} ^ {n}\right) \phi_ {n} (t) \cdot \frac {\left| S _ {1} ^ {n} \right|}{\sqrt {B} p _ {n} (t)}. \tag {100}
+$$
+
+Then, by combining (94) to (100), we have that for $l \in S_1^n$ , $i \in \mathcal{W}_{n,l}$ ,
+
+$$
+\boldsymbol {\mu} _ {1} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \tag {101}
+$$
+
+$$
+\gtrsim \zeta_ {i, 1, t} \cdot p _ {n} (t) \phi_ {n} (t) (P - | S _ {1} ^ {n} |).
+$$
+
+For $l \in S_1^n$ , $i \in \mathcal{W}_{n,l}$ , we have that for $k \neq 1,2$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {2} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \tag {102} \\ = - \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {1}. \\ \end{array}
+$$
+
+For $l \in S_1^n$ , $i \in \mathcal{W}_{n,l}$ , we have that for $k \in [M]$
+
+$$
+\begin{array}{l} \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \tag {103} \\ \cdot \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right|} \cdot \frac {\left| \mathcal {S} _ {1} ^ {n} \right| \phi_ {n} (t)}{p _ {n} (t)}. \\ \end{array}
+$$
+
+For $i\in \mathcal{U}_{n,l}$ , by the definition of $\mathcal{U}_{n,l}$ in Definition 4, we have
+
+$$
+\mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq 0 \right] = 0. \tag {104}
+$$
+
+For $i \notin \mathcal{W}_{n,l} \cup \mathcal{U}_{n,l}$ , we have that for $j \in \mathcal{W}_{n,l}, k \in [M]$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \tag {105} \\ \cdot \phi_ {n} (t) \frac {| \mathcal {S} _ {1} ^ {n} |}{\sqrt {B} p _ {n} (t)}. \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {2} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {1} (106) \\ = - \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {1}. \\ \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {1} (107) \\ \cdot \phi_ {n} (t) \frac {| \mathcal {S} _ {1} ^ {n} |}{\sqrt {B} p _ {n} (t)} \cdot \frac {| \mathcal {R} _ {k} ^ {n} |}{P - | \mathcal {S} _ {1} ^ {n} |}. \\ \end{array}
+$$
+
+When $l \notin S_1^n$ , we have that $\pmb{x}_l^{n^\top} \pmb{\mu}_1 = 0$ . If $l \in S_2^n$ , we can obtain that
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2} \tag {108} \\ \gtrsim \zeta_ {i, 1, t} \cdot \frac {p _ {n} (t) | \mathcal {S} _ {2} ^ {n} |}{| \mathcal {S} _ {1} ^ {n} |} \phi_ {n} (t) (P - | \mathcal {S} _ {1} ^ {n} |), \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {1} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2} (109) \\ = - \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2}, \\ \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2} \\ \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2} (110) \\ \cdot \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{P - \left| \mathcal {S} _ {2} ^ {n} \right|} \frac {\left| \mathcal {S} _ {1} ^ {n} \right| \phi_ {n} (t)}{p _ {n} (t)}, \\ \end{array}
+$$
+
+where $k\in [M],i\in \mathcal{U}_{n,l}$ . If $i\in \mathcal{W}_{n,l}$
+
+$$
+\mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n ^ {\top}} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq 0 \right] = 0. \tag {111}
+$$
+
+If $i \notin \mathcal{W}_{n,l} \cup \mathcal{U}_{n,l}$ , we have that for $j \in \mathcal{U}_{n,l}$ , $k \in [M]$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2} \\ \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2} \tag {112} \\ \cdot \phi_ {n} (t) \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{\sqrt {B} p _ {n} (t)}. \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {1} ^ {\top} \mathbf {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {\mu} _ {2} (113) \\ = - \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2}. \\ \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(i, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2} \\ \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2} (114) \\ \cdot \phi_ {n} (t) \frac {| \mathcal {S} _ {1} ^ {n} |}{\sqrt {B} p _ {n} (t)} \cdot \frac {| \mathcal {R} _ {k} ^ {n} |}{P - | \mathcal {S} _ {1} ^ {n} |}. \\ \end{array}
+$$
+
+If $l \in \mathcal{R}_k^n$ , $k \in [M]$ , we have that for $j \in \mathcal{W}_{n,l}$ , if $V_{(j,\cdot)} \sum_{s=1}^{P} \pmb{x}_s^n \mathrm{softmax}_l(\pmb{x}_s^{n\top} \pmb{W} \pmb{x}_l^n) > 0$ , $l' \in S_1^n$ ,
+
+$$
+\begin{array}{l} 0 \leq \boldsymbol {\mu} _ {1} ^ {\top} \mathbf {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {v} _ {k} \tag {115} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime}} (\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime}} (\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l ^ {\prime}} ^ {n} ^ {\top} \boldsymbol {\mu} _ {1}, \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \boldsymbol {\mu} _ {2} ^ {\top} \mathbf {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {v} _ {k} (116) \\ = - \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \cdot (\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} (\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \boldsymbol {x} _ {r} ^ {n}) \boldsymbol {x} _ {l} ^ {n} ^ {\top} \boldsymbol {v} _ {k}, \\ \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {v} _ {k} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime}} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime}} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l ^ {\prime}} ^ {n \top} \boldsymbol {\mu} _ {1} (117) \\ \cdot \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right|}. \\ \end{array}
+$$
+
+Likewise, if $l \in \mathcal{R}_k^n$ , $k \in [M]$ , $\pmb{V}_{(j,\cdot)}\sum_{s=1}^{P}\pmb{x}_s^n\mathrm{softmax}_l(\pmb{x}_s^{n^\top}\pmb{W}\pmb{x}_l^n) > 0$ , $j \in \mathcal{U}_{n,l}$ , $l' \in S_1^n$ , $l'' \in S_2^n$ ,
+
+$$
+\begin{array}{l} 0 \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {v} _ {k} \\ \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2}, \tag {118} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} 0 \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {v} _ {k} \\ \leq \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {r} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n} ^ {\top} \boldsymbol {\mu} _ {2}, (118) \\ \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {v} _ {k} \\ = - \boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime \prime}} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l ^ {\prime \prime}} ^ {n \top} \boldsymbol {\mu} _ {2}, (119) \\ \boldsymbol {v} _ {k} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l} ^ {n \top} \boldsymbol {v} _ {k} \\ \leq \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {V} _ {(j, \cdot)} \sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l ^ {\prime}} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}\right) \cdot \left(\boldsymbol {x} _ {s} ^ {n} - \sum_ {r = 1} ^ {P} \operatorname {s o f t m a x} _ {l ^ {\prime}} \left(\boldsymbol {x} _ {r} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l ^ {\prime}} ^ {n}\right) \boldsymbol {x} _ {r} ^ {n}\right) \boldsymbol {x} _ {l ^ {\prime}} ^ {n \top} \boldsymbol {\mu} _ {1} (120) \\ \cdot \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right|}. \\ \end{array}
+$$
+
+Therefore, by the update rule, we know
+
+$$
+\begin{array}{l} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1} = \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {1} - \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {W} ^ {(t)}} \boldsymbol {\mu} _ {1} \tag {121} \\ = \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {1} + K (t) \boldsymbol {\mu} _ {1} + \sum_ {l = 2} ^ {M} \iota_ {l} ^ {\prime} \boldsymbol {\mu} _ {l}, \\ \end{array}
+$$
+
+where
+
+$$
+K (t) \gtrsim \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {m \left| \mathcal {S} _ {1} ^ {n} \right|}{a P} \zeta_ {1, t} p _ {n} (t) \phi_ {n} (t) (P - \left| \mathcal {S} _ {1} ^ {n} \right|), \tag {122}
+$$
+
+$$
+\iota_ {l} ^ {\prime} \leq K (t) \cdot \max _ {n} \left\{\frac {| S _ {1} ^ {n} | \phi_ {n} (t)}{p _ {n} (t)} \right\} \leq K (t) \cdot e ^ {- q _ {1} (t)}. \tag {123}
+$$
+
+We know that
+
+$$
+\boldsymbol {W} ^ {(0)} \boldsymbol {\mu} _ {1} \approx 0. \tag {124}
+$$
+
+Then,
+
+$$
+\begin{array}{l} q _ {1} (t + 1) = \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1} \\ = \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {1} + K (t) \\ = q _ {1} (t) + K (t) \tag {125} \\ = \sum_ {b = 0} ^ {t} K (b). \\ \end{array}
+$$
+
+Similarly,
+
+$$
+\boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {2} = \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {2} - \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {W} ^ {(t)}} \boldsymbol {\mu} _ {2} \tag {126}
+$$
+
+$$
+= \boldsymbol {W} ^ {(t)} \boldsymbol {\mu} _ {2} + K (t) \boldsymbol {\mu} _ {2} + \sum_ {l \neq 2} \iota_ {l} ^ {\prime} \boldsymbol {\mu} _ {l}.
+$$
+
+$$
+\boldsymbol {\mu} _ {2} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {2} = \sum_ {b = 0} ^ {t} K (b). \tag {127}
+$$
+
+For $k\in [M]$
+
+$$
+\boldsymbol {W} ^ {(t + 1)} \boldsymbol {v} _ {k} = \boldsymbol {W} ^ {(t)} \boldsymbol {v} _ {k} + J _ {1} (t) \boldsymbol {\mu} _ {1} + J _ {2} (t) \boldsymbol {\mu} _ {2} + \sum_ {l = 1} ^ {M} \iota_ {l} ^ {\prime} \boldsymbol {v} _ {l}. \tag {128}
+$$
+
+By Hoeffding's inequality (15), with high probability,
+
+$$
+\left\| \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {v} _ {k} \right\| \leq \Theta (1) \cdot \sqrt {\frac {\log B}{B}} \sum_ {b = 0} ^ {t} K (b) \lesssim \epsilon \cdot \sum_ {b = 0} ^ {t} K (b), \tag {129}
+$$
+
+where the second step holds if $B \geq \epsilon^{-2} \log M$ . And for $j \neq k$ , $j \in [M]$
+
+$$
+\left\| \boldsymbol {v} _ {j} ^ {\top} \boldsymbol {W} ^ {(t)} \boldsymbol {v} _ {k} \right\| \leq K (t) e ^ {- q _ {1} (t)}. \tag {130}
+$$
+
+For any $\pmb{\mu}'$ such that $\pmb{\mu}_1^\top \pmb{\mu}' = \alpha$ and $\pmb{\mu}' \perp \{v_1, v_2, \dots, v_M\}$ , we can write $\pmb{\mu}'$ as $\alpha \pmb{\mu}_1 \pm \sqrt{1 - \alpha^2} \pmb{\mu}_\perp$ for some $\pmb{\mu}_\perp \perp \{\pmb{\mu}_1, v_1, v_2, \dots, v_M\}$ . Therefore,
+
+$$
+\begin{array}{l} \boldsymbol {\mu} ^ {\prime} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} ^ {\prime} = \left(\alpha \boldsymbol {\mu} _ {1} \pm \sqrt {1 - \alpha^ {2}} \boldsymbol {\mu} _ {\perp}\right) ^ {\top} \boldsymbol {W} ^ {(t + 1)} \left(\alpha \boldsymbol {\mu} _ {1} \pm \sqrt {1 - \alpha^ {2}} \boldsymbol {\mu} _ {\perp}\right) \tag {131} \\ = \alpha^ {2} \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1} \pm \Theta (\epsilon) \cdot \boldsymbol {\mu} _ {1} ^ {\top} \boldsymbol {W} ^ {(t + 1)} \boldsymbol {\mu} _ {1}. \\ \end{array}
+$$
+
+# E.2 PROOF OF LEMMA 4
+
+For ease of presentation, we sometimes use $\pmb{\mu}_{2}$ to represent $-\pmb{\mu}_{1}$ in the proof.
+
+$$
+\begin{array}{l} \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {V} _ {(i , .)}} \\ = \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial f \left(\boldsymbol {X} ^ {n} ; \Psi\right)} \frac {f \left(\boldsymbol {X} ^ {n} ; \Psi\right)}{\partial \boldsymbol {V} _ {(i , .)}} \tag {132} \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {V} _ {(i , .)}} \\ = \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial f \left(\boldsymbol {X} ^ {n} ; \Psi\right)} \frac {f \left(\boldsymbol {X} ^ {n} ; \Psi\right)}{\partial \boldsymbol {V} _ {(i , .)}} \tag {132} \\ = \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} (- y ^ {n}) \frac {1}{P} \sum_ {l = 1} ^ {P} a _ {(l) _ {i}} \mathbb {1} [ \boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} (\boldsymbol {X} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}) \geq 0 ] \\ \cdot \left(\sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right)\right). \\ \end{array}
+$$
+
+For $n$ such that $y^{n} = +1$ and $i\in \mathcal{W}_{n,l}$ , we have that
+
+$$
+\mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) \geq 0 \right] = 1, \tag {133}
+$$
+
+and for $l\in S_1^n$
+
+$$
+\sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) = p _ {n} (t) \boldsymbol {\mu} _ {1} + \sum_ {l = 1} ^ {M _ {2}} \iota_ {l} ^ {\prime} \boldsymbol {v} _ {l} + \iota_ {M _ {2} + 1} ^ {\prime} \boldsymbol {\mu} _ {2}, \tag {134}
+$$
+
+where
+
+$$
+\iota_ {l} ^ {\prime} \leq (1 - p _ {n} (t)) \cdot \frac {\left| \mathcal {R} _ {k} ^ {l} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right|}. \tag {135}
+$$
+
+If $l\in \mathcal{S}_2^n$ , we have
+
+$$
+\sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) = p _ {n} ^ {\prime} (t) \boldsymbol {\mu} _ {2} + \sum_ {l = 1} ^ {M _ {2}} \kappa_ {l} ^ {\prime} \boldsymbol {v} _ {l} + \kappa_ {M _ {2} + 1} ^ {\prime} \boldsymbol {\mu} _ {2}, \tag {136}
+$$
+
+where
+
+$$
+p _ {n} ^ {\prime} (t) \leq p _ {n} (t), \tag {137}
+$$
+
+$$
+\kappa_ {l} ^ {\prime} \leq (1 - p _ {n} (t)) \cdot \frac {\left| \mathcal {R} _ {k} ^ {l} \right|}{P - \left| \mathcal {S} _ {2} ^ {n} \right|}. \tag {138}
+$$
+
+If $l\in \mathcal{R}_k^n$ $k\in [M]$ , we have
+
+$$
+\sum_ {s = 1} ^ {P} \boldsymbol {x} _ {s} ^ {n} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right) = p _ {n} ^ {\prime} (t) \boldsymbol {\mu} _ {1} + p _ {n} ^ {\prime \prime} (t) \boldsymbol {\mu} _ {2} + o _ {n} (t) \boldsymbol {v} _ {k} + \sum_ {l \neq k} u _ {l} ^ {\prime} \boldsymbol {v} _ {l}, \tag {139}
+$$
+
+where
+
+$$
+p _ {n} ^ {\prime} (t) \leq \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{P} \cdot p _ {n} (t), \tag {140}
+$$
+
+$$
+p _ {n} ^ {\prime \prime} (t) \leq \frac {\left| \mathcal {S} _ {2} ^ {n} \right|}{P} \cdot p _ {n} (t), \tag {141}
+$$
+
+$$
+o _ {n} (t) \leq \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{P} \cdot p _ {n} (t) \tag {142}
+$$
+
+$$
+u _ {l} ^ {\prime} \leq \left(1 - \frac {\left| \mathcal {S} _ {1} ^ {n} \right| + \left| \mathcal {S} _ {2} ^ {n} \right| + \left| \mathcal {R} _ {k} ^ {n} \right|}{\left| \mathcal {S} _ {1} ^ {n} \right|} \cdot p _ {n} (t)\right) \cdot \frac {\left| \mathcal {R} _ {k} ^ {l} \right|}{P - \left| \mathcal {S} _ {1} ^ {n} \right| - \left| \mathcal {S} _ {2} ^ {n} \right| - \left| \mathcal {R} _ {k} ^ {n} \right|}. \tag {143}
+$$
+
+Therefore, we have
+
+$$
+- \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell \left(\boldsymbol {X} ^ {n} , y ^ {n} ; \Psi\right)}{\partial \boldsymbol {V}} = \sum_ {l = 1} ^ {M} u _ {l} ^ {\prime} \boldsymbol {v} _ {l} + q _ {n} (t) \boldsymbol {\mu} _ {1} + q _ {n} ^ {\prime} (t) \boldsymbol {\mu} _ {2}, \tag {144}
+$$
+
+where
+
+$$
+q _ {n} (t) ^ {\prime} \gtrsim \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P} \cdot p _ {n} (t), \tag {145}
+$$
+
+$$
+\left| q _ {n} ^ {\prime} (t) \right| \lesssim \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {2} ^ {n} \right|}{a P} \cdot p _ {n} (t), \tag {146}
+$$
+
+$$
+\left| u _ {k} ^ {\prime} \right| \lesssim \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {R} _ {k} ^ {n} \right|}{a P} \cdot (1 - p _ {n} (t)) \frac {1}{M}. \tag {147}
+$$
+
+Then,
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} \geq \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| S _ {1} ^ {n} \right|}{a P} \cdot p _ {n} (b), \tag {148}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {2} = - \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1}, \tag {149}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \leq \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P M}, \tag {150}
+$$
+
+for $k\in [M]$ . For $i\in \mathcal{U}_{n,l}$ , we similarly have
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {2} \geq \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| S _ {2} ^ {n} \right|}{a P} \cdot p _ {n} (b), \tag {151}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} = - \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {2}, \tag {152}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \leq \eta \sum_ {b = 0} ^ {t - 1} \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{a P M}, \tag {153}
+$$
+
+for some $k\in [M]$ . For $i\notin \mathcal{W}_{n,l}\cup \mathcal{U}_{n,l}$ , we have that
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {v} _ {k} \leq \sqrt {\frac {\log B}{B}} \boldsymbol {V} _ {(j, \cdot)} ^ {(t)} \boldsymbol {v} _ {k}, \tag {154}
+$$
+
+$$
+\boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1} \leq \sqrt {\frac {\log B}{B}} \boldsymbol {V} _ {(j, \cdot)} ^ {(t)} \boldsymbol {\mu} _ {1}, \tag {155}
+$$
+
+where $k\in [M],j\in \mathcal{W}_{n,l}\cup \mathcal{U}_{n,l}$
+
+# E.3 PROOF OF LEMMA 1
+
+We know that by Lemma 3 and 4 in (Li et al., 2023a), for $i \in \mathcal{W}_{n,l}(0)$ and $l \in S_1^n$ , we have that
+
+$$
+\mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {R} _ {l} ^ {n} (t) \right] = 1, \tag {156}
+$$
+
+and for $i\in \mathcal{U}_{n,l}(0)$ and $l\in S_2^n$ , we have that
+
+$$
+\mathbb {1} \left[ \boldsymbol {V} _ {(i, \cdot)} ^ {(t)} \boldsymbol {R} _ {l} ^ {n} (t) \right] = 1. \tag {157}
+$$
+
