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data/sampled_jsons/'Insufficient_Exploration'_failure_mode_AI_agents_web_vulnerability_exploitation.jsonl

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+{"idx": 0, "title": "CVE-Bench: A Benchmark for AI Agents' Ability to Exploit ...", "date": "", "ddg_snippet": "by Y Zhu · 2025 · Cited by 12 — Insufficient Exploration: Agents fail to explore all possi- ble attacks or endpoints , leading to missed opportunities. • Tool Misuse ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2503.17332", "content": "by Y Zhu · 2025 · Cited by 12 — Insufficient Exploration: Agents fail to explore all possi- ble attacks or endpoints , leading to missed opportunities. • Tool Misuse ..."}
+{"idx": 1, "title": "CVE-Bench: A Benchmark for AI Agents' Ability to Exploit ...", "date": "", "ddg_snippet": "by Y Zhu · Cited by 12 — As shown, all agents are bottlenecked by insufficient exploration , meaning that they failed to identify the vulnerability endpoint of applications, even when ...", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/forum?id=3pk0p4NGmQ", "content": "by Y Zhu · Cited by 12 — As shown, all agents are bottlenecked by insufficient exploration , meaning that they failed to identify the vulnerability endpoint of applications, even when ..."}
+{"idx": 2, "title": "CVE-Bench: A Benchmark for AI Agents' Ability to Exploit ...", "date": "", "ddg_snippet": "In CVE-Bench, we design a sandbox framework that enables LLM agents to exploit vulnerable web applications in scenarios that mimic real-world conditions.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/html/2503.17332v4", "content": "In CVE-Bench, we design a sandbox framework that enables LLM agents to exploit vulnerable web applications in scenarios that mimic real-world conditions."}
+{"idx": 3, "title": "CVE-Bench: A Benchmark for AI Agents' Ability to Exploit Real- ...", "date": "", "ddg_snippet": "Insufficient exploration emerged as the dominant failure mode , affecting 67.5% to 80% of zero-day attempts and 37.5% to 55% of one-day attempts across all ...", "subpage_snippet": "", "source": "www.alphaxiv.org", "link": "https://www.alphaxiv.org/overview/2503.17332", "content": "Insufficient exploration emerged as the dominant failure mode , affecting 67.5% to 80% of zero-day attempts and 37.5% to 55% of one-day attempts across all ..."}
+{"idx": 4, "title": "AI Agents Under Threat: A Survey of Key Security ...", "date": "", "ddg_snippet": "21 Feb 2025 — This survey delves into the emerging security threats faced by AI agents , categorizing them into four critical knowledge gaps.", "subpage_snippet": "", "source": "dl.acm.org", "link": "https://dl.acm.org/doi/10.1145/3716628", "content": "21 Feb 2025 — This survey delves into the emerging security threats faced by AI agents , categorizing them into four critical knowledge gaps."}
+{"idx": 5, "title": "Addressing the New Security Risks of AI Agents", "date": "", "ddg_snippet": "10 Sept 2025 — Adversarial Exploitation Vulnerability : Agents that directly interact with humans remain inherently vulnerable to adversarial exploitation .", "subpage_snippet": "", "source": "www.lumenova.ai", "link": "https://www.lumenova.ai/blog/ai-agents-securing-the-swarm/", "content": "10 Sept 2025 — Adversarial Exploitation Vulnerability : Agents that directly interact with humans remain inherently vulnerable to adversarial exploitation ."}
+{"idx": 6, "title": "Mitigating the Top 10 Vulnerabilities in AI Agents", "date": "", "ddg_snippet": "11 Apr 2025 — Explore vulnerabilities in AI agents , addressing risks like adversarial attacks, data poisoning, bias, and explainability challenges.", "subpage_snippet": "", "source": "www.xenonstack.com", "link": "https://www.xenonstack.com/blog/vulnerabilities-in-ai-agents", "content": "11 Apr 2025 — Explore vulnerabilities in AI agents , addressing risks like adversarial attacks, data poisoning, bias, and explainability challenges."}
+{"idx": 7, "title": "Hacking a $150 Billion Giant: Why AI Agents Need Strict ...", "date": "", "ddg_snippet": "In this article, I want to candidly share how I pulled off that exploit , and more importantly, how we can prevent such incidents by implementing strict access ...", "subpage_snippet": "", "source": "www.linkedin.com", "link": "https://www.linkedin.com/pulse/hacking-150-billion-giant-why-ai-agents-need-strict-access-fernández-uynuf", "content": "In this article, I want to candidly share how I pulled off that exploit , and more importantly, how we can prevent such incidents by implementing strict access ..."}
+{"idx": 8, "title": "Agent Red-Teaming: Exposing Vulnerabilities in ...", "date": "", "ddg_snippet": "6 May 2025 — Eight primary vulnerability categories targeted in the Enkrypt AI's red-teaming of agentic systems, each representing a distinct failure mode in ...", "subpage_snippet": "", "source": "www.enkryptai.com", "link": "https://www.enkryptai.com/blog/agent-red-teaming-exposing-vulnerabilities-in-autonomous-financial-ai-systems", "content": "6 May 2025 — Eight primary vulnerability categories targeted in the Enkrypt AI's red-teaming of agentic systems, each representing a distinct failure mode in ..."}
+{"idx": 9, "title": "Transforming cybersecurity with agentic AI to combat ...", "date": "", "ddg_snippet": "by N Kshetri · 2025 · Cited by 17 — This paper investigates the transformative potential of agentic AI in cybersecurity, specifically addressing how it can enhance practices in response to ...", "subpage_snippet": "", "source": "www.sciencedirect.com", "link": "https://www.sciencedirect.com/science/article/pii/S0308596125000734", "content": "by N Kshetri · 2025 · Cited by 17 — This paper investigates the transformative potential of agentic AI in cybersecurity, specifically addressing how it can enhance practices in response to ..."}

data/sampled_jsons/0.95_0.15_Contrastive_CRL_MCC_synthetic_ablation.jsonl

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+{"idx": 0, "title": "GitHub - simonbing/CRLSanityCheck", "date": "", "ddg_snippet": "Contrastive CRL .root_dir /path/to/save/output. To run the experiment on the synthetic ablation data, set --dataset contrast_semi_synth_decoder and optionally set the --task flag to a name that reflects this change. Multiview CRL.", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/simonbing/CRLSanityCheck", "content": "Contrastive CRL .root_dir /path/to/save/output. To run the experiment on the synthetic ablation data, set --dataset contrast_semi_synth_decoder and optionally set the --task flag to a name that reflects this change. Multiview CRL."}
+{"idx": 1, "title": "Sanity Checking Causal Representation Learning on a Simple...", "date": "", "ddg_snippet": "Synthetic ablation . 3 Results. 3.1 Contrastive CRL . Background.Spearman, C. The proof and measurement of association between two things. The American Journal of Psychology, 15(1):72–101, 1904. Squires et al.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/html/2502.20099v1", "content": "Synthetic ablation . 3 Results. 3.1 Contrastive CRL . Background.Spearman, C. The proof and measurement of association between two things. The American Journal of Psychology, 15(1):72–101, 1904. Squires et al."}
+{"idx": 2, "title": "identification of nonparametric dynamic causal structure ...", "date": "", "ddg_snippet": "by M Fu · 2025 — The MCC and R2 results for the Independent and Sparse settings demonstrate that our model achieves component-wise iden- tifiability (Theorem 3.3) ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2501.12500", "content": "by M Fu · 2025 — The MCC and R2 results for the Independent and Sparse settings demonstrate that our model achieves component-wise iden- tifiability (Theorem 3.3) ..."}
+{"idx": 3, "title": "Material", "date": "", "ddg_snippet": "Current synthetic materials used to replace synthetic cartilage do not mimic costal cartilage, which should be addressed in the future. Keywords. Auricular ...", "subpage_snippet": "", "source": "www.biomomentum.com", "link": "https://www.biomomentum.com/material/", "content": "Current synthetic materials used to replace synthetic cartilage do not mimic costal cartilage, which should be addressed in the future. Keywords. Auricular ..."}
+{"idx": 4, "title": "", "date": "", "ddg_snippet": "", "subpage_snippet": "", "source": "", "link": "", "content": ""}
+{"idx": 5, "title": "Autophagy-Dependent Generation of Free Fatty Acids Is ...", "date": "", "ddg_snippet": "by T Riffelmacher · 2017 · Cited by 316 — Autophagy-dependent generation of free fatty acids is critical for normal neutrophil differentiation.", "subpage_snippet": "", "source": "pmc.ncbi.nlm.nih.gov", "link": "https://pmc.ncbi.nlm.nih.gov/articles/PMC5610174/", "content": "by T Riffelmacher · 2017 · Cited by 316 — Autophagy-dependent generation of free fatty acids is critical for normal neutrophil differentiation."}
+{"idx": 6, "title": "The 76th Annual Congress of the Japan Society of ...", "date": "", "ddg_snippet": "We recently described reduced fetal growth in physiologically and nutritionally stable extremely preterm ovine fetuses undergo- ing artificial placenta therapy.", "subpage_snippet": "", "source": "obgyn.onlinelibrary.wiley.com", "link": "https://obgyn.onlinelibrary.wiley.com/doi/pdf/10.1111/jog.16282", "content": "We recently described reduced fetal growth in physiologically and nutritionally stable extremely preterm ovine fetuses undergo- ing artificial placenta therapy."}
+{"idx": 7, "title": "World Molecular Imaging Congress 2022 - PMC", "date": "", "ddg_snippet": "by J Uddin · 2022 — 2022 World Molecular Imaging Congress Program. Title: World Molecular Imaging Congress 2022. Date: September 28-October 1, 2022. Location: Miami, FL, USA.", "subpage_snippet": "", "source": "pmc.ncbi.nlm.nih.gov", "link": "https://pmc.ncbi.nlm.nih.gov/articles/PMC9844952/", "content": "by J Uddin · 2022 — 2022 World Molecular Imaging Congress Program. Title: World Molecular Imaging Congress 2022. Date: September 28-October 1, 2022. Location: Miami, FL, USA."}
+{"idx": 8, "title": "Recent Advances of Novel Pharmaceutical Designs for Anti ...", "date": "", "ddg_snippet": "0.95 ± 0.15 . 5a. 3.9 ± 0.71. 1.5 ± 0.17. 5b. >40. 29.8 ± 0.52. 5c. >40. 27.33 ± 0.83. 5d. >40. 29.2 ± 0.52. 5e. >40. 28.13 ± 0.83. 5f. >40. 4.5 ± 0.28. 5g. 23.2 ...", "subpage_snippet": "", "source": "mdpi-res.com", "link": "https://mdpi-res.com/bookfiles/book/7693/Recent_Advances_of_Novel_Pharmaceutical_Designs_for_Anticancer_Therapies.pdf?v=1718125915", "content": "0.95 ± 0.15 . 5a. 3.9 ± 0.71. 1.5 ± 0.17. 5b. >40. 29.8 ± 0.52. 5c. >40. 27.33 ± 0.83. 5d. >40. 29.2 ± 0.52. 5e. >40. 28.13 ± 0.83. 5f. >40. 4.5 ± 0.28. 5g. 23.2 ..."}
+{"idx": 9, "title": "Capillary function in patients with chronic venous insufficiency", "date": "", "ddg_snippet": "It has been demonstrated in normal subjects that persistently raised venous pressure results in trapping of leucocytes in the peripheral circulation'.", "subpage_snippet": "", "source": "academic.oup.com", "link": "https://academic.oup.com/bjs/article-pdf/75/6/597/59386982/bjs1800750635.pdf", "content": "It has been demonstrated in normal subjects that persistently raised venous pressure results in trapping of leucocytes in the peripheral circulation'."}

data/sampled_jsons/0A4Y9qRnu9_Leveraging Per-Instance Privacy for Machine Unlearning/auto/0A4Y9qRnu9_Leveraging Per-Instance Privacy for Machine Unlearning.md

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+# Leveraging Per-Instance Privacy for Machine Unlearning
+
+Nazanin Mohammadi Sepahvand \* 1 2 Anvith Thudi \* 3 4 Berivan Isik 5 Ashmita Bhattacharyya 3 4   
+Nicolas Papernot 3 4 Eleni Triantafillou 6 Daniel M. Roy 4 7 Gintare Karolina Dziugaite 8 2 9
+
+# Abstract
+
+We present a principled, per-instance approach to quantifying the difficulty of unlearning via finetuning. We begin by sharpening an analysis of noisy gradient descent for unlearning (Chien et al., 2024), obtaining a better utility–unlearning tradeoff by replacing worst-case privacy loss bounds with per-instance privacy losses (Thudi et al., 2024), each of which bounds the (Renyi) diver-´ gence to retraining without an individual data point. To demonstrate the practical applicability of our theory, we present empirical results showing that our theoretical predictions are born out both for Stochastic Gradient Langevin Dynamics (SGLD) as well as for standard fine-tuning without explicit noise. We further demonstrate that per-instance privacy losses correlate well with several existing data difficulty metrics, while also identifying harder groups of data points, and introduce novel evaluation methods based on loss barriers. All together, our findings provide a foundation for more efficient and adaptive unlearning strategies tailored to the unique properties of individual data points.
+
+# 1. Introduction
+
+et al., 2024), mitigating the impact of poisoning attacks (Liu et al., 2024b), or updating out-of-date information.
+
+In modern large-scale AI training regimes, exact unlearning (i.e., anything equivalent to retraining the model from scratch, without the forget set) is prohibitively expensive. To meet this challenge, a range of approximate unlearning methods have been developed.
+
+Machine unlearning aims to efficiently remove the influence of specific subsets of training data, called forget sets. The need for unlearning arises in many scenarios, such as handling requests for data deletion (Mantelero, 2013; Cooper
+
+Approaches based on ideas from differential privacy (DP) offer meaningful worst-case guarantees (Guo et al., 2020; Sekhari et al., 2021). These will be our focus. Unfortunately, for non-convex models, like neural networks, DP-based unlearning guarantees have so far come with higher error than their counterparts not designed to support unlearning, limiting their practical applicability. One of the key problems is the mismatch between the worst-case, data-agnostic nature of DP and the data-dependent nature of unlearning. Our work attempts to bridge this gap.
+
+Alongside work on theoretical guarantees, research into heuristics has flourished. The aesthetics of such research is to combine strong utility with empirical evaluations on metrics inspired by DP-based notions of approximate unlearning. It is not uncommon, however, for state of the art heuristics to be caught out by new attacks, revealing that they do not meet stringent theoretical criteria (Hayes et al., 2024; Pawelczyk et al., 2024). These challenges highlight the need for methods that can balance strong theoretical guarantees with strong practical performance.
+
+Towards understanding failure modes of unlearning, recent empirical work has shown that the behavior of unlearning varies considerably across individual data points (Zhao et al., 2024; Baluta et al., 2024). While prior work has attempted to incorporate these insights into methodology and evaluation, we lack a theoretical basis to 1) explain how individual data influence unlearning and 2) exploit this effect optimally.
+
+In this work, we introduce a principled approach to estimating the unlearning difficulty of individual data points. Our approach applies to learning with noisy gradient descent. Using recent results in differential privacy with Renyi ´ divergences (Thudi et al., 2024), our proposed measures— per-instance privacy losses—bound the Renyi divergence´ between a model trained with and without an individual data point. Critically, per-instance privacy losses can be efficiently estimated during training, based on the norms of gradient associated to individual data points.
+
+Armed with per-instance privacy losses, we revisit Chien et al.’s 2024 theoretical analysis of noisy gradient descent as an unlearning scheme (coined “Langevin unlearning” a.k.a “noisy fine-tuning”), based on training without the forget set. Our analysis provides a theoretical foundation for understanding how the number of iterations needed to approximately unlearn scales with per-instance privacy losses. Empirical validation demonstrates that our estimates of perinstance privacy losses serve as reliable predictors of unlearning difficulty under noisy fine-tuning.
+
+While noisy fine-tuning is not widely used, standard (noiseless) fine-tuning is a surprisingly effective and simple approximate unlearning approach, and serves as a building block for SOTA methods, such as L1-Sparse fine-tuning (Hayes et al., 2024; Liu et al., 2024a). We demonstrate the applicability of our theoretical insights, showing they extend empirically to standard fine-tuning-based unlearning, even in the absence of explicit noise injection. Empirically, we find that per-instance privacy losses continue to predict unlearning difficulty for fine-tuning across multiple approximate unlearning metrics, datasets, and architectures. Beyond fine-tuning, results with the L1-Sparse method (Liu et al., 2024a; Hayes et al., 2024) show privacy losses once again predict the number of steps to unlearn.
+
+Privacy losses may also offer a more refined measure of data difficulty for unlearning. We show that privacy losses are strongly correlated with some (cheaper-to-compute) proxy measures of data difficulty studied in prior work. On the other hand, we show that privacy losses identify more “difficult” data, which may be useful for evaluation more broadly.
+
+Besides standard empirical unlearning metrics, we evaluate privacy losses on a novel loss-landscape-based metric, based on linear connectivity and loss barriers (Frankle et al., 2020; Fort et al., 2020). While past unlearning metrics look at the output (e.g., loss) of an individual model, loss barriers capture some of the geometry of the loss landscape. Under this stringent test, fine-tuning is able to successfully unlearn. Interestingly, loss barriers also give insight into the differences between difficult and easy data points: points with higher privacy losses start off with larger loss barriers which need to be overcome to unlearn.
+
+Our contributions can be summarized as follows:
+
+• Per-Instance Theoretical Analysis of Unlearning. We revise Chien et al.’s analysis of noisy gradient descent for unlearning-via-fine-tuning, replacing a worstcase Renyi-DP bound with recent ´ per-instance privacy losses (Thudi et al., 2024). By exploiting that typical data points may have far less influence on the learned model, our per-instance analysis uncovers an improved trade-off between unlearning and utility compared to prior worst-case analyses.
+
+• Empirical Validation of Unlearning Difficulty. We demonstrate the practical applicability of our theoretical insights, showing that per-instance privacy losses reliably predict unlearning difficulty in experiments, both for noisy gradient descent and standard fine-tuning.
+
+• Efficient Proxies for Privacy Losses. We demonstrate that cheap proxy measures of data difficulty are strongly correlated with per-instance privacy losses. This identifies practical and scalable alternatives for assessing unlearning quality and forget set difficulty, particularly in resource-constrained settings where full privacy loss computation might be prohibitive.
+
+• Improved Identification of Difficult Data. Unlike existing heuristics for capturing aspects of unlearning difficulty, we show that privacy losses identify harder groups of data points, offering a versatile and theoretically justified measure of unlearning difficulty.
+
+• Loss Barrier Analysis of Unlearning. We introduce loss barriers (modulo permutation) as a way to evaluate unlearning. We find that the loss barrier is significantly reduced after unlearning, reaching levels comparable to baseline levels.
+
+• Broader Implications for Unlearning Methodology. Our empirical findings suggest that per-instance privacy losses may be useful in adapting other fine-tuningbased algorithms, such as L1-sparse.
+
+# 2. Related Work
+
+Unlearning Unlearning (Cao & Yang, 2015) aims to remove the influence of training data. In the context of neural networks and non-convex learning problems, even highly optimized exact approaches (Bourtoule et al., 2021) are computationally intensive. Going beyond exact unlearning, Ginart et al. (2019) introduced a notion of approximate unlearning, inspired by differential privacy (Dwork et al., 2014). This approach relies on approximate statistical indistinguishability between retraining and unlearning, in exchange for better efficiency or utility compared to exact unlearning. Guo et al. (2020); Neel et al. (2021) studied approximate unlearning algorithms for convex models based on gradient descent using all the training data except those data to be forgotten (a.k.a., fine-tuning), followed by the addition of noise (often via the Gaussian mechanism, Dwork et al. 2006) to obtain statistical indistinguishability. Neel et al. (2021) proved that, under various assumptions, their approach achieves approximate unlearning in a sequential framework with fixed per-deletion runtime.
+
+For non-convex models, a plethora of approximate unlearning methods have been proposed with empirical validation, rather than theoretical guarantees (Graves et al., 2021; Goel et al., 2022; Thudi et al., 2022; Kurmanji et al., 2024; Liu et al., 2024a; Fan et al., 2023). While these approaches are shown to work on some metrics, recent work shows that several unlearning methods fail against more sophisticated attacks for either privacy or poisoning (Hayes et al., 2024; Pawelczyk et al., 2024).
+
+Given the heuristic nature of many unlearning methods, a line of work has attempted to obtain a better understanding of their failure modes. Zhao et al. (2024); Baluta et al. (2024); Fan et al. (2025); Barbulescu & Triantafillou (2024) identified properties of forget sets that influence the behaviors of approximate unlearning algorithms. Zhao et al. (2024); Barbulescu & Triantafillou (2024) also derived improved unlearning methods based on these insights, for vision classifiers and LLMs, respectively. However, we lack a theoretical understanding of the underlying relationship between the identified factors and unlearning difficulty.
+
+Recently, Chien et al. (2024) studied Langevin dynamics for unlearning and introduced an approximate method with privacy guarantees for non-convex models that we build upon in this work, replacing a worst-case bound by a recent per-instance privacy bound. Their method leverages noisy gradient descent, for both the training algorithm, as well as during (noisy) fine-tuning at unlearning time. While our theoretical analysis considers that same setup, we empirically also investigate standard training without noise addition.
+
+Per-Instance Differential Privacy Differential privacy (DP) is a standard approach to privacy-preserving data analysis (Dwork et al., 2006; 2014) and machine learning (Chaudhuri et al., 2011; Abadi et al., 2016). DP methodology provides an upper bound on the divergence between output distributions under neighbouring datasets, i.e., when only one individual data point is altered. Since this upper bound needs to hold for every dataset and all of its neighbors, DP is a worst-case privacy notion with a single privacy parameter, shared among all individual data points. However, in practice, different individual points have different effects when training a machine learning model (Thudi et al., 2024; Yu et al., 2023). For instance, some data points have a much smaller gradient norm than others, making the worstcase privacy analysis for these points over-conservative, and leads to unnecessary degradations in privacy-utility tradeoffs. To mitigate such degradations, Ghosh & Roth (2011) and Cummings & Durfee (2020) have analyzed individual sensitivity rather than worst-case sensitivity, which led to an alternative but weaker notion of privacy, called per-instance (or individual, or personalized) DP. Per-instance DP assigns a different privacy loss for different data points in a dataset, i.e., it is not worst-case, and it is less pessimistic than standard DP for points that do not affect the output distribution as much as the worst-case points. Building on per-instance DP, Ebadi et al. (2015) and Feldman & Zrnic (2021) have introduced algorithms that filter out data points once their per-instance DP loss exceeds a budget in database query problems and neural network training, respectively. For iterative algorithms like SGD, per-instance DP requires new composition theorems as the privacy loss is adaptive to the individual data points and is different at each iteration (Wang, 2019). Feldman & Zrnic (2021); Thudi et al. (2024) have filled this gap by providing new privacy composition analyses using the associated divergences computed at an individual level.
+
+Motivated by the empirical observation that per-instance privacy loss provides much more promising guarantees than the worst-case DP analysis, we propose to port this per-instance approach to unlearning. We will see that this approach will allow us to provide unlearning guarantees and uncover properties of data that makes a forget set easy to unlearn.
+
+# 3. Preliminaries and Problem Setup
+
+Given a dataset $\mathcal { D } = \{ x _ { i } \} _ { i = 1 } ^ { n }$ of $n$ points, we are interested in estimating the difficulty of unlearning as a function of the particular forget set $\mathcal { D } _ { F } \subset \mathcal { D }$ we are aiming to unlearn.
+
+We are interested in settings where retraining the model from scratch on the retain set $\mathcal { D } ^ { \prime } = \mathcal { D } \setminus \mathcal { D } _ { F }$ is too costly, and so we consider approximate unlearning. In this work, we adopt a notion of approximate unlearning based on Renyi´ divergences. For $\alpha > 0$ and $\alpha \neq 1$ , recall that the $\alpha$ -Renyi´ divergence of $\mu$ relative to $\nu \gg \mu$ , is defined to be
+
+$$
+D _ { \alpha } ( \mu \| \nu ) = \frac { 1 } { \alpha - 1 } \log \left( \mathbb { E } _ { x \sim \nu } \left[ \frac { \mu ( x ) } { \nu ( x ) } \right] ^ { \alpha } \right) .
+$$
+
+(Here, one can interpret µ(x)ν(x) as a Radon–Nikodym derivative more generally.)
+
+For a fixed training set and forget set, the following definition captures the goal of unlearning in Renyi divergence: ´
+
+Definition 3.1. Fix a dataset $\mathcal { D }$ and forget set $\mathcal { D } _ { F } \subset \mathcal { D }$ . Let $\nu$ be the distribution of the output of $\boldsymbol { \mathcal { A } } ( \mathcal { D } ^ { \prime } )$ , i.e., the learning algorithm on the retain set $\mathcal { D } ^ { \prime } = \mathcal { D } \setminus \mathcal { D } _ { F }$ , and let $\mu$ be the distribution of the output of $\mathcal { U } ( A ( \mathcal { D } ) , \mathcal { D } ^ { \prime } )$ , i.e., the unlearning algorithm on the learned model $\mathcal { A } ( \mathcal { D } )$ and $\mathcal { D } ^ { \prime }$ . For $\alpha > 1$ , we say $\mathcal { U } \left( \alpha , \varepsilon \right)$ -Renyi unlearns ´ $\mathcal { D } _ { F }$ from $\mathcal A ( \mathcal D )$ when $D _ { \alpha } ( \mu \| \nu ) \leq \varepsilon$ .
+
+This leads to a uniform notion of Renyi unlearning, which ´ offers the same guarantee for all datasets and all forget sets, which is analogous to standard notions.
+
+Definition 3.2 (Renyi Unlearning) ´ . For $\alpha > 1$ , an unlearning algorithm $\mathcal { U }$ is an $( \alpha , \varepsilon )$ -Renyi unlearner (for ´ $\mathcal { A }$ ) when, for all dataset $\mathcal { D }$ and forget sets $\mathcal { D } _ { F } \subseteq \mathcal { D } , \mathcal { U } \left( \alpha , \varepsilon \right) .$ -Renyi´ unlearns $\mathcal { D } _ { F }$ from $\mathcal { A } ( \mathcal { D } )$ in the sense of Definition 3.1.
+
+Note that when this definition holds, one can immediately derive (standard) approximate DP-based unlearning guarantees (Ginart et al., 2019; Guo et al., 2020; Neel et al., 2021; Sekhari et al., 2021), using standard reductions (Proposition 10, Mironov, 2017).1
+
+# 3.1. Learning–Unlearning with Noisy Gradient Descent
+
+We consider a learning–unlearning setup based on using projected noisy gradient descent during both learning and unlearning, as studied recently by Chien et al. (2024). In this section, we present theoretical results that rely on noise. In Section 6, we present empirical evidence from multiple benchmarks that the overall trends extend to standard variants of gradient descent.
+
+Formally, our learning algorithm $\mathcal { A } = \mathcal { A } _ { T }$ is $T$ steps of noisy gradient descent on $\mathcal { D }$ , starting from a random initialization. Let $\nu _ { T , D }$ denote the distribution on weights obtained after $T$ steps of training on $\mathcal { D }$ , and let $\nu _ { D } = \nu _ { \infty , D }$ denote the stationary distribution, to which the output distribution converges as $T \to \infty$ .
+
+Let $\mathcal { D } ^ { \prime } = \mathcal { D } \setminus \mathcal { D } _ { F }$ denote the retain set of an unlearning request. Assuming we trained for $T$ iterations, exact unlearning would produce weights whose (marginal) distribution was $\nu _ { T , \mathcal { D } ^ { \prime } }$ , i.e., the distribution as if we had trained on $\mathcal { D } ^ { \prime }$ from scratch for $T$ iterations.
+
+Our unlearning algorithm $\mathcal { U } = \mathcal { U } _ { k }$ runs $k$ steps of noisy gradient descent on $\mathcal { D } ^ { \prime }$ . We denote by $\rho _ { { D ^ { \prime } } } ^ { k } ( \nu _ { { T } , { \mathcal { D } } } )$ the output distribution of unlearning, i.e., of running $k$ steps of projected noisy gradient descent on $\mathcal { D } ^ { \prime }$ , initialized at a sample from $\nu _ { T , \mathcal { D } }$ , i.e., first training on $\mathcal { D }$ .