+We also have that the size of $\mathcal{W}_{n,l}$ and $\mathcal{V}_{n,l}$ are larger than $\Omega(m)$ . Therefore, for $y^n = +1$ , by Lemma 4 and 3, we have
+
+$$
+\begin{array}{l} f \left(\boldsymbol {X} ^ {n}; \Psi\right) = \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i \in \mathcal {W} _ {l, n} (0)} \frac {1}{a} \operatorname {R e l u} \left(\boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n ^ {\top}} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right)\right) \\ + \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i \notin \mathcal {W} _ {l, n} (0), a _ {(l) _ {i}} > 0} \frac {1}{a} \operatorname {R e l u} \left(\boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n ^ {\top}} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right)\right) \tag {158} \\ - \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i: a _ {(l) _ {i}} < 0} \frac {1}{a} \operatorname {R e l u} \left(\boldsymbol {V} _ {(i, \cdot)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n} ^ {\top} \boldsymbol {W} \boldsymbol {x} _ {l} ^ {n}\right)\right). \\ \end{array}
+$$
+
+We know that
+
+$$
+\begin{array}{l} \frac {1}{P} \sum_ {l = 1} ^ {P} \sum_ {i \in \mathcal {W} _ {l, n} (0)} \frac {1}{a} \operatorname {R e l u} \left(\boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {X} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {X} ^ {n ^ {\top}} \boldsymbol {W} ^ {(T)} \boldsymbol {x} _ {l} ^ {n}\right)\right) \\ \gtrsim \frac {\left| S _ {1} ^ {n} \right|}{P} \cdot \frac {m}{a} \cdot \zeta_ {T} \cdot p _ {n} (T) \tag {159} \\ \gtrsim \frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{P} \cdot \frac {m}{a ^ {2}} \cdot \eta \sum_ {b = 0} ^ {T - 1} \frac {1}{B} \sum_ {h \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {h} \right|}{P} p _ {h} (b) \cdot p _ {n} (T). \\ \end{array}
+$$
+
+We can derive that
+
+$$
+\begin{array}{l} q _ {1} (T) = \sum_ {b = 0} ^ {T - 1} K (b) \\ \geq \sum_ {b = 0} ^ {T - 1} \eta \frac {1}{B} \sum_ {n \in \mathcal {B} _ {b}} \frac {m \left| \mathcal {S} _ {1} ^ {n} \right|}{a P} p _ {n} (b) \phi_ {n} (b) (P - \left| \mathcal {S} _ {1} ^ {n} \right|) \eta \sum_ {c = 0} ^ {b - 1} \frac {1}{B} \sum_ {h \in \mathcal {B} _ {c}} \frac {\left| \mathcal {S} _ {1} ^ {h} \right|}{a P} p _ {h} (c) \tag {160} \\ \gtrsim \delta_ {*} ^ {4} \eta \sum_ {b = 0} ^ {T - 1} \frac {1}{e ^ {q _ {1} (b)}}. \\ \end{array}
+$$
+
+Therefore, we have that when $q_{1}(T) \leq O(1)$ or $q_{1}(T) \geq \Theta(T^{c})$ for $c = \Theta(1)$ , (160) does not hold. When $q_{1}(T) = \Theta(\log T)$ , we have that (160) holds. In this case,
+
+$$
+p _ {n} (T) \geq \frac {\delta_ {*} T ^ {C}}{\delta_ {*} T ^ {C} + 1 - \delta_ {*}} \geq 1 - \frac {1 - \delta_ {*}}{\delta_ {*}} T ^ {- C}, \tag {161}
+$$
+
+where $C > 1$ . Meanwhile, for $l \in \mathcal{R}_k^n$ , $k \in [M]$ , and any $s \in [P]$
+
+$$
+\operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {n \top} \boldsymbol {W} ^ {(T)} \boldsymbol {x} _ {l} ^ {n}\right) = \Theta \left(\frac {1}{P}\right). \tag {162}
+$$
+
+We can then derive that as long as
+
+$$
+T \gtrsim \eta^ {- 1} \delta_ {*} ^ {- 2}, \tag {163}
+$$
+
+we have
+
+$$
+\frac {\left| \mathcal {S} _ {1} ^ {n} \right|}{P} \cdot \frac {m}{a ^ {2}} \cdot \eta \sum_ {b = 0} ^ {T - 1} \frac {1}{B} \sum_ {h \in \mathcal {B} _ {b}} \frac {\left| \mathcal {S} _ {1} ^ {h} \right|}{P} p _ {h} (b) \cdot p _ {n} (T) \geq 1. \tag {164}
+$$
+
+Then,
+
+$$
+f \left(\boldsymbol {X} ^ {n}; \Psi\right) \geq 1, \ell \left(\boldsymbol {X} ^ {n}, y ^ {n}; \Psi\right) = 0. \tag {165}
+$$
+
+With (163), we can also derive that
+
+$$
+\sum_ {k = 1} ^ {M} \left\| \boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {v} _ {k} \right\| ^ {2} \lesssim \frac {1}{M} \left\| \boldsymbol {V} _ {(i, \cdot)} ^ {(T)} \boldsymbol {\mu} _ {1} \right\| ^ {2}, \tag {166}
+$$
+
+which means that for $i \in \mathcal{W}_{n,l}$ with $l \in S_1^n$ , $V_{(i,\cdot)}^{(T)}$ is mainly in the direction of $\pmb{\mu}_1$ . This verifies condition (B) of Lemma 1. Therefore, by Hoeffding's inequality (15), for any $W' \in \Psi$ ,
+
+$$
+\Pr \left( \right.\left\| \frac {1}{| \mathcal {B} _ {b} |} \sum_ {n \in \mathcal {B} _ {b}} \frac {\partial \ell (\Psi ; \boldsymbol {P} ^ {n} , z ^ {n})}{\partial \boldsymbol {W} ^ {\prime}} - \mathbb {E} \left[ \frac {\partial \ell (\Psi ; \boldsymbol {P} ^ {n} , z ^ {n})}{\partial \boldsymbol {W} ^ {\prime}} \right]\right\| \geq \left| \right. \mathbb {E} \left[ \frac {\partial \ell (\Psi ; \boldsymbol {P} ^ {n} , z ^ {n})}{\partial \boldsymbol {W} ^ {\prime}} \right] \epsilon\left. \right) \tag {167}
+$$
+
+$$
+\leq e ^ {- B \epsilon^ {2}} \leq M ^ {- C},
+$$
+
+as long as
+
+$$
+B \gtrsim \epsilon^ {- 2} \log M. \tag {168}
+$$
+
+Then,
+
+$$
+\mathbb {E} _ {(\boldsymbol {X}, y) \sim \mathcal {D} _ {\tau}} \ell (\boldsymbol {X}, y; \Psi) \leq \epsilon . \tag {169}
+$$
+
+# F EXTENSION TO MULTI-CLASSIFICATION
+
+Define that a $2^{c}$ -classification is achieved by $c$ times of binary classification with the orthonormal set $\{\pmb{\mu}_{\mathcal{T}}^{(1)}, \dots, \pmb{\mu}_{\mathcal{T}}^{(c)}\}$ as the discriminative patterns for the task $\mathcal{T}$ . We have $\pmb{\mu}_{\mathcal{T}}^{(i)} \perp \pmb{v}_m$ , $m \in [M]$ , $i \in [c]$ . The label $\pmb{y}$ is $c$ -dimensional with each entry chosen from $\{+1, -1\}$ . Specifically, each $(X \in \mathbb{R}^{d \times P}, y \in \mathbb{R}^c) \sim \mathcal{D}_{\mathcal{T}}$ is generated as follows:
+
+- Randomly generate the $k$ -th entry $y_{k}, k \in [c]$ of the label $\mathbf{y}$ from $\{+1, -1\}$ with an equal probability.
+- Each token is randomly chosen from $\{\pmb{\mu}_{\mathcal{T}}^{(i)}, - \pmb{\mu}_{\mathcal{T}}^{(i)}\}_{i = 1}^{c}\cup \{\pmb{v}_1,\dots ,\pmb{v}_M\}$ . If $y_{k} = 1$ (or $-1$ ), the number of tokens corresponding to $\pmb{\mu}_{\mathcal{T}_k}$ (or $-\pmb{\mu}_{\mathcal{T}_k}$ ) is larger than that of $-\pmb{\mu}_{\mathcal{T}_k}$ (or $\pmb{\mu}_{\mathcal{T}_k}$ ). $\pmb{\mu}_{\mathcal{T}}^{(i)}$ and $-\pmb{\mu}_{\mathcal{T}}^{(i)}$ (or “ $-\pmb{\mu}_{\mathcal{T}}^{(i)}$ and $\pmb{\mu}_{\mathcal{T}}^{(i),}$ ” are referred to label-relevant and confusion patterns for $y_{k} = 1$ (or $y_{k} = -1$ ), respectively. The average fractions of label-relevant and confusion tokens of $\pmb{\mu}_{\mathcal{T}}^{(i)}$ are $\delta_{*}^{(i)}$ and $\delta_{\#}^{(i)}$ , respectively.
+
+We then need $c$ sets of our binary model (4) to generate the output for $2^{c}$ -classification, i.e.,
+
+$$
+f (\boldsymbol {X}; \Psi) = \left(f _ {1} (\boldsymbol {X}; \Psi), f _ {2} (\boldsymbol {X}; \Psi), \dots , f _ {c} (\boldsymbol {X}; \Psi)\right)
+$$
+
+$$
+f _ {i} (\boldsymbol {X}; \Psi) = \frac {1}{P} \sum_ {l = 1} ^ {P} \boldsymbol {a} _ {(l) _ {i}} ^ {\top} \operatorname {R e l u} \left(\boldsymbol {W} _ {O _ {i}} \sum_ {s = 1} ^ {P} \boldsymbol {W} _ {V _ {i}} \boldsymbol {x} _ {s} \operatorname {s o f t m a x} _ {l} \left(\boldsymbol {x} _ {s} ^ {\top} \boldsymbol {W} _ {K _ {i}} ^ {\top} \boldsymbol {W} _ {Q _ {i}} \boldsymbol {x} _ {l}\right)\right), \tag {170}
+$$
+
+with $\Psi = \{\{a_{(l)i}\}_{l=1}^{P}, W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}\}_{i=1}^{c}$ . The dimensions of $W_{O_i}, W_{V_i}, W_{K_i}, W_{Q_i}$ , $i \in [c]$ follow Section 3.2.
+
+The learning process is then $c$ independent and parallel binary classification problems for each entry of the $c$ -dimensional output. After fine-tuning, the trained model of each output entry has a similar property to Lemma 1 for single binary classification. $\delta_{*}^{(i)}$ , the fraction of label-relevant pattern $\mu_{\mathcal{T}}^{(i)}$ , $i \in [c]$ , may decrease by $c$ times in average from the binary classification scenario. Therefore, by condition (iii) of Theorem 1, the number of iterations and samples increases by $c^2$ times, which is a polynomial of log scale of the number of classes $2^c$ . Then, for the disriminative patterns $\{\pmb{\mu}_{\mathcal{T}_1}^{(i)}\}_{i=1}^c$ of task $\mathcal{T}_1$ and $\{\pmb{\mu}_{\mathcal{T}_2}^{(i)}\}_{i=1}^c$ and $\mathcal{T}_2$ of task $\mathcal{T}_2$ , if for any $\pmb{\mu}_{\mathcal{T}_1}^{(i)}$ , there exists a unique $\pmb{\mu}_{\mathcal{T}_2}^{(i)}$ close to be orthogonal to $\pmb{\mu}_{\mathcal{T}_1}^{(i)}$ , then $\mathcal{T}_1$ and $\mathcal{T}_2$ are irrelevant. If for any $\pmb{\mu}_{\mathcal{T}_1}^{(i)}$ , there exists a unique $\pmb{\mu}_{\mathcal{T}_2}^{(i)}$ with a small angle to (or almost opposite to) $\pmb{\mu}_{\mathcal{T}_1}^{(i)}$ , then $\mathcal{T}_1$ and $\mathcal{T}_2$ are aligned (or contradictory). We can then derive similar conclusions as our Theorems 1 and 2 by combining the results of all the output entries.
\ No newline at end of file
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+ "content": "Task arithmetic refers to editing the pre-trained model by adding a weighted sum of task vectors, each of which is the weight update from the pre-trained model to fine-tuned models for certain tasks. This approach recently gained attention as a computationally efficient inference method for model editing, e.g., multi-task learning, forgetting, and out-of-domain generalization capabilities. However, the theoretical understanding of why task vectors can execute various conceptual operations remains limited, due to the highly non-convexity of training Transformer-based models. To the best of our knowledge, this paper provides the first theoretical characterization of the generalization guarantees of task vector methods on nonlinear Transformers. We consider a conceptual learning setting, where each task is a binary classification problem based on a discriminative pattern. We theoretically prove the effectiveness of task addition in simultaneously learning a set of irrelevant or aligned tasks, as well as the success of task negation in unlearning one task from irrelevant or contradictory tasks. Moreover, we prove the proper selection of linear coefficients for task arithmetic to achieve guaranteed generalization to out-of-domain tasks. All of our theoretical results hold for both dense-weight parameters and their low-rank approximations. Although established in a conceptual setting, our theoretical findings were validated on a practical machine unlearning task using the large language model Phi-1.5 (1.3B)."
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+ "content": "*Corresponding author. Email: wangm7@rpi.edu."
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+ "content": "rank and/or sparse task vectors can further improve efficiency while maintaining the performance (Yadav et al., 2023; Chitale et al., 2023; Yu et al., 2024; He et al., 2025)."
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+ "content": "Note that experiments in (Ilharco et al., 2022a) only summarize the empirical findings when tasks are almost \"orthogonal\" to each other, while non-orthogonal cases are less explored. Therefore, in Table 1, we further construct binary classification tasks on the parity of digits of Colored-MNIST"
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+ "content": "(Arjovsky et al., 2019; Chapel et al., 2020). We control the colors of digits to generate a pair of two datasets so that the parity classification tasks on different pairs of datasets are conceptually \"irrelevant,\" \"aligned,\" or \"contradictory\" to each other, respectively."
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+ "html": "| Notations | Annotation |
| X, xi, Xn, yn | X is the input data, which contains P tokens. xi is the i-th token of X. Xn is the n-th input data with yn as the corresponding label. |
| Ψ | Ψ = {{a(l)}Pl=1, WO, WV, WK, WQ} denotes the set of all the model parameters. a(l) ∈ Rm and WO ∈ Rm×ma are the weights in the MLP layer. WV ∈ Rma×d, WK, WQ ∈ Rmb×d are weights in the self-attention layer. |
| Ψ(0), ΨT*, ΔΨT | Ψ(0) is the pre-trained model. ΨT* is the fine-tuned model on a given task T. ΔΨT is the task vector of the task T, which is computed as ΔΨT = ΨT* - Ψ(0). |
| μT, vj | μT is the discriminative pattern of the task T. vj is the j-th task-irrelevant pattern, j ∈ [M]. |
| δ*, δ# | δ* is the average fraction of label-relevant pattern in the input data. δ# is the average fraction of confusion pattern in the input data. |
| q1(t),ζ1,t, pn(t) | q1(t) = μ1T W(t) μ1 denotes the value of the product, where the patterns on both sides of W(t) are the same.ζ1,t denotes the modified value embedding of μ1 at the t-th iteration. pn(t) refers to the summation of attention weights where the key and the query are the same discriminative pattern. |
| Wn,l,Un,l | Wn,l and Un,l respectively represent of sets of positive or negative neurons so that the Relu activation is activated with xln as the query. |
| Bb | Bb is the SGD batch at the b-th iteration. |
| O(), Ω(), Θ() | We follow the convention that f(x) = O(g(x)) (or Ω(g(x)), Θ(g(x))) means that f(x) increases at most, at least, or in the order of g(x), respectively. |
| a | a = |a(l)i| = 1/√m for i ∈ [m]. |
| ≥, ≤ | f(x) ≥ g(x) (or f(x) ≤ g(x)) means that f(x) ≥ Ω(g(x)) (or f(x) ≤ O(g(x))). |
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+ {
+ "type": "text",
+ "text": "Zero-Shot Whole-Body Humanoid Control via Behavioral Foundation Models",
+ "text_level": 1,
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+ {
+ "type": "text",
+ "text": "Andrea Tirinzoni $^{1,\\ast}$ , Ahmed Touati $^{1,\\ast}$ , Jesse Farebrother $^{2, + }$ , Mateusz Guzek $^{1}$ , Anssi Kanervisto $^{1}$ , Yingchen Xu $^{1,3}$ , Alessandro Lazaric $^{1,\\dagger}$ , Matteo Pirotta $^{1,\\dagger}$",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "$^{1}$ FAIR at Meta, $^{2}$ Mila, McGill University, $^{3}$ UCL",
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "*Joint first author, ${}^{ + }$ Work done at Meta, ${}^{ \\dagger }$ Joint last author",
+ "bbox": [
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+ "page_idx": 0
+ },
+ {
+ "type": "text",
+ "text": "Unsupervised reinforcement learning (RL) aims at pre-training agents that can solve a wide range of downstream tasks in complex environments. Despite recent advancements, existing approaches suffer from several limitations: they may require running an RL process on each downstream task to achieve a satisfactory performance, they may need access to datasets with good coverage or well-curated task-specific samples, or they may pre-train policies with unsupervised losses that are poorly correlated with the downstream tasks of interest. In this paper, we introduce a novel algorithm regularizing unsupervised RL towards imitating trajectories from unlabeled behavior datasets. The key technical novelty of our method, called Forward-Backward Representations with Conditional-Policy Regularization, is to train forward-backward representations to embed the unlabeled trajectories to the same latent space used to represent states, rewards, and policies, and use a latent-conditional discriminator to encourage policies to \"cover\" the states in the unlabeled behavior dataset. As a result, we can learn policies that are well aligned with the behaviors in the dataset, while retaining zero-shot generalization capabilities for reward-based and imitation tasks. We demonstrate the effectiveness of this new approach in a challenging humanoid control problem: leveraging observation-only motion capture datasets, we train META MOTIVO, the first humanoid behavioral foundation model that can be prompted to solve a variety of whole-body tasks, including motion tracking, goal reaching, and reward optimization. The resulting model is capable of expressing human-like behaviors and it achieves competitive performance with task-specific methods while outperforming state-of-the-art unsupervised RL and model-based baselines.",
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+ "text": "Code: https://github.com/facebookresearch/metamotivo",
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+ {
+ "type": "text",
+ "text": "Website: https://metamotivo.metademolab.com",
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+ {
+ "type": "text",
+ "text": "Meta",
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+ "img_path": "images/fea861cb7f1dbcfafe2f911ea26c71dde60d73a75003e88303a0104eaee57457.jpg",
+ "image_caption": [
+ "Figure 1 META MOTIVO is the first behavioral foundation model for humanoid agents that can solve whole-body control tasks such as tracking, pose-reaching, and reward optimization through zero-shot inference. META MOTIVO is trained with a novel unsupervised reinforcement learning algorithm regularizing zero-shot forward-backward policy learning with imitation of unlabeled motions."
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+ "text": "arXiv:2504.11054v1 [cs.LG] 15 Apr 2025",
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+ "type": "text",
+ "text": "1 Introduction",
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+ {
+ "type": "text",
+ "text": "Foundation models pre-trained on vast amounts of unlabeled data have emerged as the state-of-the-art approach for developing AI systems that can be applied to a wide range of use cases and solve complex tasks by responding to specific prompts (e.g., Anil et al., 2023; OpenAI et al., 2024; Dubey et al., 2024). A natural step forward is to extend this approach beyond language and visual domains, towards behavioral foundation models (BFMs) for agents interacting with dynamic environments through actions. In this paper, we aim to develop BFMs for humanoid agents and we focus on whole-body control from proprioceptive observations, a long-standing challenge due to the high-dimensionality and intrinsic instability of the system (Peng et al., 2021; Won et al., 2022; Luo et al., 2024a). Our goal is to learn BFMs that can express a diverse range of behaviors in response to various prompts, including behaviors to imitate, goals to achieve, or rewards to optimize. By doing so, we could significantly simplify the creation of general-purpose humanoid agents for robotics (Cheng et al., 2024), virtual avatars, and non-player characters (Kwiatkowski et al., 2022).",
+ "bbox": [
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+ "text": "While recent advancements in unsupervised reinforcement learning (RL) have demonstrated the potential of BFMs, several limitations still exist. Pre-trained policies or representations (e.g., Eysenbach et al., 2019; Schwarzer et al., 2021) still require training an RL agent to solve any given downstream task. Unsupervised zero-shot RL (e.g., Touati et al., 2023; Frans et al., 2024) addresses this limitation by pre-training policies that are *promptable* (e.g., by rewards or goals) without additional learning or planning. However, this approach relies on 1) access to large and diverse datasets of transitions collected through some *unsupervised exploration* strategy, and 2) optimize unsupervised losses that aim at learning as many and diverse policies as possible, but provide limited inductive bias on which ones to favor. As a result, zero-shot RL performs well in simple environments (e.g., low-dimensional continuous control), while struggle in complex scenarios with high-dimensional control and unstable dynamics, where unsupervised exploration is unlikely to yield useful samples and unsupervised learning may lead to policies that are not well aligned with the tasks of interest.",
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+ "text": "An alternative approach is to train sequence models (e.g., transformer- or diffusion-based) from large demonstration datasets to clone or imitate existing behaviors and rely on their generalization capabilities and prompt conditioning to obtain different behaviors (e.g., Schmidhuber, 2019; Chen et al., 2021; Wu et al., 2023). This approach is particularly effective when high-quality task-oriented data are available, but it tends to generate models that are limited to reproducing the policies demonstrated in the training datasets and struggle to generalize to unseen tasks (Brandfonbrener et al., 2022). Recently, several methods (e.g., Peng et al., 2022; Gehring et al., 2023; Luo et al., 2024b) integrate demonstrations into an RL routine to learn \"regularized\" policies that preserve RL generalization capabilities while avoiding the issues related to complete unsupervised learning. While the resulting policies can serve as behavior priors, a full hierarchical RL process is often needed to solve any specific downstream task. See App. A for a full review of other related works.",
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+ "type": "text",
+ "text": "In this paper, we aim at leveraging an unlabeled dataset of trajectories to ground zero-shot RL algorithms towards BFMs that not only express useful behaviors but also retain the capability of solving a wide range of tasks in a zero-shot fashion. Our main contributions in this direction are:",
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+ "- We introduce FB-CPR (Forward-Backward representations with Conditional Policy Regularization) a novel online unsupervised RL algorithm that grounds the unsupervised policy learning of forward-backward (FB) representations (Touati and Ollivier, 2021) towards imitating observation-only unlabeled behaviors. The key technical novelty of FB-CPR is to leverage the FB representation to embed unlabeled trajectories to the same latent space used to represent policies and use a latent-conditional discriminator to encourage policies to \"cover\" the states in the dataset.",
+ "- We demonstrate the effectiveness of FB-CPR by training a BFM for whole-body control of a humanoid that can solve a wide range of tasks (i.e., motion tracking, goal reaching, reward optimization) in zero-shot. We consider a humanoid agent built on the SMPL skeleton (Loper et al., 2015), which is widely used in the virtual character animation community for its human-like structure, and we use the AMASS dataset (Mahmood et al., 2019), a large collection of uncurated motion capture data, for regularization. Through an extensive quantitative and qualitative evaluation, we show that our model expresses behaviors that are \"human-like\" and it is competitive with ad-hoc methods trained for specific tasks while outperforming unsupervised RL as well as model-based baselines. Furthermore, we confirm the effectiveness of our regularization scheme in additional ablations in the bipedal walker (App. F) and ant maze domains (App. G). Finally, in order to ensure reproducibility, we release the environment $^{1}$ , code $^{2}$ , and pre-trained models."
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+ "text": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| Regularization coefficient α | 0.01 |
| F network | second column of Tab. 6, output dim 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| B network | fourth column of Tab. 5, output dim 256 |
| Discriminator | Tab. 7 |
| Learning rate for F | 10-4 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for B | 10-5 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -encoding of unlabeled trajectories | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| FB penalty coefficient | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
| Coefficient for Fz-regularization loss | 0.1 |
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+ "text": "We implemented an off-policy version of ASE to be consistent with the training protocol of FB-CPR. In particular, we use a TD3-based scheme to optimize all networks instead of PPO as in the original implementation of Peng et al. (2022). As for FB-CPR, we fit a critic to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10), and another critic to predict $\\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^{t}\\phi(s_{t+1})^{\\top}z|s, a, \\pi_{z}]$ , where $\\phi$ is the representation learned by the DIAYN-based (Eysenbach et al., 2019) skill discovery part of the algorithm. We train such representation by an off-policy version of Eq. 13 in (Peng et al., 2022), where we sample couples $(s', z)$ from the online buffer and",
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+ {
+ "type": "page_number",
+ "text": "28",
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+ {
+ "type": "text",
+ "text": "maximize $\\mathbb{E}_{(s',z)\\sim \\mathcal{D}_{\\mathrm{online}}}\\left[\\phi (s')^T z\\right]$ . Note that this is consistent with the original off-policy implementation of DIAYN (Eysenbach et al., 2019). The output of $\\phi$ is normalized on the hypersphere of radius $\\sqrt{d}$ . We also add an othornormality loss (same as the one used by FB) as we found this to be essential for preventing collapse of the encoder.",
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+ "text": "Inference methods. For reward-based and goal-based inference we use the same methods as FB-CPR, with B replaced with $\\phi$ . For tracking we use $z_{t} \\propto B(s_{t+1})$ for each timestep $t$ in the target motion.",
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+ {
+ "type": "table",
+ "img_path": "images/38e9299167d4156ac620a1ac75ad9a871c986c5b9dcc4d1673c4d71b9fc48cd5.jpg",
+ "table_caption": [
+ "Table 10 Hyperparameters used for ASE pretraining."
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 64 |
| Regularization coefficient α | 0.01 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 64 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-8 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -goals from unlabeled dataset | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Coefficient for diversity loss (Eq. 15 in (Peng et al., 2022)) | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
",
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+ {
+ "type": "text",
+ "text": "C.5.5 CALM",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "As for ASE, we implemented an off-policy TD3-based version of CALM to be consistent with the training protocol of FB-CPR. We fit a critic $Q(s,a,z)$ to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10). We also train a sequence encoder $\\phi(\\tau)$ which embeds a sub-trajectory $\\tau$ from the unlabeled dataset into $z$ space through a transformer. The encoder and the actor are trained end-to-end by maximizing $Q(s,\\pi(s,z = \\phi(\\tau)),z = \\phi(\\tau))$ , plus the constrastive regularization loss designed to prevent the encoder from collapsing (Eq. 5,6 in (Tessler et al., 2023)). The transformer interleaves attention and feed-forward blocks. The former uses a layernorm followed by multi-head self-attention plus a residual connection, while the latter uses a layernorm followed by two linear layers interleaved by a GELU activation. Its output is normalized on the hypersphere of radius $\\sqrt{d}$ .",
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+ {
+ "type": "text",
+ "text": "Inference methods. We use the same methods as FB-CPR for goal-based and tracking inference.",
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+ "type": "page_number",
+ "text": "29",
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+ "type": "table",
+ "img_path": "images/8b3cf555669931330648291135d7d8173f3f1cdf578bb9b68d8350bf6c7a967f.jpg",
+ "table_caption": [
+ "Table 11 Hyperparameters used for CALM pretraining."
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| φ encoder network | transformer (see text above) |
| -attention blocks | 2 |
| -embedding dim | 256 |
| -MLP first linear layer | 256x1024 |
| -MLP second linear layer | 1024x256 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-7 |
| Coefficient for constrastive loss | 0.1 |
| z distributionν | |
| -encoding of unlabeled trajectories | 100% |
| -goals from the online buffer | 0% |
| -uniform on unit sphere | 0% |
| Probability of relabeling zs) | 1 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
",
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "C.5.6 PHC",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 29
+ },
+ {
+ "type": "text",
+ "text": "PHC is similar to a goal-conditioned algorithm except that the goal is \"forced\" to be the next state in the motion. This makes PHC an algorithm specifically designed for one-step tracking. We use a TD3-based variant of the original implementation (Luo et al., 2023). Concretely the implementation is exactly the same of TD3 but we changed the underlying environment. In this tracking environment the state is defined as the concatenation of the current state $s$ and the state $g$ to track. The resulting state space is $\\mathbb{R}^{716}$ . At the beginning of an episode, we sample a motion $m$ from the motion set (either $\\mathcal{M}$ or $\\mathcal{D}_{\\mathrm{test}}$ ) and we initialize the agent to a randomly selected state of the motion. Let $\\bar{t}$ being the randomly selected initial step of the motion, then at any episode step $t \\in [1, \\mathrm{len}(m) - \\bar{t} - 1]$ the target state $g_{t}$ correspond to the motion state $m_{\\bar{t} + t + 1}$ . We use the negative distance in position/orientation as reward function, i.e., $r((s, g), a, (s', g')) = -d_{\\mathrm{smp1}}(g, s')$ .",
+ "bbox": [
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+ {
+ "type": "text",
+ "text": "Inference methods. By being a goal-conditioned algorithm we just need to pass the desired goal as target reference and can be evaluated for goal and tracking tasks.",
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+ {
+ "type": "table",
+ "img_path": "images/e3eb39adf8403c686e7f554c47836f49c607d831033019228b181604fa859451.jpg",
+ "table_caption": [
+ "Table 12 Hyperparameters used for PHC pretraining."
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 50% |
| -goals from the online buffer | 50% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
",
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "C.5.8 GOAL-TD3",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We closely follow the implementation of Pirotta et al. (2024). For reaching each goal $g$ , we use the reward function $r(s', g) = -\\|\\mathrm{pos}(s') - \\mathrm{pos}(g)\\|_2$ , where $\\mathrm{pos}(\\cdot)$ extracts only the position of each joint, ignoring their velocities. We then train a goal-conditioned TD3 agent to optimize such a reward for all $g$ . We sample a percentage of training goals from the unlabeled dataset, and a percentage using hindsight experience replay (HER, Andrychowicz et al., 2017) on trajectories from the online buffer.",
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "Inference methods. We use the same methods as ASE for goal-based and tracking inference.",
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "table",
+ "img_path": "images/7a9dd717614245c126a5cd7f5212d05595fb69d6023b0ea5bf32847794564cfe.jpg",
+ "table_caption": [
+ "Table 14 Hyperparameters used for GOAL-TD3 pretraining."
+ ],
+ "table_footnote": [],
+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for HER sampling | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 100% |
| -goals from the online buffer (HER) | 0% |
| Probability of relabeling zs | 0.5 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
",
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "C.5.9 MPPI",
+ "text_level": 1,
+ "bbox": [
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+ "page_idx": 30
+ },
+ {
+ "type": "text",
+ "text": "We use MPPI with the real dynamic and real reward function for each task. For each evaluation state, action plans are sampled according to a factorized Gaussian distribution. Initially, mean and standard variation of the Gaussian are set with 0 and 1, respectively. actions plans are evaluated by deploying them in the real dynamics and computed the cumulative return over some planning horizon. Subsequently, the Gaussian parameters are updated using the top- $k$ most rewarding plans. For goal-reaching tasks, we use the reward $r(s', g) = -\\|\\mathrm{pos}(s') - \\mathrm{pos}(g)\\|_2$",
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+ {
+ "type": "table",
+ "img_path": "images/89d769132203031aba7bf2c5e143a64ac2be8edf29e2bc9a0fe4faf324cbe75b.jpg",
+ "table_caption": [
+ "Table 15 Hyperparameters used for MPPI planning."
+ ],
+ "table_footnote": [],
+ "table_body": "| Task | TD3 | ORACLE MPPI Normalized | DIFFUSER Normalized | ASE Normalized | FB-CPR Normalized |
| move-ego-0-2-raisearms-l-1 | 191.13 | 168.22 | 0.88 | 148.10 (0.47) | 0.77 (0.00) | 145.78 (7.59) | 0.76 (0.04) | 145.59 (4.38) | 0.76 (0.02) |
| move-ego-0-2-raisearms-l-m | 174.97 | 194.84 | 1.11 | 125.14 (2.16) | 0.72 (0.01) | 109.36 (30.34) | 0.63 (0.17) | 143.90 (7.09) | 0.82 (0.04) |
| move-ego-0-2-raisearms-l-h | 194.72 | 114.30 | 0.59 | 103.11 (1.22) | 0.53 (0.01) | 129.21 (31.41) | 0.66 (0.16) | 123.14 (15.90) | 0.63 (0.08) |
| move-ego-0-2-raisearms-m-l | 179.42 | 199.26 | 1.11 | 124.31 (4.28) | 0.69 (0.02) | 125.39 (5.79) | 0.70 (0.03) | 136.74 (2.40) | 0.76 (0.01) |
| move-ego-0-2-raisearms-m-m | 178.42 | 155.28 | 0.87 | 121.55 (3.97) | 0.68 (0.02) | 60.19 (24.89) | 0.34 (0.14) | 139.19 (18.63) | 0.78 (0.10) |
| move-ego-0-2-raisearms-m-h | 179.02 | 129.99 | 0.73 | 116.50 (3.88) | 0.65 (0.02) | 123.84 (6.10) | 0.69 (0.03) | 128.15 (0.86) | 0.72 (0.00) |
| move-ego-0-2-raisearms-h-l | 191.00 | 115.25 | 0.60 | 101.58 (2.72) | 0.53 (0.01) | 85.89 (7.09) | 0.45 (0.04) | 111.92 (1.20) | 0.59 (0.01) |
| move-ego-0-2-raisearms-h-m | 175.72 | 130.86 | 0.74 | 113.81 (3.34) | 0.65 (0.02) | 121.19 (4.20) | 0.69 (0.02) | 128.10 (0.78) | 0.73 (0.00) |
| move-ego-0-2-raisearms-h-h | 165.19 | 112.35 | 0.68 | 102.09 (3.56) | 0.62 (0.02) | 133.96 (14.35) | 0.81 (0.09) | 143.83 (14.21) | 0.87 (0.09) |
| Average | 181.06 | 146.70 | 0.81 | 117.36 | 0.65 | 114.98 | 0.64 | 133.40 | 0.74 |
| Median | 179.02 | 130.86 | 0.74 | 116.50 | 0.65 | 123.84 | 0.69 | 136.74 | 0.76 |
",
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+ "page_idx": 41
+ },
+ {
+ "type": "text",
+ "text": "Table 25 Average return for each task in the composite reward evaluation. These tasks combine between locomotion and arm-raising behaviors",
+ "bbox": [
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+ "page_idx": 41
+ },
+ {
+ "type": "text",
+ "text": "assessed as human-like by raters, even in tasks when they consider it failed completing the task. Interestingly, while the human-likeness of FB-CPR may come at the cost of lower reward scores, it does not affect the perceived success in accomplishing the assigned goal tasks and FB-CPR has better success rate than TD3 for those tasks.",
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+ {
+ "type": "text",
+ "text": "More in detail, per-task success rate scores are presented in Fig. 9 and Fig. 10.",
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+ {
+ "type": "text",
+ "text": "D.3.2 Reward-based tasks",
+ "text_level": 1,
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+ },
+ {
+ "type": "text",
+ "text": "We provide a further investigation of the performance of our FB-CPR agent on tasks that are i) a combination of tasks used for the main evaluation; and ii) highly under-specified.",
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+ "type": "text",
+ "text": "The objective $i$ is to evaluate the ability of FB-CPR of composing behaviors. We thus created a new category of reward-based tasks by combining locomotion and arm-raising tasks. Specifically, we pair the medium-speed forward locomotion task (with an angle of zero and speed of 2) with all possible arm-raising tasks. Since these two types of tasks have conflicting objectives - locomotion requires movement, while arm-raising rewards stillness - we define a composite reward function that balances the two. This is achieved by taking a weighted average of the individual task rewards, where the weighting varies depending on the specific task combination. Tab. 25 reports the performance of the algorithms on these \"combined\" tasks. We can see that FB-CPR is able to achieve $74\\%$ of the performance of TD3 trained on each individual task. Despite the higher performance, even in this case, TD3 generates unnatural",
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+ "text": "42",
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+ {
+ "type": "image",
+ "img_path": "images/3b7e9fc56687b4a83383c37f058a0ddd7e158d17a3296a978bad85922fc41874.jpg",
+ "image_caption": [
+ "Figure 9 Human-likeness and success rate scores of algorithms per goal task sorted by FB-CPR performance."
+ ],
+ "image_footnote": [],
+ "bbox": [
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+ "page_idx": 42
+ },
+ {
+ "type": "text",
+ "text": "behaviors. The higher quality of FB-CPR is evident in Fig. 11 where we report a few frames of an episode for the task move-ego-0-2-raisearms-m-m. Similarly, almost the totality (about $98\\%$ ) of human evaluators rated FB-CPR as more natural than TD3 on these tasks.",
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+ "type": "text",
+ "text": "The objective of ii) is to evaluate the ability of our model to solve task with a human-like bias. To show this, we designed a few reward functions inspired by the way human person would describe a task.",
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+ },
+ {
+ "type": "text",
+ "text": "Run. The simplest way to describe running is \"move with high speed\". Let $v_{x}$ and $v_{y}$ the horizontal velocities of the center of mass at the pelvis joint. Then, we define the reward for the task $\\mathrm{RUN}_{\\mathrm{eq}}$ as",
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+ "page_idx": 42
+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nr (s ^ {\\prime}) = \\mathbb {I} (v _ {x} ^ {2} + v _ {y} ^ {2} > 2)\n$$\n",
+ "text_format": "latex",
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+ {
+ "type": "text",
+ "text": "Walking with left hand up. This task has two components: walking requires moving with low speed; raising the hand means having the hand $z$ -coordinate above a certain threshold. Then, we define the reward for the task WALK-LAMeq as",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nr (s ^ {\\prime}) = \\mathbb {I} \\Big [ 1 < (v _ {x} ^ {2} + v _ {y} ^ {2}) < 1. 5 \\Big ] \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {l e f t w r i s t}} > 1. 2 \\Big ]\n$$\n",
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+ {
+ "type": "text",
+ "text": "Standing with right foot up. This is the most complex task. We define standing at being in upright position with the head z-coordinate above a certain threshold and zero velocity. Similar to before, we ask the right ankle to be above a certain threshold. Then, we define the reward for the tasks $\\mathrm{STAND - RTM_{eq}}$ ( $\\beta = 0.5$ ) and $\\mathrm{STAND - RTH_{eq}}$ ( $\\beta = 1.2$ ) as",
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+ },
+ {
+ "type": "equation",
+ "text": "\n$$\nr (s ^ {\\prime}) = \\mathbb {I} \\Big [ \\mathrm {u p} > 0. 9 \\Big ] \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {h e a d}} > 1. 4 \\Big ] \\cdot \\exp \\Big (- \\sqrt {v _ {x} ^ {2} + v _ {y} ^ {2}} \\Big) \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {r i g h t a n k l e}} > \\beta \\Big ]\n$$\n",
+ "text_format": "latex",
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+ "page_idx": 42
+ },
+ {
+ "type": "text",
+ "text": "It is evident to any expert in Reinforcement Learning (RL) that the reward functions in question are not optimal for learning from scratch. These reward functions are too vague, and a traditional RL algorithm would likely derive a",
+ "bbox": [
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+ "page_idx": 42
+ },
+ {
+ "type": "page_number",
+ "text": "43",
+ "bbox": [
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+ "page_idx": 42
+ },
+ {
+ "type": "image",
+ "img_path": "images/f3658bb605758e567a75f5b980b49eaa6ee59a4fe977b77241241538a3be851a.jpg",
+ "image_caption": [
+ "Figure 10 Human-likeness and success rate scores of algorithms per reward task sorted by FB-CPR performance."