+
+# 4. Per-Instance Unlearning Difficulty Analysis
+
+Our goal is to bound the number of steps, $k$ , of unlearning by noisy fine-tuning needed to achieve unlearning in a way that adapts to the influence each data point has on the learned model. We do so by building on two pieces of work: a convergence analysis of Langevin dynamics by Chien et al. (2024) and a per-instance Renyi-differential privacy analysis ´ by Thudi et al. (2024).
+
+# 4.1. Convergence Analysis
+
+Fix a dataset $\mathcal { D }$ and retain set $\mathcal { D } ^ { \prime } \subset \mathcal { D }$ . We begin with the following corollary of (Thm. 3.2, Chien et al., 2024), which highlights the role of an initial bound on $D _ { \alpha } ( \nu _ { T , \mathcal { D } } | | \nu _ { T , \mathcal { D } ^ { \prime } } )$ In the following, we assume the loss is Lipschitz continuous and smooth, and that the step sizes used by $\mathcal { A }$ and $\mathcal { U }$ have been set according to (Thm. 3.2, Chien et al., 2024).
+
+Corollary 4.1. Fix $\mathcal { D } , \mathcal { D } ^ { \prime }$ . Let $\{ \varepsilon _ { \alpha } , \varepsilon _ { \alpha } ^ { \prime } \} _ { \alpha \ge 1 }$ satisfy $\begin{array} { r l r } { \operatorname* { m a x } \{ D _ { \alpha } ( \nu _ { D ^ { \prime } } \| \nu _ { T , \mathcal { D } ^ { \prime } } ) , D _ { \alpha } ( \nu _ { T , \mathcal { D } ^ { \prime } } \| \nu _ { \mathcal { D } ^ { \prime } } ) \} } & { \leq } & { \varepsilon _ { \alpha } } \end{array}$ and $D _ { \alpha } ( \nu _ { T , \mathcal { D } } | | \nu _ { T , \mathcal { D } ^ { \prime } } ) \leq \varepsilon _ { \alpha } ^ { \prime }$ for all $\alpha$ . Then, there exists a constant $C > 0$ such that, for all $\alpha > 1 , \mathcal { U } _ { k } \left( \alpha , \epsilon ^ { * } \right)$ -Renyi ´ unlearns $\mathcal { D } \backslash \mathcal { D ^ { \prime } }$ from $\mathcal { A } ( \mathcal { D } )$ in $k$ steps, where $\epsilon ^ { * }$ is
+
+$$
+\Bigl ( \frac { 2 \alpha - 1 / 2 } { 2 \alpha - 2 } \varepsilon _ { 4 \alpha } ^ { \prime } + \frac { 2 \alpha - 1 } { 2 \alpha - 2 } \varepsilon _ { 4 \alpha - 1 } \Bigr ) \exp \Big ( - \frac { C k } { 2 \alpha } \Big ) + \varepsilon _ { 2 \alpha - 1 } .
+$$
+
+The proof is in Appendix A.1, and follows from (Thm. 3.2, Chien et al., 2024) and the weak triangle inequality for Renyi divergences.´
+
+The key observation is that the first term decays exponentially fast in the number of steps $k$ , with the initial value determined by the per-instance guarantee and distance to stationarity and the rate determined by the desired value of $\alpha$ and problem parameters (like the Lipschitz and smoothness constants, behind $C$ ). The second term, $\varepsilon _ { 2 \alpha - 1 }$ , however, does not vanish.
+
+The irreducible term captures the distance to stationarity after $T$ steps of training, measured in a higher order $( 2 \alpha -$ 1)-divergence. This term exists as a consequence of our unlearning algorithm not attempting to correct for the $k$ steps of extra training.
+
+# 4.2. Unlearning Individual Data Points
+
+The following definition is adapted from a differential privacy result by Thudi et al. (2024) to our unlearning setting:
+
+Definition 4.2 (Per-Instance Privacy Loss). Recall that $\nu _ { T , D }$ is the distribution of the model weights after $T$ iterations of noisy gradient descent on $\mathcal { D }$ . Let $g ( x ^ { * } , w ) = \nabla _ { w } \ell ( w , x ^ { * } )$ be the contribution to the weight-gradient at $w$ , coming from one data point $x ^ { * }$ . The per-instance privacy loss for $x ^ { * }$ is:
+
+$$
+\begin{array} { r } { P ( \boldsymbol { x } , \alpha ) : = \sum _ { t = 1 } ^ { T } C _ { t , \alpha } \ln \mathbb { E } _ { \boldsymbol { w } \sim \nu _ { t , D } } f _ { t , \alpha } ( \| \boldsymbol { g } ( \boldsymbol { x } ^ { * } , \boldsymbol { w } ) \| _ { 2 } ) , } \end{array}
+$$
+
+where Ct,α = 1α−1 (p−1)t+1pt+1 a nd $\ln f _ { t , \alpha } ( g )$ is
+
+$$
+p \ln \bigg ( \sum _ { k = 0 } ^ { o _ { p } ^ { i } ( \alpha ) } { \binom { o _ { p } ^ { t } ( \alpha ) } { k } } \mathbb { P } _ { x ^ { * } } ( 1 ) ^ { k } \overline { { \mathbb { P } _ { x ^ { * } } ( 1 ) } } ^ { o _ { p } ^ { t } ( \alpha ) - k } e ^ { \frac { g ^ { 2 } ( k ^ { 2 } - k ) } { 2 \sigma ^ { 2 } } } \bigg ) ,
+$$
+
+with $\begin{array} { r } { o _ { p } ( \alpha ) \ = \ \frac { p } { p - 1 } \alpha - \frac { 1 } { p } } \end{array}$ and $O _ { p } ^ { t }$ is $o _ { p }$ composed $t$ times, $\mathbb { P } _ { x ^ { * } } ( 1 ) = 1 - \overline { { \mathbb { P } _ { x ^ { * } } ( 1 ) } }$ is the sampling probability of the data point (batch size over dataset size), and $p$ is a free parameter we set to $3 T$ , following (Fact 3.4, Thudi et al., 2024).
+
+This quantity, which we show how to estimate efficiently in Section 5.1, yields a bound on a data point’s sensitivity.
+
+Theorem 4.3. Fix $\mathcal { D }$ , let $x \in \mathcal { D }$ , and put ${ \mathcal { D } } ^ { \prime } = { \mathcal { D } } \setminus \{ x \}$ .   
+Then $D _ { \alpha } ( \nu _ { T , \mathcal { D } } | | \nu _ { T , \mathcal { D ^ { \prime } } } ) \leq P ( x , \alpha )$ .
+
+Proof. This follows by applying the per-instance momentbased composition theorem (Thm. 3.3, Thudi et al., 2024) with the per-step divergence bound for single data points (Thm. 3.2, Thudi et al., 2024), using the post-processing inequality to conclude that the divergence after projections is bounded by that before projections.
+
+As a consequence of Theorem 4.3 applied to Corollary 4.1, we have unlearning individual data points by noisy gradient descent depends logarithmically on the privacy loss. The following corollary is a direct substitution of Theorem 4.3 into Corollary 4.1.
+
+Corollary 4.4. Under the assumptions of Corollary 4.1 and Theorem 4.3, for all $\mathcal { D }$ and ${ \mathcal { D } } ^ { \prime } = { \mathcal { D } } \setminus \{ x \}$ for $x \in \mathcal { D }$ , for all $\alpha > 1$ , there exist constants $A _ { \alpha } , B _ { \alpha } , C _ { \alpha } > 0$ such that, for all $\delta > \varepsilon _ { 2 \alpha - 1 }$ , running noisy gradient descent for
+
+$$
+k \geq A _ { \alpha } \ln \left( \frac { B _ { \alpha } P ( x , 4 \alpha ) + C _ { \alpha } \varepsilon _ { 4 \alpha - 1 } } { \delta - \varepsilon _ { 2 \alpha - 1 } } \right)
+$$
+
+steps $( \alpha , \delta )$ -unlearns $\{ x \}$ from $\mathcal A ( \mathcal D )$ .
+
+# 4.3. Limitations of Existing Group Unlearning Analyses
+
+The above analysis focuses on forgetting one data point. How about bounding the work to unlearn multiple data points simultaneously? The group unlearning bound of (Chien et al., 2024, Cor. 3.4) requires that the order of $\alpha$ grows with each data point to unlearn. This is problematic, as the Renyi divergence between, e.g., Gaussians, ´ grows linearly with $\alpha$ (Prop. 7, Mironov, 2017). In contrast, (Thms. 3.3 and 3.6, Thudi et al., 2024) imply that the Renyi divergence ´ $D _ { \alpha } ( \nu _ { T , \mathcal { D } } | | \nu _ { T , \mathcal { D } \backslash \mathcal { D } _ { F } } )$ , and hence steps to unlearn, does not necessarily grow with $\mathcal { D } _ { F }$ ; instead, Thudi et al. bound the divergence by comparing the distribution of gradients under $\mathcal { D }$ and under $\mathcal { D } \backslash \mathcal { D } _ { F }$ . Implementing their group privacy accounting (described in Appendix B), however, we found evidence the bounds were likely too loose, as they did not differentiate forget sets that we empirically knew to require different numbers of steps to unlearn (see Figure 4 in appendix). In contrast, we found the rankings provided by privacy losses meaningfully differentiate the number of steps needed to unlearn, as we will describe in Section 6.
+
+We conclude that the current state of analysis for group unlearning is not tight enough to capture the behavior we observe in practice. The problem of obtaining a tight analysis remains an open problem.
+
+# 5. Methodology
+
+Our empirical methodology is structured around two key objectives driven by the theoretical role of per-instance privacy losses in unlearning. First, we aim to validate that privacy losses effectively predict the relative difficulty of unlearning data points. Second, we seek to understand the factors contributing to this difficulty by investigating the relationship between privacy losses, the loss landscape, and existing metrics of data difficulty. In the following sections, we present our experimental design to address these questions.
+
+# 5.1. Empirically Validating Unlearning Difficulty
+
+Unlearning Algorithms Our investigation primarily focuses on unlearning via fine-tuning on the retain set. We also run unlearning experiments with L1-Sparse (Liu et al., 2024a), a regularized version of fine-tuning that is widely recognized as one of the most effective unlearning methods. Hayes et al. (2024) has demonstrated that L1-Sparse outperforms a number of other methods in defending against a basic Membership Inference Attack (MIA), as well as other attacks of varying strengths.
+
+Per-instance Privacy loss We compute the terms in the privacy loss $P ( x , \alpha )$ stated in Definition 4.2 by taking a Monte-Carlo estimate from a single training run with checkpoints $w _ { 0 } , w _ { s _ { 1 } } , w _ { s _ { 2 } } , \cdot \cdot \cdot , w _ { s _ { N } }$ , i.e.,
+
+$$
+\begin{array} { r } { \hat { P } _ { s _ { i } } ( x , \alpha ) = C _ { s _ { i } , \alpha } \ln f _ { s _ { i } , \alpha } ( \| g ( x ^ { * } , w _ { s _ { i } } ) \| _ { 2 } ) , } \end{array}
+$$
+
+where $C _ { s _ { i } , \alpha } , f _ { s _ { i } , \alpha } , g$ are defined in Definition 4.2.
+
+We then approximate the area under the per-step privacy curve (i.e., sum over $t = 0 , 1 , \cdots , s _ { N } )$ by using the right hand rule: keeping $\hat { P } _ { s _ { i } }$ constant between the checkpointing intervals $\left( { { s } _ { i - 1 } } , { { s } _ { i } } \right]$ . This gives our approximate privacy loss:
+
+$$
+\begin{array} { r } { \hat { P } ( x , \alpha ) = \sum _ { i = 1 } ^ { N } \hat { P } _ { s _ { i } } ( x ) ( s _ { i } - s _ { i - 1 } ) . } \end{array}
+$$
+
+Throughout the paper we take $N = 3 5$ and have the checkpoints $s _ { i }$ evenly spaced throughout training. In the case of SGD, without explicit noise, we approximate these scores by assuming a negligible amount of noise is present. In particular we take $\sigma \leq 0 . 1$ , which matches the trends we observe with SGLD. We also found the privacy losses rankings for SGD to be stable to the implicit $\sigma$ used for privacy losses computation (see Appendix F). We note that past work has looked into quantifying the noise inherent in training, due to hardware and software nondeterminism (Jia et al., 2021; Zhuang et al., 2022). It is an open problem to exploit software and hardware nondeterminism to offer a theoretical justification to our approach here.
+
+Forget Set Difficulty We rank examples by privacy losses and form 5 forget sets of varying difficulty by taking evenly spaced sequences of 1000 data points. The size of the forget set was chosen to be in a similar range as prior work (e.g., (Zhao et al., 2024)). Our theory suggests that higher privacy losses should lead to longer, thus more difficult, unlearning. We thus call a forget set more difficult than another forget set if its average privacy loss is higher. In our analysis, each forget set is represented by the average privacy loss of all samples within that set. Further methodological details are provided in Appendix D.
+
+Unlearning Evaluation We evaluate unlearning efficacy using three metrics: (1) accuracy, measured separately on the retain set (RA), test set (utility), and on the forget set (FA). We report $\mathrm { U A } = 1 - \mathrm { F A }$ , indicating how “accurate” unlearning is, as done in (Fan et al., 2023); (2) membership inference attack (MIA): we train a logistic regression classifier to identify training samples and report the fraction of forget set samples incorrectly classified as test samples, thus indicating successful forgetting; and (3) Gaussian Unlearning Score (GUS) (Pawelczyk et al., 2024), which employs Gaussian input poisoning attacks to reveal if the unlearned model still encodes noise patterns associated with the poisoned forget set data. See Appendix E for more details.
+
+These metrics are monitored during unlearning and compared with the oracle model to evaluate the effectiveness of the unlearning methods. Recall that the oracle model is obtained by training from scratch using the retain set only. We choose these metrics due to their common use in machine unlearning research (Fan et al., 2025; Zhao et al., 2024; Jia et al., 2023; Deeb & Roger, 2024; Fan et al., 2024; Pawelczyk et al., 2024) and their computational simplicity, enabling us to compute them at every step during unlearning. Additional details on their computation and associated parameter choices can be found in Appendix E.
+
+Datasets and models Our experiments are performed on the SVHN (Netzer et al., 2011) and CIFAR-10 (Alex, 2009) datasets, with a ResNet-18 architecture (He et al., 2016). Appendix F presents results for ViT-small (Dosovitskiy, 2020). Appendix D describes other details for reproducibility.
+
+Each experimental configuration (oracle training, or training on all data and unlearning) is run 10 times, and the average performance across these runs is reported.
+
+# 5.2. Loss Landscape Analysis
+
+While per-instance privacy losses provide a quantitative measure of unlearning difficulty based on training dynamics, they do not directly reveal the underlying geometric properties of the loss landscape that contribute to this difficulty. To gain a deeper understanding of why certain data points are harder to unlearn, we complement our privacy loss analysis with an investigation of the loss landscape.
+
+Specifically, we employ the concepts of (linear) loss barri ers (Frankle et al., 2020; Nagarajan & Kolter, 2019). Loss barriers characterize the “flatness” or “curvature” of the loss surface between different model parameter configurations.
+
+The loss barrier $\mathrm { e r r } ( w , w ^ { \prime } ; \mathcal { D } )$ is the deviation in cross entropy $\mathcal { L }$ on the data $\mathcal { D }$ along the linear path in weight space connecting $w$ to $w ^ { \prime }$ . Let $\bar { \alpha } = 1 - \alpha$ . Then $\mathrm { e r r } ( w , w ^ { \prime } ; S )$ is
+
+$$
+\operatorname* { m a x } _ { \alpha \in [ 0 , 1 ] } \left[ \mathcal { L } ( \alpha w + \bar { \alpha } w ^ { \prime } ; S ) - \alpha \mathcal { L } ( w ; S ) - \bar { \alpha } \mathcal { L } ( w ^ { \prime } ; S ) \right] .
+$$
+
+To account for the permutation invariance of neural networks, we compute these loss barriers modulo permutation, as detailed in (Entezari et al., 2022; Sharma et al., 2024).
+
+Loss barriers provide insight into the geometric properties of high-dimensional loss surfaces. In our experiments, we compute loss barriers between oracle models (trained without the forget set) and models before and after unlearning forget sets with various average privacy losses. This allows us to examine if forget sets with higher privacy losses (higher predicted unlearning difficulty) exhibit distinct loss landscape characteristics, particularly in terms of loss barriers.
+
+# 5.3. Comparison to Alternative Data Difficulty Metrics
+
+While the per-instance privacy losses provide valuable insights into the unlearning process, their computation requires storing gradients throughout training, leading to considerable computational overhead. Therefore, we explore alternative metrics that could serve as proxies for these scores, offering a more efficient way to estimate forget set difficulty. Furthermore, we investigate the relationship between these proxies and fine-tuning-based unlearning difficulty, shedding light on their underlying mechanisms.
+
+We evaluate five proxies: (1) the gradient norm of individual data points at a single mid-training iteration; (2) the gradient norm at the end of training; (3) the average gradient norm across all training iterations; (4) C-Proxy used by Zhao et al. (2024) to approximate memorization scores from (Feldman et al., 2018); adapted from Jiang et al. (2020). This proxy computes prediction confidence: the entry in the softmax vector corresponding to the ground truth class, averaged throughout the training trajectory; and (5) a single-trajectory variant of the EL2N score (Paul et al., 2021). Normally, EL2N score is computed by averaging error signals over multiple training trajectories at a fixed time point. However, for computational efficiency and direct comparability with our gradient-based proxies, we compute a single-trajectory EL2N score at a mid-training checkpoint. See Appendix C for a discussion on connections among the scores.
+
+![](images/f643d81abd9a6b7961e949f9183d2a3b60fd582bcb509684e0ebbc249e97d7f0.jpg)  
+Figure 1. CIFAR-10 dataset results. Left: SGLD unlearning with varying levels of noise $( \sigma )$ . Forget set difficulty $\mathbf { \widetilde { x } }$ -axis), as measured by the privacy loss, against time to unlearn (y-axis). Time to unlearn is measured in terms of epochs needed to get within $5 \%$ of the unlearning metric (e.g., UA or MIA) measured on the oracle model. Middle: SGD unlearning. Time to unlearn measured across three evaluation metrics. Right: Error barrier between the oracle and the unlearned model before and after unlearning for forget sets with different privacy losses. Baseline corresponds to the loss barrier between two oracles.
+
+# 6. Experimental Results
+
+We empirically validate the predictive capabilities of perinstance privacy losses in forecasting unlearning difficulty. Our primary finding is that privacy losses accurately rank datapoints according to the number of steps needed to unlearn. This is tested for two settings of interest: (1) SGLD, which aligns directly with the assumptions of our theoretical upper bounds; and (2) SGD. For SGD, while lacking explicit noise, we adapt privacy losses by assuming a small, implicit noise component.
+
+To probe the geometric origins of unlearning difficulty as captured by privacy losses, we analyze the loss landscape. Our investigation reveals a consistent trend: data points with higher privacy losses exhibit larger loss barriers on a linear path (in the weight space) to the oracle model.
+
+We also assess the practical utility of privacy losses by comparing them to computationally efficient proxy metrics. Our results demonstrate a strong correlation between privacy losses and existing proxy metrics, including those employed in previous studies to estimate forget set difficulty (C-Proxy in (Zhao et al., 2024)). However, a key advantage emerges: privacy losses provide superior precision in identifying truly difficult-to-unlearn data points within the context of finetuning-based unlearning. The forget sets ranked as most difficult by privacy losses consistently take longer to unlearn than those identified by these established proxies.
+
+# 6.1. Time to Unlearn Depends on Per-Instance Privacy
+
+We find that across a variety of unlearning metrics, privacy losses accurately separate data points by the number of steps needed to unlearn in both SGLD and SGD. Appendix F.2 shows the same trends for L1-sparse unlearning.
+
+SGLD Training To evaluate the predictions outlined in Section 4, we examine the relationship between the number of steps needed to unlearn using noisy fine-tuning (SGLD), and the average privacy loss within the forget set. Figure 1 (left) shows that as forget set privacy losses increase, a higher number of fine-tuning steps is needed to reach a $5 \%$ error margin relative to the oracle, validating our prediction.
+
+SGD Training While SGLD offers theoretical advantages for privacy analysis and bounding Renyi unlearning, SGD ´ remains the dominant training paradigm in practice. Therefore, we investigate whether our theoretical framework, developed in the context of Langevin dynamics, can also predict unlearning time for models trained with SGD.
+
+As shown in Figure 1 (middle), we observe a similar trend as in the SGLD experiments across all evaluation metrics: unlearning more difficult forget sets, characterized by higher average privacy losses, requires more fine-tuning steps.
+
+This suggests that even without explicit noise injection during training, the concept of per-instance privacy losses derived from our theoretical analysis can provide valuable insights into the unlearning process for SGD-trained models. That is, SGD seems to be well-approximated by low noise SGLD. These results are replicated across additional datasets, architectures, and forget set sizes, with full details and qualitative examples in Appendix F.
+
+# 6.2. More Difficult Forget Sets, Larger Loss Barriers
+
+We characterize difficult to unlearn data points by their loss landscape, in particular, loss barriers (see Section 5).
+
+Figure 1 (right) depicts the loss barrier between oracle models and unlearned models, while varying the difficulty of the forget set, as measured by the average privacy loss.
+
+We observe two key takeaways from this analysis: (1) Comparing our results to the baseline loss barriers between independently trained oracle models, we find that fine-tuning achieves comparable levels after unlearning. This can be seen as a measure of unlearning efficacy based on loss barriers: if the (distribution of the) loss barrier is different from that of the baseline between oracles, unlearning was unsuccessful; (2) Higher privacy loss values correspond to larger initial barriers, providing a geometric interpretation for privacy losses; data points that require more steps to unlearn have to overcome larger loss barriers to the oracle.
+
+Overall, our findings provide additional evidence for the utility of privacy losses in predicting unlearning difficulty, and point to the effectiveness of loss barriers as both a diagnostic and an evaluation tool for machine unlearning.
+
+# 6.3. Existing Metrics Correlate with Privacy Losses
+
+We now investigate how privacy losses compare to other data difficulty metrics, as described in Section 5. Our results in Figure 2 reveal that all these proxies exhibit high correlation with the actual privacy loss. As expected, the best proxy is the average gradient norm throughout training, but it is also the most expensive proxy as it requires computing gradient norms for each data point at every iteration, versus once. Other proxies (with the exception of C-Proxy) only require a gradient/error computation at a single checkpoint, yet still provide a reasonable approximation for categorizing examples into broad difficulty groups.
+
+These findings suggest that, in scenarios where computational resources are limited, utilizing these proxies can offer a practical alternative for estimating forget set difficulty and predicting unlearning time. We refer to recent work by Kwok et al. (2024) for an in-depth comparison of different data difficulty metrics that may serve as good proxies.
+
+Recall that C-Proxy has been used in prior work by Zhao et al. (2024) to identify difficult to unlearn forget sets for certain unlearning algorithms. Figure 2 shows that privacy losses are highly correlated with this heuristic. Our work thus offers theoretical grounding to previously proposed heuristics for identifying difficult to unlearn forget sets.
+
+Finally, note that among metrics relying on averaging over training, our method is more efficient, requiring only 35 evenly spaced checkpoints (approximately $20 \%$ of training time), compared to C-Proxy and average gradient norm, which store values at every epoch (150 in total). As shown in Section 6.4, it is also more effective at identifying hard-tounlearn samples than all other metrics, regardless of whether they rely on averaging.
+
+![](images/fdc0a71cf8d1ea8bb5d7e12b63d29805546210233bb5547f3702a64475662a19.jpg)  
+Figure 2. Correlation between privacy losses $\mathbf { \dot { x } }$ -axis) and various proxy metrics (y-axis). The values of all proxy metrics are normalized to their maximum value for better visual clarity. For improved readability, the data is binned into 30 bins.
+
+![](images/caaaf3ffdbb7de347fe47c1a164b0fe030f6a75ab30e0cd8a8a77a153c9b6a1c.jpg)  
+Figure 3. Comparison of the time needed to unlearn (y-axis) the most difficult forget sets as identified by privacy losses (ours), C-Proxy, average gradient norm and EL2N, across different forget set sizes ( $\mathbf { \dot { x } }$ -axis).
+
+# 6.4. Privacy Losses Identify Harder Data
+
+In the previous subsection, we provided evidence that existing data difficulty metrics correlate with per-instance privacy losses. In this section, we provide evidence that, in fact, privacy losses are able to identify harder data. In Figure 3, we compare our per-instance privacy losses to several data difficulty metrics, including C-Proxy (described in Section 5), average gradient norm, and EL2N scores.
+
+For forget sets of size $s \in \{ 6 0 0 , 1 0 0 0 , 2 0 0 0 \}$ , we look at the top $s$ data points as ranked by C-Proxy, average gradient norm, EL2N scores, and privacy losses. What we see is that, across the board, the data picked out by privacy losses are harder to unlearn, as measured both by the number of iterations to reach $5 \%$ unlearning accuracy or reach $5 \%$ excess membership inference attack error. By identifying more difficult examples, privacy losses open up new empirical approaches to evaluating unlearning performance.
+
+# 7. Conclusion
+
+In this work, we have introduced a principled approach to quantifying machine unlearning difficulty at the level of individual data points in terms of per-instance privacy losses, which bound the Renyi divergence between training with´ and without a datapoint. Our theoretical analysis provides a foundation for understanding how unlearning scales with the properties of specific data points, particularly in the context of Langevin dynamics.
+
+We have shown that per-instance privacy losses, estimated from training statistics, reliably predict unlearning difficulty in fine-tuning-based unlearning algorithms, across different architectures and datasets, even in the absence of explicit noise injection during training. Our empirical results demonstrate that these privacy losses offer a precise and actionable measure of unlearning difficulty. Our work also offers a theoretical grounding for previous work suggesting that certain forget sets are harder to unlearn, with privacy losses capturing similar aspects of difficulty for fine-tuningbased unlearning as previously proposed heuristics (Zhao & Triantafillou, 2024; Baluta et al., 2024; Zhao et al., 2024). Moreover, we show privacy losses identify harder forget sets than previous methods.
+
+Our findings have broader implications for unlearning methodology, suggesting that per-instance divergence analysis can guide the development of new, more efficient unlearning algorithms tailored to specific data characteristics. Extending our theoretical framework to other unlearning methods beyond fine-tuning and exploring the use of privacy losses in designing adaptive unlearning strategies is a promising direction for future work.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of machine unlearning, which is specifically oriented to improve the trustworthiness of machine learning, by supporting requests to remove the influence of training data. There are many potential positive societal consequences of our work, none which we feel must be specifically highlighted here.
+
+# Acknowledgments
+
+We thank Ioannis Mitliagkas and Ilia Shumailov for feedback on a draft of this work. Anvith Thudi, Ashmita Bhattacharyya, and Nicolas Papernot would like to acknowledge the sponsors of the CleverHans lab, who support our research with financial and in-kind contributions: CIFAR through the Canada CIFAR AI Chair, NSERC through the Discovery Grant, the Ontario Early Researcher Award, the Schmidt Sciences foundation through the AI2050 Early Career Fellow program, and the Sloan Foundation. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. Anvith Thudi is also supported by a Vanier Fellowship from NSERC. Daniel M. Roy is supported by the funding through
+
+NSERC Discovery Grant and Canada CIFAR AI Chair at the Vector Institute. We also thank Mila – Quebec AI Insti tute and Google DeepMind for providing the computational resources that supported this work.
+
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+
+![](images/1f29428a7f4dba131222a4978b1d3f0e97fcfd8b72247591a5ade0f01ee64657.jpg)  
+Figure 4. We compared our estimates of the group privacy guarantees (y-axis) across forget sets determined by rankings of privacy losses $\mathbf { \widetilde { x } }$ -axis), and found the group privacy guarantees did not change. This was despite these forget sets leading to consistent differences in the number of steps to unlearn. We report the mean over 20 estimates of the group privacy values, and one standard deviation. We conclude the theory for group unlearning is currently not sharp enough to capture trends seen in practice.