+ ],
+ "image_footnote": [],
+ "bbox": [
+ 109,
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+ "page_idx": 43
+ },
+ {
+ "type": "text",
+ "text": "high-performing policy that deviates significantly from the natural \"behavioral\" biases. For instance, with TD3, we observe completely unnatural behaviors. In stark contrast, FB-CPR manages to address the tasks in a manner that closely resembles human behavior (refer to Fig. 13). Intriguingly, FB-CPR appears to identify the \"simplest\" policy necessary to solve a task. It effectively distinguishes between two different policies, $\\mathrm{STAND - RTM_{eq}}$ and $\\mathrm{STAND - RTH_{eq}}$ , even though the policy designed for the higher task would suffice for the medium task, provided that the foot remains above a certain threshold. It is also evident the data bias. For example, we do not specify the direction of movement in run, just the high speed. FB-CPR recovers a perfect forward movement probably because the majority of run motions in $\\mathcal{M}$ show this behavior. ASE is not able to solve all the tasks.",
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+ {
+ "type": "page_number",
+ "text": "44",
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+ "page_idx": 43
+ },
+ {
+ "type": "image",
+ "img_path": "images/c3b4d7c94e8b7ecc4f9a85768ee03aa8cd6dbc17b11619a30e25069f1fb7f2dc.jpg",
+ "image_caption": [
+ "Figure 11 Example of combination of locomotion and arm raising tasks (move-ego-0-2-raisearms-m-m). Our FB-CPR (top) agent produces natural human motions while TD3 (bottom) learns high-performing but unnatural behaviors. ASE (middle) has a natural behavior but it is not correctly aligned with the tasks (arms are in the high position not medium)."
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+ "Figure 12 Human-evaluation on locomotion combined with arm raising. Left figure reports the percentage of times a behavior solved a reward-based task (tasks are independently evaluated). Right figure reports the score for human-likeness by direct comparison of the two algorithms."
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+ "Figure 13 Example of behaviors inferred by FB-CPR from under-specified reward equations."
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+ "Figure 14 Rollouts of policies learned by different variants of METRA on Humanoid. Each line corresponds to a trajectory in $(x, y, z)$ space generated by a policy $\\pi_z$ with $z$ uniformly sampled from the unit sphere. (left) The original METRA algorithm trained from scratch (no unlabeled data) with representation $\\phi$ taking as input the full observation vector. (middle) The original METRA algorithm trained from scratch (no unlabeled data) with representation $\\phi$ taking as input only the linear velocities of the robot's pelvis along the x,y,z axes. (right) The ASE algorithm trained within the same setting as in Table 1 but with METRA replacing DIAYN as the skill discovery component."
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+ "table_caption": [],
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+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 16 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 16, 2 hidden layers |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-6 |
| Constraint slack ε | 10-3 |
| Initial Lagrange multiplier λ | 30 |
| z distributionν | uniform on unit sphere |
| Probability of relabeling zs | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
",
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+ "page_idx": 47
+ },
+ {
+ "type": "text",
+ "text": "Inference methods. For goal-based inference, we follow the zero-shot scheme proposed by Park et al. (2024c): when given a goal state $g$ to reach from state $s$ , we set $z = (\\phi(g) - \\phi(s)) / \\|\\phi(g) - \\phi(s)\\|_2$ . Similarly, for tracking we set $z_t = (\\phi(g_{t+1}) - \\phi(s_t)) / \\|\\phi(g_{t+1}) - \\phi(s_t)\\|_2$ at each step $t$ of the episode, where $g_{t+1}$ is the next state in the trajectory to be tracked, while $s_t$ is current agent state. Finally, for reward inference, given a dataset of transitions $(s, s', r)$ sampled from the train buffer and labeled with the corresponding reward $r$ , we infer $z$ through linear regression on top of features $\\phi(s') - \\phi(s)$ . This is motivated by the fact that METRA's actor is pretrained to maximize a self-supervised reward function given by $r(s, s', z) := (\\phi(s') - \\phi(s))^T z$ . Notice, however, that we do not expect this to work well since such a reward, up to discounting, yields a telescopic sum which eventually makes the agent care only about the reward received at the end of an episode instead of the cumulative sum. Thus we report its performance for completeness.",
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+ "type": "text",
+ "text": "Results. We test METRA in the same setting as Table 1. The results are reported in the first row of Table 26, where we find that METRA achieves near zero performance in all tasks. After a deeper investigation, we found that in all runs, and with all hyperparameters we tested, the agent simply learned to fall on the floor and remain still in different positions, as shown in Figure 14 (left). Interestingly, this happens despite all the objectives, and in particular the \"diversity loss\" for representation learning, are well optimized during pre-training. This is due to the fact that, from the agent perspective, lying still on the floor in different positions can be regarded as displaying diverse behaviors, and no extra inductive bias would push the agent to learn more complicated skills (e.g., locomotion ones). On the other hand, we believe that METRA manages to learn few of such skills in the Humanoid experiments of Park et al. (2024c) given that it is pretrained on pixel-based observations (instead of proprioception) with a color map on the ground and very small dimension of the latent space $(d = 2)$ . This may provide an implicit inductive bias towards locomotion behaviors that make the robot move around the x,y coordinates, which are likely to be the observation variables that can be maximally spread out by the agent's controls. On the other hand, we do not have any such bias in our setup, where each joint has roughly the same \"controllability\" and the agent thus learns the simplest way to display diverse behaviors.",
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+ "type": "text",
+ "text": "To verify this last conjecture, we retrained METRA with the same parameters except that we make the representation $\\phi$ only a function of the linear velocities of the robot's pelvis along the three x,y,z directions. Intuitively, this should provide an inductive bias that makes the agent focus on controlling those variables alone, thus learning locomotion behaviors to move around the x,y,z space. This is confirmed in Figure 14 (middle), where we see that the learned skills do not collapse anymore but rather move around different directions of the space.",
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+ "type": "text",
+ "text": "METRA with ASE regularization. Finally, we tried to combine METRA with the same policy regularization on top of unlabeled data as used by ASE. As we recall that ASE (Peng et al., 2022) combines a USD algorithm (DIAYN) with an unconditional policy regularization term, we simply replace DIAYN with METRA and keep all other components the same. The results are shown in Table 26, where we see that the ASE regularization improves the performance of METRA significantly on goal reaching and tracking. Moreover, METRA-ASE achieves competitive performance w.r.t. the original DIAYN-based ASE, improving its success rate in those tasks. Both DIAYN-ASE and METRA-ASE perform, however, significantly worse than FB-CPR. Finally, we note from Figure 14 (right) that METRA-ASE learns to navigate along different directions, though less far than plain METRA trained only on the pelvis' velocities. This is likely due to the regularization w.r.t. unlabeled data, which makes the agent focus on human-like behaviors, thus",
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+ "type": "page_number",
+ "text": "48",
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+ {
+ "type": "text",
+ "text": "avoiding over-actuated movements that would be otherwise learned when naively trying to maximize controls of a subset of the observation variables.",
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+ "page_idx": 48
+ },
+ {
+ "type": "text",
+ "text": "E Understanding the Behavioral Latent Space",
+ "text_level": 1,
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+ "text": "In this section, we summarize results from a qualitative investigation aimed at better understanding the structure of the latent space learned by FB-CPR. We recall that the latent space $Z$ works at the same time as a state embedding through $B(s)$ , a trajectory embedding through $\\mathrm{ER}_{\\mathrm{FB}}$ , and a policy embedding through $\\pi_z$ .",
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+ "type": "text",
+ "text": "E.1 Diversity, Dataset Coverage and Transitions",
+ "text_level": 1,
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+ "text": "In this section we intend to further investigate the behaviors learned by FB-CPR beyond its performance in solving downstream tasks.",
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+ "img_path": "images/cdeb6841a7f004b50f80553ff9864c0ea3270b60d24902d31ada42e09a4374de.jpg",
+ "image_caption": [
+ "Figure 15 Distribution of EMD distance between trajectories generated by two randomly sampled policies $\\pi_z$ and $\\pi_{z'}$ ."
+ ],
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+ "type": "table",
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+ "table_footnote": [],
+ "table_body": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| Regularization coefficient α | 0.01 |
| F network | second column of Tab. 6, output dim 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| B network | fourth column of Tab. 5, output dim 256 |
| Discriminator | Tab. 7 |
| Learning rate for F | 10-4 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for B | 10-5 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -encoding of unlabeled trajectories | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| FB penalty coefficient | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
| Coefficient for Fz-regularization loss | 0.1 |
"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "C.5.4 ASE"
+ },
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+ "bbox": [
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+ "content": "We implemented an off-policy version of ASE to be consistent with the training protocol of FB-CPR. In particular, we use a TD3-based scheme to optimize all networks instead of PPO as in the original implementation of Peng et al. (2022). As for FB-CPR, we fit a critic to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10), and another critic to predict \\(\\mathbb{E}[\\sum_{t=0}^{\\infty} \\gamma^{t}\\phi(s_{t+1})^{\\top}z|s, a, \\pi_{z}]\\), where \\(\\phi\\) is the representation learned by the DIAYN-based (Eysenbach et al., 2019) skill discovery part of the algorithm. We train such representation by an off-policy version of Eq. 13 in (Peng et al., 2022), where we sample couples \\((s', z)\\) from the online buffer and"
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+ {
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+ "content": "28"
+ }
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+ [
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "maximize \\(\\mathbb{E}_{(s',z)\\sim \\mathcal{D}_{\\mathrm{online}}}\\left[\\phi (s')^T z\\right]\\). Note that this is consistent with the original off-policy implementation of DIAYN (Eysenbach et al., 2019). The output of \\(\\phi\\) is normalized on the hypersphere of radius \\(\\sqrt{d}\\). We also add an othornormality loss (same as the one used by FB) as we found this to be essential for preventing collapse of the encoder."
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+ "content": "Inference methods. For reward-based and goal-based inference we use the same methods as FB-CPR, with B replaced with \\(\\phi\\). For tracking we use \\(z_{t} \\propto B(s_{t+1})\\) for each timestep \\(t\\) in the target motion."
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+ {
+ "type": "table_caption",
+ "bbox": [
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+ "content": "Table 10 Hyperparameters used for ASE pretraining."
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+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 64 |
| Regularization coefficient α | 0.01 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 64 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-8 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -goals from unlabeled dataset | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Coefficient for diversity loss (Eq. 15 in (Peng et al., 2022)) | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
"
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "C.5.5 CALM"
+ },
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "As for ASE, we implemented an off-policy TD3-based version of CALM to be consistent with the training protocol of FB-CPR. We fit a critic \\( Q(s,a,z) \\) to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10). We also train a sequence encoder \\( \\phi(\\tau) \\) which embeds a sub-trajectory \\( \\tau \\) from the unlabeled dataset into \\( z \\) space through a transformer. The encoder and the actor are trained end-to-end by maximizing \\( Q(s,\\pi(s,z = \\phi(\\tau)),z = \\phi(\\tau)) \\), plus the constrastive regularization loss designed to prevent the encoder from collapsing (Eq. 5,6 in (Tessler et al., 2023)). The transformer interleaves attention and feed-forward blocks. The former uses a layernorm followed by multi-head self-attention plus a residual connection, while the latter uses a layernorm followed by two linear layers interleaved by a GELU activation. Its output is normalized on the hypersphere of radius \\( \\sqrt{d} \\)."
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+ "angle": 0,
+ "content": "Inference methods. We use the same methods as FB-CPR for goal-based and tracking inference."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "content": "29"
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+ "content": "Table 11 Hyperparameters used for CALM pretraining."
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+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| φ encoder network | transformer (see text above) |
| -attention blocks | 2 |
| -embedding dim | 256 |
| -MLP first linear layer | 256x1024 |
| -MLP second linear layer | 1024x256 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-7 |
| Coefficient for constrastive loss | 0.1 |
| z distributionν | |
| -encoding of unlabeled trajectories | 100% |
| -goals from the online buffer | 0% |
| -uniform on unit sphere | 0% |
| Probability of relabeling zs) | 1 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
"
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+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "C.5.6 PHC"
+ },
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+ "type": "text",
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+ "content": "PHC is similar to a goal-conditioned algorithm except that the goal is \"forced\" to be the next state in the motion. This makes PHC an algorithm specifically designed for one-step tracking. We use a TD3-based variant of the original implementation (Luo et al., 2023). Concretely the implementation is exactly the same of TD3 but we changed the underlying environment. In this tracking environment the state is defined as the concatenation of the current state \\( s \\) and the state \\( g \\) to track. The resulting state space is \\( \\mathbb{R}^{716} \\). At the beginning of an episode, we sample a motion \\( m \\) from the motion set (either \\( \\mathcal{M} \\) or \\( \\mathcal{D}_{\\mathrm{test}} \\)) and we initialize the agent to a randomly selected state of the motion. Let \\( \\bar{t} \\) being the randomly selected initial step of the motion, then at any episode step \\( t \\in [1, \\mathrm{len}(m) - \\bar{t} - 1] \\) the target state \\( g_{t} \\) correspond to the motion state \\( m_{\\bar{t} + t + 1} \\). We use the negative distance in position/orientation as reward function, i.e., \\( r((s, g), a, (s', g')) = -d_{\\mathrm{smp1}}(g, s') \\)."
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+ "content": "Inference methods. By being a goal-conditioned algorithm we just need to pass the desired goal as target reference and can be evaluated for goal and tracking tasks."
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
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+ "content": "Table 12 Hyperparameters used for PHC pretraining."
+ },
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+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 50% |
| -goals from the online buffer | 50% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
"
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "C.5.8 GOAL-TD3"
+ },
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+ "bbox": [
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+ "content": "We closely follow the implementation of Pirotta et al. (2024). For reaching each goal \\( g \\), we use the reward function \\( r(s', g) = -\\|\\mathrm{pos}(s') - \\mathrm{pos}(g)\\|_2 \\), where \\( \\mathrm{pos}(\\cdot) \\) extracts only the position of each joint, ignoring their velocities. We then train a goal-conditioned TD3 agent to optimize such a reward for all \\( g \\). We sample a percentage of training goals from the unlabeled dataset, and a percentage using hindsight experience replay (HER, Andrychowicz et al., 2017) on trajectories from the online buffer."
+ },
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+ "content": "Inference methods. We use the same methods as ASE for goal-based and tracking inference."
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
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+ "angle": 0,
+ "content": "Table 14 Hyperparameters used for GOAL-TD3 pretraining."
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+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for HER sampling | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 100% |
| -goals from the online buffer (HER) | 0% |
| Probability of relabeling zs | 0.5 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
"
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "C.5.9 MPPI"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.685,
+ 0.888,
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+ ],
+ "angle": 0,
+ "content": "We use MPPI with the real dynamic and real reward function for each task. For each evaluation state, action plans are sampled according to a factorized Gaussian distribution. Initially, mean and standard variation of the Gaussian are set with 0 and 1, respectively. actions plans are evaluated by deploying them in the real dynamics and computed the cumulative return over some planning horizon. Subsequently, the Gaussian parameters are updated using the top-\\(k\\) most rewarding plans. For goal-reaching tasks, we use the reward \\(r(s', g) = -\\|\\mathrm{pos}(s') - \\mathrm{pos}(g)\\|_2\\)"
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
+ 0.111,
+ 0.775,
+ 0.431,
+ 0.79
+ ],
+ "angle": 0,
+ "content": "Table 15 Hyperparameters used for MPPI planning."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.316,
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+ "angle": 0,
+ "content": "| Task | TD3 | ORACLE MPPI Normalized | DIFFUSER Normalized | ASE Normalized | FB-CPR Normalized |
| move-ego-0-2-raisearms-l-1 | 191.13 | 168.22 | 0.88 | 148.10 (0.47) | 0.77 (0.00) | 145.78 (7.59) | 0.76 (0.04) | 145.59 (4.38) | 0.76 (0.02) |
| move-ego-0-2-raisearms-l-m | 174.97 | 194.84 | 1.11 | 125.14 (2.16) | 0.72 (0.01) | 109.36 (30.34) | 0.63 (0.17) | 143.90 (7.09) | 0.82 (0.04) |
| move-ego-0-2-raisearms-l-h | 194.72 | 114.30 | 0.59 | 103.11 (1.22) | 0.53 (0.01) | 129.21 (31.41) | 0.66 (0.16) | 123.14 (15.90) | 0.63 (0.08) |
| move-ego-0-2-raisearms-m-l | 179.42 | 199.26 | 1.11 | 124.31 (4.28) | 0.69 (0.02) | 125.39 (5.79) | 0.70 (0.03) | 136.74 (2.40) | 0.76 (0.01) |
| move-ego-0-2-raisearms-m-m | 178.42 | 155.28 | 0.87 | 121.55 (3.97) | 0.68 (0.02) | 60.19 (24.89) | 0.34 (0.14) | 139.19 (18.63) | 0.78 (0.10) |
| move-ego-0-2-raisearms-m-h | 179.02 | 129.99 | 0.73 | 116.50 (3.88) | 0.65 (0.02) | 123.84 (6.10) | 0.69 (0.03) | 128.15 (0.86) | 0.72 (0.00) |
| move-ego-0-2-raisearms-h-l | 191.00 | 115.25 | 0.60 | 101.58 (2.72) | 0.53 (0.01) | 85.89 (7.09) | 0.45 (0.04) | 111.92 (1.20) | 0.59 (0.01) |
| move-ego-0-2-raisearms-h-m | 175.72 | 130.86 | 0.74 | 113.81 (3.34) | 0.65 (0.02) | 121.19 (4.20) | 0.69 (0.02) | 128.10 (0.78) | 0.73 (0.00) |
| move-ego-0-2-raisearms-h-h | 165.19 | 112.35 | 0.68 | 102.09 (3.56) | 0.62 (0.02) | 133.96 (14.35) | 0.81 (0.09) | 143.83 (14.21) | 0.87 (0.09) |
| Average | 181.06 | 146.70 | 0.81 | 117.36 | 0.65 | 114.98 | 0.64 | 133.40 | 0.74 |
| Median | 179.02 | 130.86 | 0.74 | 116.50 | 0.65 | 123.84 | 0.69 | 136.74 | 0.76 |
"
+ },
+ {
+ "type": "table_caption",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Table 25 Average return for each task in the composite reward evaluation. These tasks combine between locomotion and arm-raising behaviors"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "assessed as human-like by raters, even in tasks when they consider it failed completing the task. Interestingly, while the human-likeness of FB-CPR may come at the cost of lower reward scores, it does not affect the perceived success in accomplishing the assigned goal tasks and FB-CPR has better success rate than TD3 for those tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.111,
+ 0.696,
+ 0.628,
+ 0.712
+ ],
+ "angle": 0,
+ "content": "More in detail, per-task success rate scores are presented in Fig. 9 and Fig. 10."
+ },
+ {
+ "type": "title",
+ "bbox": [
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+ "angle": 0,
+ "content": "D.3.2 Reward-based tasks"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "We provide a further investigation of the performance of our FB-CPR agent on tasks that are i) a combination of tasks used for the main evaluation; and ii) highly under-specified."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The objective \\( i \\) is to evaluate the ability of FB-CPR of composing behaviors. We thus created a new category of reward-based tasks by combining locomotion and arm-raising tasks. Specifically, we pair the medium-speed forward locomotion task (with an angle of zero and speed of 2) with all possible arm-raising tasks. Since these two types of tasks have conflicting objectives - locomotion requires movement, while arm-raising rewards stillness - we define a composite reward function that balances the two. This is achieved by taking a weighted average of the individual task rewards, where the weighting varies depending on the specific task combination. Tab. 25 reports the performance of the algorithms on these \"combined\" tasks. We can see that FB-CPR is able to achieve \\( 74\\% \\) of the performance of TD3 trained on each individual task. Despite the higher performance, even in this case, TD3 generates unnatural"
+ },
+ {
+ "type": "page_number",
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+ "angle": 0,
+ "content": "42"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.108,
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+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Figure 9 Human-likeness and success rate scores of algorithms per goal task sorted by FB-CPR performance."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.537,
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+ "angle": 0,
+ "content": "behaviors. The higher quality of FB-CPR is evident in Fig. 11 where we report a few frames of an episode for the task move-ego-0-2-raisearms-m-m. Similarly, almost the totality (about \\(98\\%\\)) of human evaluators rated FB-CPR as more natural than TD3 on these tasks."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "The objective of ii) is to evaluate the ability of our model to solve task with a human-like bias. To show this, we designed a few reward functions inspired by the way human person would describe a task."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Run. The simplest way to describe running is \"move with high speed\". Let \\( v_{x} \\) and \\( v_{y} \\) the horizontal velocities of the center of mass at the pelvis joint. Then, we define the reward for the task \\( \\mathrm{RUN}_{\\mathrm{eq}} \\) as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.417,
+ 0.675,
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+ "angle": 0,
+ "content": "\\[\nr (s ^ {\\prime}) = \\mathbb {I} (v _ {x} ^ {2} + v _ {y} ^ {2} > 2)\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
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+ 0.887,
+ 0.741
+ ],
+ "angle": 0,
+ "content": "Walking with left hand up. This task has two components: walking requires moving with low speed; raising the hand means having the hand \\( z \\)-coordinate above a certain threshold. Then, we define the reward for the task WALK-LAMeq as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.321,
+ 0.748,
+ 0.673,
+ 0.775
+ ],
+ "angle": 0,
+ "content": "\\[\nr (s ^ {\\prime}) = \\mathbb {I} \\Big [ 1 < (v _ {x} ^ {2} + v _ {y} ^ {2}) < 1. 5 \\Big ] \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {l e f t w r i s t}} > 1. 2 \\Big ]\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Standing with right foot up. This is the most complex task. We define standing at being in upright position with the head z-coordinate above a certain threshold and zero velocity. Similar to before, we ask the right ankle to be above a certain threshold. Then, we define the reward for the tasks \\(\\mathrm{STAND - RTM_{eq}}\\) (\\(\\beta = 0.5\\)) and \\(\\mathrm{STAND - RTH_{eq}}\\) (\\(\\beta = 1.2\\)) as"
+ },
+ {
+ "type": "equation",
+ "bbox": [
+ 0.23,
+ 0.843,
+ 0.765,
+ 0.871
+ ],
+ "angle": 0,
+ "content": "\\[\nr (s ^ {\\prime}) = \\mathbb {I} \\Big [ \\mathrm {u p} > 0. 9 \\Big ] \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {h e a d}} > 1. 4 \\Big ] \\cdot \\exp \\Big (- \\sqrt {v _ {x} ^ {2} + v _ {y} ^ {2}} \\Big) \\cdot \\mathbb {I} \\Big [ z _ {\\mathrm {r i g h t a n k l e}} > \\beta \\Big ]\n\\]"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
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+ 0.887,
+ 0.915
+ ],
+ "angle": 0,
+ "content": "It is evident to any expert in Reinforcement Learning (RL) that the reward functions in question are not optimal for learning from scratch. These reward functions are too vague, and a traditional RL algorithm would likely derive a"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "43"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.111,
+ 0.082,
+ 0.852,
+ 0.482
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.111,
+ 0.497,
+ 0.796,
+ 0.513
+ ],
+ "angle": 0,
+ "content": "Figure 10 Human-likeness and success rate scores of algorithms per reward task sorted by FB-CPR performance."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.539,
+ 0.888,
+ 0.66
+ ],
+ "angle": 0,
+ "content": "high-performing policy that deviates significantly from the natural \"behavioral\" biases. For instance, with TD3, we observe completely unnatural behaviors. In stark contrast, FB-CPR manages to address the tasks in a manner that closely resembles human behavior (refer to Fig. 13). Intriguingly, FB-CPR appears to identify the \"simplest\" policy necessary to solve a task. It effectively distinguishes between two different policies, \\(\\mathrm{STAND - RTM_{eq}}\\) and \\(\\mathrm{STAND - RTH_{eq}}\\), even though the policy designed for the higher task would suffice for the medium task, provided that the foot remains above a certain threshold. It is also evident the data bias. For example, we do not specify the direction of movement in run, just the high speed. FB-CPR recovers a perfect forward movement probably because the majority of run motions in \\(\\mathcal{M}\\) show this behavior. ASE is not able to solve all the tasks."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ "angle": 0,
+ "content": "44"
+ }
+ ],
+ [
+ {
+ "type": "image",
+ "bbox": [
+ 0.117,
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+ 0.878,
+ 0.407
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.11,
+ 0.418,
+ 0.889,
+ 0.463
+ ],
+ "angle": 0,
+ "content": "Figure 11 Example of combination of locomotion and arm raising tasks (move-ego-0-2-raisearms-m-m). Our FB-CPR (top) agent produces natural human motions while TD3 (bottom) learns high-performing but unnatural behaviors. ASE (middle) has a natural behavior but it is not correctly aligned with the tasks (arms are in the high position not medium)."
+ },
+ {
+ "type": "image",
+ "bbox": [
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+ {
+ "type": "image_caption",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Figure 12 Human-evaluation on locomotion combined with arm raising. Left figure reports the percentage of times a behavior solved a reward-based task (tasks are independently evaluated). Right figure reports the score for human-likeness by direct comparison of the two algorithms."
+ },
+ {
+ "type": "page_number",
+ "bbox": [
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+ [
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 13 Example of behaviors inferred by FB-CPR from under-specified reward equations."
+ },
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+ "bbox": [
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+ ],
+ [
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+ {
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+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.11,
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+ ],
+ "angle": 0,
+ "content": "Figure 14 Rollouts of policies learned by different variants of METRA on Humanoid. Each line corresponds to a trajectory in \\((x, y, z)\\) space generated by a policy \\(\\pi_z\\) with \\(z\\) uniformly sampled from the unit sphere. (left) The original METRA algorithm trained from scratch (no unlabeled data) with representation \\(\\phi\\) taking as input the full observation vector. (middle) The original METRA algorithm trained from scratch (no unlabeled data) with representation \\(\\phi\\) taking as input only the linear velocities of the robot's pelvis along the x,y,z axes. (right) The ASE algorithm trained within the same setting as in Table 1 but with METRA replacing DIAYN as the skill discovery component."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.136,
+ 0.42,
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+ 0.498
+ ],
+ "angle": 0,
+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 16 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 16, 2 hidden layers |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-6 |
| Constraint slack ε | 10-3 |
| Initial Lagrange multiplier λ | 30 |
| z distributionν | uniform on unit sphere |
| Probability of relabeling zs | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.348,
+ 0.888,
+ 0.485
+ ],
+ "angle": 0,
+ "content": "Inference methods. For goal-based inference, we follow the zero-shot scheme proposed by Park et al. (2024c): when given a goal state \\( g \\) to reach from state \\( s \\), we set \\( z = (\\phi(g) - \\phi(s)) / \\|\\phi(g) - \\phi(s)\\|_2 \\). Similarly, for tracking we set \\( z_t = (\\phi(g_{t+1}) - \\phi(s_t)) / \\|\\phi(g_{t+1}) - \\phi(s_t)\\|_2 \\) at each step \\( t \\) of the episode, where \\( g_{t+1} \\) is the next state in the trajectory to be tracked, while \\( s_t \\) is current agent state. Finally, for reward inference, given a dataset of transitions \\( (s, s', r) \\) sampled from the train buffer and labeled with the corresponding reward \\( r \\), we infer \\( z \\) through linear regression on top of features \\( \\phi(s') - \\phi(s) \\). This is motivated by the fact that METRA's actor is pretrained to maximize a self-supervised reward function given by \\( r(s, s', z) := (\\phi(s') - \\phi(s))^T z \\). Notice, however, that we do not expect this to work well since such a reward, up to discounting, yields a telescopic sum which eventually makes the agent care only about the reward received at the end of an episode instead of the cumulative sum. Thus we report its performance for completeness."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.491,
+ 0.889,
+ 0.689
+ ],
+ "angle": 0,
+ "content": "Results. We test METRA in the same setting as Table 1. The results are reported in the first row of Table 26, where we find that METRA achieves near zero performance in all tasks. After a deeper investigation, we found that in all runs, and with all hyperparameters we tested, the agent simply learned to fall on the floor and remain still in different positions, as shown in Figure 14 (left). Interestingly, this happens despite all the objectives, and in particular the \"diversity loss\" for representation learning, are well optimized during pre-training. This is due to the fact that, from the agent perspective, lying still on the floor in different positions can be regarded as displaying diverse behaviors, and no extra inductive bias would push the agent to learn more complicated skills (e.g., locomotion ones). On the other hand, we believe that METRA manages to learn few of such skills in the Humanoid experiments of Park et al. (2024c) given that it is pretrained on pixel-based observations (instead of proprioception) with a color map on the ground and very small dimension of the latent space \\((d = 2)\\). This may provide an implicit inductive bias towards locomotion behaviors that make the robot move around the x,y coordinates, which are likely to be the observation variables that can be maximally spread out by the agent's controls. On the other hand, we do not have any such bias in our setup, where each joint has roughly the same \"controllability\" and the agent thus learns the simplest way to display diverse behaviors."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.695,
+ 0.888,
+ 0.773
+ ],
+ "angle": 0,
+ "content": "To verify this last conjecture, we retrained METRA with the same parameters except that we make the representation \\(\\phi\\) only a function of the linear velocities of the robot's pelvis along the three x,y,z directions. Intuitively, this should provide an inductive bias that makes the agent focus on controlling those variables alone, thus learning locomotion behaviors to move around the x,y,z space. This is confirmed in Figure 14 (middle), where we see that the learned skills do not collapse anymore but rather move around different directions of the space."
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.777,
+ 0.888,
+ 0.915
+ ],
+ "angle": 0,
+ "content": "METRA with ASE regularization. Finally, we tried to combine METRA with the same policy regularization on top of unlabeled data as used by ASE. As we recall that ASE (Peng et al., 2022) combines a USD algorithm (DIAYN) with an unconditional policy regularization term, we simply replace DIAYN with METRA and keep all other components the same. The results are shown in Table 26, where we see that the ASE regularization improves the performance of METRA significantly on goal reaching and tracking. Moreover, METRA-ASE achieves competitive performance w.r.t. the original DIAYN-based ASE, improving its success rate in those tasks. Both DIAYN-ASE and METRA-ASE perform, however, significantly worse than FB-CPR. Finally, we note from Figure 14 (right) that METRA-ASE learns to navigate along different directions, though less far than plain METRA trained only on the pelvis' velocities. This is likely due to the regularization w.r.t. unlabeled data, which makes the agent focus on human-like behaviors, thus"
+ },
+ {
+ "type": "page_number",
+ "bbox": [
+ 0.49,
+ 0.938,
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+ ],
+ "angle": 0,
+ "content": "48"
+ }
+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.082,
+ 0.888,
+ 0.112
+ ],
+ "angle": 0,
+ "content": "avoiding over-actuated movements that would be otherwise learned when naively trying to maximize controls of a subset of the observation variables."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.111,
+ 0.133,
+ 0.643,
+ 0.156
+ ],
+ "angle": 0,
+ "content": "E Understanding the Behavioral Latent Space"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.167,
+ 0.888,
+ 0.214
+ ],
+ "angle": 0,
+ "content": "In this section, we summarize results from a qualitative investigation aimed at better understanding the structure of the latent space learned by FB-CPR. We recall that the latent space \\( Z \\) works at the same time as a state embedding through \\( B(s) \\), a trajectory embedding through \\( \\mathrm{ER}_{\\mathrm{FB}} \\), and a policy embedding through \\( \\pi_z \\)."
+ },
+ {
+ "type": "title",
+ "bbox": [
+ 0.111,
+ 0.23,
+ 0.573,
+ 0.248
+ ],
+ "angle": 0,
+ "content": "E.1 Diversity, Dataset Coverage and Transitions"
+ },
+ {
+ "type": "text",
+ "bbox": [
+ 0.11,
+ 0.256,
+ 0.886,
+ 0.286
+ ],
+ "angle": 0,
+ "content": "In this section we intend to further investigate the behaviors learned by FB-CPR beyond its performance in solving downstream tasks."
+ },
+ {
+ "type": "image",
+ "bbox": [
+ 0.174,
+ 0.312,
+ 0.538,
+ 0.527
+ ],
+ "angle": 0,
+ "content": null
+ },
+ {
+ "type": "image_caption",
+ "bbox": [
+ 0.153,
+ 0.542,
+ 0.585,
+ 0.571
+ ],
+ "angle": 0,
+ "content": "Figure 15 Distribution of EMD distance between trajectories generated by two randomly sampled policies \\(\\pi_z\\) and \\(\\pi_{z'}\\)."
+ },
+ {
+ "type": "table",
+ "bbox": [
+ 0.619,
+ 0.381,
+ 0.804,
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+ ],
+ "angle": 0,
+ "content": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| Regularization coefficient α | 0.01 |
| F network | second column of Tab. 6, output dim 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| B network | fourth column of Tab. 5, output dim 256 |
| Discriminator | Tab. 7 |
| Learning rate for F | 10-4 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for B | 10-5 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -encoding of unlabeled trajectories | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| FB penalty coefficient | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
| Coefficient for Fz-regularization loss | 0.1 |
+
+# C.5.4 ASE
+
+We implemented an off-policy version of ASE to be consistent with the training protocol of FB-CPR. In particular, we use a TD3-based scheme to optimize all networks instead of PPO as in the original implementation of Peng et al. (2022). As for FB-CPR, we fit a critic to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10), and another critic to predict $\mathbb{E}[\sum_{t=0}^{\infty} \gamma^{t}\phi(s_{t+1})^{\top}z|s, a, \pi_{z}]$ , where $\phi$ is the representation learned by the DIAYN-based (Eysenbach et al., 2019) skill discovery part of the algorithm. We train such representation by an off-policy version of Eq. 13 in (Peng et al., 2022), where we sample couples $(s', z)$ from the online buffer and
+
+maximize $\mathbb{E}_{(s',z)\sim \mathcal{D}_{\mathrm{online}}}\left[\phi (s')^T z\right]$ . Note that this is consistent with the original off-policy implementation of DIAYN (Eysenbach et al., 2019). The output of $\phi$ is normalized on the hypersphere of radius $\sqrt{d}$ . We also add an othornormality loss (same as the one used by FB) as we found this to be essential for preventing collapse of the encoder.
+
+Inference methods. For reward-based and goal-based inference we use the same methods as FB-CPR, with B replaced with $\phi$ . For tracking we use $z_{t} \propto B(s_{t+1})$ for each timestep $t$ in the target motion.
+
+Table 10 Hyperparameters used for ASE pretraining.
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 64 |
| Regularization coefficient α | 0.01 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 64 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-8 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -goals from unlabeled dataset | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Coefficient for diversity loss (Eq. 15 in (Peng et al., 2022)) | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
+
+# C.5.5 CALM
+
+As for ASE, we implemented an off-policy TD3-based version of CALM to be consistent with the training protocol of FB-CPR. We fit a critic $Q(s,a,z)$ to predict the expected discounted sum of rewards from the discriminator by temporal difference (see Eq. 10). We also train a sequence encoder $\phi(\tau)$ which embeds a sub-trajectory $\tau$ from the unlabeled dataset into $z$ space through a transformer. The encoder and the actor are trained end-to-end by maximizing $Q(s,\pi(s,z = \phi(\tau)),z = \phi(\tau))$ , plus the constrastive regularization loss designed to prevent the encoder from collapsing (Eq. 5,6 in (Tessler et al., 2023)). The transformer interleaves attention and feed-forward blocks. The former uses a layernorm followed by multi-head self-attention plus a residual connection, while the latter uses a layernorm followed by two linear layers interleaved by a GELU activation. Its output is normalized on the hypersphere of radius $\sqrt{d}$ .
+
+Inference methods. We use the same methods as FB-CPR for goal-based and tracking inference.
+
+Table 11 Hyperparameters used for CALM pretraining.
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| φ encoder network | transformer (see text above) |
| -attention blocks | 2 |
| -embedding dim | 256 |
| -MLP first linear layer | 256x1024 |
| -MLP second linear layer | 1024x256 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-7 |
| Coefficient for constrastive loss | 0.1 |
| z distributionν | |
| -encoding of unlabeled trajectories | 100% |
| -goals from the online buffer | 0% |
| -uniform on unit sphere | 0% |
| Probability of relabeling zs) | 1 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
+
+# C.5.6 PHC
+
+PHC is similar to a goal-conditioned algorithm except that the goal is "forced" to be the next state in the motion. This makes PHC an algorithm specifically designed for one-step tracking. We use a TD3-based variant of the original implementation (Luo et al., 2023). Concretely the implementation is exactly the same of TD3 but we changed the underlying environment. In this tracking environment the state is defined as the concatenation of the current state $s$ and the state $g$ to track. The resulting state space is $\mathbb{R}^{716}$ . At the beginning of an episode, we sample a motion $m$ from the motion set (either $\mathcal{M}$ or $\mathcal{D}_{\mathrm{test}}$ ) and we initialize the agent to a randomly selected state of the motion. Let $\bar{t}$ being the randomly selected initial step of the motion, then at any episode step $t \in [1, \mathrm{len}(m) - \bar{t} - 1]$ the target state $g_{t}$ correspond to the motion state $m_{\bar{t} + t + 1}$ . We use the negative distance in position/orientation as reward function, i.e., $r((s, g), a, (s', g')) = -d_{\mathrm{smp1}}(g, s')$ .
+
+Inference methods. By being a goal-conditioned algorithm we just need to pass the desired goal as target reference and can be evaluated for goal and tracking tasks.
+
+Table 12 Hyperparameters used for PHC pretraining.
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 50% |
| -goals from the online buffer | 50% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
+
+# C.5.8 GOAL-TD3
+
+We closely follow the implementation of Pirotta et al. (2024). For reaching each goal $g$ , we use the reward function $r(s', g) = -\|\mathrm{pos}(s') - \mathrm{pos}(g)\|_2$ , where $\mathrm{pos}(\cdot)$ extracts only the position of each joint, ignoring their velocities. We then train a goal-conditioned TD3 agent to optimize such a reward for all $g$ . We sample a percentage of training goals from the unlabeled dataset, and a percentage using hindsight experience replay (HER, Andrychowicz et al., 2017) on trajectories from the online buffer.
+
+Inference methods. We use the same methods as ASE for goal-based and tracking inference.
+
+Table 14 Hyperparameters used for GOAL-TD3 pretraining.