+
+# A. Proofs
+
+# A.1. Corollary 4.1
+
+Proof. By (Theorem 3.2, Chien et al., 2024), for all $\alpha > 1$ , there exists a constant $C > 0$ such that
+
+$$
+D _ { 2 \alpha } ( \rho _ { \mathcal { D ^ { \prime } } } ^ { k } ( \nu _ { T , \mathcal { D } } ) \| \nu _ { \mathcal { D ^ { \prime } } } ) \leq e ^ { - \frac { C k } { 2 \alpha } } D _ { 2 \alpha } ( \nu _ { T , \mathcal { D } } \| \nu _ { \mathcal { D ^ { \prime } } } ) ,
+$$
+
+where $D _ { 2 \alpha } ( \nu _ { T , \mathcal { D } } | | v _ { D ^ { \prime } } )$ is the initial $2 \alpha$ -Renyi divergence to stationarity on ´ $\mathcal { D } ^ { \prime }$ , after training on $\mathcal { D }$ . By the weak triangle inequality (Proposition 11, Mironov, 2017), this is bounded by
+
+$$
+\frac { 2 \alpha - 1 / 2 } { 2 \alpha - 1 } D _ { 4 \alpha } ( \nu _ { T , \mathcal { D } } \| \nu _ { T , \mathcal { D ^ { \prime } } } ) + D _ { 4 \alpha - 1 } ( \nu _ { T , \mathcal { D ^ { \prime } } } \| \nu _ { \mathcal { D ^ { \prime } } } ) \leq \frac { 2 \alpha - 1 / 2 } { 2 \alpha - 1 } \varepsilon _ { 4 \alpha } ^ { \prime } + \varepsilon _ { 4 \alpha - 1 } ,
+$$
+
+where the second inequality follows from our hypotheses.
+
+Finally, applying the weak triangle inequality once more, $D _ { \alpha } ( \rho _ { D ^ { \prime } } ^ { k } ( \nu _ { T , D } ) \Vert \nu _ { T , D ^ { \prime } } )$ is bounded by
+
+$$
+\frac { \alpha - 1 / 2 } { \alpha - 1 } D _ { 2 \alpha } ( \rho _ { \mathcal { D ^ { \prime } } } ^ { k } ( \nu _ { T , \mathcal { D } } ) \Vert \nu _ { \mathcal { D ^ { \prime } } } ) + D _ { 2 \alpha - 1 } ( \nu _ { \mathcal { D ^ { \prime } } } \Vert \nu _ { T , \mathcal { D ^ { \prime } } } ) ,
+$$
+
+which yields the claimed bound from our hypotheses, after substituting Equations (3) and (4).
+
+# B. Group Privacy Analysis and Methodology
+
+The following theorem comes from the results of Thudi et al. (2024).
+
+Theorem B.1. Suppose we train with noisy gradient descent for $T$ steps. Then for an arbitrary $\mathcal { D } , \mathcal { D } ^ { \prime } = \mathcal { D } \setminus \mathcal { D } _ { F }$ , we have:
+
+$$
+D _ { \alpha } ( \nu _ { T , \mathcal { D } } \| \nu _ { T , \mathcal { D } ^ { \prime } } ) \leq \sum _ { t = 1 } ^ { T } C _ { t , \alpha } \ln \mathbb { E } _ { w \sim \nu _ { t , \mathcal { D } } } G _ { t , \alpha } ( \mathcal { D } , \mathcal { D } ^ { \prime } , w ) .
+$$
+
+where Ct,α = 1α−1 (p−1)t+1pt+1 and
+
+![](images/1960511be62d24b4f193f22e125a101876974b6bf5e1a889ae929e809922d11f.jpg)  
+Figure 5. Unlearning results for accuracy metrics (top) and MIA success rate (bottom). The $\mathbf { X }$ -axis represents the number of epochs. In each plot, lines of different colors represent forget sets of varying difficulty, while the dashed line indicates the oracle’s performance.
+
+$$
+\ln G _ { t , \alpha } ( \mathcal { D } , \mathcal { D } ^ { \prime } , w ) = p \mathbb { E } _ { \mathcal { D } _ { B } ^ { \prime } } \ln \mathbb { E } _ { \mathcal { D } _ { \mathbf { B } } ^ { \mathbf { o f } } ( \alpha ) } \left( e ^ { \frac { - 1 } { 2 \sigma ^ { 2 } } \Delta _ { o _ { p } ^ { t } ( \alpha ) } \left( \mathcal { D } _ { \mathbf { B } } ^ { \mathbf { o f } } ( \alpha ) , \mathcal { D } _ { B } ^ { \prime } , w \right) } \right)
+$$
+
+where $\mathcal { D } _ { \mathbf { B } } ^ { \alpha } = \{ \mathcal { D } _ { B } ^ { 1 } , \cdot \cdot \cdot , \mathcal { D } _ { B } ^ { \alpha } \}$ is a random sample of α minibatches from $\mathcal { D }$ , $\mathcal { D } _ { B } ^ { \prime }$ is a single random minibatch, $o _ { p } ( \alpha ) =$ $\textstyle { \frac { p } { p - 1 } } \alpha - { \frac { 1 } { p } }$ and $O _ { p } ^ { t }$ is $o _ { p }$ composed $t$ times, and $p$ is a free parameter we set to $3 T$ following (Fact 3.4, Thudi et al., 2024), and letting $U ( \mathcal { D } _ { B } , w ) = \nabla _ { w } \ell ( w , \mathcal { D } _ { B } )$ be the mini-batch gradient:
+
+$$
+\gamma _ { \mathbf { B } } ^ { \mathbf { a } ^ { \alpha } } , \mathcal { D } _ { B } ^ { \prime } , w ) : = \sum _ { i } | | U ( \mathcal { D } _ { B } { } ^ { i } , w ) | | _ { 2 } ^ { 2 } - ( \alpha - 1 ) | | U ( \mathcal { D } _ { B } ^ { \prime } , w ) | | _ { 2 } ^ { 2 } - | | \sum _ { i } U ( \mathcal { D } _ { B } { } ^ { i } , w ) - ( \alpha - 1 ) U ( \mathcal { D } _ { B } ^ { \prime } , w ) | | _ { 2 } ^ { 2 } ,
+$$
+
+Proof. A direct consequence of applying (Theorem 3.3, Thudi et al., 2024) with the general update per-step divergence bound of (Theorem 3.6, Thudi et al., 2024), and noting the divergence after applying the projections is bounded by the divergence before applying the projections by the post-processing inequality.
+
+# B.1. Methodology for privacy losses computation
+
+To estimate the guarantees we take a Monte-Carlo sample of checkpoints from a single training run at steps $s _ { 0 } , s _ { 1 } , \cdots , s _ { N }$ , and estimate the $\ln G _ { t , \alpha } ( D , { \mathcal { D } } ^ { \prime } , w )$ term at each step by sampling a single random mini-batch from $\mathcal { D } ^ { \prime }$ and $o _ { p } ^ { s _ { i } } ( \alpha )$ minibatches from $\mathcal { D }$ to estimate the expectations. We then compute the sum using the right hand rule, analogous to our estimate of privacy losses described in Section 5.
+
+In Figure 4 we took checkpoints from an SGD training run on CIFAR10 with ResNet18, and used $\sigma = 0 . 1$ , and took $\alpha = 8$ to compute the group privacy scores. We report the mean over 20 estimates (given the stochasticity in our estimates for the per-step terms $\ln G _ { t , \alpha } ( \mathcal { D } , \mathcal { D } ^ { \prime } , w ) )$ and shaded in one standard deviation.
+
+# C. Example Difficulty Related Work
+
+Paul et al. (2021) propose EL2N score to capture how much an example contributes to learning a high accuracy predictor, high score meaning high importance during training to achieve high accuracy. At the same time, the authors find that high scoring examples tend to be difficult to learn, are often memorized at the end of training and are outliers. This score has been shown to correlate with a number of other example difficulty and memorization metrics proposed in the literature (Kwok et al., 2024; Paul et al., 2021), some of which have been shown to also capture unlearning difficulty for a number of unlearning algorithms (Zhao et al., 2024; Baluta et al., 2024; Zhao & Triantafillou, 2024).
+
+![](images/e586c20a5148948799e65556a836da70bd85cacc521ae169cd1717fac34b47da.jpg)  
+Figure 6. Unlearning time for forget sets with different privacy loss values using the L1-sparse method. Time is measured by the number of steps required for the unlearning method to reach a $5 \%$ margin of error, where error is defined as the difference between the unlearned model’s UA and the oracle’s UA for the given forget set.
+
+# D. Additional Experimental Details
+
+Constructing Difficulty-Based Forget Sets To create forget sets with varying difficulty levels, the training dataset is partitioned into five subsets based on privacy scores. First, the samples are sorted in ascending order by their scores. Recursive splits are then performed to identify key thresholds: the lower quartile (Q1), the median (Q2), and the upper quartile (Q3). Using these thresholds, five subsets are constructed: (1) the first 1000 samples, (2) intervals centered around Q1 $\mathrm { Q 1 } \pm 5 0 0$ samples), (3) intervals centered around Q2 $\mathrm { \Delta } Q 2 \pm 5 0 0$ samples), (4) intervals centered around Q3 $( Q 3 \pm 5 0 0$ samples), and (5) the last 1000 samples. This approach provides a systematic stratification of the dataset, enabling the evaluation of unlearning performance across varying levels of difficulty as determined by privacy scores.
+
+Learning rates and training times for SGD The original model, which serves as the starting point for all unlearning techniques (not for SGLD), is trained for 150 epochs using an initial learning rate of 0.01, a weight decay of 0.0005, and a learning rate schedule that reduces the learning rate by an order of magnitude at epochs 80 and 120. Each unlearning method is subsequently fine-tuned for 25 epochs.
+
+Additional details for SGLD At every step we added $N ( 0 , \sigma ^ { 2 } )$ Gaussian noise to the minibatch gradient, where we vary $\sigma$ for ablations. All other hyperparameters were kept the same as SGD. In particular we do not do any additional gradient clipping.
+
+Additional details for L1-sparse L1-sparse is an unlearning method inspired by the observation that pruning aids unlearning (Liu et al., 2024a). Its objective function closely resembles that of fine-tuning but includes an additional $L _ { 1 }$ regularization term, weighted by a hyperparameter $\alpha$ , which encourages sparsity in model parameters to facilitate unlearning.
+
+Hyperparameter tuning We perform hyperparameter tuning (HPT) for the unlearning methods using the Bayesian optimization method on a random forget set. While fine-tuning involves a single hyperparameter–the learning rate– L1- Sparse additionally optimizes $\alpha$ . To determine the best hyperparameters for each method, we employ Bayesian optimization to find configurations that achieve an optimal balance between privacy and utility. Additionally, to ensure that the selected hyperparameters are also optimized with respect to the number of steps required for unlearning, we identify hyperparameter sets that fall within a $5 \%$ margin of error for this trade-off. Among these, we select the configuration that converges the fastest.
+
+![](images/ab4fc962bb7902acfa526248ac713c5fa7d03c4434bdda7f144d3c236740387a.jpg)  
+Figure 7. SGD unlearning. Unlearning time vs. privacy score for ResNet-18 on SVHN (left) and ViT-small on CIFAR-10 (right) Unlearning time is measured in steps required to reach a $5 \%$ UA error margin.
+
+Compute resources Experiments were conducted using L40 and RTX8000 GPUs, and AMD EPYC 7452 CPUs.
+
+# E. Evaluation metrics
+
+Membership Inference Attack Membership inference attacks (MIA) aim to determine whether a given sample was part of a model’s training data by analyzing differences in the model’s responses. In our approach, we train a logistic regression classifier using the model’s confidence values on training and test samples as inputs. The attacker then attempts to classify the forget samples, with success measured by the percentage of forget samples labeled as test, indicating effective unlearning.
+
+Gaussian Unlearning Score We use the Gaussian Unlearning Score (GUS), introduced by Pawelczyk et al. (2024), to quantify the impact of poisoned samples on the model. To compute GUS, each sample in the forget set is perturbed with zero-mean Gaussian noise with a standard deviation of $\sigma$ . GUS is then computed by averaging (over the forget set) the per-example inner product between the gradient of the loss with respect to the clean (non-poisoned) sample and the stored Gaussian noise used for poisoning, normalized by the L2 norm of the gradient. The effectiveness of an unlearning method is then assessed by how well it mitigates the influence of these poisoned samples. Specifically, the change in GUS before and after unlearning serves as a measure of unlearning success.
+
+For the CIFAR-10 and ResNet-18 setup, the original work recommends a variance value of 0.32. However, in our experiments, we explored different values and found that smaller variances were better suited for our setup. Based on these empirical findings, we set $\sigma ^ { 2 } = 0 . 0 6 2$ .
+
+# F. Additional Experimental Results
+
+# F.1. Unlearning trends during fine-tuning
+
+In this experiment, we apply the fine-tuning method to unlearn a ResNet-18 model trained on the full CIFAR-10 dataset. The forget set varies in difficulty, with five different levels determined by the proxy losses of the samples. The UA, utility and RA values during unlearning is depicted in Figure 5 (top), while the MIA results are depicted in Figure 5 (bottom). For UA and MIA, we see the most difficult forget sets do indeed take longer to unlearn. We see utility and RA are similar across difficulty levels.
+
+# F.2. L1-sparse fine-tuning
+
+The unlearning results for the L1-sparse method are presented in Figure 6. Similar to our observations for unlearning with SGD or SGLD fine-tuning, unlearning with L1-sparse takes longer to forget samples with higher privacy scores. In fact, the more challenging the forget sets, the longer the unlearning.
+
+![](images/85866e60b7ce4fb809a77d33e6f5c5f38e7122b5da0900a8f4c8829141b9382e.jpg)  
+Figure 8. Unlearning time for forget sets of size 100 (left), 5,000 (middle), and 10,000 (right), evaluated on CIFAR-10. Unlearning time is reported as the number of training steps required to achieve a UA error within $5 \%$ .
+
+# F.3. Additional datasets/architectures
+
+In addition to ResNet-18 on CIFAR-10, we conduct experiments with additional dataset-architecture pairs. Specifically, we evaluate ResNet-18 on SVHN and ViT-small on CIFAR-10. ViT-small is a Vision Transformer model that applies self-attention mechanisms to sequences of image patches, enabling effective global feature extraction. Figure 7 presents the unlearning results for ResNet-18 on SVHN (left) and ViT-small on CIFAR-10 (right). The results suggest that, consistent with our previous findings, unlearning takes longer for forget sets with a higher average privacy loss.
+
+Rankings across noise levels We ran experiments to test sensitivity of the ranking for SGD to the noise values used in estimating privacy losses. For SVHN with ResNet-18 we found: (1) Spearman correlation between rankings at $\sigma = 0 . 0 1$ and $\sigma = 0 . 0 0 1$ was $0 . 7 0 ( p = 0 . 0 )$ . (2) Between $\sigma = 0 . 0 0 1$ and $\sigma = 0 . 0 0 0 5$ was $0 . 9 9 ( p = 0 . 0 )$ . (3) Between $\sigma = 0 . 0 0 0 5$ and $\sigma = 0 . 0 0 0 1$ was $0 . 9 9 ( p = 0 . 0 )$ . These results suggest that rankings are largely noise-invariant, as long as some noise is present. Past work has shown evidence of observable noise during training due to software and hardware nondeterminism (Jia et al., 2021).
+
+![](images/50865345780c9bc437f982852cc515ddae8f1f88a10ccd7c66e5c96853b3c6e9.jpg)  
+Figure 9. Qualitative examples of forgetting difficulty on CIFAR-10. We show 4 examples each from the easy-to-forget (top row) and hard-to-forget (bottom row) subsets, identified from a forget set of 1,000 samples.
+
+# F.4. Varying Forget Set Sizes
+
+We conducted additional experiments varying the size of the forget set (100, 5,000, and 10,000 samples). The results are presented in Figure 8. Our findings indicate that privacy loss consistently distinguishes between easy and hard-to-forget subsets, even when the forget set is small. Interestingly, for very large forget sets (e.g., 10,000 samples), the separation begins to diminish. This observation is intuitive, as the variance in average privacy loss decreases with increasing subset size.
+
+# F.5. Qualitative Results
+
+Qualitative examples of easy and hard to forget CIFAR-10 samples are provided in Figure 9.

data/sampled_jsons/0A4Y9qRnu9_Leveraging Per-Instance Privacy for Machine Unlearning/auto/0A4Y9qRnu9_Leveraging Per-Instance Privacy for Machine Unlearning_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "Leveraging Per-Instance Privacy for Machine Unlearning ",
+        "text_level": 1,
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nazanin Mohammadi Sepahvand \\* 1 2 Anvith Thudi \\* 3 4 Berivan Isik 5 Ashmita Bhattacharyya 3 4   \nNicolas Papernot 3 4 Eleni Triantafillou 6 Daniel M. Roy 4 7 Gintare Karolina Dziugaite 8 2 9 ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present a principled, per-instance approach to quantifying the difficulty of unlearning via finetuning. We begin by sharpening an analysis of noisy gradient descent for unlearning (Chien et al., 2024), obtaining a better utility–unlearning tradeoff by replacing worst-case privacy loss bounds with per-instance privacy losses (Thudi et al., 2024), each of which bounds the (Renyi) diver-´ gence to retraining without an individual data point. To demonstrate the practical applicability of our theory, we present empirical results showing that our theoretical predictions are born out both for Stochastic Gradient Langevin Dynamics (SGLD) as well as for standard fine-tuning without explicit noise. We further demonstrate that per-instance privacy losses correlate well with several existing data difficulty metrics, while also identifying harder groups of data points, and introduce novel evaluation methods based on loss barriers. All together, our findings provide a foundation for more efficient and adaptive unlearning strategies tailored to the unique properties of individual data points. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1. Introduction ",
+        "text_level": 1,
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "et al., 2024), mitigating the impact of poisoning attacks (Liu et al., 2024b), or updating out-of-date information. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In modern large-scale AI training regimes, exact unlearning (i.e., anything equivalent to retraining the model from scratch, without the forget set) is prohibitively expensive. To meet this challenge, a range of approximate unlearning methods have been developed. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine unlearning aims to efficiently remove the influence of specific subsets of training data, called forget sets. The need for unlearning arises in many scenarios, such as handling requests for data deletion (Mantelero, 2013; Cooper ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Approaches based on ideas from differential privacy (DP) offer meaningful worst-case guarantees (Guo et al., 2020; Sekhari et al., 2021). These will be our focus. Unfortunately, for non-convex models, like neural networks, DP-based unlearning guarantees have so far come with higher error than their counterparts not designed to support unlearning, limiting their practical applicability. One of the key problems is the mismatch between the worst-case, data-agnostic nature of DP and the data-dependent nature of unlearning. Our work attempts to bridge this gap. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Alongside work on theoretical guarantees, research into heuristics has flourished. The aesthetics of such research is to combine strong utility with empirical evaluations on metrics inspired by DP-based notions of approximate unlearning. It is not uncommon, however, for state of the art heuristics to be caught out by new attacks, revealing that they do not meet stringent theoretical criteria (Hayes et al., 2024; Pawelczyk et al., 2024). These challenges highlight the need for methods that can balance strong theoretical guarantees with strong practical performance. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Towards understanding failure modes of unlearning, recent empirical work has shown that the behavior of unlearning varies considerably across individual data points (Zhao et al., 2024; Baluta et al., 2024). While prior work has attempted to incorporate these insights into methodology and evaluation, we lack a theoretical basis to 1) explain how individual data influence unlearning and 2) exploit this effect optimally. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce a principled approach to estimating the unlearning difficulty of individual data points. Our approach applies to learning with noisy gradient descent. Using recent results in differential privacy with Renyi ´ divergences (Thudi et al., 2024), our proposed measures— per-instance privacy losses—bound the Renyi divergence´ between a model trained with and without an individual data point. Critically, per-instance privacy losses can be efficiently estimated during training, based on the norms of gradient associated to individual data points. ",
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Armed with per-instance privacy losses, we revisit Chien et al.’s 2024 theoretical analysis of noisy gradient descent as an unlearning scheme (coined “Langevin unlearning” a.k.a “noisy fine-tuning”), based on training without the forget set. Our analysis provides a theoretical foundation for understanding how the number of iterations needed to approximately unlearn scales with per-instance privacy losses. Empirical validation demonstrates that our estimates of perinstance privacy losses serve as reliable predictors of unlearning difficulty under noisy fine-tuning. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "While noisy fine-tuning is not widely used, standard (noiseless) fine-tuning is a surprisingly effective and simple approximate unlearning approach, and serves as a building block for SOTA methods, such as L1-Sparse fine-tuning (Hayes et al., 2024; Liu et al., 2024a). We demonstrate the applicability of our theoretical insights, showing they extend empirically to standard fine-tuning-based unlearning, even in the absence of explicit noise injection. Empirically, we find that per-instance privacy losses continue to predict unlearning difficulty for fine-tuning across multiple approximate unlearning metrics, datasets, and architectures. Beyond fine-tuning, results with the L1-Sparse method (Liu et al., 2024a; Hayes et al., 2024) show privacy losses once again predict the number of steps to unlearn. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Privacy losses may also offer a more refined measure of data difficulty for unlearning. We show that privacy losses are strongly correlated with some (cheaper-to-compute) proxy measures of data difficulty studied in prior work. On the other hand, we show that privacy losses identify more “difficult” data, which may be useful for evaluation more broadly. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Besides standard empirical unlearning metrics, we evaluate privacy losses on a novel loss-landscape-based metric, based on linear connectivity and loss barriers (Frankle et al., 2020; Fort et al., 2020). While past unlearning metrics look at the output (e.g., loss) of an individual model, loss barriers capture some of the geometry of the loss landscape. Under this stringent test, fine-tuning is able to successfully unlearn. Interestingly, loss barriers also give insight into the differences between difficult and easy data points: points with higher privacy losses start off with larger loss barriers which need to be overcome to unlearn. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions can be summarized as follows: ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Per-Instance Theoretical Analysis of Unlearning. We revise Chien et al.’s analysis of noisy gradient descent for unlearning-via-fine-tuning, replacing a worstcase Renyi-DP bound with recent ´ per-instance privacy losses (Thudi et al., 2024). By exploiting that typical data points may have far less influence on the learned model, our per-instance analysis uncovers an improved trade-off between unlearning and utility compared to prior worst-case analyses. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Empirical Validation of Unlearning Difficulty. We demonstrate the practical applicability of our theoretical insights, showing that per-instance privacy losses reliably predict unlearning difficulty in experiments, both for noisy gradient descent and standard fine-tuning. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Efficient Proxies for Privacy Losses. We demonstrate that cheap proxy measures of data difficulty are strongly correlated with per-instance privacy losses. This identifies practical and scalable alternatives for assessing unlearning quality and forget set difficulty, particularly in resource-constrained settings where full privacy loss computation might be prohibitive. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Improved Identification of Difficult Data. Unlike existing heuristics for capturing aspects of unlearning difficulty, we show that privacy losses identify harder groups of data points, offering a versatile and theoretically justified measure of unlearning difficulty. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Loss Barrier Analysis of Unlearning. We introduce loss barriers (modulo permutation) as a way to evaluate unlearning. We find that the loss barrier is significantly reduced after unlearning, reaching levels comparable to baseline levels. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Broader Implications for Unlearning Methodology. Our empirical findings suggest that per-instance privacy losses may be useful in adapting other fine-tuningbased algorithms, such as L1-sparse. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2. Related Work ",
+        "text_level": 1,
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unlearning Unlearning (Cao & Yang, 2015) aims to remove the influence of training data. In the context of neural networks and non-convex learning problems, even highly optimized exact approaches (Bourtoule et al., 2021) are computationally intensive. Going beyond exact unlearning, Ginart et al. (2019) introduced a notion of approximate unlearning, inspired by differential privacy (Dwork et al., 2014). This approach relies on approximate statistical indistinguishability between retraining and unlearning, in exchange for better efficiency or utility compared to exact unlearning. Guo et al. (2020); Neel et al. (2021) studied approximate unlearning algorithms for convex models based on gradient descent using all the training data except those data to be forgotten (a.k.a., fine-tuning), followed by the addition of noise (often via the Gaussian mechanism, Dwork et al. 2006) to obtain statistical indistinguishability. Neel et al. (2021) proved that, under various assumptions, their approach achieves approximate unlearning in a sequential framework with fixed per-deletion runtime. ",
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For non-convex models, a plethora of approximate unlearning methods have been proposed with empirical validation, rather than theoretical guarantees (Graves et al., 2021; Goel et al., 2022; Thudi et al., 2022; Kurmanji et al., 2024; Liu et al., 2024a; Fan et al., 2023). While these approaches are shown to work on some metrics, recent work shows that several unlearning methods fail against more sophisticated attacks for either privacy or poisoning (Hayes et al., 2024; Pawelczyk et al., 2024). ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given the heuristic nature of many unlearning methods, a line of work has attempted to obtain a better understanding of their failure modes. Zhao et al. (2024); Baluta et al. (2024); Fan et al. (2025); Barbulescu & Triantafillou (2024) identified properties of forget sets that influence the behaviors of approximate unlearning algorithms. Zhao et al. (2024); Barbulescu & Triantafillou (2024) also derived improved unlearning methods based on these insights, for vision classifiers and LLMs, respectively. However, we lack a theoretical understanding of the underlying relationship between the identified factors and unlearning difficulty. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, Chien et al. (2024) studied Langevin dynamics for unlearning and introduced an approximate method with privacy guarantees for non-convex models that we build upon in this work, replacing a worst-case bound by a recent per-instance privacy bound. Their method leverages noisy gradient descent, for both the training algorithm, as well as during (noisy) fine-tuning at unlearning time. While our theoretical analysis considers that same setup, we empirically also investigate standard training without noise addition. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Per-Instance Differential Privacy Differential privacy (DP) is a standard approach to privacy-preserving data analysis (Dwork et al., 2006; 2014) and machine learning (Chaudhuri et al., 2011; Abadi et al., 2016). DP methodology provides an upper bound on the divergence between output distributions under neighbouring datasets, i.e., when only one individual data point is altered. Since this upper bound needs to hold for every dataset and all of its neighbors, DP is a worst-case privacy notion with a single privacy parameter, shared among all individual data points. However, in practice, different individual points have different effects when training a machine learning model (Thudi et al., 2024; Yu et al., 2023). For instance, some data points have a much smaller gradient norm than others, making the worstcase privacy analysis for these points over-conservative, and leads to unnecessary degradations in privacy-utility tradeoffs. To mitigate such degradations, Ghosh & Roth (2011) and Cummings & Durfee (2020) have analyzed individual sensitivity rather than worst-case sensitivity, which led to an alternative but weaker notion of privacy, called per-instance (or individual, or personalized) DP. Per-instance DP assigns a different privacy loss for different data points in a dataset, i.e., it is not worst-case, and it is less pessimistic than standard DP for points that do not affect the output distribution as much as the worst-case points. Building on per-instance DP, Ebadi et al. (2015) and Feldman & Zrnic (2021) have introduced algorithms that filter out data points once their per-instance DP loss exceeds a budget in database query problems and neural network training, respectively. For iterative algorithms like SGD, per-instance DP requires new composition theorems as the privacy loss is adaptive to the individual data points and is different at each iteration (Wang, 2019). Feldman & Zrnic (2021); Thudi et al. (2024) have filled this gap by providing new privacy composition analyses using the associated divergences computed at an individual level. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Motivated by the empirical observation that per-instance privacy loss provides much more promising guarantees than the worst-case DP analysis, we propose to port this per-instance approach to unlearning. We will see that this approach will allow us to provide unlearning guarantees and uncover properties of data that makes a forget set easy to unlearn. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Preliminaries and Problem Setup ",
+        "text_level": 1,
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given a dataset $\\mathcal { D } = \\{ x _ { i } \\} _ { i = 1 } ^ { n }$ of $n$ points, we are interested in estimating the difficulty of unlearning as a function of the particular forget set $\\mathcal { D } _ { F } \\subset \\mathcal { D }$ we are aiming to unlearn. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are interested in settings where retraining the model from scratch on the retain set $\\mathcal { D } ^ { \\prime } = \\mathcal { D } \\setminus \\mathcal { D } _ { F }$ is too costly, and so we consider approximate unlearning. In this work, we adopt a notion of approximate unlearning based on Renyi´ divergences. For $\\alpha > 0$ and $\\alpha \\neq 1$ , recall that the $\\alpha$ -Renyi´ divergence of $\\mu$ relative to $\\nu \\gg \\mu$ , is defined to be ",
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b2f19fe47a5010af418c3f7d08c0a1e1870a0b23b9291a46ad5aea3a88d77c3d.jpg",
+        "text": "$$\nD _ { \\alpha } ( \\mu \\| \\nu ) = \\frac { 1 } { \\alpha - 1 } \\log \\left( \\mathbb { E } _ { x \\sim \\nu } \\left[ \\frac { \\mu ( x ) } { \\nu ( x ) } \\right] ^ { \\alpha } \\right) .\n$$",
+        "text_format": "latex",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(Here, one can interpret µ(x)ν(x) as a Radon–Nikodym derivative more generally.) ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a fixed training set and forget set, the following definition captures the goal of unlearning in Renyi divergence: ´ ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.1. Fix a dataset $\\mathcal { D }$ and forget set $\\mathcal { D } _ { F } \\subset \\mathcal { D }$ . Let $\\nu$ be the distribution of the output of $\\boldsymbol { \\mathcal { A } } ( \\mathcal { D } ^ { \\prime } )$ , i.e., the learning algorithm on the retain set $\\mathcal { D } ^ { \\prime } = \\mathcal { D } \\setminus \\mathcal { D } _ { F }$ , and let $\\mu$ be the distribution of the output of $\\mathcal { U } ( A ( \\mathcal { D } ) , \\mathcal { D } ^ { \\prime } )$ , i.e., the unlearning algorithm on the learned model $\\mathcal { A } ( \\mathcal { D } )$ and $\\mathcal { D } ^ { \\prime }$ . For $\\alpha > 1$ , we say $\\mathcal { U } \\left( \\alpha , \\varepsilon \\right)$ -Renyi unlearns ´ $\\mathcal { D } _ { F }$ from $\\mathcal A ( \\mathcal D )$ when $D _ { \\alpha } ( \\mu \\| \\nu ) \\leq \\varepsilon$ . ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This leads to a uniform notion of Renyi unlearning, which ´ offers the same guarantee for all datasets and all forget sets, which is analogous to standard notions. ",
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 3.2 (Renyi Unlearning) ´ . For $\\alpha > 1$ , an unlearning algorithm $\\mathcal { U }$ is an $( \\alpha , \\varepsilon )$ -Renyi unlearner (for ´ $\\mathcal { A }$ ) when, for all dataset $\\mathcal { D }$ and forget sets $\\mathcal { D } _ { F } \\subseteq \\mathcal { D } , \\mathcal { U } \\left( \\alpha , \\varepsilon \\right) .$ -Renyi´ unlearns $\\mathcal { D } _ { F }$ from $\\mathcal { A } ( \\mathcal { D } )$ in the sense of Definition 3.1. ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that when this definition holds, one can immediately derive (standard) approximate DP-based unlearning guarantees (Ginart et al., 2019; Guo et al., 2020; Neel et al., 2021; Sekhari et al., 2021), using standard reductions (Proposition 10, Mironov, 2017).1 ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1. Learning–Unlearning with Noisy Gradient Descent ",
+        "text_level": 1,
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider a learning–unlearning setup based on using projected noisy gradient descent during both learning and unlearning, as studied recently by Chien et al. (2024). In this section, we present theoretical results that rely on noise. In Section 6, we present empirical evidence from multiple benchmarks that the overall trends extend to standard variants of gradient descent. ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Formally, our learning algorithm $\\mathcal { A } = \\mathcal { A } _ { T }$ is $T$ steps of noisy gradient descent on $\\mathcal { D }$ , starting from a random initialization. Let $\\nu _ { T , D }$ denote the distribution on weights obtained after $T$ steps of training on $\\mathcal { D }$ , and let $\\nu _ { D } = \\nu _ { \\infty , D }$ denote the stationary distribution, to which the output distribution converges as $T \\to \\infty$ . ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { D } ^ { \\prime } = \\mathcal { D } \\setminus \\mathcal { D } _ { F }$ denote the retain set of an unlearning request. Assuming we trained for $T$ iterations, exact unlearning would produce weights whose (marginal) distribution was $\\nu _ { T , \\mathcal { D } ^ { \\prime } }$ , i.e., the distribution as if we had trained on $\\mathcal { D } ^ { \\prime }$ from scratch for $T$ iterations. ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our unlearning algorithm $\\mathcal { U } = \\mathcal { U } _ { k }$ runs $k$ steps of noisy gradient descent on $\\mathcal { D } ^ { \\prime }$ . We denote by $\\rho _ { { D ^ { \\prime } } } ^ { k } ( \\nu _ { { T } , { \\mathcal { D } } } )$ the output distribution of unlearning, i.e., of running $k$ steps of projected noisy gradient descent on $\\mathcal { D } ^ { \\prime }$ , initialized at a sample from $\\nu _ { T , \\mathcal { D } }$ , i.e., first training on $\\mathcal { D }$ . ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4. Per-Instance Unlearning Difficulty Analysis ",
+        "text_level": 1,
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to bound the number of steps, $k$ , of unlearning by noisy fine-tuning needed to achieve unlearning in a way that adapts to the influence each data point has on the learned model. We do so by building on two pieces of work: a convergence analysis of Langevin dynamics by Chien et al. (2024) and a per-instance Renyi-differential privacy analysis ´ by Thudi et al. (2024). ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1. Convergence Analysis ",
+        "text_level": 1,
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fix a dataset $\\mathcal { D }$ and retain set $\\mathcal { D } ^ { \\prime } \\subset \\mathcal { D }$ . We begin with the following corollary of (Thm. 3.2, Chien et al., 2024), which highlights the role of an initial bound on $D _ { \\alpha } ( \\nu _ { T , \\mathcal { D } } | | \\nu _ { T , \\mathcal { D } ^ { \\prime } } )$ In the following, we assume the loss is Lipschitz continuous and smooth, and that the step sizes used by $\\mathcal { A }$ and $\\mathcal { U }$ have been set according to (Thm. 3.2, Chien et al., 2024). ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.1. Fix $\\mathcal { D } , \\mathcal { D } ^ { \\prime }$ . Let $\\{ \\varepsilon _ { \\alpha } , \\varepsilon _ { \\alpha } ^ { \\prime } \\} _ { \\alpha \\ge 1 }$ satisfy $\\begin{array} { r l r } { \\operatorname* { m a x } \\{ D _ { \\alpha } ( \\nu _ { D ^ { \\prime } } \\| \\nu _ { T , \\mathcal { D } ^ { \\prime } } ) , D _ { \\alpha } ( \\nu _ { T , \\mathcal { D } ^ { \\prime } } \\| \\nu _ { \\mathcal { D } ^ { \\prime } } ) \\} } & { \\leq } & { \\varepsilon _ { \\alpha } } \\end{array}$ and $D _ { \\alpha } ( \\nu _ { T , \\mathcal { D } } | | \\nu _ { T , \\mathcal { D } ^ { \\prime } } ) \\leq \\varepsilon _ { \\alpha } ^ { \\prime }$ for all $\\alpha$ . Then, there exists a constant $C > 0$ such that, for all $\\alpha > 1 , \\mathcal { U } _ { k } \\left( \\alpha , \\epsilon ^ { * } \\right)$ -Renyi ´ unlearns $\\mathcal { D } \\backslash \\mathcal { D ^ { \\prime } }$ from $\\mathcal { A } ( \\mathcal { D } )$ in $k$ steps, where $\\epsilon ^ { * }$ is ",
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e03d5c3e930f0a17c7c4b34714c0edb3a8aca6670d20b8012117804bcaaf4ec7.jpg",
+        "text": "$$\n\\Bigl ( \\frac { 2 \\alpha - 1 / 2 } { 2 \\alpha - 2 } \\varepsilon _ { 4 \\alpha } ^ { \\prime } + \\frac { 2 \\alpha - 1 } { 2 \\alpha - 2 } \\varepsilon _ { 4 \\alpha - 1 } \\Bigr ) \\exp \\Big ( - \\frac { C k } { 2 \\alpha } \\Big ) + \\varepsilon _ { 2 \\alpha - 1 } .\n$$",
+        "text_format": "latex",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The proof is in Appendix A.1, and follows from (Thm. 3.2, Chien et al., 2024) and the weak triangle inequality for Renyi divergences.´ ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The key observation is that the first term decays exponentially fast in the number of steps $k$ , with the initial value determined by the per-instance guarantee and distance to stationarity and the rate determined by the desired value of $\\alpha$ and problem parameters (like the Lipschitz and smoothness constants, behind $C$ ). The second term, $\\varepsilon _ { 2 \\alpha - 1 }$ , however, does not vanish. ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The irreducible term captures the distance to stationarity after $T$ steps of training, measured in a higher order $( 2 \\alpha -$ 1)-divergence. This term exists as a consequence of our unlearning algorithm not attempting to correct for the $k$ steps of extra training. ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2. Unlearning Individual Data Points ",
+        "text_level": 1,
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The following definition is adapted from a differential privacy result by Thudi et al. (2024) to our unlearning setting: ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 4.2 (Per-Instance Privacy Loss). Recall that $\\nu _ { T , D }$ is the distribution of the model weights after $T$ iterations of noisy gradient descent on $\\mathcal { D }$ . Let $g ( x ^ { * } , w ) = \\nabla _ { w } \\ell ( w , x ^ { * } )$ be the contribution to the weight-gradient at $w$ , coming from one data point $x ^ { * }$ . The per-instance privacy loss for $x ^ { * }$ is: ",
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e0d75eee0f5bb270c1f1d724994ea153606955cf69e556cdbcc6f562ad88e8ce.jpg",
+        "text": "$$\n\\begin{array} { r } { P ( \\boldsymbol { x } , \\alpha ) : = \\sum _ { t = 1 } ^ { T } C _ { t , \\alpha } \\ln \\mathbb { E } _ { \\boldsymbol { w } \\sim \\nu _ { t , D } } f _ { t , \\alpha } ( \\| \\boldsymbol { g } ( \\boldsymbol { x } ^ { * } , \\boldsymbol { w } ) \\| _ { 2 } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where Ct,α = 1α−1 (p−1)t+1pt+1 a nd $\\ln f _ { t , \\alpha } ( g )$ is ",
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23f12cc923d11a08754d8143007d6a222cbae46d96eef61f075a280d54d84d11.jpg",
+        "text": "$$\np \\ln \\bigg ( \\sum _ { k = 0 } ^ { o _ { p } ^ { i } ( \\alpha ) } { \\binom { o _ { p } ^ { t } ( \\alpha ) } { k } } \\mathbb { P } _ { x ^ { * } } ( 1 ) ^ { k } \\overline { { \\mathbb { P } _ { x ^ { * } } ( 1 ) } } ^ { o _ { p } ^ { t } ( \\alpha ) - k } e ^ { \\frac { g ^ { 2 } ( k ^ { 2 } - k ) } { 2 \\sigma ^ { 2 } } } \\bigg ) ,\n$$",
+        "text_format": "latex",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $\\begin{array} { r } { o _ { p } ( \\alpha ) \\ = \\ \\frac { p } { p - 1 } \\alpha - \\frac { 1 } { p } } \\end{array}$ and $O _ { p } ^ { t }$ is $o _ { p }$ composed $t$ times, $\\mathbb { P } _ { x ^ { * } } ( 1 ) = 1 - \\overline { { \\mathbb { P } _ { x ^ { * } } ( 1 ) } }$ is the sampling probability of the data point (batch size over dataset size), and $p$ is a free parameter we set to $3 T$ , following (Fact 3.4, Thudi et al., 2024). ",
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This quantity, which we show how to estimate efficiently in Section 5.1, yields a bound on a data point’s sensitivity. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.3. Fix $\\mathcal { D }$ , let $x \\in \\mathcal { D }$ , and put ${ \\mathcal { D } } ^ { \\prime } = { \\mathcal { D } } \\setminus \\{ x \\}$ .   \nThen $D _ { \\alpha } ( \\nu _ { T , \\mathcal { D } } | | \\nu _ { T , \\mathcal { D ^ { \\prime } } } ) \\leq P ( x , \\alpha )$ . ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. This follows by applying the per-instance momentbased composition theorem (Thm. 3.3, Thudi et al., 2024) with the per-step divergence bound for single data points (Thm. 3.2, Thudi et al., 2024), using the post-processing inequality to conclude that the divergence after projections is bounded by that before projections. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As a consequence of Theorem 4.3 applied to Corollary 4.1, we have unlearning individual data points by noisy gradient descent depends logarithmically on the privacy loss. The following corollary is a direct substitution of Theorem 4.3 into Corollary 4.1. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Corollary 4.4. Under the assumptions of Corollary 4.1 and Theorem 4.3, for all $\\mathcal { D }$ and ${ \\mathcal { D } } ^ { \\prime } = { \\mathcal { D } } \\setminus \\{ x \\}$ for $x \\in \\mathcal { D }$ , for all $\\alpha > 1$ , there exist constants $A _ { \\alpha } , B _ { \\alpha } , C _ { \\alpha } > 0$ such that, for all $\\delta > \\varepsilon _ { 2 \\alpha - 1 }$ , running noisy gradient descent for ",
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d4e933c924746fa46d05d2ee2b97fc64dcd0cf958239c5f9ef137607ee147670.jpg",
+        "text": "$$\nk \\geq A _ { \\alpha } \\ln \\left( \\frac { B _ { \\alpha } P ( x , 4 \\alpha ) + C _ { \\alpha } \\varepsilon _ { 4 \\alpha - 1 } } { \\delta - \\varepsilon _ { 2 \\alpha - 1 } } \\right)\n$$",
+        "text_format": "latex",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "steps $( \\alpha , \\delta )$ -unlearns $\\{ x \\}$ from $\\mathcal A ( \\mathcal D )$ . ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3. Limitations of Existing Group Unlearning Analyses ",
+        "text_level": 1,
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The above analysis focuses on forgetting one data point. How about bounding the work to unlearn multiple data points simultaneously? The group unlearning bound of (Chien et al., 2024, Cor. 3.4) requires that the order of $\\alpha$ grows with each data point to unlearn. This is problematic, as the Renyi divergence between, e.g., Gaussians, ´ grows linearly with $\\alpha$ (Prop. 7, Mironov, 2017). In contrast, (Thms. 3.3 and 3.6, Thudi et al., 2024) imply that the Renyi divergence ´ $D _ { \\alpha } ( \\nu _ { T , \\mathcal { D } } | | \\nu _ { T , \\mathcal { D } \\backslash \\mathcal { D } _ { F } } )$ , and hence steps to unlearn, does not necessarily grow with $\\mathcal { D } _ { F }$ ; instead, Thudi et al. bound the divergence by comparing the distribution of gradients under $\\mathcal { D }$ and under $\\mathcal { D } \\backslash \\mathcal { D } _ { F }$ . Implementing their group privacy accounting (described in Appendix B), however, we found evidence the bounds were likely too loose, as they did not differentiate forget sets that we empirically knew to require different numbers of steps to unlearn (see Figure 4 in appendix). In contrast, we found the rankings provided by privacy losses meaningfully differentiate the number of steps needed to unlearn, as we will describe in Section 6. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We conclude that the current state of analysis for group unlearning is not tight enough to capture the behavior we observe in practice. The problem of obtaining a tight analysis remains an open problem. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5. Methodology ",
+        "text_level": 1,
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our empirical methodology is structured around two key objectives driven by the theoretical role of per-instance privacy losses in unlearning. First, we aim to validate that privacy losses effectively predict the relative difficulty of unlearning data points. Second, we seek to understand the factors contributing to this difficulty by investigating the relationship between privacy losses, the loss landscape, and existing metrics of data difficulty. In the following sections, we present our experimental design to address these questions. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1. Empirically Validating Unlearning Difficulty ",
+        "text_level": 1,
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Unlearning Algorithms Our investigation primarily focuses on unlearning via fine-tuning on the retain set. We also run unlearning experiments with L1-Sparse (Liu et al., 2024a), a regularized version of fine-tuning that is widely recognized as one of the most effective unlearning methods. Hayes et al. (2024) has demonstrated that L1-Sparse outperforms a number of other methods in defending against a basic Membership Inference Attack (MIA), as well as other attacks of varying strengths. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Per-instance Privacy loss We compute the terms in the privacy loss $P ( x , \\alpha )$ stated in Definition 4.2 by taking a Monte-Carlo estimate from a single training run with checkpoints $w _ { 0 } , w _ { s _ { 1 } } , w _ { s _ { 2 } } , \\cdot \\cdot \\cdot , w _ { s _ { N } }$ , i.e., ",
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/75099358bca6a43bde070417e27bf4847528b5cff6335891030b8bdb897a2a21.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { P } _ { s _ { i } } ( x , \\alpha ) = C _ { s _ { i } , \\alpha } \\ln f _ { s _ { i } , \\alpha } ( \\| g ( x ^ { * } , w _ { s _ { i } } ) \\| _ { 2 } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $C _ { s _ { i } , \\alpha } , f _ { s _ { i } , \\alpha } , g$ are defined in Definition 4.2. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We then approximate the area under the per-step privacy curve (i.e., sum over $t = 0 , 1 , \\cdots , s _ { N } )$ by using the right hand rule: keeping $\\hat { P } _ { s _ { i } }$ constant between the checkpointing intervals $\\left( { { s } _ { i - 1 } } , { { s } _ { i } } \\right]$ . This gives our approximate privacy loss: ",
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8bbf435beb28b5ee3c92e0acc44a6504ff883f20f10cea1964be9cec34d107c2.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { P } ( x , \\alpha ) = \\sum _ { i = 1 } ^ { N } \\hat { P } _ { s _ { i } } ( x ) ( s _ { i } - s _ { i - 1 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Throughout the paper we take $N = 3 5$ and have the checkpoints $s _ { i }$ evenly spaced throughout training. In the case of SGD, without explicit noise, we approximate these scores by assuming a negligible amount of noise is present. In particular we take $\\sigma \\leq 0 . 1$ , which matches the trends we observe with SGLD. We also found the privacy losses rankings for SGD to be stable to the implicit $\\sigma$ used for privacy losses computation (see Appendix F). We note that past work has looked into quantifying the noise inherent in training, due to hardware and software nondeterminism (Jia et al., 2021; Zhuang et al., 2022). It is an open problem to exploit software and hardware nondeterminism to offer a theoretical justification to our approach here. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Forget Set Difficulty We rank examples by privacy losses and form 5 forget sets of varying difficulty by taking evenly spaced sequences of 1000 data points. The size of the forget set was chosen to be in a similar range as prior work (e.g., (Zhao et al., 2024)). Our theory suggests that higher privacy losses should lead to longer, thus more difficult, unlearning. We thus call a forget set more difficult than another forget set if its average privacy loss is higher. In our analysis, each forget set is represented by the average privacy loss of all samples within that set. Further methodological details are provided in Appendix D. ",
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Unlearning Evaluation We evaluate unlearning efficacy using three metrics: (1) accuracy, measured separately on the retain set (RA), test set (utility), and on the forget set (FA). We report $\\mathrm { U A } = 1 - \\mathrm { F A }$ , indicating how “accurate” unlearning is, as done in (Fan et al., 2023); (2) membership inference attack (MIA): we train a logistic regression classifier to identify training samples and report the fraction of forget set samples incorrectly classified as test samples, thus indicating successful forgetting; and (3) Gaussian Unlearning Score (GUS) (Pawelczyk et al., 2024), which employs Gaussian input poisoning attacks to reveal if the unlearned model still encodes noise patterns associated with the poisoned forget set data. See Appendix E for more details. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "These metrics are monitored during unlearning and compared with the oracle model to evaluate the effectiveness of the unlearning methods. Recall that the oracle model is obtained by training from scratch using the retain set only. We choose these metrics due to their common use in machine unlearning research (Fan et al., 2025; Zhao et al., 2024; Jia et al., 2023; Deeb & Roger, 2024; Fan et al., 2024; Pawelczyk et al., 2024) and their computational simplicity, enabling us to compute them at every step during unlearning. Additional details on their computation and associated parameter choices can be found in Appendix E. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets and models Our experiments are performed on the SVHN (Netzer et al., 2011) and CIFAR-10 (Alex, 2009) datasets, with a ResNet-18 architecture (He et al., 2016). Appendix F presents results for ViT-small (Dosovitskiy, 2020). Appendix D describes other details for reproducibility. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Each experimental configuration (oracle training, or training on all data and unlearning) is run 10 times, and the average performance across these runs is reported. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2. Loss Landscape Analysis ",
+        "text_level": 1,
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While per-instance privacy losses provide a quantitative measure of unlearning difficulty based on training dynamics, they do not directly reveal the underlying geometric properties of the loss landscape that contribute to this difficulty. To gain a deeper understanding of why certain data points are harder to unlearn, we complement our privacy loss analysis with an investigation of the loss landscape. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Specifically, we employ the concepts of (linear) loss barri ers (Frankle et al., 2020; Nagarajan & Kolter, 2019). Loss barriers characterize the “flatness” or “curvature” of the loss surface between different model parameter configurations. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The loss barrier $\\mathrm { e r r } ( w , w ^ { \\prime } ; \\mathcal { D } )$ is the deviation in cross entropy $\\mathcal { L }$ on the data $\\mathcal { D }$ along the linear path in weight space connecting $w$ to $w ^ { \\prime }$ . Let $\\bar { \\alpha } = 1 - \\alpha$ . Then $\\mathrm { e r r } ( w , w ^ { \\prime } ; S )$ is ",
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a10207cd48b4eadc008d4cc39cde464b1b81e40614e93e6c61d11d82ad4b2d17.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\alpha \\in [ 0 , 1 ] } \\left[ \\mathcal { L } ( \\alpha w + \\bar { \\alpha } w ^ { \\prime } ; S ) - \\alpha \\mathcal { L } ( w ; S ) - \\bar { \\alpha } \\mathcal { L } ( w ^ { \\prime } ; S ) \\right] .\n$$",
+        "text_format": "latex",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To account for the permutation invariance of neural networks, we compute these loss barriers modulo permutation, as detailed in (Entezari et al., 2022; Sharma et al., 2024). ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Loss barriers provide insight into the geometric properties of high-dimensional loss surfaces. In our experiments, we compute loss barriers between oracle models (trained without the forget set) and models before and after unlearning forget sets with various average privacy losses. This allows us to examine if forget sets with higher privacy losses (higher predicted unlearning difficulty) exhibit distinct loss landscape characteristics, particularly in terms of loss barriers. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3. Comparison to Alternative Data Difficulty Metrics ",
+        "text_level": 1,
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While the per-instance privacy losses provide valuable insights into the unlearning process, their computation requires storing gradients throughout training, leading to considerable computational overhead. Therefore, we explore alternative metrics that could serve as proxies for these scores, offering a more efficient way to estimate forget set difficulty. Furthermore, we investigate the relationship between these proxies and fine-tuning-based unlearning difficulty, shedding light on their underlying mechanisms. ",
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We evaluate five proxies: (1) the gradient norm of individual data points at a single mid-training iteration; (2) the gradient norm at the end of training; (3) the average gradient norm across all training iterations; (4) C-Proxy used by Zhao et al. (2024) to approximate memorization scores from (Feldman et al., 2018); adapted from Jiang et al. (2020). This proxy computes prediction confidence: the entry in the softmax vector corresponding to the ground truth class, averaged throughout the training trajectory; and (5) a single-trajectory variant of the EL2N score (Paul et al., 2021). Normally, EL2N score is computed by averaging error signals over multiple training trajectories at a fixed time point. However, for computational efficiency and direct comparability with our gradient-based proxies, we compute a single-trajectory EL2N score at a mid-training checkpoint. See Appendix C for a discussion on connections among the scores. ",
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f643d81abd9a6b7961e949f9183d2a3b60fd582bcb509684e0ebbc249e97d7f0.jpg",
+        "image_caption": [
+            "Figure 1. CIFAR-10 dataset results. Left: SGLD unlearning with varying levels of noise $( \\sigma )$ . Forget set difficulty $\\mathbf { \\widetilde { x } }$ -axis), as measured by the privacy loss, against time to unlearn (y-axis). Time to unlearn is measured in terms of epochs needed to get within $5 \\%$ of the unlearning metric (e.g., UA or MIA) measured on the oracle model. Middle: SGD unlearning. Time to unlearn measured across three evaluation metrics. Right: Error barrier between the oracle and the unlearned model before and after unlearning for forget sets with different privacy losses. Baseline corresponds to the loss barrier between two oracles. "
+        ],
+        "image_footnote": [],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6. Experimental Results ",
+        "text_level": 1,
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We empirically validate the predictive capabilities of perinstance privacy losses in forecasting unlearning difficulty. Our primary finding is that privacy losses accurately rank datapoints according to the number of steps needed to unlearn. This is tested for two settings of interest: (1) SGLD, which aligns directly with the assumptions of our theoretical upper bounds; and (2) SGD. For SGD, while lacking explicit noise, we adapt privacy losses by assuming a small, implicit noise component. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To probe the geometric origins of unlearning difficulty as captured by privacy losses, we analyze the loss landscape. Our investigation reveals a consistent trend: data points with higher privacy losses exhibit larger loss barriers on a linear path (in the weight space) to the oracle model. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also assess the practical utility of privacy losses by comparing them to computationally efficient proxy metrics. Our results demonstrate a strong correlation between privacy losses and existing proxy metrics, including those employed in previous studies to estimate forget set difficulty (C-Proxy in (Zhao et al., 2024)). However, a key advantage emerges: privacy losses provide superior precision in identifying truly difficult-to-unlearn data points within the context of finetuning-based unlearning. The forget sets ranked as most difficult by privacy losses consistently take longer to unlearn than those identified by these established proxies. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1. Time to Unlearn Depends on Per-Instance Privacy ",
+        "text_level": 1,
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We find that across a variety of unlearning metrics, privacy losses accurately separate data points by the number of steps needed to unlearn in both SGLD and SGD. Appendix F.2 shows the same trends for L1-sparse unlearning. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SGLD Training To evaluate the predictions outlined in Section 4, we examine the relationship between the number of steps needed to unlearn using noisy fine-tuning (SGLD), and the average privacy loss within the forget set. Figure 1 (left) shows that as forget set privacy losses increase, a higher number of fine-tuning steps is needed to reach a $5 \\%$ error margin relative to the oracle, validating our prediction. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "SGD Training While SGLD offers theoretical advantages for privacy analysis and bounding Renyi unlearning, SGD ´ remains the dominant training paradigm in practice. Therefore, we investigate whether our theoretical framework, developed in the context of Langevin dynamics, can also predict unlearning time for models trained with SGD. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 1 (middle), we observe a similar trend as in the SGLD experiments across all evaluation metrics: unlearning more difficult forget sets, characterized by higher average privacy losses, requires more fine-tuning steps. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This suggests that even without explicit noise injection during training, the concept of per-instance privacy losses derived from our theoretical analysis can provide valuable insights into the unlearning process for SGD-trained models. That is, SGD seems to be well-approximated by low noise SGLD. These results are replicated across additional datasets, architectures, and forget set sizes, with full details and qualitative examples in Appendix F. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.2. More Difficult Forget Sets, Larger Loss Barriers ",
+        "text_level": 1,
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We characterize difficult to unlearn data points by their loss landscape, in particular, loss barriers (see Section 5). ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 (right) depicts the loss barrier between oracle models and unlearned models, while varying the difficulty of the forget set, as measured by the average privacy loss. ",
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We observe two key takeaways from this analysis: (1) Comparing our results to the baseline loss barriers between independently trained oracle models, we find that fine-tuning achieves comparable levels after unlearning. This can be seen as a measure of unlearning efficacy based on loss barriers: if the (distribution of the) loss barrier is different from that of the baseline between oracles, unlearning was unsuccessful; (2) Higher privacy loss values correspond to larger initial barriers, providing a geometric interpretation for privacy losses; data points that require more steps to unlearn have to overcome larger loss barriers to the oracle. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Overall, our findings provide additional evidence for the utility of privacy losses in predicting unlearning difficulty, and point to the effectiveness of loss barriers as both a diagnostic and an evaluation tool for machine unlearning. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.3. Existing Metrics Correlate with Privacy Losses ",
+        "text_level": 1,
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We now investigate how privacy losses compare to other data difficulty metrics, as described in Section 5. Our results in Figure 2 reveal that all these proxies exhibit high correlation with the actual privacy loss. As expected, the best proxy is the average gradient norm throughout training, but it is also the most expensive proxy as it requires computing gradient norms for each data point at every iteration, versus once. Other proxies (with the exception of C-Proxy) only require a gradient/error computation at a single checkpoint, yet still provide a reasonable approximation for categorizing examples into broad difficulty groups. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "These findings suggest that, in scenarios where computational resources are limited, utilizing these proxies can offer a practical alternative for estimating forget set difficulty and predicting unlearning time. We refer to recent work by Kwok et al. (2024) for an in-depth comparison of different data difficulty metrics that may serve as good proxies. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Recall that C-Proxy has been used in prior work by Zhao et al. (2024) to identify difficult to unlearn forget sets for certain unlearning algorithms. Figure 2 shows that privacy losses are highly correlated with this heuristic. Our work thus offers theoretical grounding to previously proposed heuristics for identifying difficult to unlearn forget sets. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, note that among metrics relying on averaging over training, our method is more efficient, requiring only 35 evenly spaced checkpoints (approximately $20 \\%$ of training time), compared to C-Proxy and average gradient norm, which store values at every epoch (150 in total). As shown in Section 6.4, it is also more effective at identifying hard-tounlearn samples than all other metrics, regardless of whether they rely on averaging. ",