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for HER sampling | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 100% |
| -goals from the online buffer (HER) | 0% |
| Probability of relabeling zs | 0.5 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
+
+# C.5.9 MPPI
+
+We use MPPI with the real dynamic and real reward function for each task. For each evaluation state, action plans are sampled according to a factorized Gaussian distribution. Initially, mean and standard variation of the Gaussian are set with 0 and 1, respectively. actions plans are evaluated by deploying them in the real dynamics and computed the cumulative return over some planning horizon. Subsequently, the Gaussian parameters are updated using the top- $k$ most rewarding plans. For goal-reaching tasks, we use the reward $r(s', g) = -\|\mathrm{pos}(s') - \mathrm{pos}(g)\|_2$
+
+Table 15 Hyperparameters used for MPPI planning.
+
+| Task | TD3 | ORACLE MPPI Normalized | DIFFUSER Normalized | ASE Normalized | FB-CPR Normalized |
| move-ego-0-2-raisearms-l-1 | 191.13 | 168.22 | 0.88 | 148.10 (0.47) | 0.77 (0.00) | 145.78 (7.59) | 0.76 (0.04) | 145.59 (4.38) | 0.76 (0.02) |
| move-ego-0-2-raisearms-l-m | 174.97 | 194.84 | 1.11 | 125.14 (2.16) | 0.72 (0.01) | 109.36 (30.34) | 0.63 (0.17) | 143.90 (7.09) | 0.82 (0.04) |
| move-ego-0-2-raisearms-l-h | 194.72 | 114.30 | 0.59 | 103.11 (1.22) | 0.53 (0.01) | 129.21 (31.41) | 0.66 (0.16) | 123.14 (15.90) | 0.63 (0.08) |
| move-ego-0-2-raisearms-m-l | 179.42 | 199.26 | 1.11 | 124.31 (4.28) | 0.69 (0.02) | 125.39 (5.79) | 0.70 (0.03) | 136.74 (2.40) | 0.76 (0.01) |
| move-ego-0-2-raisearms-m-m | 178.42 | 155.28 | 0.87 | 121.55 (3.97) | 0.68 (0.02) | 60.19 (24.89) | 0.34 (0.14) | 139.19 (18.63) | 0.78 (0.10) |
| move-ego-0-2-raisearms-m-h | 179.02 | 129.99 | 0.73 | 116.50 (3.88) | 0.65 (0.02) | 123.84 (6.10) | 0.69 (0.03) | 128.15 (0.86) | 0.72 (0.00) |
| move-ego-0-2-raisearms-h-l | 191.00 | 115.25 | 0.60 | 101.58 (2.72) | 0.53 (0.01) | 85.89 (7.09) | 0.45 (0.04) | 111.92 (1.20) | 0.59 (0.01) |
| move-ego-0-2-raisearms-h-m | 175.72 | 130.86 | 0.74 | 113.81 (3.34) | 0.65 (0.02) | 121.19 (4.20) | 0.69 (0.02) | 128.10 (0.78) | 0.73 (0.00) |
| move-ego-0-2-raisearms-h-h | 165.19 | 112.35 | 0.68 | 102.09 (3.56) | 0.62 (0.02) | 133.96 (14.35) | 0.81 (0.09) | 143.83 (14.21) | 0.87 (0.09) |
| Average | 181.06 | 146.70 | 0.81 | 117.36 | 0.65 | 114.98 | 0.64 | 133.40 | 0.74 |
| Median | 179.02 | 130.86 | 0.74 | 116.50 | 0.65 | 123.84 | 0.69 | 136.74 | 0.76 |
+
+Table 25 Average return for each task in the composite reward evaluation. These tasks combine between locomotion and arm-raising behaviors
+
+assessed as human-like by raters, even in tasks when they consider it failed completing the task. Interestingly, while the human-likeness of FB-CPR may come at the cost of lower reward scores, it does not affect the perceived success in accomplishing the assigned goal tasks and FB-CPR has better success rate than TD3 for those tasks.
+
+More in detail, per-task success rate scores are presented in Fig. 9 and Fig. 10.
+
+# D.3.2 Reward-based tasks
+
+We provide a further investigation of the performance of our FB-CPR agent on tasks that are i) a combination of tasks used for the main evaluation; and ii) highly under-specified.
+
+The objective $i$ is to evaluate the ability of FB-CPR of composing behaviors. We thus created a new category of reward-based tasks by combining locomotion and arm-raising tasks. Specifically, we pair the medium-speed forward locomotion task (with an angle of zero and speed of 2) with all possible arm-raising tasks. Since these two types of tasks have conflicting objectives - locomotion requires movement, while arm-raising rewards stillness - we define a composite reward function that balances the two. This is achieved by taking a weighted average of the individual task rewards, where the weighting varies depending on the specific task combination. Tab. 25 reports the performance of the algorithms on these "combined" tasks. We can see that FB-CPR is able to achieve $74\%$ of the performance of TD3 trained on each individual task. Despite the higher performance, even in this case, TD3 generates unnatural
+
+
+Figure 9 Human-likeness and success rate scores of algorithms per goal task sorted by FB-CPR performance.
+
+behaviors. The higher quality of FB-CPR is evident in Fig. 11 where we report a few frames of an episode for the task move-ego-0-2-raisearms-m-m. Similarly, almost the totality (about $98\%$ ) of human evaluators rated FB-CPR as more natural than TD3 on these tasks.
+
+The objective of ii) is to evaluate the ability of our model to solve task with a human-like bias. To show this, we designed a few reward functions inspired by the way human person would describe a task.
+
+Run. The simplest way to describe running is "move with high speed". Let $v_{x}$ and $v_{y}$ the horizontal velocities of the center of mass at the pelvis joint. Then, we define the reward for the task $\mathrm{RUN}_{\mathrm{eq}}$ as
+
+$$
+r (s ^ {\prime}) = \mathbb {I} (v _ {x} ^ {2} + v _ {y} ^ {2} > 2)
+$$
+
+Walking with left hand up. This task has two components: walking requires moving with low speed; raising the hand means having the hand $z$ -coordinate above a certain threshold. Then, we define the reward for the task WALK-LAMeq as
+
+$$
+r (s ^ {\prime}) = \mathbb {I} \Big [ 1 < (v _ {x} ^ {2} + v _ {y} ^ {2}) < 1. 5 \Big ] \cdot \mathbb {I} \Big [ z _ {\mathrm {l e f t w r i s t}} > 1. 2 \Big ]
+$$
+
+Standing with right foot up. This is the most complex task. We define standing at being in upright position with the head z-coordinate above a certain threshold and zero velocity. Similar to before, we ask the right ankle to be above a certain threshold. Then, we define the reward for the tasks $\mathrm{STAND - RTM_{eq}}$ ( $\beta = 0.5$ ) and $\mathrm{STAND - RTH_{eq}}$ ( $\beta = 1.2$ ) as
+
+$$
+r (s ^ {\prime}) = \mathbb {I} \Big [ \mathrm {u p} > 0. 9 \Big ] \cdot \mathbb {I} \Big [ z _ {\mathrm {h e a d}} > 1. 4 \Big ] \cdot \exp \Big (- \sqrt {v _ {x} ^ {2} + v _ {y} ^ {2}} \Big) \cdot \mathbb {I} \Big [ z _ {\mathrm {r i g h t a n k l e}} > \beta \Big ]
+$$
+
+It is evident to any expert in Reinforcement Learning (RL) that the reward functions in question are not optimal for learning from scratch. These reward functions are too vague, and a traditional RL algorithm would likely derive a
+
+
+Figure 10 Human-likeness and success rate scores of algorithms per reward task sorted by FB-CPR performance.
+
+high-performing policy that deviates significantly from the natural "behavioral" biases. For instance, with TD3, we observe completely unnatural behaviors. In stark contrast, FB-CPR manages to address the tasks in a manner that closely resembles human behavior (refer to Fig. 13). Intriguingly, FB-CPR appears to identify the "simplest" policy necessary to solve a task. It effectively distinguishes between two different policies, $\mathrm{STAND - RTM_{eq}}$ and $\mathrm{STAND - RTH_{eq}}$ , even though the policy designed for the higher task would suffice for the medium task, provided that the foot remains above a certain threshold. It is also evident the data bias. For example, we do not specify the direction of movement in run, just the high speed. FB-CPR recovers a perfect forward movement probably because the majority of run motions in $\mathcal{M}$ show this behavior. ASE is not able to solve all the tasks.
+
+
+Figure 11 Example of combination of locomotion and arm raising tasks (move-ego-0-2-raisearms-m-m). Our FB-CPR (top) agent produces natural human motions while TD3 (bottom) learns high-performing but unnatural behaviors. ASE (middle) has a natural behavior but it is not correctly aligned with the tasks (arms are in the high position not medium).
+
+
+Figure 12 Human-evaluation on locomotion combined with arm raising. Left figure reports the percentage of times a behavior solved a reward-based task (tasks are independently evaluated). Right figure reports the score for human-likeness by direct comparison of the two algorithms.
+
+
+
+
+Figure 13 Example of behaviors inferred by FB-CPR from under-specified reward equations.
+
+
+Figure 14 Rollouts of policies learned by different variants of METRA on Humanoid. Each line corresponds to a trajectory in $(x, y, z)$ space generated by a policy $\pi_z$ with $z$ uniformly sampled from the unit sphere. (left) The original METRA algorithm trained from scratch (no unlabeled data) with representation $\phi$ taking as input the full observation vector. (middle) The original METRA algorithm trained from scratch (no unlabeled data) with representation $\phi$ taking as input only the linear velocities of the robot's pelvis along the x,y,z axes. (right) The ASE algorithm trained within the same setting as in Table 1 but with METRA replacing DIAYN as the skill discovery component.
+
+
+
+
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 16 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 16, 2 hidden layers |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-6 |
| Constraint slack ε | 10-3 |
| Initial Lagrange multiplier λ | 30 |
| z distributionν | uniform on unit sphere |
| Probability of relabeling zs | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
+
+Inference methods. For goal-based inference, we follow the zero-shot scheme proposed by Park et al. (2024c): when given a goal state $g$ to reach from state $s$ , we set $z = (\phi(g) - \phi(s)) / \|\phi(g) - \phi(s)\|_2$ . Similarly, for tracking we set $z_t = (\phi(g_{t+1}) - \phi(s_t)) / \|\phi(g_{t+1}) - \phi(s_t)\|_2$ at each step $t$ of the episode, where $g_{t+1}$ is the next state in the trajectory to be tracked, while $s_t$ is current agent state. Finally, for reward inference, given a dataset of transitions $(s, s', r)$ sampled from the train buffer and labeled with the corresponding reward $r$ , we infer $z$ through linear regression on top of features $\phi(s') - \phi(s)$ . This is motivated by the fact that METRA's actor is pretrained to maximize a self-supervised reward function given by $r(s, s', z) := (\phi(s') - \phi(s))^T z$ . Notice, however, that we do not expect this to work well since such a reward, up to discounting, yields a telescopic sum which eventually makes the agent care only about the reward received at the end of an episode instead of the cumulative sum. Thus we report its performance for completeness.
+
+Results. We test METRA in the same setting as Table 1. The results are reported in the first row of Table 26, where we find that METRA achieves near zero performance in all tasks. After a deeper investigation, we found that in all runs, and with all hyperparameters we tested, the agent simply learned to fall on the floor and remain still in different positions, as shown in Figure 14 (left). Interestingly, this happens despite all the objectives, and in particular the "diversity loss" for representation learning, are well optimized during pre-training. This is due to the fact that, from the agent perspective, lying still on the floor in different positions can be regarded as displaying diverse behaviors, and no extra inductive bias would push the agent to learn more complicated skills (e.g., locomotion ones). On the other hand, we believe that METRA manages to learn few of such skills in the Humanoid experiments of Park et al. (2024c) given that it is pretrained on pixel-based observations (instead of proprioception) with a color map on the ground and very small dimension of the latent space $(d = 2)$ . This may provide an implicit inductive bias towards locomotion behaviors that make the robot move around the x,y coordinates, which are likely to be the observation variables that can be maximally spread out by the agent's controls. On the other hand, we do not have any such bias in our setup, where each joint has roughly the same "controllability" and the agent thus learns the simplest way to display diverse behaviors.
+
+To verify this last conjecture, we retrained METRA with the same parameters except that we make the representation $\phi$ only a function of the linear velocities of the robot's pelvis along the three x,y,z directions. Intuitively, this should provide an inductive bias that makes the agent focus on controlling those variables alone, thus learning locomotion behaviors to move around the x,y,z space. This is confirmed in Figure 14 (middle), where we see that the learned skills do not collapse anymore but rather move around different directions of the space.
+
+METRA with ASE regularization. Finally, we tried to combine METRA with the same policy regularization on top of unlabeled data as used by ASE. As we recall that ASE (Peng et al., 2022) combines a USD algorithm (DIAYN) with an unconditional policy regularization term, we simply replace DIAYN with METRA and keep all other components the same. The results are shown in Table 26, where we see that the ASE regularization improves the performance of METRA significantly on goal reaching and tracking. Moreover, METRA-ASE achieves competitive performance w.r.t. the original DIAYN-based ASE, improving its success rate in those tasks. Both DIAYN-ASE and METRA-ASE perform, however, significantly worse than FB-CPR. Finally, we note from Figure 14 (right) that METRA-ASE learns to navigate along different directions, though less far than plain METRA trained only on the pelvis' velocities. This is likely due to the regularization w.r.t. unlabeled data, which makes the agent focus on human-like behaviors, thus
+
+avoiding over-actuated movements that would be otherwise learned when naively trying to maximize controls of a subset of the observation variables.
+
+# E Understanding the Behavioral Latent Space
+
+In this section, we summarize results from a qualitative investigation aimed at better understanding the structure of the latent space learned by FB-CPR. We recall that the latent space $Z$ works at the same time as a state embedding through $B(s)$ , a trajectory embedding through $\mathrm{ER}_{\mathrm{FB}}$ , and a policy embedding through $\pi_z$ .
+
+# E.1 Diversity, Dataset Coverage and Transitions
+
+In this section we intend to further investigate the behaviors learned by FB-CPR beyond its performance in solving downstream tasks.
+
+
+Figure 15 Distribution of EMD distance between trajectories generated by two randomly sampled policies $\pi_z$ and $\pi_{z'}$ .
+
+| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| Regularization coefficient α | 0.01 |
| F network | second column of Tab. 6, output dim 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| B network | fourth column of Tab. 5, output dim 256 |
| Discriminator | Tab. 7 |
| Learning rate for F | 10-4 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for B | 10-5 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -encoding of unlabeled trajectories | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| FB penalty coefficient | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
| Coefficient for Fz-regularization loss | 0.1 |
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| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 64 |
| Regularization coefficient α | 0.01 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 64 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-8 |
| Coefficient for orthonormality loss | 100 |
| z distributionν | |
| -goals from unlabeled dataset | 60% |
| -goals from the online buffer | 20% |
| -uniform on unit sphere | 20% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Coefficient for diversity loss (Eq. 15 in (Peng et al., 2022)) | 0 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
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| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 256 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| φ encoder network | transformer (see text above) |
| -attention blocks | 2 |
| -embedding dim | 256 |
| -MLP first linear layer | 256x1024 |
| -MLP second linear layer | 1024x256 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-7 |
| Coefficient for constrastive loss | 0.1 |
| z distributionν | |
| -encoding of unlabeled trajectories | 100% |
| -goals from the online buffer | 0% |
| -uniform on unit sphere | 0% |
| Probability of relabeling zs) | 1 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
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| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for trajectory sampling from D | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Discriminator | Tab. 7 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 50% |
| -goals from the online buffer | 50% |
| Probability of relabeling zs) | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
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+ "html": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| Sequence length for HER sampling | 8 |
| goal update frequency during rollouts | once every 150 steps |
| actor network | third column of Tab. 6, output dim = action dim |
| critic network | second column of Tab. 6, output dim 1 |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| goal sampling distribution | |
| -goals from the unlabeled dataset | 100% |
| -goals from the online buffer (HER) | 0% |
| Probability of relabeling zs | 0.5 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
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+ "html": "| Task | TD3 | ORACLE MPPI Normalized | DIFFUSER Normalized | ASE Normalized | FB-CPR Normalized |
| move-ego-0-2-raisearms-l-1 | 191.13 | 168.22 | 0.88 | 148.10 (0.47) | 0.77 (0.00) | 145.78 (7.59) | 0.76 (0.04) | 145.59 (4.38) | 0.76 (0.02) |
| move-ego-0-2-raisearms-l-m | 174.97 | 194.84 | 1.11 | 125.14 (2.16) | 0.72 (0.01) | 109.36 (30.34) | 0.63 (0.17) | 143.90 (7.09) | 0.82 (0.04) |
| move-ego-0-2-raisearms-l-h | 194.72 | 114.30 | 0.59 | 103.11 (1.22) | 0.53 (0.01) | 129.21 (31.41) | 0.66 (0.16) | 123.14 (15.90) | 0.63 (0.08) |
| move-ego-0-2-raisearms-m-l | 179.42 | 199.26 | 1.11 | 124.31 (4.28) | 0.69 (0.02) | 125.39 (5.79) | 0.70 (0.03) | 136.74 (2.40) | 0.76 (0.01) |
| move-ego-0-2-raisearms-m-m | 178.42 | 155.28 | 0.87 | 121.55 (3.97) | 0.68 (0.02) | 60.19 (24.89) | 0.34 (0.14) | 139.19 (18.63) | 0.78 (0.10) |
| move-ego-0-2-raisearms-m-h | 179.02 | 129.99 | 0.73 | 116.50 (3.88) | 0.65 (0.02) | 123.84 (6.10) | 0.69 (0.03) | 128.15 (0.86) | 0.72 (0.00) |
| move-ego-0-2-raisearms-h-l | 191.00 | 115.25 | 0.60 | 101.58 (2.72) | 0.53 (0.01) | 85.89 (7.09) | 0.45 (0.04) | 111.92 (1.20) | 0.59 (0.01) |
| move-ego-0-2-raisearms-h-m | 175.72 | 130.86 | 0.74 | 113.81 (3.34) | 0.65 (0.02) | 121.19 (4.20) | 0.69 (0.02) | 128.10 (0.78) | 0.73 (0.00) |
| move-ego-0-2-raisearms-h-h | 165.19 | 112.35 | 0.68 | 102.09 (3.56) | 0.62 (0.02) | 133.96 (14.35) | 0.81 (0.09) | 143.83 (14.21) | 0.87 (0.09) |
| Average | 181.06 | 146.70 | 0.81 | 117.36 | 0.65 | 114.98 | 0.64 | 133.40 | 0.74 |
| Median | 179.02 | 130.86 | 0.74 | 116.50 | 0.65 | 123.84 | 0.69 | 136.74 | 0.76 |
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+ "html": "| Hyperparameter | Value |
| General training parameters | See Tab. 3 |
| General prioritization parameters | See Tab. 4 |
| z update frequency during rollouts | once every 150 steps |
| z dimension d | 16 |
| actor network | third column of Tab. 6, output dim = action dim |
| critic networks | second column of Tab. 6, output dim 1 |
| φ encoder network | fourth column of Tab. 5, output dim 16, 2 hidden layers |
| Learning rate for actor | 10-4 |
| Learning rate for critic | 10-4 |
| Learning rate for φ | 10-6 |
| Constraint slack ε | 10-3 |
| Initial Lagrange multiplier λ | 30 |
| z distributionν | uniform on unit sphere |
| Probability of relabeling zs | 0.8 |
| Polyak coefficient for target network update | 0.005 |
| Actor penalty coefficient | 0.5 |
| Critic penalty coefficient | 0.5 |
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+ "content": ". This may provide an implicit inductive bias towards locomotion behaviors that make the robot move around the x,y coordinates, which are likely to be the observation variables that can be maximally spread out by the agent's controls. On the other hand, we do not have any such bias in our setup, where each joint has roughly the same \"controllability\" and the agent thus learns the simplest way to display diverse behaviors."