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/fdc0a71cf8d1ea8bb5d7e12b63d29805546210233bb5547f3702a64475662a19.jpg",
+        "image_caption": [
+            "Figure 2. Correlation between privacy losses $\\mathbf { \\dot { x } }$ -axis) and various proxy metrics (y-axis). The values of all proxy metrics are normalized to their maximum value for better visual clarity. For improved readability, the data is binned into 30 bins. "
+        ],
+        "image_footnote": [],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/caaaf3ffdbb7de347fe47c1a164b0fe030f6a75ab30e0cd8a8a77a153c9b6a1c.jpg",
+        "image_caption": [
+            "Figure 3. Comparison of the time needed to unlearn (y-axis) the most difficult forget sets as identified by privacy losses (ours), C-Proxy, average gradient norm and EL2N, across different forget set sizes ( $\\mathbf { \\dot { x } }$ -axis). "
+        ],
+        "image_footnote": [],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.4. Privacy Losses Identify Harder Data ",
+        "text_level": 1,
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In the previous subsection, we provided evidence that existing data difficulty metrics correlate with per-instance privacy losses. In this section, we provide evidence that, in fact, privacy losses are able to identify harder data. In Figure 3, we compare our per-instance privacy losses to several data difficulty metrics, including C-Proxy (described in Section 5), average gradient norm, and EL2N scores. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For forget sets of size $s \\in \\{ 6 0 0 , 1 0 0 0 , 2 0 0 0 \\}$ , we look at the top $s$ data points as ranked by C-Proxy, average gradient norm, EL2N scores, and privacy losses. What we see is that, across the board, the data picked out by privacy losses are harder to unlearn, as measured both by the number of iterations to reach $5 \\%$ unlearning accuracy or reach $5 \\%$ excess membership inference attack error. By identifying more difficult examples, privacy losses open up new empirical approaches to evaluating unlearning performance. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7. Conclusion ",
+        "text_level": 1,
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have introduced a principled approach to quantifying machine unlearning difficulty at the level of individual data points in terms of per-instance privacy losses, which bound the Renyi divergence between training with´ and without a datapoint. Our theoretical analysis provides a foundation for understanding how unlearning scales with the properties of specific data points, particularly in the context of Langevin dynamics. ",
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have shown that per-instance privacy losses, estimated from training statistics, reliably predict unlearning difficulty in fine-tuning-based unlearning algorithms, across different architectures and datasets, even in the absence of explicit noise injection during training. Our empirical results demonstrate that these privacy losses offer a precise and actionable measure of unlearning difficulty. Our work also offers a theoretical grounding for previous work suggesting that certain forget sets are harder to unlearn, with privacy losses capturing similar aspects of difficulty for fine-tuningbased unlearning as previously proposed heuristics (Zhao & Triantafillou, 2024; Baluta et al., 2024; Zhao et al., 2024). Moreover, we show privacy losses identify harder forget sets than previous methods. ",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our findings have broader implications for unlearning methodology, suggesting that per-instance divergence analysis can guide the development of new, more efficient unlearning algorithms tailored to specific data characteristics. Extending our theoretical framework to other unlearning methods beyond fine-tuning and exploring the use of privacy losses in designing adaptive unlearning strategies is a promising direction for future work. ",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Impact Statement ",
+        "text_level": 1,
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper presents work whose goal is to advance the field of machine unlearning, which is specifically oriented to improve the trustworthiness of machine learning, by supporting requests to remove the influence of training data. There are many potential positive societal consequences of our work, none which we feel must be specifically highlighted here. ",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Ioannis Mitliagkas and Ilia Shumailov for feedback on a draft of this work. Anvith Thudi, Ashmita Bhattacharyya, and Nicolas Papernot would like to acknowledge the sponsors of the CleverHans lab, who support our research with financial and in-kind contributions: CIFAR through the Canada CIFAR AI Chair, NSERC through the Discovery Grant, the Ontario Early Researcher Award, the Schmidt Sciences foundation through the AI2050 Early Career Fellow program, and the Sloan Foundation. Resources used in preparing this research were provided, in part, by the Province of Ontario, the Government of Canada through CIFAR, and companies sponsoring the Vector Institute. Anvith Thudi is also supported by a Vanier Fellowship from NSERC. Daniel M. Roy is supported by the funding through ",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "NSERC Discovery Grant and Canada CIFAR AI Chair at the Vector Institute. We also thank Mila – Quebec AI Insti tute and Google DeepMind for providing the computational resources that supported this work. ",
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \nAbadi, M., Chu, A., Goodfellow, I., McMahan, H. B., Mironov, I., Talwar, K., and Zhang, L. Deep learning with differential privacy. In ACM SIGSAC Conf. Computer and Communications Security, pp. 308–318, 2016.   \nAlex, K. Learning multiple layers of features from tiny images, 2009. URL https://www.cs.toronto.edu/ kriz/learning-features-2009-TR.pdf.   \nBaluta, T., Lamblin, P., Tarlow, D., Pedregosa, F., and Dziugaite, G. K. Unlearning in-vs. out-of-distribution data in LLMs under gradient-based method. arXiv:2411.04388, 2024.   \nBarbulescu, G.-O. and Triantafillou, P. To each (textual sequence) its own: Improving memorized-data unlearning in large language models. arXiv:2405.03097, 2024.   \nBourtoule, L., Chandrasekaran, V., Choquette-Choo, C. A., Jia, H., Travers, A., Zhang, B., Lie, D., and Papernot, N. Machine unlearning. In IEEE Symp. Security and Privacy $( S P )$ , pp. 141–159. IEEE, 2021.   \nCao, Y. and Yang, J. Towards making systems forget with machine unlearning. In IEEE Symposium on Security and Privacy, pp. 463–480. IEEE, 2015.   \nChaudhuri, K., Monteleoni, C., and Sarwate, A. D. Differentially private empirical risk minimization. Journal of Machine Learning Research, 12(3), 2011.   \nChien, E., Wang, H. P., Chen, Z., and Li, P. Langevin unlearning: A new perspective of noisy gradient descent for machine unlearning. In Advances in Neural Information Processing Systems, 2024. URL https: //openreview.net/forum?id $\\mid =$ 3LKuC8rbyV.   \nCooper, A. F., Choquette-Choo, C. A., Bogen, M., Jagielski, M., Filippova, K., Liu, K. Z., Chouldechova, A., Hayes, J., Huang, Y., Mireshghallah, N., et al. Machine unlearning doesn’t do what you think: Lessons for generative AI policy, research, and practice. arXiv:2412.06966, 2024.   \nCummings, R. and Durfee, D. Individual sensitivity preprocessing for data privacy. In ACM-SIAM Symposium on Discrete Algorithms, pp. 528–547. SIAM, 2020.   \nDeeb, A. and Roger, F. Do unlearning methods remove information from language model weights? arXiv:2410.08827, 2024.   \nDosovitskiy, A. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.   \nDwork, C., McSherry, F., Nissim, K., and Smith, A. Calibrating noise to sensitivity in private data analysis. In Theory of Cryptography: Third Theory of Cryptography Conference, pp. 265–284. Springer, 2006.   \nDwork, C., Roth, A., et al. The algorithmic foundations of differential privacy. Foundations and Trends in Theoretical Computer Science, 9(3–4):211–407, 2014.   \nEbadi, H., Sands, D., and Schneider, G. Differential privacy: Now it’s getting personal. ACM SIGPLAN Notices, 50 (1):69–81, 2015.   \nEntezari, R., Sedghi, H., Saukh, O., and Neyshabur, B. The role of permutation invariance in linear mode connectivity of neural networks. In International Conference on Learning Representations, 2022. URL https: //openreview.net/forum?id=dNigytemkL.   \nFan, C., Liu, J., Zhang, Y., Wong, E., Wei, D., and Liu, S. Salun: Empowering machine unlearning via gradientbased weight saliency in both image classification and generation. arXiv:2310.12508, 2023.   \nFan, C., Liu, J., Zhang, Y., Wong, E., Wei, D., and Liu, S. SalUn: Empowering machine unlearning via gradientbased weight saliency in both image classification and generation. In International Conference on Learning Representations, 2024. URL https://openreview.net/ forum?id=gn0mIhQGNM.   \nFan, C., Liu, J., Hero, A., and Liu, S. Challenging forgets: Unveiling the worst-case forget sets in machine unlearning. In European Conference on Computer Vision, pp. 278–297. Springer, 2025.   \nFeldman, V. and Zrnic, T. Individual privacy accounting via a renyi filter. ´ Advances in Neural Information Processing Systems, 34:28080–28091, 2021.   \nFeldman, V., Mironov, I., Talwar, K., and Thakurta, A. Privacy amplification by iteration. In IEEE Symp. Foundations of Computer Science, pp. 521–532. IEEE, 2018.   \nFort, S., Dziugaite, G. K., Paul, M., Kharaghani, S., Roy, D. M., and Ganguli, S. Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the neural tangent kernel. Advances in Neural Information Processing Systems, 33: 5850–5861, 2020.   \nFrankle, J., Dziugaite, G. K., Roy, D., and Carbin, M. Lin",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/1f29428a7f4dba131222a4978b1d3f0e97fcfd8b72247591a5ade0f01ee64657.jpg",
+        "image_caption": [
+            "Figure 4. We compared our estimates of the group privacy guarantees (y-axis) across forget sets determined by rankings of privacy losses $\\mathbf { \\widetilde { x } }$ -axis), and found the group privacy guarantees did not change. This was despite these forget sets leading to consistent differences in the number of steps to unlearn. We report the mean over 20 estimates of the group privacy values, and one standard deviation. We conclude the theory for group unlearning is currently not sharp enough to capture trends seen in practice. "
+        ],
+        "image_footnote": [],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A. Proofs ",
+        "text_level": 1,
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1. Corollary 4.1 ",
+        "text_level": 1,
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. By (Theorem 3.2, Chien et al., 2024), for all $\\alpha > 1$ , there exists a constant $C > 0$ such that ",
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/963572dc2df4f8ff50f011498363b5ddf5bcc760b8669c666b4d5c29da1af30e.jpg",
+        "text": "$$\nD _ { 2 \\alpha } ( \\rho _ { \\mathcal { D ^ { \\prime } } } ^ { k } ( \\nu _ { T , \\mathcal { D } } ) \\| \\nu _ { \\mathcal { D ^ { \\prime } } } ) \\leq e ^ { - \\frac { C k } { 2 \\alpha } } D _ { 2 \\alpha } ( \\nu _ { T , \\mathcal { D } } \\| \\nu _ { \\mathcal { D ^ { \\prime } } } ) ,\n$$",
+        "text_format": "latex",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { 2 \\alpha } ( \\nu _ { T , \\mathcal { D } } | | v _ { D ^ { \\prime } } )$ is the initial $2 \\alpha$ -Renyi divergence to stationarity on ´ $\\mathcal { D } ^ { \\prime }$ , after training on $\\mathcal { D }$ . By the weak triangle inequality (Proposition 11, Mironov, 2017), this is bounded by ",
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afaa61cef54c747a0d6436dd81639492ee8ffb42bf0bebb020acc30bde76f166.jpg",
+        "text": "$$\n\\frac { 2 \\alpha - 1 / 2 } { 2 \\alpha - 1 } D _ { 4 \\alpha } ( \\nu _ { T , \\mathcal { D } } \\| \\nu _ { T , \\mathcal { D ^ { \\prime } } } ) + D _ { 4 \\alpha - 1 } ( \\nu _ { T , \\mathcal { D ^ { \\prime } } } \\| \\nu _ { \\mathcal { D ^ { \\prime } } } ) \\leq \\frac { 2 \\alpha - 1 / 2 } { 2 \\alpha - 1 } \\varepsilon _ { 4 \\alpha } ^ { \\prime } + \\varepsilon _ { 4 \\alpha - 1 } ,\n$$",
+        "text_format": "latex",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where the second inequality follows from our hypotheses. ",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, applying the weak triangle inequality once more, $D _ { \\alpha } ( \\rho _ { D ^ { \\prime } } ^ { k } ( \\nu _ { T , D } ) \\Vert \\nu _ { T , D ^ { \\prime } } )$ is bounded by ",
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d60475ba5747989bf9bdb2f7e3c42974b4066f49bc6ad69e2173f6423020eba.jpg",
+        "text": "$$\n\\frac { \\alpha - 1 / 2 } { \\alpha - 1 } D _ { 2 \\alpha } ( \\rho _ { \\mathcal { D ^ { \\prime } } } ^ { k } ( \\nu _ { T , \\mathcal { D } } ) \\Vert \\nu _ { \\mathcal { D ^ { \\prime } } } ) + D _ { 2 \\alpha - 1 } ( \\nu _ { \\mathcal { D ^ { \\prime } } } \\Vert \\nu _ { T , \\mathcal { D ^ { \\prime } } } ) ,\n$$",
+        "text_format": "latex",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "which yields the claimed bound from our hypotheses, after substituting Equations (3) and (4). ",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B. Group Privacy Analysis and Methodology ",
+        "text_level": 1,
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The following theorem comes from the results of Thudi et al. (2024). ",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Theorem B.1. Suppose we train with noisy gradient descent for $T$ steps. Then for an arbitrary $\\mathcal { D } , \\mathcal { D } ^ { \\prime } = \\mathcal { D } \\setminus \\mathcal { D } _ { F }$ , we have: ",
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4afee4974d47ac2202d7b3962c86308e6f6ac842fbaad3bdc769b047314a340b.jpg",
+        "text": "$$\nD _ { \\alpha } ( \\nu _ { T , \\mathcal { D } } \\| \\nu _ { T , \\mathcal { D } ^ { \\prime } } ) \\leq \\sum _ { t = 1 } ^ { T } C _ { t , \\alpha } \\ln \\mathbb { E } _ { w \\sim \\nu _ { t , \\mathcal { D } } } G _ { t , \\alpha } ( \\mathcal { D } , \\mathcal { D } ^ { \\prime } , w ) .\n$$",
+        "text_format": "latex",
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where Ct,α = 1α−1 (p−1)t+1pt+1 and ",
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/1960511be62d24b4f193f22e125a101876974b6bf5e1a889ae929e809922d11f.jpg",
+        "image_caption": [
+            "Figure 5. Unlearning results for accuracy metrics (top) and MIA success rate (bottom). The $\\mathbf { X }$ -axis represents the number of epochs. In each plot, lines of different colors represent forget sets of varying difficulty, while the dashed line indicates the oracle’s performance. "
+        ],
+        "image_footnote": [],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b50a35ab23b83fc8fe02b67759ac497b129f5d83d033a999f771f9a99f6f2b4d.jpg",
+        "text": "$$\n\\ln G _ { t , \\alpha } ( \\mathcal { D } , \\mathcal { D } ^ { \\prime } , w ) = p \\mathbb { E } _ { \\mathcal { D } _ { B } ^ { \\prime } } \\ln \\mathbb { E } _ { \\mathcal { D } _ { \\mathbf { B } } ^ { \\mathbf { o f } } ( \\alpha ) } \\left( e ^ { \\frac { - 1 } { 2 \\sigma ^ { 2 } } \\Delta _ { o _ { p } ^ { t } ( \\alpha ) } \\left( \\mathcal { D } _ { \\mathbf { B } } ^ { \\mathbf { o f } } ( \\alpha ) , \\mathcal { D } _ { B } ^ { \\prime } , w \\right) } \\right)\n$$",
+        "text_format": "latex",
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D } _ { \\mathbf { B } } ^ { \\alpha } = \\{ \\mathcal { D } _ { B } ^ { 1 } , \\cdot \\cdot \\cdot , \\mathcal { D } _ { B } ^ { \\alpha } \\}$ is a random sample of α minibatches from $\\mathcal { D }$ , $\\mathcal { D } _ { B } ^ { \\prime }$ is a single random minibatch, $o _ { p } ( \\alpha ) =$ $\\textstyle { \\frac { p } { p - 1 } } \\alpha - { \\frac { 1 } { p } }$ and $O _ { p } ^ { t }$ is $o _ { p }$ composed $t$ times, and $p$ is a free parameter we set to $3 T$ following (Fact 3.4, Thudi et al., 2024), and letting $U ( \\mathcal { D } _ { B } , w ) = \\nabla _ { w } \\ell ( w , \\mathcal { D } _ { B } )$ be the mini-batch gradient: ",
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/78f8150b1995ca9e4f518fe5753023d2b7e0ebad4e0c175e33550f15a7869d68.jpg",
+        "text": "$$\n\\gamma _ { \\mathbf { B } } ^ { \\mathbf { a } ^ { \\alpha } } , \\mathcal { D } _ { B } ^ { \\prime } , w ) : = \\sum _ { i } | | U ( \\mathcal { D } _ { B } { } ^ { i } , w ) | | _ { 2 } ^ { 2 } - ( \\alpha - 1 ) | | U ( \\mathcal { D } _ { B } ^ { \\prime } , w ) | | _ { 2 } ^ { 2 } - | | \\sum _ { i } U ( \\mathcal { D } _ { B } { } ^ { i } , w ) - ( \\alpha - 1 ) U ( \\mathcal { D } _ { B } ^ { \\prime } , w ) | | _ { 2 } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. A direct consequence of applying (Theorem 3.3, Thudi et al., 2024) with the general update per-step divergence bound of (Theorem 3.6, Thudi et al., 2024), and noting the divergence after applying the projections is bounded by the divergence before applying the projections by the post-processing inequality. ",
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1. Methodology for privacy losses computation ",
+        "text_level": 1,
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To estimate the guarantees we take a Monte-Carlo sample of checkpoints from a single training run at steps $s _ { 0 } , s _ { 1 } , \\cdots , s _ { N }$ , and estimate the $\\ln G _ { t , \\alpha } ( D , { \\mathcal { D } } ^ { \\prime } , w )$ term at each step by sampling a single random mini-batch from $\\mathcal { D } ^ { \\prime }$ and $o _ { p } ^ { s _ { i } } ( \\alpha )$ minibatches from $\\mathcal { D }$ to estimate the expectations. We then compute the sum using the right hand rule, analogous to our estimate of privacy losses described in Section 5. ",
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Figure 4 we took checkpoints from an SGD training run on CIFAR10 with ResNet18, and used $\\sigma = 0 . 1$ , and took $\\alpha = 8$ to compute the group privacy scores. We report the mean over 20 estimates (given the stochasticity in our estimates for the per-step terms $\\ln G _ { t , \\alpha } ( \\mathcal { D } , \\mathcal { D } ^ { \\prime } , w ) )$ and shaded in one standard deviation. ",
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C. Example Difficulty Related Work ",
+        "text_level": 1,
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Paul et al. (2021) propose EL2N score to capture how much an example contributes to learning a high accuracy predictor, high score meaning high importance during training to achieve high accuracy. At the same time, the authors find that high scoring examples tend to be difficult to learn, are often memorized at the end of training and are outliers. This score has been shown to correlate with a number of other example difficulty and memorization metrics proposed in the literature (Kwok et al., 2024; Paul et al., 2021), some of which have been shown to also capture unlearning difficulty for a number of unlearning algorithms (Zhao et al., 2024; Baluta et al., 2024; Zhao & Triantafillou, 2024). ",
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/e586c20a5148948799e65556a836da70bd85cacc521ae169cd1717fac34b47da.jpg",
+        "image_caption": [
+            "Figure 6. Unlearning time for forget sets with different privacy loss values using the L1-sparse method. Time is measured by the number of steps required for the unlearning method to reach a $5 \\%$ margin of error, where error is defined as the difference between the unlearned model’s UA and the oracle’s UA for the given forget set. "
+        ],
+        "image_footnote": [],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D. Additional Experimental Details ",
+        "text_level": 1,
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Constructing Difficulty-Based Forget Sets To create forget sets with varying difficulty levels, the training dataset is partitioned into five subsets based on privacy scores. First, the samples are sorted in ascending order by their scores. Recursive splits are then performed to identify key thresholds: the lower quartile (Q1), the median (Q2), and the upper quartile (Q3). Using these thresholds, five subsets are constructed: (1) the first 1000 samples, (2) intervals centered around Q1 $\\mathrm { Q 1 } \\pm 5 0 0$ samples), (3) intervals centered around Q2 $\\mathrm { \\Delta } Q 2 \\pm 5 0 0$ samples), (4) intervals centered around Q3 $( Q 3 \\pm 5 0 0$ samples), and (5) the last 1000 samples. This approach provides a systematic stratification of the dataset, enabling the evaluation of unlearning performance across varying levels of difficulty as determined by privacy scores. ",
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Learning rates and training times for SGD The original model, which serves as the starting point for all unlearning techniques (not for SGLD), is trained for 150 epochs using an initial learning rate of 0.01, a weight decay of 0.0005, and a learning rate schedule that reduces the learning rate by an order of magnitude at epochs 80 and 120. Each unlearning method is subsequently fine-tuned for 25 epochs. ",
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Additional details for SGLD At every step we added $N ( 0 , \\sigma ^ { 2 } )$ Gaussian noise to the minibatch gradient, where we vary $\\sigma$ for ablations. All other hyperparameters were kept the same as SGD. In particular we do not do any additional gradient clipping. ",
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Additional details for L1-sparse L1-sparse is an unlearning method inspired by the observation that pruning aids unlearning (Liu et al., 2024a). Its objective function closely resembles that of fine-tuning but includes an additional $L _ { 1 }$ regularization term, weighted by a hyperparameter $\\alpha$ , which encourages sparsity in model parameters to facilitate unlearning. ",
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter tuning We perform hyperparameter tuning (HPT) for the unlearning methods using the Bayesian optimization method on a random forget set. While fine-tuning involves a single hyperparameter–the learning rate– L1- Sparse additionally optimizes $\\alpha$ . To determine the best hyperparameters for each method, we employ Bayesian optimization to find configurations that achieve an optimal balance between privacy and utility. Additionally, to ensure that the selected hyperparameters are also optimized with respect to the number of steps required for unlearning, we identify hyperparameter sets that fall within a $5 \\%$ margin of error for this trade-off. Among these, we select the configuration that converges the fastest. ",
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab4fc962bb7902acfa526248ac713c5fa7d03c4434bdda7f144d3c236740387a.jpg",
+        "image_caption": [
+            "Figure 7. SGD unlearning. Unlearning time vs. privacy score for ResNet-18 on SVHN (left) and ViT-small on CIFAR-10 (right) Unlearning time is measured in steps required to reach a $5 \\%$ UA error margin. "
+        ],
+        "image_footnote": [],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Compute resources Experiments were conducted using L40 and RTX8000 GPUs, and AMD EPYC 7452 CPUs. ",
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E. Evaluation metrics ",
+        "text_level": 1,
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Membership Inference Attack Membership inference attacks (MIA) aim to determine whether a given sample was part of a model’s training data by analyzing differences in the model’s responses. In our approach, we train a logistic regression classifier using the model’s confidence values on training and test samples as inputs. The attacker then attempts to classify the forget samples, with success measured by the percentage of forget samples labeled as test, indicating effective unlearning. ",
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Gaussian Unlearning Score We use the Gaussian Unlearning Score (GUS), introduced by Pawelczyk et al. (2024), to quantify the impact of poisoned samples on the model. To compute GUS, each sample in the forget set is perturbed with zero-mean Gaussian noise with a standard deviation of $\\sigma$ . GUS is then computed by averaging (over the forget set) the per-example inner product between the gradient of the loss with respect to the clean (non-poisoned) sample and the stored Gaussian noise used for poisoning, normalized by the L2 norm of the gradient. The effectiveness of an unlearning method is then assessed by how well it mitigates the influence of these poisoned samples. Specifically, the change in GUS before and after unlearning serves as a measure of unlearning success. ",
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For the CIFAR-10 and ResNet-18 setup, the original work recommends a variance value of 0.32. However, in our experiments, we explored different values and found that smaller variances were better suited for our setup. Based on these empirical findings, we set $\\sigma ^ { 2 } = 0 . 0 6 2$ . ",
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F. Additional Experimental Results ",
+        "text_level": 1,
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F.1. Unlearning trends during fine-tuning ",
+        "text_level": 1,
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we apply the fine-tuning method to unlearn a ResNet-18 model trained on the full CIFAR-10 dataset. The forget set varies in difficulty, with five different levels determined by the proxy losses of the samples. The UA, utility and RA values during unlearning is depicted in Figure 5 (top), while the MIA results are depicted in Figure 5 (bottom). For UA and MIA, we see the most difficult forget sets do indeed take longer to unlearn. We see utility and RA are similar across difficulty levels. ",
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F.2. L1-sparse fine-tuning ",
+        "text_level": 1,
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The unlearning results for the L1-sparse method are presented in Figure 6. Similar to our observations for unlearning with SGD or SGLD fine-tuning, unlearning with L1-sparse takes longer to forget samples with higher privacy scores. In fact, the more challenging the forget sets, the longer the unlearning. ",
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/85866e60b7ce4fb809a77d33e6f5c5f38e7122b5da0900a8f4c8829141b9382e.jpg",
+        "image_caption": [
+            "Figure 8. Unlearning time for forget sets of size 100 (left), 5,000 (middle), and 10,000 (right), evaluated on CIFAR-10. Unlearning time is reported as the number of training steps required to achieve a UA error within $5 \\%$ . "
+        ],
+        "image_footnote": [],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F.3. Additional datasets/architectures ",
+        "text_level": 1,
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In addition to ResNet-18 on CIFAR-10, we conduct experiments with additional dataset-architecture pairs. Specifically, we evaluate ResNet-18 on SVHN and ViT-small on CIFAR-10. ViT-small is a Vision Transformer model that applies self-attention mechanisms to sequences of image patches, enabling effective global feature extraction. Figure 7 presents the unlearning results for ResNet-18 on SVHN (left) and ViT-small on CIFAR-10 (right). The results suggest that, consistent with our previous findings, unlearning takes longer for forget sets with a higher average privacy loss. ",
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Rankings across noise levels We ran experiments to test sensitivity of the ranking for SGD to the noise values used in estimating privacy losses. For SVHN with ResNet-18 we found: (1) Spearman correlation between rankings at $\\sigma = 0 . 0 1$ and $\\sigma = 0 . 0 0 1$ was $0 . 7 0 ( p = 0 . 0 )$ . (2) Between $\\sigma = 0 . 0 0 1$ and $\\sigma = 0 . 0 0 0 5$ was $0 . 9 9 ( p = 0 . 0 )$ . (3) Between $\\sigma = 0 . 0 0 0 5$ and $\\sigma = 0 . 0 0 0 1$ was $0 . 9 9 ( p = 0 . 0 )$ . These results suggest that rankings are largely noise-invariant, as long as some noise is present. Past work has shown evidence of observable noise during training due to software and hardware nondeterminism (Jia et al., 2021). ",
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/50865345780c9bc437f982852cc515ddae8f1f88a10ccd7c66e5c96853b3c6e9.jpg",
+        "image_caption": [
+            "Figure 9. Qualitative examples of forgetting difficulty on CIFAR-10. We show 4 examples each from the easy-to-forget (top row) and hard-to-forget (bottom row) subsets, identified from a forget set of 1,000 samples. "
+        ],
+        "image_footnote": [],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F.4. Varying Forget Set Sizes ",
+        "text_level": 1,
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "We conducted additional experiments varying the size of the forget set (100, 5,000, and 10,000 samples). The results are presented in Figure 8. Our findings indicate that privacy loss consistently distinguishes between easy and hard-to-forget subsets, even when the forget set is small. Interestingly, for very large forget sets (e.g., 10,000 samples), the separation begins to diminish. This observation is intuitive, as the variance in average privacy loss decreases with increasing subset size. ",
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "F.5. Qualitative Results ",
+        "text_level": 1,
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Qualitative examples of easy and hard to forget CIFAR-10 samples are provided in Figure 9. ",
+        "page_idx": 16
+    }
+]

data/sampled_jsons/0ObGn4e1IS_Gumiho- A Hybrid Architecture to Prioritize Early Tokens in Speculative Decoding/auto/0ObGn4e1IS_Gumiho- A Hybrid Architecture to Prioritize Early Tokens in Speculative Decoding.md

@@ -0,0 +1,632 @@
+# Gumiho: A Hybrid Architecture to Prioritize Early Tokens in Speculative Decoding
+
+Jinze Li 1 2 Yixing $\mathbf { X } \mathbf { u } ^ { 1 }$ Haiduo Huang 1 3 Xuanwu Yin 1 Dong Li 1 Edith C.H. Ngai∗ 2 Emad Barsoum 1
+
+lijinze-hku@connect.hku.hk, huanghd@stu.xjtu.edu.cn, chngai@eee.hku.hk {yixing.xu, Xuanwu.Yin, d.li, emad.barsoum}@amd.com
+
+# Abstract
+
+# 1. Introduction
+
+Speculative decoding (SPD) aims to accelerate the auto-regressive token generation process of a target Large Language Model (LLM). Some approaches employ a draft model with multiple heads to predict a sequence of future tokens, where each head handles a token in the sequence. The target LLM verifies the predicted sequence and accepts aligned tokens, enabling efficient multi-token generation. However, existing methods assume that all tokens within a sequence are equally important, employing identical head structures and relying on a single-generation paradigm, either serial or parallel. To this end, we theoretically demonstrate that initial tokens in the draft sequence are more important than later ones. Building on this insight, we propose Gumiho, a hybrid model combining serial and parallel heads. Specifically, given the critical importance of early tokens, we employ a sophisticated Transformer architecture for the early draft heads in a serial configuration to improve accuracy. For later tokens, we utilize multiple lightweight MLP heads operating in parallel to enhance efficiency. By allocating more advanced model structures and longer running times to the early heads, Gumiho achieves improved overall performance. The experimental results demonstrate that our method outperforms existing approaches, fully validating its effectiveness. Our code is available at https://github.com/AMD-AIG-AIMA/Gumiho.
+
+While Large Language Models (LLMs) (Achiam et al., 2023; Touvron et al., 2023) have demonstrated impressive capabilities, their auto-regressive inference introduces significant latency challenges, especially as the number of parameters continues to increase. Speculative decoding (SPD) (Leviathan et al., 2023; Chen et al., 2023) has emerged as a promising solution to this problem. It leverages smaller models to efficiently propose draft tokens for future steps, which are then verified in parallel by the LLM. Specifically, in each draft round, the draft model generates a sequence of multiple draft tokens, and the target LLM verifies these tokens in parallel. This generation-verification process constitutes a draft round. These draft tokens are accepted only if they match the LLM’s original output. If a mismatch occurs, starting from the first divergent token, all subsequent draft tokens are discarded.
+
+Recent advances (Cai et al., 2024; Li et al., 2024a;b; Ankner et al., 2024) have shown that smaller models, which leverage the hidden states of the LLM itself, can achieve substantial speedups in inference while maintaining output quality. Medusa (Cai et al., 2024) predicts multiple future tokens in parallel using MLPs with the last verified token’s hidden state, while Hydra (Ankner et al., 2024) and Eagle (Li et al., 2024a) generate tokens serially.
+
+Although Medusa’s parallel prediction paradigm runs faster through simultaneous draft token predictions, it relies solely on hidden states from previously verified tokens, making it blind to earlier unverified predictions within the current draft round. Serial methods like Eagle, Eagle-2 and Hydra can fully utilize previously generated draft tokens, but their sequential paradigm tends to slow down the drafting process, making it less efficient. More critically, all approaches treat all tokens within a draft round as equally important, which is unsuitable for SPD. In our view, the tokens generated earlier in each draft round hold more importance than those generated later. This is because when the first incorrect token is encountered, it causes both that token and all subsequent draft tokens to be discarded, even if the following tokens are correct. We will formally present a theorem and mathematically prove that prioritizing the initial tokens of a sequence consistently improves the mean accepted tokens $( \tau )$ in each draft round.
+
+Motivated by this, we propose Gumiho, a hybrid model that prioritizes the initial positions in the draft sequence, combining both sequential and parallel architectures. We employ a serial structure comprising a two-layer Transformer, enabling comprehensive modeling of token dependencies and context for early tokens. The subsequent tokens, which are relatively less critical, are predicted in parallel through simple MLPs, thereby enhancing computational efficiency. By combining the serial and parallel architectures, our hybrid design achieves higher acceptance lengths and less processing time in each draft-verification cycle at the same time. The key idea of our hybrid model lies in two folds: (1) it allocates more parameters and leverages serial processing for crucial early token predictions, maximizing accuracy where it matters most, and (2) it employs efficient parallel computation with simple architecture for later tokens, reducing the overall computational cost.
+
+In addition, we enhance Eagle-2’s dynamic tree mechanism by introducing Full Tree Attention (FTA). While Eagle-2’s dynamic tree selectively expands promising nodes and reranks draft tokens for optimal verification, this re-ranking process can result in shorter candidate sequences when later tokens are discarded due to low scores, potentially reducing the mean accepted tokens $( \tau )$ . We observed that tokens generated by parallel heads exhibit correlations, as they share both input and purpose. Specifically, each $n$ -th head is designed to predict the token that should appear $n$ positions away from the input token. Inspired by this, our full tree attention mechanism supplements shorter paths with tokens from longer paths at corresponding positions, thereby increasing the mean accepted tokens $( \tau )$ of shorter candidate paths. Since these supplementary tokens come from existing longer candidates, their $q , k$ , and $v$ have already been computed, incurring no additional computational overhead.
+
+Our contributions are summarized as follows:
+
+• We propose Gumiho, a hybrid structure model for SPD, inspired by the observation that earlier tokens have more impact on the overall sequence length accepted, while later tokens are relatively less critical. We prioritize the generation of earlier tokens by allocating more computational resources and using a serial approach to enhance accuracy, while simpler models are employed in parallel for the later tokens to improve computational efficiency. • We demonstrate through theoretical analysis that tokens appearing earlier in the draft sequence have a more significant impact on the overall accepted length. • We propose a full tree attention mechanism for tree candidates, allowing tokens from longer candidates to augment shorter ones. This approach further increases the acceptance length without incurring additional computational overhead.
+
+• We conduct comprehensive experiments to demonstrate Gumiho’s superior performance compared to existing methods.
+
+# 2. Related Works
+
+With the widespread adoption of large language models (LLMs), significant research has been devoted to accelerating their inference through techniques such as distillation (Hinton, 2015; Bercovich et al., 2024; Zhao et al., 2024; Fu et al., 2024), low-bit quantization (Hubara et al., 2018; Shen et al., 2020; Kim et al., 2021; Zadeh et al., 2020; Zafrir et al., 2019), pruning (Gale et al., 2019; Sanh et al., 2020; Kurtic et al., 2022; Voita et al., 2019), and innovative network architecture designs (Gu & Dao, 2023; Wu et al., 2020). These approaches aim to reduce the computational cost of each forward pass to improve efficiency. However, they often involve a trade-off (Donisch et al., 2024), as these optimizations can partially compromise model performance, requiring a balance between generation quality and computational overhead. Speculative decoding, a draft-then-verify paradigm (Xia et al., 2023), achieves lossless acceleration by leveraging the original LLM for verification.
+
+Drafting approaches can be broadly categorized into two paradigms (Xia et al., 2024): independent drafting which employs draft models that can be deployed without additional training, and self-drafting which requires dedicated training processes to develop effective draft models.
+
+Independent drafting typically uses a separate, smaller model to generate multiple future tokens concurrently, thereby enhancing the efficiency of speculative decoding. For example, using T5-small to accelerate T5- XXL (Leviathan et al., 2023). These off-the-shelf drafters do not need extra training or architectural modifications and benefit from the inherent alignment in prediction behaviors due to shared tokenizers and pretraining processes. However, independent drafting requires additional work to find or train a compatible model that matches the target LLM. This becomes more challenging when smaller versions of the LLM don’t exist.
+
+Orthogonal to independent drafting, self-drafting typically uses the target LLM itself (Liu et al., 2024a; Du et al., 2024; Elhoushi et al., 2024; Gloeckle et al., 2024; Cai et al., 2024; Li et al., 2024a; Zimmer et al., 2024; Xiao et al., 2024; Zhang et al., 2024; Brown et al., 2024; Liu et al., 2024b), utilizing features like its hidden states for more efficient drafting. Medusa (Cai et al., 2024) is one of the studies to leverage the hidden state of the original LLM as input for draft models. It employs $K$ distinct MLPs as draft models, each predicting one of $K$ future tokens. Since these $K$ draft models operate independently, they can execute in parallel, enabling Medusa to generate $K$ tokens in a single forward pass. Hydra (Ankner et al., 2024) is a sequential variant of Medusa, transforming the parallel MLPs into a serial architecture. It feeds the unverified tokens from draft models as input to subsequent ones. This sequential approach enables each draft model to leverage more contextual information when predicting later tokens, enhancing the quality of successive predictions. Eagle (Li et al., 2024a) advances the architecture by converting the serial MLPs into serial Transformers and introducing concatenated token-hidden state pairs as input. Eagle-2 (Li et al., 2024b) further innovates by implementing a dynamic tree candidate selection mechanism to enhance token prediction efficiency.
+
+Our method falls within the self-drafting category. Different from previous self-drafting methods, we utilize the fact that the importance of a token decreases as its position moves back and propose different architectures for front-positioned tokens and later ones.
+
+# 3. Method
+
+In this section, we begin by introducing the preliminaries of speculative decoding (SPD). We then present a theorem and provide a rigorous mathematical proof to validate that tokens at the beginning of the sequence are more crucial than those at the end. Finally, we propose Gumiho, a novel method derived from our theorem.
+
+# 3.1. Preliminaries
+
+LLM Decoding The process of generating text from LLMs is termed decoding: the sequential production of tokens in response to an input prompt. This generation process follows an auto-regressive pattern, where each token $y _ { t }$ is generated by sampling from a probability distribution conditioned on both the initial prompt $z$ and all previously generated tokens $y _ { < t }$ . In practice, the key-value cache $\left( k v _ { < t } \right)$ of previously generated tokens is maintained, with the model taking both $k v _ { < t }$ and the current token $y _ { t }$ as inputs for efficient generation.
+
+Let $\mathcal { M } _ { \mathrm { L } }$ denote the Large Language Model, which comprises two components: the decoder layer $f _ { \mathrm { L } } ( \cdot )$ and LM head that maps the embeddings back to the vocabulary with size $| \nu |$ . The vanilla decoding process can be formulated as:
+
+$$
+\begin{array} { r l } & { k v _ { < t + 1 } , h _ { t + 1 } = f _ { \mathrm { L } } ( k v _ { < t } , e ( y _ { t } ) ) , } \\ & { y _ { t + 1 } \sim \mathrm { S o f t m a x } ( L M _ { \mathrm { - } h e a d ( h _ { t + 1 } ) } ) , } \end{array}
+$$
+
+where $h _ { t + 1 }$ denotes the hidden states of the final decoder layer, and $e ( \cdot )$ is an embedding function that maps tokens to their corresponding vector representations. In the following, we define $y _ { t + 1 } = \mathcal { M } _ { \mathrm { L } } ( y _ { t } ) \triangleq \mathcal { M } _ { \mathrm { L } } ( y _ { t } , k v _ { < t } )$ and ignore $k v _ { < t }$ for simplicity.
+
+Speculative Decoding Auto-regressive text generation in LLMs is time-consuming. Speculative decoding addresses this limitation by employing a smaller, faster draft model $\mathcal { M } _ { \mathrm { S } }$ to generate candidate tokens ahead of time. These candidate tokens, commonly referred to as drafts, are then verified in parallel by the target LLM $\mathcal { M } _ { \mathrm { L } }$ using rejection sampling (Leviathan et al., 2023), i.e., if any token in the draft sequence is rejected, all subsequent tokens are discarded, and the draft-verification process resumes from the last accepted token.
+
+During each draft-verification iteration, $\mathcal { M } _ { \mathrm { S } }$ generates a sequence of $D$ draft tokens $\{ \hat { y } _ { t + i } \} _ { i = 1 } ^ { D }$ . Subsequently, $\mathcal { M } _ { \mathrm { L } }$ verifies these drafts in parallel according to Eq.(1):
+
+$$
+\begin{array} { r l } & { y _ { t + 1 } \sim \mathcal { M } _ { \mathrm { L } } ( y _ { t } ) , } \\ & { y _ { t + 2 } \sim \mathcal { M } _ { \mathrm { L } } ( \hat { y } _ { t + 1 } ) , } \\ & { ~ \vdots } \\ & { ~ y _ { t + D } \sim \mathcal { M } _ { \mathrm { L } } ( \hat { y } _ { t + D - 1 } ) . } \end{array}
+$$
+
+The number of accepted draft tokens for each iteration is determined by comparing the draft sequence $\{ \hat { y } _ { t + 1 } , . . . , \hat { y } _ { t + D } \}$ with the verified sequence $\{ y _ { t + 1 } , . . . , y _ { t + D } \}$ using rejection sampling.
+
+Metrics We assess the method’s performance by measuring its acceleration effect. Specifically, we use the speedup ratio as a metric, which is calculated by dividing the speed of our proposed method by the speed of standard (vanilla) decoding:
+
+$$
+\mathrm { S p e e d u p r a t i o } = \frac { \mathrm { s p e e d } _ { \mathrm { G u m i h o } } } { \mathrm { s p e e d } _ { \mathrm { v a n i l l a } } }
+$$
+
+The speed of each method is calculated by dividing the total number of generated tokens by the total processing time:
+
+$$
+{ \begin{array} { r l } & { { \mathrm { s p e e d } } = { \frac { \mathrm { t o t a l ~ t o k e n s } } { \mathrm { t o t a l ~ t i m e } } } } \\ & { \qquad = { \frac { { \mathrm { m e a n ~ a c c e p t e d ~ t o k e n s } } \times { \mathrm { d r a f t ~ r o u n d s } } } { \mathrm { a v e r a g e ~ t i m e } \times { \mathrm { d r a f t ~ r o u n d s } } } } } \\ & { \qquad = { \frac { { \mathrm { m e a n ~ a c c e p t e d ~ t o k e n s } } } { \mathrm { a v e r a g e ~ t i m e } } } } \end{array} }
+$$
+
+Following existing works (Li et al., 2024b; Ankner et al., 2024), we primarily use mean accepted tokens $( \tau )$ as the main metric. We also present the differences in draft time across different methods to demonstrate the effectiveness of our approach in improving efficiency.
+
+# 3.2. Theoretical Analysis
+
+In this section, we prove that tokens at the beginning of the draft sequence are more crucial than those at the end.
+
+![](images/025d498f0239206759607359d538d41ff6032d8c9efe513fa13815ac4daf6e31.jpg)  
+Figure 1. Left: Differences between our proposed Gumiho and existing methods: Unlike existing approaches that use similar models to predict every token in a sequence, we propose that initial tokens are more critical than later ones. So we employ a larger model with a serial structure to generate the early tokens, while leveraging smaller parallel models for the later ones. Right: Overview of Gumiho. Given an LLM input Describe a beautiful scene., Gumiho predicts the next 7 draft tokens (sun rose above the mountains through the). The first two tokens (sun and rose) are deemed critical and are produced sequentially using the Transformer $\mathcal { M } _ { \mathrm { T } }$ for higher accuracy. The remaining tokens are generated simultaneously through the MLP heads, optimizing for computational efficiency.
+
+Consider a draft model that predicts 3 tokens at a time with a uniform acceptance probability of 0.8 at each position. The expected length $\mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$ of accepted tokens per draft round is:
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } = 1 \times P ( L = 1 ) + 2 \times P ( L = 2 ) + 3 \times P ( L = 3 ) } \\ { \displaystyle \qquad = \sum _ { i = 1 } ^ { 3 } P ( L \geq i ) } \\ { \displaystyle \qquad = 0 . 8 + 0 . 8 ^ { 2 } + 0 . 8 ^ { 3 } = 1 . 9 5 , } \end{array}
+$$
+
+where $L$ represents the accepted length determined by the target LLM’s verification after each draft round. $\mathbb { E } [ \cdot ]$ denotes the expectation operator, and $P ( L = i )$ represents the probability that the accepted length $L$ is equal to $i$ .
+
+Now, suppose we redistribute the model’s parameters to prioritize earlier positions, i.e., allocating more parameters to predict the first position and fewer for subsequent positions. Assume this improved structure creates positiondependent acceptance probabilities of 0.85, 0.8, and 0.75 for the first, second, and third positions respectively. The expected length $\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } }$ becomes:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } = \sum _ { i = 1 } ^ { 3 } P ( L \geq i )
+$$
+
+This example empirically demonstrates that when the overall token accuracy remains constant, improving the accuracy of the initial token can increase the mean accepted tokens $( \tau )$ .
+
+In the following, we provide a theorem to generalize the above example to a broader scenario. Given a draft sequence with length $D$ , we first have an original setting with acceptance probabilities $\{ p _ { i } \} _ { i = 1 } ^ { D }$ and denote $\mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$ as its mean accepted tokens $( \tau )$ . Since errors accumulate when predicting a sequence, the acceptance probability of later tokens tends to be lower than that of earlier tokens in practical scenarios. Based on this observation, we have:
+
+$$
+1 \geq p _ { 1 } \geq p _ { 2 } \geq \cdot \cdot \cdot \geq p _ { D } \geq 0 .
+$$
+
+Then, we define an improved setting whose sequence is separated by index $d$ with $1 < d \bar { < } D$ . In this setting, acceptance probabilities ${ \tilde { p } } _ { i }$ are modified as follows:
+
+$$
+\begin{array} { r l } & { \tilde { p } _ { i } = \left\{ { p } _ { i } + { \zeta } _ { i } , 1 \leq i \leq d \atop p _ { i } - { \zeta } _ { i } , d < i \leq D \right.} ,   \\ & { \left. s . t . 0 \leq \{ { \zeta } _ { i } \} _ { i = 1 } ^ { D } \leq 1 , 0 \leq \{ { \tilde { p } } _ { i } \} _ { i = 1 } ^ { D } \leq 1 , \right. } \\ & { \left. \sum _ { i = 1 } ^ { d } { \zeta } _ { i } = \displaystyle \sum _ { j = d + 1 } ^ { D } { \zeta } _ { j } , \zeta _ { i } < p _ { i } . \right. } \end{array}
+$$
+
+In this improved setting, we increase the acceptance probabilities for the first $d$ tokens by a small amount $\zeta _ { i }$ and decrease those for the remaining tokens by the same total amount. In this way, the sum of the acceptance probabilities remains unchanged. We denote the mean accepted tokens $( \tau )$ in this improved setting as $\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } }$ . With these definitions above, we can derive the following theorem:
+
+Theorem 3.1. The mean accepted tokens $( \tau )$ under the improved probability distribution exceeds that of the original distribution:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } .
+$$
+
+This theorem shows that redistributing the acceptance probabilities across sequence tokens by increasing the accuracies of the initial tokens and decreasing those of the later tokens can improve the overall expected performance. A detailed proof of this theorem is provided in Appendix A.
+
+# 3.3. Gumiho
+
+Inspired by Theorem 3.1, we propose Gumiho, a model that prioritizes the front part of the draft sequence. Gumiho consists of two main components: heads for generating draft tokens and a full tree attention mechanism for verification.
+
+Gumiho Heads. As shown in Fig. 1, Gumiho introduces a hybrid architecture that distinguishes itself from existing methods. Unlike approaches that rely solely on a single serial or parallel structure and employ uniform head size across all positions, Gumiho combines large serial heads with small parallel heads to enhance accuracy and efficiency.
+
+The serial component aims to increase the accuracy of the initial tokens and comprises a two-layer Transformer $\mathcal { M } _ { \mathrm { T } }$ that predicts initial draft tokens sequentially. Specifically, $\mathcal { M } _ { \mathrm { T } }$ generates the first two tokens of the draft sequence autoregressively. Similar to Eagle-2, our method concatenates the hidden state $h _ { t } \in \mathbb { R } ^ { d }$ with the corresponding output token embedding $e ( y _ { t } ) \in \mathbb { R } ^ { d }$ generated by LLM $\mathcal { M } _ { L }$ at time step $t$ , and employs a fully connected layer FC to reduce the dimension from $2 d$ to $d$ :
+
+$$
+o _ { t } = \mathrm { F C } ( \operatorname { c a t } ( e ( y _ { t } ) , h _ { t } ) ) ,
+$$
+
+where cat denotes the concatenation operation, and $o _ { t } \in$ $\mathbb { R } ^ { d }$ . Then, the concatenated result $o _ { t }$ is fed into the serial component $\mathcal { M } _ { \mathrm { T } }$ , which sequentially generates the first two drafts:
+
+$$
+\begin{array} { r } { \hat { h } _ { t + 1 } = \mathcal { M } _ { \mathrm { T } } ( o _ { t } ) , \hat { o } _ { t + 1 } = \mathrm { F C } ( \cot ( e ( \hat { y } _ { t + 1 } ) , \hat { h } _ { t + 1 } ) ) , } \\ { \hat { h } _ { t + 2 } = \mathcal { M } _ { \mathrm { T } } ( \hat { o } _ { t + 1 } ) , \hat { o } _ { t + 2 } = \mathrm { F C } ( \cot ( e ( \hat { y } _ { t + 2 } ) , \hat { h } _ { t + 2 } ) ) . } \end{array}
+$$
+
+For simplicity, we omit the input and output of KV cache in $\mathcal { M } _ { \mathrm { T } }$ , and also the step of using Softmax to obtain $\hat { y } _ { t + 1 }$ , $\hat { y } _ { t + 2 }$ , which is similar to Eq. (1).
+
+The parallel component aims to speed up the generation of the remaining tokens whileconsists of five different MLPs $\{ \mathcal { M } _ { \mathrm { M } } ^ { i } \} _ { i = 1 } ^ { 5 }$ ng accuracy andrunning concurrently. These MLPs share the same architecture, consisting of two fully connected (FC) layers with a ReLU activation function in between. They also share the same input, i.e., $\cot ( \hat { o } _ { t + 1 } , \hat { o } _ { t + 2 } )$ which concatenate the two outputs generated by the serial model $\mathcal { M } _ { \mathrm { T } }$ . The outputs of MLPs repre sent the draft tokens at the following five positions:
+
+![](images/bc8a5c01c7f5dc8d8478a03e4baeca9ea2ef0eab2d75f7193c875d22dc578e6e.jpg)
+
+Select top 8 tokens as candidates by score   
+Figure 2. Our proposed Full Tree Attention enhances shorter candidate paths by borrowing tokens from other tree nodes, thereby increasing the likelihood that candidates achieve longer acceptance lengths. Note that for each depth in the tree, we only have $s = 3$ different tokens.   
+
+<table><tr><td colspan="4">Tree Attention</td><td colspan="4">Full Tree Attention</td></tr><tr><td>The sun</td><td>shines</td><td> warmly</td><td rowspan="6">now</td><td rowspan="2">The sun</td><td rowspan="2"> shines</td><td rowspan="2">warmly</td><td rowspan="2">now</td></tr><tr><td></td><td>shines</td><td></td></tr><tr><td>The sun</td><td></td><td>brightly</td><td>The sun</td><td>shines</td><td>brightly</td><td>now. now.</td></tr><tr><td>The sun</td><td>shines</td><td>intensely</td><td>The sun The sun</td><td> shines</td><td>intensely</td><td>.now.</td></tr><tr><td>The sun</td><td>glows</td><td> warmly</td><td></td><td>glows</td><td> warmly</td><td>now</td></tr><tr><td>The sun</td><td>radiates</td><td></td><td>The sun</td><td>radiates</td><td>warmly</td><td>-- --.</td></tr></table>
+
+$$
+\hat { h } _ { t + 2 + i } = \mathcal { M } _ { \mathrm { M } } ^ { i } ( c a t ( \hat { o } _ { t + 1 } , \hat { o } _ { t + 2 } ) ) , i = 1 , . . . , 5 .
+$$
+
+Given the hidden states $\{ \hat { h } _ { t + i } \} _ { i = 1 } ^ { 7 }$ , we obtain the draft tokens $\{ \hat { y } _ { t + i } \} _ { i = 1 } ^ { 7 }$ using Eq. (1).
+
+Full Tree Attention (FTA). A key distinction between our Gumiho and Eagle-2 lies in the use of parallel heads for generating subsequent tokens. To fully leverage this parallel paradigm, we introduce the full tree attention mechanism, which enhances the existing Tree Attention mechanism employed by Eagle-2.
+
+In Eagle-2, tokens at each position are generated in an autoregressive manner, meaning that each subsequent token is entirely dependent on the tokens generated in the previous. Conversely, our parallel heads $\mathcal { M } _ { \mathrm { M } } ^ { i }$ generate tokens for all positions simultaneously. This parallel generation paradigm removes dependencies between these tokens, as they are determined solely by the outputs of the preceding serial outputs generated by $\mathcal { M } _ { \mathrm { T } }$ . The independence between tokens enables us to perform a full traversal connection operation on the tokens generated in parallel. Specifically, any two tokens generated by two different $\mathcal { M } _ { \mathrm { M } } ^ { i }$ can be connected to form a candidate path. As illustrated in Fig. 2, after the serial heads output the tokens the and sun, the three parallel heads generate $s$ subsequent tokens for each position $s = 3$ in Fig. 2). These $s$ tokens at each position can be arbitrarily combined with tokens from other positions, resulting in a total of $s ^ { 3 }$ candidate paths with only $3 s$ different tokens by the time we complete the third MLP head. In Fig. 2, the score for each token is displayed, calculated by multiplying the score of the preceding token with the confidence of the current MLP in generating specific tokens.
+
+To select candidate paths for verification, we choose the top eight tokens with the highest scores. As shown in Fig. 2, traditional tree attention often results in some candidate paths being very short because tokens on early positions usually have higher scores. To address this issue, our proposed FTA mechanism supplements shorter paths with tokens from corresponding positions in longer paths. This is reasonable since any two tokens from different positions among the parallel-generated tokens can be combined, and the borrowed tokens from longer candidate paths do not conflict with the original tokens in shorter paths. This ensures the coherence of the final candidate paths, maintaining the integrity of the generated sequence. Consequently, this approach increases the average length of candidate paths, enhancing the overall performance of the model.
+
+It is worth noting that FTA incurs no additional computational overhead, as the borrowed tokens already exist in other candidate paths within the tree, with their query, key, and value computations already completed. In the original attention tree, these tokens were simply discarded, while our approach unblocks them and allows shorter candidates to access and utilize them. The rejection sampling method (Leviathan et al., 2023) for token selection ensures that the appending of borrowed tokens does not impact previously selected tokens. Consequently, after applying FTA, each candidate’s acceptance length equals or exceeds that of the original implementation.
+
+# 4. Experiments
+
+In this section, we compare our proposed Gumiho with other SOTA methods to show the priority of our approach. Then, we conduct several ablation studies to validate the effectiveness of each part of our method.
+
+# 4.1. Experimental Setup
+
+We conduct experiments using seven target LLMs: Vicuna-7B/13B (Chiang et al., 2023), Llama2-chat7B/13B/70B (Touvron et al., 2023), and Llama3-instruct8B/70B (Meta, 2024). Target LLMs are fixed during training, with only the draft heads being trained. Following Eagle and Eagle-2 (Li et al., 2024a;b), we train our draft model on the ShareGPT dataset. Our Gumiho model comprises a Transformer model and five MLPs to predict the next seven draft tokens: the Transformer autoregressively generates the first two tokens, and the remaining five are predicted in parallel by the MLPs. Training details and hyperparameters can be found in Appendix C.