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+ "html": "| Model | Modalities | Any-to-Any Generation | Multi-Scale Features |
| RemoteCLIP | optical, text | X | X |
| CROMA | optical, radar | X | X |
| AnySat | aerial, optical, radar, NAIP | X | X |
| DeCUR | optical, radar | X | X |
| DOFA | optical, radar, hyperspectral, NAIP | X | X |
| MetaEarth | optical (unimodal) | X | ✓ |
| Galileo | optical, radar, elevation, weather, location, population, ... | X | ✓ |
| TerraMind | optical, radar, land use, elevation, vegetation index, location, text | ✓ | ✓ |
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+ "text": "Table 1. Comparison of TerraMind to other model architectures. TerraMind represents a first-of-its-kind generative, multimodal model.",
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+ "text": "be open-sourced to the community. TerraMesh builds on existing datasets, which we expand by adding modalities from external data sources or by applying pseudo-labeling. We provide an overview of the aligned image modalities and a detailed dataset description in the supplementary material.",
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+ "text": "Base datasets. TerraMesh is based on SSL4EO-S12 [6, 65] and MajorTOM-Core [23], two unlabeled remote sensing datasets containing co-aligned radar and optical imagery from Sentinel-1 and Sentinel-2 satellites. SSL4EO-S12 has lower geographic coverage but is multi-seasonal. MajorTOM-Core covers most of the Earth's land surface at a single timestamp. For MajorTOM-Core, we apply a subsampling scheme based on LULC classes and ecoregions. TerraMesh includes a total of approximately 9 million globally distributed samples from both Sentinel-1 and Sentinel-2,",
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+ "text": "Additional modalities. We obtain co-aligned yearly LULC maps by ESRI with nine land use classes. Additionally, we leverage SEnSeI v2 [22] as a cloud and ice annotation model and update the ESRI LULC classes for better spatiotemporal alignment. NDVI maps are computed using the corresponding spectral bands from Sentinel-2. DEM is extracted from the Copernicus DEM 30m dataset [2], which provides global coverage of the Earth's elevation at a 30m resolution. Captions are generated synthetically by constructing RGB images from Sentinel-2 patches using the corresponding spectral bands and processing them with LLaVANext [37]. A tailored prompt guides the model to describe the content of each image as described in [47]. For geolocations, we round latitude and longitude from the center of each patch to the nearest quarter degree and store the discretized coordinates as strings in a pre-defined format.",
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+ "text": "TerraMind pretraining is two-staged following [52]. We first pretrain unimodal tokenizer models, tokenize the modalities, and then leverage token-level and pixel-level input to pretrain the TerraMind encoder-decoder architecture. We describe those individual stages in the following.",
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+ "text": "We develop modality-specific tokenizers to encode each modality into a sequence of discrete tokens for pretraining and decode token sequences back to images. Thus, TerraMind is in principle compatible with any modality, as long as it can be tokenized and aligned with other modalities. For reasons of space, we delegate most experiments related to the tokenizer performances to the supplementary material.",
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+ "text": "Image-like modalities. We train autoencoder-based architectures with a quantization step in the bottleneck for image-like modalities such as S-1, S-2, LULC, NDVI, and DEM. Tokenizer encoders process an input image and generate a latent representation for each $16 \\times 16$ patch, which is then discretized with finite-scalar-quantization (FSQ) [51] into one of $N$ codewords. All tokenizers use a vocabulary size of 16K besides the simpler LULC modality for which we use 4K. These codewords are then used by the diffusion decoder to reconstruct the original image. The benefit of leveraging diffusion decoders lies in facilitating cross-modal generation in TerraMind by transforming tokens back into images. By mapping each codeword to a unique integer in $\\{0, 1, \\dots, N - 1\\}$ , we obtain discrete tokens for each image patch. We pretrain the tokenizers in a self-supervised setting. FSQ as quantization method enhances training stability [51] compared to vector quantization [63] by eliminating the need for codebook-related loss terms. Notably, FSQ is",
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+ "text": "heavily influenced by ideas of neural compression [27]. For example, on 12-band S-2 images, we achieve compression rates of over $3000\\mathrm{x}$ by applying quantization. We summarize the architecture of our tokenizers in Figure 4. The main objective of the overall tokenizer is to encode image patches consistently into discrete tokens based on semantic similarity to enable cross-modal correlation learning. Therefore, the loss of some details is an expected trade-off, since the focus is on grouping similar patches rather than preserving all fine-grained features. Naturally, more accurate reconstructions facilitate cross-modal generation, however the main focus of the pretraining lies on consistent cross-modal correlation learning. We provided further details on the pretraining of the tokenizers in the supplementary material.",
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+ "text": "Sequence-like modalities. We treat both captions and geolocations as text and use a single text tokenizer to process both modalities. By discretizing the geographic coordinates and representing them as strings, we introduce special coordinate tokens into the vocabulary. This allows us to encode geolocations as a sequence of discrete tokens, beginning with a latitude token followed by a longitude token. For textual data, we modify the existing WordPiece tokenizer [33].",
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+ "text": "Architecture. TerraMind uses a symmetric Transformer-based encoder-decoder architecture proposed in [52], which is designed to process sequences of multimodal tokens. In addition to discrete tokens, TerraMind accepts pixel-level inputs, specifically satellite imagery and digital elevation maps. For pixel-level inputs, we apply learnable patch-wise linear projections to generate patch embeddings for each $16 \\times 16$ patch, similar to the approach used in ViT [17].",
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+ "text": "Dual-scale early fusion. In contrast to [52], we not only embed token-level data but additionally leverage pixel-level data across a range of input modalities to introduce a dual-scale feature representation to support the structuring of the embedding space. Both tokens and patches represent a 16x16 pixel area. Tokens represent this area via a single discrete integer value, while the image patches describe the same area with the actual floating point sensor data. Thus, during pretraining, the model not only learns a correlation between modalities (i.e., cross-modal learning) but also between dif",
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+ "text": "ferent levels of abstraction within the same modality. The low-level token information enables cross-modal correlation learning, while adding pixel level input accounts for spatial nuances. Based on dual-scale features the model further learns to better structure pixel-level data in the embedding space via the corresponding information from the discrete token. We illustrate the pretraining paradigm in Figure 5. The model is agnostic to processing tokens or patches in the input space, while the target is generally token-level data. We use six pixel-level modalities and eight token-level modalities.",
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+ "Figure 5. Illustration of the pre-training task. Given an encoded multimodal sample of random subsets of patches and input tokens, the decoder predicts target tokens for the masked input."
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+ "text": "Masking strategy. TerraMind applies a masked modeling approach in the token space following [52]. The model leverages a set of randomly selected target tokens that have to be reconstructed from a randomly selected set of input tokens and pixel-level data. During pre-training, we sample input and target data from a Dirichlet distribution.",
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+ "text": "We opt for masked token reconstruction to familiarize the model with the absence of entire modalities, which is crucial for a high usability of a multimodal model in Earth observation. During pre-training, the model learns an internal representation of unseen modalities which is expected to benefit a range of downstream applications. In addition, sampling input and target tokens improves the computational efficiency of the pre-training, as each token is a compressed representation of a patch with compression factors of between 250x and 3000x depending on the modality. Finally, without tokenized representations of the image-like modalities, it is challenging to learn the correlation to sequence-like modalities. The overall training objective of TerraMind boils down to a cross-modal patch-level classification problem optimized via a cross entropy loss:",
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+ "text": "\n$$\n\\mathcal {L} _ {\\mathrm {C E}} = - \\sum_ {i = 1} ^ {N} y _ {i} \\log \\left(p _ {i}\\right), \\tag {1}\n$$\n",
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+ "text": "where $y_{i}$ is the one-hot encoded true class of token $i$ , $p_{i}$ is the predicted probability for token $i$ , $N$ is the total number of possible tokens. Interestingly, we can infer an upper bound loss for a random model where the cross entropy loss will collapse to the natural logarithm of the vocabulary size $\\mathcal{L}_{\\mathrm{CE,random}} = -\\sum_{i=1}^{N} y_{i} \\log \\left( \\frac{1}{N} \\right) = \\log(N)$ .",
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+ "text": "Scaling. We trained three versions of TerraMind scaling across model size, compute, and data. In addition, we pretrain different versions of TerraMind with respect to the number of dual-scale features. TerraMindv1-B is pre-trained on 500B tokens for 6 days on 32 NVIDIA A100 GPUs. The model uses dual-scale features from both token-level and pixel-level. During initial experiments, we observed significant improvements from scaling model size when switching from a tiny backbone to a small backbone to a base backbone. Therefore, we pre-trained TerraMindv1-L on a large backbone with 500B tokens on 32 NVIDIA A100 GPUs trained for 10 days. Finally, to better understand the effect of scaling across the dual-scale feature representation, we pre-train TerraMindv1-B-single as a single-scale model on primarily token-level data with optical S-2 L2A data as only pixel-level input (compared to pixel-level S-2 L1C, S-2 RGB, S-1 GRD, S-1 RTC, and DEM in TerraMindv1-B and -L). TerraMindv1-B-single is pretrained on 500B tokens from over one million samples for 6 days on 32 NVIDIA A100 GPUs. We summarize the scaling behavior in model size, compute, and data in Figure 9 of the supplementary material. We additionally provide final validation losses in Table 9 comparing v1-B and v1-L with the theoretical random loss.",
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+ "type": "text",
+ "text": "4.3. Generation",
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+ "text": "Once pretrained, TerraMind can generate tokens for any modality, conditioned on any subset of input modalities. The generative capabilities unlock various zero-shot tasks, such as water body segmentation. For the generation of image-like modalities, the decoder receives mask tokens for the modality to be generated and predicts the corresponding tokens based on the encoded input. For sequence-like modalities, the decoder generates the output autoregressively. After generating tokens from the target modality, the corresponding tokenizer decoder allows to map from token-space to image or text space. TerraMind further supports chained generation which ensures consistency across generated modalities. The chained generation represents a conditional probability distribution where the prior probability distribution is determined by the input modality, and all subsequent modalities are generated conditioned on the input modality and potentially other generated modalities.",
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+ "text": "4.4. Thinking-in-Modalities",
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+ "text": "Thinking in Modalities (TiM) is a recursive fine-tuning and inference technique designed to enhance multimodal learning by leveraging the generative capabilities of the model itself. Given an input $x \\in \\mathcal{X}$ (e.g., an optical satellite image), the model first generates additional synthetic modalities $\\tilde{x} = f_{\\mathrm{gen}}(x)$ on a token-level using a learned generative function $f_{\\mathrm{gen}}$ . These generated tokens are then concatenated with the original input and jointly processed by the downstream",
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+ "type": "text",
+ "text": "model $f$ (e.g., TerraMind encoder with a segmentation head), yielding the final output $y = f(x, f_{\\mathrm{gen}}(x))$ . This formulation allows the model to reason over both observed and inferred modalities, effectively enriching the input space. TiM can leverage multiple generated modalities which are then generated in a chained approach. For example, for $k$ modalities, the input is augmented with newly generated modalities:",
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+ "text": "\n$$\n\\tilde {x} ^ {(k + 1)} = \\tilde {x} ^ {(k)} \\cup f _ {\\text {g e n}} (\\tilde {x} ^ {(k)}), \\tag {2}\n$$\n",
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+ "text": "and the final model output is described by:",
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+ "text": "\n$$\ny = f \\left(\\tilde {x} ^ {(K)}\\right). \\tag {3}\n$$\n",
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+ "text": "This recursive augmentation mimics a chain-of-thought process, enabling the model to iteratively refine its internal representation, particularly in scenarios with missing modalities.",
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+ "text": "5. Experiments",
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+ "text": "In this section, we describe the performance gains resulting from TerraMind and experiment with the unlocked capabilities of any-to-any generation and Thinking-in-Modalities.",
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+ "text": "5.1. Foundational experiments",
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+ "text": "Multimodality vs. unimodality. As a first motivational experiment, we outline the benefit of using multimodal data in Earth observation at the example of water body mapping. Specifically, we leverage the ViT-B encoders from the unimodal tokenizer models for S-1, S-2, and LULC, concatenate their embeddings, and train a segmentation head with four ConvNeXt [43] blocks as a late fusion approach. The results in Table 2 (left) suggest that regardless of which modalities we combine, the combination of two modalities always outperforms each unimodal model. Combining all three modalities achieves the best overall performance.",
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+ "type": "table",
+ "img_path": "images/10e82d7081183d67d8d8d2f7890ae2cc11feda557a3eb9cc3cc13bf64d1265c0.jpg",
+ "table_caption": [],
+ "table_footnote": [],
+ "table_body": "| Model | Modalities | Any-to-Any Generation | Multi-Scale Features |
| RemoteCLIP | optical, text | X | X |
| CROMA | optical, radar | X | X |
| AnySat | aerial, optical, radar, NAIP | X | X |
| DeCUR | optical, radar | X | X |
| DOFA | optical, radar, hyperspectral, NAIP | X | X |
| MetaEarth | optical (unimodal) | X | ✓ |
| Galileo | optical, radar, elevation, weather, location, population, ... | X | ✓ |
| TerraMind | optical, radar, land use, elevation, vegetation index, location, text | ✓ | ✓ |
"
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+ "type": "table_caption",
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+ "angle": 0,
+ "content": "Table 1. Comparison of TerraMind to other model architectures. TerraMind represents a first-of-its-kind generative, multimodal model."
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+ "type": "text",
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+ "angle": 0,
+ "content": "be open-sourced to the community. TerraMesh builds on existing datasets, which we expand by adding modalities from external data sources or by applying pseudo-labeling. We provide an overview of the aligned image modalities and a detailed dataset description in the supplementary material."
+ },
+ {
+ "type": "text",
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+ "content": "Base datasets. TerraMesh is based on SSL4EO-S12 [6, 65] and MajorTOM-Core [23], two unlabeled remote sensing datasets containing co-aligned radar and optical imagery from Sentinel-1 and Sentinel-2 satellites. SSL4EO-S12 has lower geographic coverage but is multi-seasonal. MajorTOM-Core covers most of the Earth's land surface at a single timestamp. For MajorTOM-Core, we apply a subsampling scheme based on LULC classes and ecoregions. TerraMesh includes a total of approximately 9 million globally distributed samples from both Sentinel-1 and Sentinel-2,"
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+ "content": "each measuring \\(264 \\times 264\\) pixels at \\(10\\mathrm{m}\\) resolution."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "content": "Additional modalities. We obtain co-aligned yearly LULC maps by ESRI with nine land use classes. Additionally, we leverage SEnSeI v2 [22] as a cloud and ice annotation model and update the ESRI LULC classes for better spatiotemporal alignment. NDVI maps are computed using the corresponding spectral bands from Sentinel-2. DEM is extracted from the Copernicus DEM 30m dataset [2], which provides global coverage of the Earth's elevation at a 30m resolution. Captions are generated synthetically by constructing RGB images from Sentinel-2 patches using the corresponding spectral bands and processing them with LLaVANext [37]. A tailored prompt guides the model to describe the content of each image as described in [47]. For geolocations, we round latitude and longitude from the center of each patch to the nearest quarter degree and store the discretized coordinates as strings in a pre-defined format."
+ },
+ {
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+ "content": "4. Methods"
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+ "content": "TerraMind pretraining is two-staged following [52]. We first pretrain unimodal tokenizer models, tokenize the modalities, and then leverage token-level and pixel-level input to pretrain the TerraMind encoder-decoder architecture. We describe those individual stages in the following."
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+ "content": "4.1. Tokenization"
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+ "bbox": [
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+ "content": "We develop modality-specific tokenizers to encode each modality into a sequence of discrete tokens for pretraining and decode token sequences back to images. Thus, TerraMind is in principle compatible with any modality, as long as it can be tokenized and aligned with other modalities. For reasons of space, we delegate most experiments related to the tokenizer performances to the supplementary material."
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+ "angle": 0,
+ "content": "Image-like modalities. We train autoencoder-based architectures with a quantization step in the bottleneck for image-like modalities such as S-1, S-2, LULC, NDVI, and DEM. Tokenizer encoders process an input image and generate a latent representation for each \\(16 \\times 16\\) patch, which is then discretized with finite-scalar-quantization (FSQ) [51] into one of \\(N\\) codewords. All tokenizers use a vocabulary size of 16K besides the simpler LULC modality for which we use 4K. These codewords are then used by the diffusion decoder to reconstruct the original image. The benefit of leveraging diffusion decoders lies in facilitating cross-modal generation in TerraMind by transforming tokens back into images. By mapping each codeword to a unique integer in \\(\\{0, 1, \\dots, N - 1\\}\\), we obtain discrete tokens for each image patch. We pretrain the tokenizers in a self-supervised setting. FSQ as quantization method enhances training stability [51] compared to vector quantization [63] by eliminating the need for codebook-related loss terms. Notably, FSQ is"
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "heavily influenced by ideas of neural compression [27]. For example, on 12-band S-2 images, we achieve compression rates of over \\(3000\\mathrm{x}\\) by applying quantization. We summarize the architecture of our tokenizers in Figure 4. The main objective of the overall tokenizer is to encode image patches consistently into discrete tokens based on semantic similarity to enable cross-modal correlation learning. Therefore, the loss of some details is an expected trade-off, since the focus is on grouping similar patches rather than preserving all fine-grained features. Naturally, more accurate reconstructions facilitate cross-modal generation, however the main focus of the pretraining lies on consistent cross-modal correlation learning. We provided further details on the pretraining of the tokenizers in the supplementary material."
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+ "content": "Figure 4. Tokenizer for image-like modalities combining finite-scalar quantization [51] with diffusion decoding."
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+ "content": "Sequence-like modalities. We treat both captions and geolocations as text and use a single text tokenizer to process both modalities. By discretizing the geographic coordinates and representing them as strings, we introduce special coordinate tokens into the vocabulary. This allows us to encode geolocations as a sequence of discrete tokens, beginning with a latitude token followed by a longitude token. For textual data, we modify the existing WordPiece tokenizer [33]."
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+ "content": "4.2. Pre-training"
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+ "angle": 0,
+ "content": "Architecture. TerraMind uses a symmetric Transformer-based encoder-decoder architecture proposed in [52], which is designed to process sequences of multimodal tokens. In addition to discrete tokens, TerraMind accepts pixel-level inputs, specifically satellite imagery and digital elevation maps. For pixel-level inputs, we apply learnable patch-wise linear projections to generate patch embeddings for each \\(16 \\times 16\\) patch, similar to the approach used in ViT [17]."
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+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Dual-scale early fusion. In contrast to [52], we not only embed token-level data but additionally leverage pixel-level data across a range of input modalities to introduce a dual-scale feature representation to support the structuring of the embedding space. Both tokens and patches represent a 16x16 pixel area. Tokens represent this area via a single discrete integer value, while the image patches describe the same area with the actual floating point sensor data. Thus, during pretraining, the model not only learns a correlation between modalities (i.e., cross-modal learning) but also between dif"
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+ "type": "footer",
+ "bbox": [
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+ "content": "https://planetarycomputer.microsoft.com/dataset/io-lulc-annual-v02"
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+ ],
+ [
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "ferent levels of abstraction within the same modality. The low-level token information enables cross-modal correlation learning, while adding pixel level input accounts for spatial nuances. Based on dual-scale features the model further learns to better structure pixel-level data in the embedding space via the corresponding information from the discrete token. We illustrate the pretraining paradigm in Figure 5. The model is agnostic to processing tokens or patches in the input space, while the target is generally token-level data. We use six pixel-level modalities and eight token-level modalities."