+
+We evaluate the performance across multiple benchmarks: MT-Bench (Zheng et al., 2023) for multi-turn dialogue, HumanEval (Chen et al., 2021) for code generation,
+
+![](images/022dea3fa0fcd0b5aca2e3e9b2796370af982be76b33c03e0686914687784623.jpg)  
+Figure 3. Comparison of average draft time (lower is better) with different temperatures. Both results are based on Vicuna 7B.
+
+GSM8K (Cobbe et al., 2021) for mathematical reasoning, Alpaca (Taori et al., 2023) for general instruction-following, CNN/Daily Mail (Nallapati et al., 2016) for summarization, and Natural Questions (Kwiatkowski et al., 2019) for question answering. We conduct model training using ${ 8 \times \mathbf { A M D } }$ Instinct MI250 GPUs. For evaluation, we use a single MI250 GPU for all models except the 70B variant, which requires $4 { \times } \mathbf { M } \mathbf { I } 2 5 0$ GPUs due to its larger size. Additionally, we include evaluation results using a single NVIDIA A100 GPU in Appendix B.
+
+We compare our method against several existing approaches: Medusa (Cai et al., 2024) with multiple parallel MLP heads, Hydra (Ankner et al., 2024) with sequential MLP heads, Eagle (Li et al., 2024a) and Eagle-2 (Li et al., 2024b) with sequential single-layer Transformer head. Eagle-2 shares the same model parameters as Eagle but distinguishes itself by incorporating a dynamic tree to generate candidates.
+
+In line with Eagle-2 (Li et al., 2024b), we also conduct experiments with temperature settings of 0 and 1. A temperature of 0 means that the target LLM uses a greedy sampling method, where the token with the highest probability is selected at each position. In contrast, a temperature of 1 increases the diversity of the output by applying postprocessing to the logits at the current position, rather than directly selecting the token with the highest probability. When the temperature is set to 1, Eagle-2 excludes methods like Medusa since their relaxed acceptance criteria under non-greedy sampling do not ensure lossless acceleration. We follow their experimental settings in this work.
+
+# 4.2. Performance Comparison
+
+Experimental results are shown in Tab. 1. The Speedup metric quantifies the actual end-to-end acceleration ratio of token generation speed compared to vanilla auto-regressive generation, while $\tau$ is the average number of tokens accepted by the target LLM per draft round after verification.
+
+Table 1. Speedup ratios and mean accepted tokens $( \tau )$ of different methods. V represents Vicuna, L2 represents LLaMA2-Chat, and L3 represents LLaMA3-Instruct. We present the results of different methods across six datasets. Mean represents the average performance across these six datasets.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="2">Sped-Bench</td><td colspan="2">SpeemanEval</td><td colspan="2">SpeG$M8K </td><td colspan="2"></td><td colspan="2">SpCNN/DM</td><td colspan="2">SNatral Ques.</td><td colspan="2">Speeduean</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td>Speelpaca </td><td></td><td></td><td></td><td></td><td></td><td></td><td>T</td></tr><tr><td rowspan="5">V 7B</td><td>Medusa</td><td></td><td></td><td></td><td></td><td>Temperature=0</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>1.96</td><td>2.50</td><td>2.15</td><td>2.69 2.01</td><td>2.59</td><td>1.94</td><td>2.48</td><td>1.60</td><td>2.02</td><td>1.68</td><td>2.05</td><td>1.89</td><td>2.39</td></tr><tr><td>Hydra</td><td>2.47</td><td>3.59</td><td>2.65 </td><td>3.78</td><td>2.49x 3.67</td><td>2.44</td><td>3.58</td><td>1.92x</td><td>2.70</td><td>2.01x</td><td>2.86</td><td>2.33 </td><td>3.36</td></tr><tr><td>Eagle</td><td>2.61 x</td><td>3.82</td><td>2.96</td><td>4.20 2.67 x</td><td>4.00</td><td>2.41x</td><td>3.66</td><td>2.35</td><td>3.34</td><td>2.10</td><td>3.13</td><td>2.52</td><td>3.69</td></tr><tr><td>Eagle-2</td><td>2.88</td><td>5.00</td><td>3.27</td><td>5.35</td><td>2.93 4.94</td><td>2.71</td><td>4.85</td><td>2.45</td><td>4.11</td><td>2.24</td><td>3.84</td><td>2.74</td><td>4.68</td></tr><tr><td rowspan="4">V 13B</td><td>Gumiho(ours)</td><td>3.15</td><td>5.29</td><td>3.65</td><td>5.77</td><td>3.10 5.06</td><td>2.83 </td><td>4.87</td><td>2.73</td><td>4.48</td><td>2.34</td><td>3.88</td><td>2.97</td><td>4.89</td></tr><tr><td>Medusa</td><td>2.03</td><td>2.58</td><td>2.24</td><td>2.77</td><td>2.08 2.64</td><td>2.04</td><td>2.44</td><td>1.67x</td><td>2.10</td><td>1.70</td><td>2.10</td><td>1.96</td><td>2.44</td></tr><tr><td>Hydra</td><td>2.65 </td><td>3.65</td><td>2.88</td><td>3.86</td><td>2.69 3.67</td><td>2.65 </td><td>3.49</td><td>2.08</td><td>2.82</td><td>2.16x</td><td>2.86</td><td>2.52</td><td>3.39</td></tr><tr><td>Eagle</td><td>2.87</td><td>3.90</td><td>3.25</td><td>4.29 2.88x</td><td>3.90</td><td>2.64 </td><td>3.50</td><td>2.58</td><td>3.49</td><td>2.21x</td><td>2.92</td><td>2.74</td><td>3.66</td></tr><tr><td rowspan="4">L2 7B</td><td>Eagle-2</td><td>3.16</td><td>4.93</td><td>3.68</td><td>5.42</td><td>3.19 4.82</td><td>3.01 </td><td>4.89</td><td>2.79 x</td><td>4.27</td><td>2.41 </td><td>3.69</td><td>3.04</td><td>4.67</td></tr><tr><td>Gumiho(ours)</td><td>3.36</td><td>5.16</td><td>4.11</td><td>5.97</td><td>3.39 5.04</td><td>3.07</td><td>4.88</td><td>2.91</td><td>4.41</td><td>2.52 </td><td>3.76</td><td>3.23 </td><td>4.87</td></tr><tr><td>Eagle</td><td>2.22</td><td>3.00</td><td>2.53</td><td>3.58</td><td>2.21 x 3.09</td><td>2.04</td><td>2.88</td><td>2.08</td><td>2.78</td><td>1.88</td><td>2.64</td><td></td><td>3.00</td></tr><tr><td>Eagle-2</td><td>2.91 </td><td>4.76</td><td>3.30</td><td>5.38 2.87</td><td>4.76</td><td>2.81</td><td>4.65</td><td>2.53 x</td><td>4.10</td><td>2.52</td><td>4.16</td><td>2.16x 2.82</td><td>4.64</td></tr><tr><td rowspan="3">L2 13B</td><td>Gumiho(ours)</td><td>3.07</td><td>4.90</td><td>3.55 </td><td>5.60</td><td>3.00 4.81</td><td>2.85 </td><td>4.55</td><td>2.66 </td><td>4.18</td><td>2.59</td><td>4.16</td><td>2.95</td><td>4.70</td></tr><tr><td>Eagle</td><td>2.59 3.17</td><td>3.30</td><td>2.96</td><td>3.90</td><td>2.61 3.45</td><td>2.41</td><td>3.16</td><td>2.39</td><td>3.09</td><td>2.15</td><td></td><td>2.82</td><td>3.29</td></tr><tr><td>Eagle-2</td><td></td><td>4.76 3.78</td><td>5.53</td><td>3.23</td><td>4.88</td><td>3.03 </td><td>4.62</td><td>2.84 </td><td>4.27</td><td>2.76</td><td>4.12</td><td>2.52 3.13</td><td>4.70</td></tr><tr><td rowspan="3">L2 70B</td><td>Gumiho(ours)</td><td>3.34</td><td>4.98</td><td>4.05</td><td>5.87</td><td>3.35 5.02</td><td>3.12</td><td>4.66</td><td>2.93 </td><td>4.40</td><td>2.84</td><td>4.20</td><td>3.27</td><td>4.85</td></tr><tr><td></td><td>2.83</td><td>4:97</td><td>3.38</td><td>5.43</td><td>2.03</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gumagle(ours)</td><td></td><td></td><td></td><td></td><td>4:63</td><td>3.48</td><td>4:46</td><td>32.34</td><td>3:68</td><td>3.14x</td><td>3.88</td><td>3:46</td><td>4:30</td></tr><tr><td rowspan="3">L3 8B L3 70B</td><td></td><td>2.38</td><td>4:48</td><td></td><td></td><td>4.43</td><td>3.34</td><td>4.88</td><td>2.82</td><td>3.84</td><td>1:73</td><td>3.54</td><td></td><td></td></tr><tr><td>Gumainle-ours)</td><td></td><td></td><td>2.91</td><td>5.08</td><td>3.33</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>4.46</td></tr><tr><td>Gumanle(ours)</td><td>3.9</td><td>4:27</td><td>3.68</td><td>5:9% 3.17</td><td>4:38</td><td>3.48</td><td>4:38</td><td>2.54</td><td>3.88</td><td>3.89</td><td>3.59</td><td>3.32 3.98</td><td>4:35</td></tr><tr><td colspan="14"></td></tr><tr><td rowspan="3">V 7B</td><td></td><td></td><td></td><td></td><td></td><td>Temperature=1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>2.1</td><td>4:32</td><td></td><td>4:62</td><td>2.48</td><td>4:33</td><td>3.38</td><td>4:37</td><td></td><td></td><td></td><td></td><td>4:.13</td></tr><tr><td>Gumagle-ours)</td><td></td><td></td><td>3.84x</td><td></td><td></td><td></td><td></td><td>2.138x</td><td>3:39</td><td>3.%</td><td>3:30</td><td>3.37 x</td><td></td></tr><tr><td rowspan="2">V 13B</td><td>Gumagle(o2urs)</td><td>2.83 x</td><td>4:34</td><td>3.33 </td><td>$.36</td><td>2.8 4.43</td><td>3.99</td><td>4.34</td></table>
+
+Across diverse target LLMs, model sizes, and temperature settings, Gumiho demonstrates superior performance. Overall, Gumiho surpasses the existing SOTA method EAGLE-2 by $4 . 5 \% \sim 1 5 . 8 \%$ . The performance gains are particularly pronounced with 70B model variants. At temperature 0, Gumiho outperforms EAGLE-2 by $1 1 . 7 \%$ on LLaMA2 70B and $1 5 . 8 \%$ on LLaMA3 70B. This substantial improvement is primarily attributed to enhancements in $\tau$ and a reduction in draft time. Specifically, the output hidden state for 70B models has a dimension of 8192, compared to 4096 in the 7B and 13B models. While the larger hidden state increases computational complexity, it also amplifies the benefits of our model parallelization, significantly reducing drafting time and further boosting the speedup ratio.
+
+In Fig. 3, we present the time required by different models for a draft round. From Eq. (4), it is evident that a shorter draft time leads to a higher speedup ratio, resulting in improved performance. Fig. 3 demonstrates that the draft time of Gumiho is consistently shorter than that of Eagle-2 across all datasets regarding different temperatures, highlighting the superiority of our approach. This is primarily attributed to the efficiency of the parallel MLP heads in Gumiho.
+
+# 4.3. Ablation Studies
+
+In this section, we evaluate the contribution of each component in our method to the overall performance. Specifically, we conduct ablation studies focusing on three key aspects: (1) The depth of the serial head (Transformer head), where we vary the number of Transformer layers to assess its impact; (2) The width of parallel heads (MLP heads), where we experiment with different numbers of MLP heads; and (3) the effectiveness of full tree attention (FTA). These ablation studies aim to provide a deeper understanding of the architectural design choices in our proposed method and their respective contributions to the final performance. We also present a detailed comparison of draft head accuracy in Appendix D. Unless otherwise specified, the ablation experiments are conducted using Vicuna 7B as the target LLM, with MT-Bench as the test dataset and the temperature set to 0.
+
+![](images/a5af5e3d02ee8f7eb7b11f816d0930ed6dec798e995cb69bd86a422991fcb4dc.jpg)  
+Figure 4. Ablation study on serial head depth. The serial head is a Transformer model, whose depth represents the number of layers within the Transformer architecture.
+
+Serial head depth. The serial head depth refers to the number of layers in the Transformer model. This Transformer serves as the initial head responsible for generating the first two tokens in a draft sequence. In this study, we vary the number of layers in the Transformer model to examine how the depth of the initial head affects the model’s overall performance. The experimental results shown in Fig. 4 reveal that reducing the number of Transformer layers results in a decline in $\tau$ , which underscores the significant impact of the initial heads. However, when the depth increases from 2 to 3 layers, although $\tau$ improves further, the speedup ratio decreases. This is because the overall speedup depends not only on $\tau$ but also on the time required to complete a single draft process. Using a three-layer Transformer substantially increases the drafting time, which ultimately reduces the speedup effect.
+
+It is worth noting that using varied depths to generate the first two tokens, such as a two-layer Transformer for the first token and a one-layer Transformer for the second, may seem intuitive but proves inefficient during inference. Since Transformers with different architectures cannot share their key-value (KV) caches, each head must compute its cache independently. This prevents cache reuse between heads, increasing the computational overhead. Our approach employs identical serial heads throughout the model, only trains one single Transformer model, and reuses it for autoregressive token generation during inference. This architectural uniformity enables efficient KV cache sharing across the entire generation process.
+
+Parallel Head Width. Parallel head width refers to the number of MLP heads in Gumiho, which run in parallel to generate subsequent tokens in the draft sequence. The experimental results are shown in Fig. 5. Note that increasing the number of MLP heads initially improves performance but eventually leads to a decline. This is because increasing the number of parallel heads enhances the model’s capacity to predict longer draft sequences, thereby improving its ability to generate more accepted tokens. Additionally, since MLP heads operate in parallel, increasing their number does not significantly increase runtime. A higher number of mean accepted tokens with a similar runtime leads to an improved performance. However, this improvement does not scale indefinitely. All MLP heads share the same input embedding, which is derived from the concatenation of outputs from the preceding Transformer head. During training, this shared embedding serves as the input to every MLP head and is simultaneously shaped by the back-propagated losses from all of them. As the number of MLP heads increases, the embedding must encode a growing amount of information to meet the requirements of each additional head. This leads to excessive information being compressed into the limited embedding space, which reduces the clarity and specificity of the information available to each MLP head and ultimately causes performance degradation.
+
+![](images/441d0ab6550cf9692487b6cbe61df55f366aae9a557c127ae5b5d148e55c54e7.jpg)  
+Figure 5. Ablation study on parallel head width. Parallel heads refer to the MLPs in Gumiho, and the width indicates the number of MLP models.
+
+Table 2. Ablation study on the FTA mechanism.   
+
+<table><tr><td></td><td>Speedup</td><td>T</td></tr><tr><td>w/o Full Tree Attention</td><td>3.10</td><td>5.18</td></tr><tr><td>w/ Full Tree Attention</td><td>3.15</td><td>5.29</td></tr></table>
+
+Effectiveness of Full Tree Attention (FTA). We verify the effectiveness of this component by conducting experiments with or without using FTA in our model. The experimental results are presented in Tab. 2. They indicate that removing FTA impacts the mean accepted tokens and diminishes the speedup effect.
+
+Wall Clock Time of Different Components. To provide a granular understanding of the model’s performance, we report the wall clock time of key components in the pipeline, as shown in Tab. 3.
+
+Table 3. Ablation study on the wall time.   
+
+<table><tr><td>Componentsa</td><td>Wall Time (ms)</td></tr><tr><td>1st Serial Head</td><td>2.80</td></tr><tr><td>2nd Serial Head</td><td>3.46</td></tr><tr><td>Parallel Head</td><td>2.02</td></tr><tr><td>Full Tree Attention</td><td>3.41</td></tr><tr><td>Other Computation</td><td>6.11</td></tr><tr><td>Verification</td><td>45.50</td></tr></table>
+
+# 5. Conclusion
+
+The core idea of this paper is to rigorously prove that the accuracy of early tokens in a draft sequence is more critical than that of later ones in speculative decoding. In other words, given a fixed budget for model parameter size and overall execution time, prioritizing the heads responsible for generating the initial tokens can improve overall performance. Building on this insight, we propose Gumiho, a novel approach that employs a hybrid head design. Specifically, Gumiho allocates a larger proportion of the parameter and execution time budget to the head responsible for generating the initial tokens. This head is a Transformer model with a serial structure designed for these initial tokens, ensuring higher accuracy. For those heads that generate later tokens, Gumiho employs lightweight MLPs and parallelizes their execution to produce multiple tokens simultaneously. This hybrid design achieves improved performance by balancing accuracy and efficiency: the serial Transformer enhances the accuracy of the initial tokens, while the parallel MLPs reduce overall generation time. Experimental results validate the effectiveness of our approach, demonstrating that Gumiho surpasses existing state-of-the-art methods.
+
+# 6. Limitation
+
+Our proposed method, while achieving significant speedup, utilizes a more parameter-heavy draft model compared to architectures like Eagle and Medusa. Specifically, the incorporation of a two-layer Transformer head alongside five parallel MLP heads, trained concurrently, results in increased GPU memory consumption during the training phase.
+
+# Impact Statement
+
+This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
+
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+
+# A. Detailed Proof of Theorem 3.1
+
+Given $\{ p _ { i } \} _ { i = 1 } ^ { D }$ and $\{ \tilde { p } _ { i } \} _ { i = 1 } ^ { D }$ defined in Eq. (5) and Eq. (6) in the main paper, the mean accepted tokens $( \tau )$ for the original and improved settings are expressed as:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } = \sum _ { k = 1 } ^ { D } P _ { o r i } ( L \geq k ) = \sum _ { k = 1 } ^ { D } ( \prod _ { i = 1 } ^ { k } p _ { i } ) ,
+$$
+
+and
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } = \sum _ { k = 1 } ^ { D } P _ { i m p } ( L \geq k ) = \sum _ { k = 1 } ^ { D } ( \prod _ { i = 1 } ^ { k } \tilde { p } _ { i } ) .
+$$
+
+We aim to prove that:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } .
+$$
+
+We introduce an auxiliary probability sequence $P _ { i } ^ { \prime }$ that concentrates the scattered changes $\zeta _ { i }$ at two adjacent positions $d$ and $d + 1$ . Specifically, we define:
+
+$$
+P _ { i } ^ { \prime } = \left\{ \begin{array} { l l } { p _ { i } + \zeta , } & { i = d } \\ { p _ { i } - \zeta , } & { i = d + 1 } \\ { p _ { i } , } & { o t h e r w i s e , } \end{array} \right. \quad s . t . \ \zeta = \sum _ { i = 1 } ^ { d } \zeta _ { i } = \sum _ { j = d + 1 } ^ { D } \zeta _ { j } .
+$$
+
+Here, we assume
+
+$$
+p _ { d } + \zeta \leq 1 .
+$$
+
+We will discuss the cases of $p _ { d } + \zeta > 1$ at the end.
+
+With these assumptions hold, the corresponding mean accepted tokens $( \tau )$ for this concentrated setting is:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } = \sum _ { k = 1 } ^ { D } P _ { c o n } ( L \geq k ) = \sum _ { k = 1 } ^ { D } ( \prod _ { i = 1 } ^ { k } P _ { i } ^ { \prime } ) .
+$$
+
+In the following, we will prove that:
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } , } \\ & { \mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } . } \end{array}
+$$
+
+Before proceeding with the main proof, let us examine the special case where $p _ { d + 1 } = 1$ . In this case, the ordering constraint $1 \geq p _ { 1 } \geq p _ { 2 } \geq \cdot \cdot \cdot \geq p _ { d } \geq p _ { d + 1 }$ mplies that all, we must have $p _ { 1 } = p _ { 2 } = \cdot \cdot \cdot = p _ { d } = p _ { d + 1 } = 1 \ /$ $p _ { i } + \zeta _ { i } \leq 1$ $i = \{ 1 , 2 , \cdots , d \}$ $\zeta _ { 1 } = \zeta _ { 2 } = \cdot \cdot \cdot = \zeta _ { d } = 0$ $\begin{array} { r } { \zeta = \sum _ { i = 1 } ^ { d } \zeta _ { i } = \sum _ { i = d + 1 } ^ { D } \zeta _ { i } = 0 } \end{array}$ meaning that $\{ p _ { i } \} _ { i = 1 } ^ { D }$ and $\{ \widetilde { p _ { i } } \} _ { i = 1 } ^ { D }$ are exactly the same. As a result,, $\mathbb E [ L ] _ { \mathrm { i m p r o v e d } } = \mathbb E [ L ] _ { \mathrm { o r i g i n a l } }$ .
+
+For the remainder of the proof, we assume $p _ { d + 1 } < 1$ .
+
+# A.1. The proof of $\mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$
+
+Define
+
+$$
+\Delta E = \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } - \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } = \sum _ { k = 1 } ^ { D } \Delta E _ { k } ,
+$$
+
+where $\begin{array} { r } { \Delta E _ { k } = \prod _ { i = 1 } ^ { k } P _ { i } ^ { \prime } - \prod _ { i = 1 } ^ { k } p _ { i } } \end{array}$ represents the contributions to the difference of the expected value for each position $k$ .   
+Step 1: Analyze of $\Delta E$ .
+
+We separate $\Delta E$ into three parts, where $\begin{array} { r } { \Delta E _ { 1 } = \sum _ { k = 1 } ^ { d } \Delta E _ { k } } \end{array}$ , $\Delta E _ { 2 } = \Delta E _ { d + 1 }$ and $\begin{array} { r } { \Delta E _ { 3 } = \sum _ { k = d + 2 } ^ { D } \Delta E _ { k } } \end{array}$ . For $1 \leq k < d$ , $P _ { k } ^ { \prime } = p _ { k }$ and $\Delta E _ { k } = 0$ . Therefore:
+
+$$
+\begin{array} { r l } & { \displaystyle \Delta E _ { 1 } = \Delta E _ { d } } \\ & { \quad \displaystyle = \prod _ { i = 1 } ^ { d } P _ { i } ^ { \prime } - \prod _ { i = 1 } ^ { d } p _ { i } } \\ & { \quad \displaystyle = \prod _ { i = 1 } ^ { d - 1 } p _ { i } \cdot ( p _ { d } + \zeta ) - \prod _ { i = 1 } ^ { d } p _ { i } } \\ & { \quad \quad \displaystyle = \zeta \prod _ { i = 1 } ^ { d - 1 } p _ { i } . } \end{array}
+$$
+
+For $k = d + 1$ , we have:
+
+$$
+\begin{array} { l } { \displaystyle \Delta E _ { 2 } = \Delta E _ { d + 1 } } \\ { \displaystyle = \prod _ { i = 1 } ^ { d + 1 } P _ { i } ^ { \prime } - \prod _ { i = 1 } ^ { d + 1 } p _ { i } } \\ { \displaystyle = \prod _ { i = 1 } ^ { d - 1 } p _ { i } \cdot ( p _ { d } + \zeta ) ( p _ { d + 1 } - \zeta ) - \prod _ { i = 1 } ^ { d + 1 } p _ { i } } \\ { \displaystyle = \prod _ { i = 1 } ^ { d - 1 } p _ { i } \cdot ( \zeta p _ { d + 1 } - \zeta p _ { d } - \zeta ^ { 2 } ) } \\ { \displaystyle = \zeta ( p _ { d + 1 } - p _ { d } - \zeta ) \prod _ { i = 1 } ^ { d - 1 } p _ { i } . } \end{array}
+$$
+
+For $k > d + 1$ , the difference arises from the terms $( p _ { d } + \zeta )$ and $\left( p _ { d + 1 } - \zeta \right)$ in the product. Thus:
+
+$$
+\begin{array} { l } { \displaystyle \Delta E _ { 3 } = \sum _ { k = d + 2 } ^ { D } \Delta E _ { k } } \\ { = \sum _ { k = d + 2 } ^ { D } \left( \underset { i = 1 } { \overset { k } { \prod } } P _ { i } ^ { \prime } - \underset { i = 1 } { \overset { k } { \prod } } p _ { i } \right) } \\ { = \displaystyle \sum _ { k = d + 2 } ^ { D } \left( ( \underset { i = 1 } { \overset { d - 1 } { \prod } } p _ { i } ) ( p _ { d } + \zeta ) ( p _ { d + 1 } - \zeta ) ( \underset { i = d + 2 } { \overset { k } { \prod } } p _ { i } ) - \underset { i = 1 } { \overset { k } { \prod } } p _ { i } \right) } \\ { = \displaystyle \sum _ { k = d + 2 } ^ { D } \left( ( \underset { i = 1 } { \overset { d - 1 } { \prod } } p _ { i } ) \zeta ( p _ { d + 1 } - p _ { d } - \zeta ) ( \underset { i = d + 2 } { \overset { k } { \prod } } p _ { i } ) \right) . } \end{array}
+$$
+
+By combining Eq. $( 2 0 ) \sim \mathrm { E q }$ . (22), we have:
+
+$$
+\begin{array} { l } { \Delta E = \Delta E _ { 1 } + \Delta E _ { 2 } + \Delta E _ { 3 } } \\ { \displaystyle = \zeta ( p _ { d + 1 } - p _ { d } + 1 - \zeta ) \prod _ { i = 1 } ^ { d - 1 } p _ { i } + \sum _ { k = d + 2 } ^ { D } \left( ( \prod _ { i = 1 } ^ { d - 1 } p _ { i } ) \zeta ( p _ { d + 1 } - p _ { d } - \zeta ) ( \prod _ { i = d + 2 } ^ { k } p _ { i } ) \right) . } \end{array}
+$$
+
+# Step 2: Scaling.
+
+Define $\begin{array} { r } { A = \prod _ { i = 1 } ^ { d - 1 } p _ { i } \zeta } \end{array}$ and recall that in Eq. (5) we have $1 \geq p _ { 1 } \geq p _ { 2 } \geq \cdot \cdot \cdot \geq p _ { D } \geq 0$ , then the total difference $\Delta E$ becomes:
+
+$$
+\begin{array} { l } { \displaystyle \Delta E = A ( p _ { d + 1 } - p _ { d } + 1 - \zeta ) + \sum _ { k = i + 2 } ^ { D } \left( A ( p _ { d + 1 } - p _ { d } - \zeta ) ( \displaystyle \sum _ { k = i + 2 } ^ { k } p _ { i } ) \right) } \\ { \displaystyle \qquad = A \left( 1 + ( p _ { d + 1 } - p _ { d } - \zeta ) + ( p _ { d + 1 } - p _ { d } - \zeta ) \sum _ { k = i + 2 + i + 2 } ^ { D } \prod _ { j } p _ { i } \right) } \\ { \displaystyle \qquad = A \left( 1 - ( p _ { d } - p _ { d + 1 } + \zeta ) - ( p _ { d } - p _ { d + 1 } + \zeta ) \sum _ { k = i + 2 + i + 2 } ^ { D } \prod _ { j } ^ { k } p _ { i } \right) } \\ { \displaystyle \qquad \geq A \left( 1 - ( p _ { d } - p _ { d + 1 } + \zeta ) - ( p _ { d } - p _ { d + 1 } + \zeta ) \sum _ { k = i + 2 } ^ { D } p _ { d + 1 } ^ { k - d - 1 } \right) } \\ { \displaystyle \qquad = A \left( 1 - ( p _ { d } - p _ { d + 1 } + \zeta ) ( 1 + \sum _ { k = i + 2 } ^ { D } p _ { d + 1 } ^ { k - d - 1 } ) \right) . } \end{array}
+$$
+
+# Step 3: Simplifying the sum.