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "Figure 5. Illustration of the pre-training task. Given an encoded multimodal sample of random subsets of patches and input tokens, the decoder predicts target tokens for the masked input."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "Masking strategy. TerraMind applies a masked modeling approach in the token space following [52]. The model leverages a set of randomly selected target tokens that have to be reconstructed from a randomly selected set of input tokens and pixel-level data. During pre-training, we sample input and target data from a Dirichlet distribution."
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "We opt for masked token reconstruction to familiarize the model with the absence of entire modalities, which is crucial for a high usability of a multimodal model in Earth observation. During pre-training, the model learns an internal representation of unseen modalities which is expected to benefit a range of downstream applications. In addition, sampling input and target tokens improves the computational efficiency of the pre-training, as each token is a compressed representation of a patch with compression factors of between 250x and 3000x depending on the modality. Finally, without tokenized representations of the image-like modalities, it is challenging to learn the correlation to sequence-like modalities. The overall training objective of TerraMind boils down to a cross-modal patch-level classification problem optimized via a cross entropy loss:"
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "\\[\n\\mathcal {L} _ {\\mathrm {C E}} = - \\sum_ {i = 1} ^ {N} y _ {i} \\log \\left(p _ {i}\\right), \\tag {1}\n\\]"
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+ {
+ "type": "text",
+ "bbox": [
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+ "angle": 0,
+ "content": "where \\( y_{i} \\) is the one-hot encoded true class of token \\( i \\), \\( p_{i} \\) is the predicted probability for token \\( i \\), \\( N \\) is the total number of possible tokens. Interestingly, we can infer an upper bound loss for a random model where the cross entropy loss will collapse to the natural logarithm of the vocabulary size \\( \\mathcal{L}_{\\mathrm{CE,random}} = -\\sum_{i=1}^{N} y_{i} \\log \\left( \\frac{1}{N} \\right) = \\log(N) \\)."
+ },
+ {
+ "type": "text",
+ "bbox": [
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+ ],
+ "angle": 0,
+ "content": "Scaling. We trained three versions of TerraMind scaling across model size, compute, and data. In addition, we pretrain different versions of TerraMind with respect to the number of dual-scale features. TerraMindv1-B is pre-trained on 500B tokens for 6 days on 32 NVIDIA A100 GPUs. The model uses dual-scale features from both token-level and pixel-level. During initial experiments, we observed significant improvements from scaling model size when switching from a tiny backbone to a small backbone to a base backbone. Therefore, we pre-trained TerraMindv1-L on a large backbone with 500B tokens on 32 NVIDIA A100 GPUs trained for 10 days. Finally, to better understand the effect of scaling across the dual-scale feature representation, we pre-train TerraMindv1-B-single as a single-scale model on primarily token-level data with optical S-2 L2A data as only pixel-level input (compared to pixel-level S-2 L1C, S-2 RGB, S-1 GRD, S-1 RTC, and DEM in TerraMindv1-B and -L). TerraMindv1-B-single is pretrained on 500B tokens from over one million samples for 6 days on 32 NVIDIA A100 GPUs. We summarize the scaling behavior in model size, compute, and data in Figure 9 of the supplementary material. We additionally provide final validation losses in Table 9 comparing v1-B and v1-L with the theoretical random loss."
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+ {
+ "type": "title",
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+ "angle": 0,
+ "content": "4.3. Generation"
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+ {
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+ "bbox": [
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+ "angle": 0,
+ "content": "Once pretrained, TerraMind can generate tokens for any modality, conditioned on any subset of input modalities. The generative capabilities unlock various zero-shot tasks, such as water body segmentation. For the generation of image-like modalities, the decoder receives mask tokens for the modality to be generated and predicts the corresponding tokens based on the encoded input. For sequence-like modalities, the decoder generates the output autoregressively. After generating tokens from the target modality, the corresponding tokenizer decoder allows to map from token-space to image or text space. TerraMind further supports chained generation which ensures consistency across generated modalities. The chained generation represents a conditional probability distribution where the prior probability distribution is determined by the input modality, and all subsequent modalities are generated conditioned on the input modality and potentially other generated modalities."
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+ "content": "4.4. Thinking-in-Modalities"
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+ "content": "Thinking in Modalities (TiM) is a recursive fine-tuning and inference technique designed to enhance multimodal learning by leveraging the generative capabilities of the model itself. Given an input \\( x \\in \\mathcal{X} \\) (e.g., an optical satellite image), the model first generates additional synthetic modalities \\( \\tilde{x} = f_{\\mathrm{gen}}(x) \\) on a token-level using a learned generative function \\( f_{\\mathrm{gen}} \\). These generated tokens are then concatenated with the original input and jointly processed by the downstream"
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+ "content": "model \\( f \\) (e.g., TerraMind encoder with a segmentation head), yielding the final output \\( y = f(x, f_{\\mathrm{gen}}(x)) \\). This formulation allows the model to reason over both observed and inferred modalities, effectively enriching the input space. TiM can leverage multiple generated modalities which are then generated in a chained approach. For example, for \\( k \\) modalities, the input is augmented with newly generated modalities:"
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+ "content": "\\[\n\\tilde {x} ^ {(k + 1)} = \\tilde {x} ^ {(k)} \\cup f _ {\\text {g e n}} (\\tilde {x} ^ {(k)}), \\tag {2}\n\\]"
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+ "content": "Multimodality vs. unimodality. As a first motivational experiment, we outline the benefit of using multimodal data in Earth observation at the example of water body mapping. Specifically, we leverage the ViT-B encoders from the unimodal tokenizer models for S-1, S-2, and LULC, concatenate their embeddings, and train a segmentation head with four ConvNeXt [43] blocks as a late fusion approach. The results in Table 2 (left) suggest that regardless of which modalities we combine, the combination of two modalities always outperforms each unimodal model. Combining all three modalities achieves the best overall performance."
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+ "content": "| Model | Modalities | Any-to-Any Generation | Multi-Scale Features |
| RemoteCLIP | optical, text | X | X |
| CROMA | optical, radar | X | X |
| AnySat | aerial, optical, radar, NAIP | X | X |
| DeCUR | optical, radar | X | X |
| DOFA | optical, radar, hyperspectral, NAIP | X | X |
| MetaEarth | optical (unimodal) | X | ✓ |
| Galileo | optical, radar, elevation, weather, location, population, ... | X | ✓ |
| TerraMind | optical, radar, land use, elevation, vegetation index, location, text | ✓ | ✓ |
+
+Table 1. Comparison of TerraMind to other model architectures. TerraMind represents a first-of-its-kind generative, multimodal model.
+
+be open-sourced to the community. TerraMesh builds on existing datasets, which we expand by adding modalities from external data sources or by applying pseudo-labeling. We provide an overview of the aligned image modalities and a detailed dataset description in the supplementary material.
+
+Base datasets. TerraMesh is based on SSL4EO-S12 [6, 65] and MajorTOM-Core [23], two unlabeled remote sensing datasets containing co-aligned radar and optical imagery from Sentinel-1 and Sentinel-2 satellites. SSL4EO-S12 has lower geographic coverage but is multi-seasonal. MajorTOM-Core covers most of the Earth's land surface at a single timestamp. For MajorTOM-Core, we apply a subsampling scheme based on LULC classes and ecoregions. TerraMesh includes a total of approximately 9 million globally distributed samples from both Sentinel-1 and Sentinel-2,
+
+each measuring $264 \times 264$ pixels at $10\mathrm{m}$ resolution.
+
+Additional modalities. We obtain co-aligned yearly LULC maps by ESRI with nine land use classes. Additionally, we leverage SEnSeI v2 [22] as a cloud and ice annotation model and update the ESRI LULC classes for better spatiotemporal alignment. NDVI maps are computed using the corresponding spectral bands from Sentinel-2. DEM is extracted from the Copernicus DEM 30m dataset [2], which provides global coverage of the Earth's elevation at a 30m resolution. Captions are generated synthetically by constructing RGB images from Sentinel-2 patches using the corresponding spectral bands and processing them with LLaVANext [37]. A tailored prompt guides the model to describe the content of each image as described in [47]. For geolocations, we round latitude and longitude from the center of each patch to the nearest quarter degree and store the discretized coordinates as strings in a pre-defined format.
+
+# 4. Methods
+
+TerraMind pretraining is two-staged following [52]. We first pretrain unimodal tokenizer models, tokenize the modalities, and then leverage token-level and pixel-level input to pretrain the TerraMind encoder-decoder architecture. We describe those individual stages in the following.
+
+# 4.1. Tokenization
+
+We develop modality-specific tokenizers to encode each modality into a sequence of discrete tokens for pretraining and decode token sequences back to images. Thus, TerraMind is in principle compatible with any modality, as long as it can be tokenized and aligned with other modalities. For reasons of space, we delegate most experiments related to the tokenizer performances to the supplementary material.
+
+Image-like modalities. We train autoencoder-based architectures with a quantization step in the bottleneck for image-like modalities such as S-1, S-2, LULC, NDVI, and DEM. Tokenizer encoders process an input image and generate a latent representation for each $16 \times 16$ patch, which is then discretized with finite-scalar-quantization (FSQ) [51] into one of $N$ codewords. All tokenizers use a vocabulary size of 16K besides the simpler LULC modality for which we use 4K. These codewords are then used by the diffusion decoder to reconstruct the original image. The benefit of leveraging diffusion decoders lies in facilitating cross-modal generation in TerraMind by transforming tokens back into images. By mapping each codeword to a unique integer in $\{0, 1, \dots, N - 1\}$ , we obtain discrete tokens for each image patch. We pretrain the tokenizers in a self-supervised setting. FSQ as quantization method enhances training stability [51] compared to vector quantization [63] by eliminating the need for codebook-related loss terms. Notably, FSQ is
+
+heavily influenced by ideas of neural compression [27]. For example, on 12-band S-2 images, we achieve compression rates of over $3000\mathrm{x}$ by applying quantization. We summarize the architecture of our tokenizers in Figure 4. The main objective of the overall tokenizer is to encode image patches consistently into discrete tokens based on semantic similarity to enable cross-modal correlation learning. Therefore, the loss of some details is an expected trade-off, since the focus is on grouping similar patches rather than preserving all fine-grained features. Naturally, more accurate reconstructions facilitate cross-modal generation, however the main focus of the pretraining lies on consistent cross-modal correlation learning. We provided further details on the pretraining of the tokenizers in the supplementary material.
+
+
+Figure 4. Tokenizer for image-like modalities combining finite-scalar quantization [51] with diffusion decoding.
+
+Sequence-like modalities. We treat both captions and geolocations as text and use a single text tokenizer to process both modalities. By discretizing the geographic coordinates and representing them as strings, we introduce special coordinate tokens into the vocabulary. This allows us to encode geolocations as a sequence of discrete tokens, beginning with a latitude token followed by a longitude token. For textual data, we modify the existing WordPiece tokenizer [33].
+
+# 4.2. Pre-training
+
+Architecture. TerraMind uses a symmetric Transformer-based encoder-decoder architecture proposed in [52], which is designed to process sequences of multimodal tokens. In addition to discrete tokens, TerraMind accepts pixel-level inputs, specifically satellite imagery and digital elevation maps. For pixel-level inputs, we apply learnable patch-wise linear projections to generate patch embeddings for each $16 \times 16$ patch, similar to the approach used in ViT [17].
+
+Dual-scale early fusion. In contrast to [52], we not only embed token-level data but additionally leverage pixel-level data across a range of input modalities to introduce a dual-scale feature representation to support the structuring of the embedding space. Both tokens and patches represent a 16x16 pixel area. Tokens represent this area via a single discrete integer value, while the image patches describe the same area with the actual floating point sensor data. Thus, during pretraining, the model not only learns a correlation between modalities (i.e., cross-modal learning) but also between dif
+
+ferent levels of abstraction within the same modality. The low-level token information enables cross-modal correlation learning, while adding pixel level input accounts for spatial nuances. Based on dual-scale features the model further learns to better structure pixel-level data in the embedding space via the corresponding information from the discrete token. We illustrate the pretraining paradigm in Figure 5. The model is agnostic to processing tokens or patches in the input space, while the target is generally token-level data. We use six pixel-level modalities and eight token-level modalities.
+
+
+Figure 5. Illustration of the pre-training task. Given an encoded multimodal sample of random subsets of patches and input tokens, the decoder predicts target tokens for the masked input.
+
+Masking strategy. TerraMind applies a masked modeling approach in the token space following [52]. The model leverages a set of randomly selected target tokens that have to be reconstructed from a randomly selected set of input tokens and pixel-level data. During pre-training, we sample input and target data from a Dirichlet distribution.
+
+We opt for masked token reconstruction to familiarize the model with the absence of entire modalities, which is crucial for a high usability of a multimodal model in Earth observation. During pre-training, the model learns an internal representation of unseen modalities which is expected to benefit a range of downstream applications. In addition, sampling input and target tokens improves the computational efficiency of the pre-training, as each token is a compressed representation of a patch with compression factors of between 250x and 3000x depending on the modality. Finally, without tokenized representations of the image-like modalities, it is challenging to learn the correlation to sequence-like modalities. The overall training objective of TerraMind boils down to a cross-modal patch-level classification problem optimized via a cross entropy loss:
+
+$$
+\mathcal {L} _ {\mathrm {C E}} = - \sum_ {i = 1} ^ {N} y _ {i} \log \left(p _ {i}\right), \tag {1}
+$$
+
+where $y_{i}$ is the one-hot encoded true class of token $i$ , $p_{i}$ is the predicted probability for token $i$ , $N$ is the total number of possible tokens. Interestingly, we can infer an upper bound loss for a random model where the cross entropy loss will collapse to the natural logarithm of the vocabulary size $\mathcal{L}_{\mathrm{CE,random}} = -\sum_{i=1}^{N} y_{i} \log \left( \frac{1}{N} \right) = \log(N)$ .
+
+Scaling. We trained three versions of TerraMind scaling across model size, compute, and data. In addition, we pretrain different versions of TerraMind with respect to the number of dual-scale features. TerraMindv1-B is pre-trained on 500B tokens for 6 days on 32 NVIDIA A100 GPUs. The model uses dual-scale features from both token-level and pixel-level. During initial experiments, we observed significant improvements from scaling model size when switching from a tiny backbone to a small backbone to a base backbone. Therefore, we pre-trained TerraMindv1-L on a large backbone with 500B tokens on 32 NVIDIA A100 GPUs trained for 10 days. Finally, to better understand the effect of scaling across the dual-scale feature representation, we pre-train TerraMindv1-B-single as a single-scale model on primarily token-level data with optical S-2 L2A data as only pixel-level input (compared to pixel-level S-2 L1C, S-2 RGB, S-1 GRD, S-1 RTC, and DEM in TerraMindv1-B and -L). TerraMindv1-B-single is pretrained on 500B tokens from over one million samples for 6 days on 32 NVIDIA A100 GPUs. We summarize the scaling behavior in model size, compute, and data in Figure 9 of the supplementary material. We additionally provide final validation losses in Table 9 comparing v1-B and v1-L with the theoretical random loss.
+
+# 4.3. Generation
+
+Once pretrained, TerraMind can generate tokens for any modality, conditioned on any subset of input modalities. The generative capabilities unlock various zero-shot tasks, such as water body segmentation. For the generation of image-like modalities, the decoder receives mask tokens for the modality to be generated and predicts the corresponding tokens based on the encoded input. For sequence-like modalities, the decoder generates the output autoregressively. After generating tokens from the target modality, the corresponding tokenizer decoder allows to map from token-space to image or text space. TerraMind further supports chained generation which ensures consistency across generated modalities. The chained generation represents a conditional probability distribution where the prior probability distribution is determined by the input modality, and all subsequent modalities are generated conditioned on the input modality and potentially other generated modalities.
+
+# 4.4. Thinking-in-Modalities
+
+Thinking in Modalities (TiM) is a recursive fine-tuning and inference technique designed to enhance multimodal learning by leveraging the generative capabilities of the model itself. Given an input $x \in \mathcal{X}$ (e.g., an optical satellite image), the model first generates additional synthetic modalities $\tilde{x} = f_{\mathrm{gen}}(x)$ on a token-level using a learned generative function $f_{\mathrm{gen}}$ . These generated tokens are then concatenated with the original input and jointly processed by the downstream
+
+model $f$ (e.g., TerraMind encoder with a segmentation head), yielding the final output $y = f(x, f_{\mathrm{gen}}(x))$ . This formulation allows the model to reason over both observed and inferred modalities, effectively enriching the input space. TiM can leverage multiple generated modalities which are then generated in a chained approach. For example, for $k$ modalities, the input is augmented with newly generated modalities:
+
+$$
+\tilde {x} ^ {(k + 1)} = \tilde {x} ^ {(k)} \cup f _ {\text {g e n}} (\tilde {x} ^ {(k)}), \tag {2}
+$$
+
+and the final model output is described by:
+
+$$
+y = f \left(\tilde {x} ^ {(K)}\right). \tag {3}
+$$
+
+This recursive augmentation mimics a chain-of-thought process, enabling the model to iteratively refine its internal representation, particularly in scenarios with missing modalities.
+
+# 5. Experiments
+
+In this section, we describe the performance gains resulting from TerraMind and experiment with the unlocked capabilities of any-to-any generation and Thinking-in-Modalities.
+
+# 5.1. Foundational experiments
+
+Multimodality vs. unimodality. As a first motivational experiment, we outline the benefit of using multimodal data in Earth observation at the example of water body mapping. Specifically, we leverage the ViT-B encoders from the unimodal tokenizer models for S-1, S-2, and LULC, concatenate their embeddings, and train a segmentation head with four ConvNeXt [43] blocks as a late fusion approach. The results in Table 2 (left) suggest that regardless of which modalities we combine, the combination of two modalities always outperforms each unimodal model. Combining all three modalities achieves the best overall performance.
+
+| Model | Modalities | Any-to-Any Generation | Multi-Scale Features |
| RemoteCLIP | optical, text | X | X |
| CROMA | optical, radar | X | X |
| AnySat | aerial, optical, radar, NAIP | X | X |
| DeCUR | optical, radar | X | X |
| DOFA | optical, radar, hyperspectral, NAIP | X | X |
| MetaEarth | optical (unimodal) | X | ✓ |
| Galileo | optical, radar, elevation, weather, location, population, ... | X | ✓ |
| TerraMind | optical, radar, land use, elevation, vegetation index, location, text | ✓ | ✓ |
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+ "content": "In this section, we describe the performance gains resulting from TerraMind and experiment with the unlocked capabilities of any-to-any generation and Thinking-in-Modalities."
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+ "content": "Multimodality vs. unimodality. As a first motivational experiment, we outline the benefit of using multimodal data in Earth observation at the example of water body mapping. Specifically, we leverage the ViT-B encoders from the unimodal tokenizer models for S-1, S-2, and LULC, concatenate their embeddings, and train a segmentation head with four ConvNeXt [43] blocks as a late fusion approach. The results in Table 2 (left) suggest that regardless of which modalities we combine, the combination of two modalities always outperforms each unimodal model. Combining all three modalities achieves the best overall performance."
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