+
+Note that the last term $\begin{array} { r } { 1 + \sum _ { k = d + 2 } ^ { D } p _ { d + 1 } ^ { k - d - 1 } } \end{array}$ pk−d−1d+1 in Eq. (24) is a geometric series. Therefore, we have:
+
+$$
+\begin{array} { l } { { 1 + \displaystyle \sum _ { k = d + 2 } ^ { D } p _ { d + 1 } ^ { k - d - 1 } = 1 + p _ { d + 1 } + p _ { d + 1 } ^ { 2 } + p _ { d + 1 } ^ { 3 } + \cdots + p _ { d + 1 } ^ { D - d - 1 } } } \\ { { \mathrm { } } } \\ { { \displaystyle \qquad = \frac { 1 - p _ { d + 1 } ^ { D - d } } { 1 - p _ { d + 1 } } . } } \end{array}
+$$
+
+Substitute this back into $\Delta E$ , and we have:
+
+$$
+\Delta E \ge A \left( 1 - ( p _ { d } - p _ { d + 1 } + \zeta ) ( \frac { 1 - p _ { d + 1 } ^ { D - d } } { 1 - p _ { d + 1 } } ) \right) .
+$$
+
+# Step 4: Proving $\Delta E \ge 0$
+
+To ensure $\Delta E \ge 0$ , we need:
+
+$$
+\begin{array} { c } { { ( p _ { d } - p _ { d + 1 } + \zeta ) ( \displaystyle \frac { 1 - p _ { d + 1 } ^ { D - d } } { 1 - p _ { d + 1 } } ) \leq 1 , } } \\ { { \Longleftrightarrow ( p _ { d } - p _ { d + 1 } + \zeta ) ( 1 - p _ { d + 1 } ^ { D - d } ) \leq 1 - p _ { d + 1 } , } } \\ { { \Longleftrightarrow ( p _ { d } + \zeta ) - p _ { d + 1 } ^ { D - d } ( p _ { d } - p _ { d + 1 } + \zeta ) \leq 1 . } } \end{array}
+$$
+
+Since $1 \geq p _ { d } \geq p _ { d + 1 } \geq 0$ and $\zeta \geq 0$ , it follows that $p _ { d + 1 } ^ { D - d } ( p _ { d } - p _ { d + 1 } + \zeta ) \geq 0$ . Besides, since we have $p _ { d } + \zeta \leq 1$ in Eq. (15), the inequality holds. Thus, $\Delta E \ge 0$ . Implying:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } .
+$$
+
+# A.2. Proving that $\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] ,$ concentrate
+
+We introduce a series of auxiliary sequences with the goal of proving the following inequality chain. Note that the first line and the last line in the following inequality chain represent the improved setting and the concentrate setting, respectively.
+
+Note that only two elements are different in every two consecutive lines (except for the last two lines). Specifically, we merge $\zeta _ { i }$ into $\zeta _ { d }$ one at a time for $1 \leq i \leq d - 1$ .
+
+$$
+\begin{array} { r } { \mathbb { E } ( p _ { 1 } + \zeta _ { 1 } , p _ { 2 } + \zeta _ { 2 } , p _ { 3 } + \zeta _ { 3 } , \ K \cdot \cdot , p _ { d } + \zeta _ { d } , p _ { d + 1 } - \zeta _ { d + 1 } , p _ { d + 2 } - \zeta _ { d + 2 } , \ K \cdot \cdot , p _ { D } - \zeta _ { D } ) \cdot \cdot , } \end{array}
+$$
+
+$$
+\begin{array} { r } { \mathbb { E } ( p _ { 1 } , ~ p _ { 2 } + \zeta _ { 2 } , ~ p _ { 3 } + \zeta _ { 3 } , ~ \cdots , ~ p _ { d } + \zeta _ { d } + \zeta _ { 1 } , ~ p _ { d + 1 } - \zeta _ { d + 1 } , ~ p _ { d + 2 } - \zeta _ { d + 2 } , ~ \cdots , ~ p _ { D } - \zeta _ { d } , ~ \cdots , } \end{array}
+$$
+
+$$
+\begin{array} { r } { \mathbb { E } ( p _ { 1 } , ~ p _ { 2 } , ~ p _ { 3 } + \zeta _ { 3 } , ~ \cdots , ~ p _ { d } + \zeta _ { d } + \zeta _ { 1 } + \zeta _ { 2 } , ~ p _ { d + 1 } - \zeta _ { d + 1 } , ~ p _ { d + 2 } - \zeta _ { d + 2 } , ~ \cdots , ~ p _ { D } - \zeta _ { D } + \zeta _ { d + 1 } , } \end{array}
+$$
+
+# First Part of the Inequality Chain
+
+We begin by proving the first inequality in the chain, which involves transferring $\zeta _ { 1 }$ from $p _ { 1 }$ to $p _ { d }$ :
+
+Let $\Delta E _ { 4 } ^ { 1 }$ denote the difference between the left-hand side (LHS) and the right-hand side (RHS) of the above inequality:
+
+$$
+\begin{array} { r l } &  \begin{array} { r l } &  \mathrm { S } E _ { 4 } ^ { \mathrm { i n } } = \mathbb { E } _ { ( p _ { 1 } + \zeta _ { 1 } , p _ { 2 } + \zeta _ { 2 } , p _ { 3 } + \zeta _ { 3 } , r _ { 3 } - \mu _ { 4 } + \zeta _ { 4 } , p _ { 4 + 1 } - \zeta _ { 4 } + 1 , p _ { 4 + 2 } - \zeta _ { 4 + 2 } , \cdots , p _ { 2 } - \zeta _ { D } ) } \\ & { - \mathbb { E } _ { ( p _ { 1 } , p _ { 2 } + \zeta _ { 2 } , p _ { 3 } + \zeta _ { 3 } , r _ { 3 } - \mu _ { 4 } ) \cdot \zeta _ { 4 } , \zeta _ { 1 } , p _ { 4 + 1 } - \zeta _ { 4 + 1 } , p _ { 4 + 2 } - \zeta _ { 4 + 1 } , \cdots , p _ { 2 } - \zeta _ { D } ) } \\ & { = ( p _ { 1 } + \zeta _ { 1 } - p _ { 1 } ) + \displaystyle \frac { d - 1 } { \sum _ { i = 2 } ^ { n } } [ \displaystyle \frac { \mathbb { H } } { \prod _ { i = 1 } } ( p _ { i } + \zeta _ { i } ) - p _ { 1 } \displaystyle \prod _ { i = 2 } ^ { n } ( p _ { i } + \zeta _ { i } ) ] + } \\ & { \qquad [ \displaystyle \prod _ { i = 1 } ^ { 1 } ( p _ { i } - \zeta _ { i } ) - p _ { 1 } ( \displaystyle \prod _ { i = 1 } ^ { n } ( p _ { i } + \zeta _ { i } ) ) ( p _ { i } + \zeta _ { 1 } + \zeta _ { i } ) ] [ 1 + \displaystyle \sum _ { k = 4 } ^ { D } ( \displaystyle \prod _ { i = 1 } ^ { k } ( p _ { i } - \zeta _ { i } ) ]  } \end{array} } \\ &  \qquad \begin{array} { r l } &   \sum _ { i = 1 } ^ { \infty } ( \displaystyle \prod _ { i = 1 } ^ { d } ( p _ { i } + \zeta _ { i } ) - p _ { 1 } ( \displaystyle \prod \end{array} \end{array}
+$$
+
+If $p _ { d } + \zeta _ { d } - p _ { 1 } \geq 0$ , then $\Delta E _ { 4 } ^ { 1 } \geq 0$ . This directly implies that the first inequality holds. If $p _ { d } + \zeta _ { d } - p _ { 1 } < 0$ , then
+
+$$
+\begin{array} { r l } { \Delta E _ { 4 } ^ { 1 } \geq \zeta _ { 1 } - \displaystyle \prod _ { i = 1 } ^ { d - 1 } ( p _ { i } + \zeta _ { i } ) \zeta _ { 1 } ( p _ { 1 } - p _ { d } - \zeta _ { d } ) \left[ 1 + \displaystyle \sum _ { k = i + 1 } ^ { D } ( p _ { d + 1 } ^ { k - d } ) \right] } & { } \\ { = \zeta _ { 1 } - \displaystyle \prod _ { i = 2 } ^ { d - 1 } ( p _ { i } + \zeta _ { i } ) \zeta _ { 1 } ( p _ { 1 } - p _ { d } - \zeta _ { d } ) \frac { 1 - p _ { d + 1 } ^ { D - d - 1 } } { 1 - p _ { d + 1 } } } & { } \\ { \geq \zeta _ { 1 } - \zeta _ { 1 } ( p _ { 1 } - p _ { d } - \zeta _ { d } ) \displaystyle \frac { 1 } { 1 - p _ { d + 1 } } } & { } \\ { = \zeta _ { 1 } \left[ 1 - ( p _ { 1 } - p _ { d } - \zeta _ { d } ) \displaystyle \frac { 1 } { 1 - p _ { d + 1 } } \right] } & { } \\ { = \zeta _ { 1 } \left[ \displaystyle \frac { ( 1 - p _ { d + 1 } ) - ( p _ { 1 } - 1 ) \cdot p _ { d + 1 } } { 1 - p _ { d + 1 } } - \zeta _ { d } \right] } & { } \\ { = \zeta _ { 1 } \left[ \displaystyle \frac { ( 1 - p _ { 1 } ) + ( p _ { d } + \zeta _ { d } - p _ { d + 1 } ) } { 1 - p _ { d + 1 } } \right] } & { } \\ { = \zeta _ { 1 } \left[ \displaystyle \frac { ( 1 - p _ { 1 } ) + ( p _ { d } + \zeta _ { d } - p _ { d + 1 } ) } { 1 - p _ { d + 1 } } \right] } & { } \end{array}
+$$
+
+Given that $p _ { 1 } \leq 1$ and $p _ { d + 1 } \leq p _ { d } < 1$ , the numerator $\left( 1 - p _ { 1 } \right) + \left( p _ { d } + \zeta _ { d } - p _ { d + 1 } \right)$ and the denominator $1 - p _ { d + 1 }$ are both positive. Hence, $\Delta E _ { 4 } ^ { 1 } \geq 0$ .
+
+In both cases, $\Delta E _ { 4 } ^ { 1 } \geq 0$ , thereby proving Inequality 30.
+
+Similarly, consider further transferring $\zeta _ { 2 }$ from $p _ { 2 }$ to $p _ { d }$ :
+
+Apparently, this equals to prove:
+
+which can be degenerated to Inequality 30 by removing $p _ { 1 }$ from both auxiliary sequences.
+
+Using the same method, we can iteratively transfer each $\zeta _ { i }$ from $p _ { i }$ to $p _ { d }$ for $i = 1 , 2 , \cdots , d$ , ensuring that each step maintains the inequality. Consequently, all inequalities in the chain (29) hold, except the last.
+
+To conclude, after transferring all $\zeta _ { i }$ from $i = 1$ to $d$ , we arrive at the following:
+
+$$
+\begin{array} { r } { \mathbb { E } ( p _ { 1 } , \quad p _ { 2 } , \quad p _ { 3 } , \quad \cdots , p _ { d } + \sum _ { i = 1 } ^ { d } \zeta _ { i } , \quad p _ { d + 1 } - \zeta _ { d + 1 } , \quad p _ { d + 2 } - \zeta _ { d + 2 } , \quad \cdots , p _ { D } - \zeta _ { D } ) } \end{array}
+$$
+
+This completes the proof for the first part of the inequality chain.
+
+# Second Part of the Inequality Chain
+
+We now address the final inequality in the chain (29), specifically:
+
+Let $\Delta E _ { 5 }$ denote the difference between the left-hand side and the right-hand side of the above inequality:
+
+$$
+\begin{array} { l } { { \displaystyle \Delta E _ { 5 } = \mathbb { E } ( p _ { 1 } , p _ { 2 } , p _ { 3 } , \cdots , p _ { d } + \sum _ { i = 1 } ^ { d } \zeta _ { i } , p _ { d + 1 } - \zeta _ { d + 1 } , p _ { d + 2 } - \zeta _ { d + 2 } , \cdots , p _ { D } - \zeta _ { D } ) } } \\ { { \mathrm { } } } \\ { { \mathrm { } - \mathbb { E } ( p _ { 1 } , p _ { 2 } , p _ { 3 } , \cdots , p _ { d } + \displaystyle \sum _ { i = 1 } ^ { d } \zeta _ { i } , p _ { d + 1 } - \sum _ { i = d + 1 } ^ { D } \zeta _ { i } , p _ { d + 2 } , \cdots , p _ { D } ) } } \\ { { \mathrm { } } } \\ { { \displaystyle = ( \prod _ { i = 1 } ^ { d - 1 } p _ { i } ) ( p _ { d } + \sum _ { i = 1 } ^ { d } \zeta _ { i } ) \left[ \sum _ { k = d + 1 } ^ { D } ( \prod _ { i = d + 1 } ^ { k } ( p _ { i } - \zeta _ { i } ) ) - \sum _ { k = d + 1 } ^ { D } ( \prod _ { i = d + 1 } ^ { k } P _ { i } ^ { \prime } ) \right] } } \end{array}
+$$
+
+Note that the multiplicative coefficient $\begin{array} { r } { ( \prod _ { i = 1 } ^ { d - 1 } p _ { i } ) ( p _ { d } + \sum _ { i = 1 } ^ { d } \zeta _ { i } ) } \end{array}$ is always positive, as probabilities are non-negative. Therefore, the sign of $\Delta E _ { 5 }$ depends solely on the later bracketed difference. We denote this difference as $\Delta E _ { 5 } ^ { \prime }$ :
+
+$$
+\begin{array} { r l } { \Delta E _ { s } ^ { \prime } } & { = \underset { k = d + 1 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } ( p _ { k } - \epsilon _ { i } ) ) - \underset { k = d + 1 } { \overset { D } { \sum } } ( \underset { \mathrm { i } = 1 } { \overset { \ ! } { \prod } } p _ { i } ) } \\ & { = \underset { k = d + 1 + d + 1 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } ( p _ { k } - \epsilon _ { i } ) ) - \underset { k = d + 1 } { \overset { D } { \sum } } ( \underset { k = d + 2 + m + 2 } { \overset { D } { \prod } } ( p _ { k } ) ) ( p _ { d + 1 } - \zeta ) \Bigg | } \\ & { = ( ( p _ { d + 1 } - \zeta _ { d + 1 } ) + \underset { k = d + 2 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } ( p _ { k } - \zeta _ { i } ) ) ) - \underset { k = d + 1 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } ( p _ { k + 1 } - \zeta ) + \underset { k = d + 2 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } p _ { i } ) ( p _ { d + 1 } - \zeta ) ) } \\ & { = ( \zeta - \zeta _ { d + 1 } ) + \underset { k = d + 2 } { \overset { D } { \sum } } ( \underset { n = + 1 } { \overset { \ ! } { \prod } } ( p _ { k } - \zeta ) - ( \underset { k = d + 2 } { \overset { \ ! } { \prod } } p _ { i } ) ( p _ { k + 1 } - \zeta ) ) } \\ &  = ( \zeta - \zeta _ { d + 1 } ) + \underset { k = d + 2 } { \overset { D } { \sum } } ( \underset { ( n = + 1 ) } { \overset { \ ! } { \prod } } ( p _ { k } - \zeta ) + \underset  i = \end{array}
+$$
+
+where in the last two terms of Eq. (38), we split the result of $\begin{array} { r } { \prod _ { i = d + 1 } ^ { k } ( p _ { i } - \zeta _ { i } ) - \prod _ { i = d + 1 } ^ { k } p _ { i } } \end{array}$ into two terms. The first term represents the summation of all elements with only one $\zeta _ { i }$ , and $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } )$ denotes the sum of all possible products involving two or more distinct $\zeta _ { i }$ terms from $\{ \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } \}$ . Now we want to prove that $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } ) \geq 0$ . Since $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } )$ only exists when $k \geq d + 2$ , we define it as follow:
+
+$$
+R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdots , \zeta _ { k } ) = \prod _ { i = d + 1 } ^ { k } ( p _ { i } - \zeta _ { i } ) - \left[ \prod _ { i = d + 1 } ^ { k } p _ { i } - \sum _ { i = d + 1 } ^ { k } \zeta _ { i } \prod _ { \stackrel { j = d + 1 } { j \neq i } } ^ { k } p _ { j } \right] , k \geq d + 2 .
+$$
+
+Then, we can calculate the partial derivative of the function $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } )$ with respect to $\zeta _ { m }$ :
+
+$$
+\frac { d ( R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , \cdot \cdot \cdot , \zeta _ { k } ) ) } { d ( \zeta _ { m } ) } = - \prod _ { \stackrel { i = d + 1 } { i \not = m } } ^ { k } ( p _ { i } - \zeta _ { i } ) + \prod _ { \stackrel { j = d + 1 } { j \not = m } } ^ { k } p _ { j } .
+$$
+
+Given $0 ~ \leq ~ p _ { i } - \zeta _ { i } ~ \leq ~ p _ { i } ~ \leq ~ 1$ , we observe that $\begin{array} { r l r } { \frac { d ( R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , . . . , \zeta _ { k } ) ) } { d ( \zeta _ { m } ) } } & { { } \ge } & { 0 } \end{array}$ is always true, which implies that $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , . . . , \zeta _ { k } )$ is monotonically non-decreasing with respect to $\zeta _ { m }$ for all $m \in \{ d + 1 , d + 2 , \ldots , k \}$ . Note that $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , . . . , \zeta _ { k } )$ is differentiable on $( 0 , 1 )$ and continuous on $[ 0 , 1 ]$ with respect to all $\zeta _ { m }$ . Therefore, $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , . . . , \zeta _ { k } )$ attains its minimum value when $\zeta _ { m } = 0$ for all $m \in \{ d + 1 , \ldots , k \}$ . Since $R _ { 2 } ( 0 , 0 , \ldots , 0 ) = 0$ , we conclude that $R _ { 2 } ( \zeta _ { d + 1 } , \zeta _ { d + 2 } , . . . , \zeta _ { k } ) \geq 0$ .
+
+Then
+
+$$
+\begin{array} { r l } & { \Delta E _ { 5 } ^ { \prime } \geq ( \zeta - \zeta _ { d + 1 } ) + \displaystyle \sum _ { k = d + 2 } ^ { D } \left( \zeta ( \displaystyle \prod _ { i = d + 2 } ^ { k } p _ { i } ) - \displaystyle \sum _ { i = d + 1 } ^ { k } \zeta _ { i } ( \displaystyle \prod _ { j = d + 1 } ^ { k } p _ { j } ) \right) } \\ & { \quad \quad = ( \zeta - \zeta _ { d + 1 } ) + \displaystyle \sum _ { k = d + 2 } ^ { D } \left( ( \displaystyle \sum _ { i = d + 1 } ^ { D } \zeta _ { i } ) ( \displaystyle \prod _ { i = d + 2 } ^ { k } p _ { i } ) - \displaystyle \sum _ { i = d + 1 } ^ { k } \zeta _ { i } ( \displaystyle \prod _ { j = d + 1 } ^ { k } p _ { j } ) \right) . } \end{array}
+$$
+
+When $D - d = 2$ :
+
+$$
+\begin{array} { r l } & { \Delta E _ { 5 } ^ { \prime [ 2 ] } = ( \zeta - \zeta _ { d + 1 } ) + \displaystyle \sum _ { k = d + 2 } ^ { d + 2 } \left( ( \displaystyle \prod _ { i = d + 2 } ^ { k } p _ { i } ) \zeta - \displaystyle \sum _ { i = d + 1 } ^ { k } \zeta _ { i } ( \displaystyle \prod _ { j = d + 1 } ^ { k } p _ { j } ) \right) } \\ & { \quad \quad \quad = \zeta _ { d + 2 } + p _ { d + 2 } \zeta - \zeta _ { d + 1 } p _ { d + 2 } - \zeta _ { d + 2 } p _ { d + 1 } } \\ & { \quad \quad = \zeta _ { d + 2 } ( 1 - p _ { d + 1 } ) + ( \zeta - \zeta _ { d + 1 } ) p _ { d + 2 } } \\ & { \quad \quad \quad \geq ( \zeta - \zeta _ { d + 1 } ) p _ { d + 2 } . } \end{array}
+$$
+
+Since $0 \leq p _ { i } \leq 1$ and $0 \leq \zeta _ { i } \leq 1$ , it is clear that $\Delta E _ { 5 } ^ { \prime [ 2 ] } \geq 0$ .
+
+Similarly, When $D - d = 3$ :
+
+$$
+\begin{array} { r l } & { \Delta E _ { 5 } ^ { \prime [ 3 ] } = \Delta E _ { 5 } ^ { \prime [ 2 ] } + \left( \underset { i = d + 2 } { \overset { d + 3 } { \prod } } \ p _ { i } \big ) \zeta - \underset { i = d + 1 } { \overset { d + 3 } { \sum } } \ \zeta _ { i } \underset { j = d + 1 } { \overset { d + 3 } { \prod } } \ p _ { j } \right) } \\ & { \qquad \geq \left( \zeta - \zeta _ { d + 1 } \right) p _ { d + 2 } + \left( p _ { d + 2 } p _ { d + 3 } \zeta - \zeta _ { d + 1 } p _ { d + 2 } p _ { d + 3 } - \zeta _ { d + 2 } p _ { d + 1 } p _ { d + 3 } - \zeta _ { d + 3 } p _ { d + 1 } p _ { d + 2 } \right) } \\ & { \qquad = \left( \zeta _ { d + 2 } + \zeta _ { d + 3 } \right) p _ { d + 2 } + \left( p _ { d + 2 } p _ { d + 3 } \zeta - \zeta _ { d + 1 } p _ { d + 2 } p _ { d + 3 } - \zeta _ { d + 2 } p _ { d + 1 } p _ { d + 3 } - \zeta _ { d + 3 } p _ { d + 1 } p _ { d + 2 } \right) } \\ & { \qquad = \zeta _ { d + 2 } ( p _ { d + 2 } - p _ { d + 1 } p _ { d + 3 } ) + \zeta _ { d + 3 } ( p _ { d + 2 } - p _ { d + 1 } p _ { d + 2 } ) + \left( \zeta - \zeta _ { d + 1 } \right) p _ { d + 2 } p _ { d + 3 } } \\ & { \qquad \geq \zeta _ { d + 2 } ( p _ { d + 2 } - p _ { d + 1 } p _ { d + 2 } ) + \zeta _ { d + 3 } ( p _ { d + 2 } - p _ { d + 1 } p _ { d + 2 } ) + \left( \zeta - \zeta _ { d + 1 } \right) p _ { d + 2 } p _ { d + 3 } } \\ & { \qquad \geq ( \zeta - \zeta _ { d + 1 } ) p _ { d + 2 } p _ { d + 3 } } \\ & { \qquad \geq 0 . } \end{array}
+$$
+
+Using induction, assume that for any $D - d = k \in [ 2 , \infty )$ , $\begin{array} { r } { \Delta E _ { 5 } ^ { \prime [ k ] } = \left( \zeta - \zeta _ { d + 1 } \right) \prod _ { i = d + 2 } ^ { d + k } p _ { i } \ge 0 . } \end{array}$ Then for $D - d = k + 1$ :
+
+$$
+\begin{array} { r l } & { \mathbb { E } ( ( - \zeta _ { \Psi } ) \frac { d \hat { \Pi } } { d t ^ { 2 } } ) ^ { n } + \frac { ( \hat { \Pi } + 1 ) ^ { n } } { \| \mathbf { r } \| } \rho ( \zeta _ { \Psi } ^ { - 1 } \frac { d \hat { \Pi } + 1 } { d t ^ { 2 } } ) ^ { n } } \\ & { = ( \zeta _ { \Psi } - \zeta _ { \Psi } ) \frac { d \hat { \Pi } } { d t ^ { 2 } } + ( \frac { ( \hat { \Pi } + 1 ) ^ { n } } { \| \mathbf { r } \| } \rho ( \zeta _ { \Psi } ^ { - 1 } - \zeta _ { \Psi } ) ( \frac { d \hat { \Pi } + 1 } { \rho \| \mathbf { r } \| ^ { 2 } } ) - \zeta _ { \Psi } ( \frac { d \hat { \Pi } + 1 } { \| \mathbf { r } \| ^ { 2 } } ) ^ { n } ) } \\ & { \quad - ( \zeta _ { \Psi } + \zeta _ { \Psi } + \kappa + \kappa + \kappa + \kappa ) \frac { d \hat { \Pi } } { d t ^ { 2 } } } \\ & { = ( \zeta _ { \Psi } + \kappa + \kappa + \kappa + \kappa ) \frac { d \hat { \Pi } } { d t ^ { 2 } } \frac { \rho \hat { \Pi } } { \rho \hat { \Pi } + \kappa } + \frac { \kappa ( \hat { \Pi } + 1 ) ^ { n } } { \rho \| \mathbf { r } \| } } \\ & { \quad \quad ( \zeta _ { \Psi } ^ { - 1 } \frac { d \hat { \Pi } + 1 } { \| \mathbf { r } \| ^ { 2 } } \rho ( \zeta _ { \Psi } ^ { - 1 } ) - \zeta _ { \Psi } ( \frac { d \hat { \Pi } + 1 } { \| \mathbf { r } \| ^ { 2 } } ) ^ { n } - \zeta _ { \Psi } + \kappa + ( \frac { d \hat { \Pi } + 1 } { \| \mathbf { r } \| ^ { 2 } } ) ^ { n } ) } \\ &  = \zeta _ { \Psi } + ( \frac { ( \hat { \Pi } + 1 ) ^ { n } } { \| \mathbf { r } \| ^ { 2 } } -  \end{array}
+$$
+
+For the first term in Eq. (44), we observe:
+
+$$
+\begin{array} { r l } { \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } - \displaystyle \prod _ { j = d + 2 } ^ { d + k + 1 } p _ { j } \right) } & { = \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } - \displaystyle \prod _ { j = d + 1 } ^ { d + k } p _ { j } \cdot p _ { d + k + 1 } \right) } \\ & { \ge \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } - \displaystyle \prod _ { j = d + 2 } ^ { d + k } p _ { j } \cdot p _ { d + 2 } \right) } \\ & { = \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } - \displaystyle \prod _ { j = d + 2 } ^ { d + k } p _ { j } \cdot p _ { d + 2 } \right) } \\ & { = \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } - \displaystyle \prod _ { j = d + 1 } ^ { d + k } p _ { j } \right) } \\ & { = ( 1 - p _ { d + 1 } ) \left( \displaystyle \prod _ { i = d + 2 } ^ { d + k } p _ { i } \right) \ge 0 . } \end{array}
+$$
+
+By the same reasoning, we can show that all coefficients in Eq. (44) are non-negative, and we have:
+
+$$
+\begin{array} { l } { \Delta E _ { 5 } ^ { \prime [ k + 1 ] } \geq ( \zeta - \zeta _ { d + 1 } ) \displaystyle \prod _ { i = d + 2 } ^ { d + k + 1 } p _ { i } } \\ { \geq 0 . } \end{array}
+$$
+
+Therefore, we can conclude that $\Delta E _ { 5 } ^ { \prime } \geq 0$ , and thus:
+
+$$
+\Delta E _ { 5 } \geq 0 .
+$$
+
+# Conclusion:
+
+Since:
+
+$$
+\Delta E _ { 4 } \geq 0 , \quad \Delta E _ { 5 } \geq 0 ,
+$$
+
+we conclude that the inequality chain (29) holds, which means:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } .
+$$
+
+A.3. Proving that $\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$
+
+Given that:
+
+$$
+\begin{array} { r } { \mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } , \quad \mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } , } \end{array}
+$$
+
+we establish that:
+
+$$
+\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } } .
+$$
+
+# A.4. Cases of $p _ { d } + \zeta > 1$
+
+At the beginning of the proof, we assume $p _ { d } + \zeta \leq 1$ . In cases where $p _ { d } + \zeta > 1$ , the parameter $\zeta$ can be distributed across multiple positions to satisfy the constraints. Specifically, we divide $\zeta$ into $n$ parts:
+
+$$
+\zeta = \hat { \zeta } _ { d } + \hat { \zeta } _ { d - 1 } + \hat { \zeta } _ { d - 2 } + \cdot \cdot \cdot + \hat { \zeta } _ { d - n + 1 } ,
+$$
+
+where
+
+$$
+\begin{array} { r } { p _ { d } + \hat { \zeta } _ { d } = 1 , } \\ { p _ { d - 1 } + \hat { \zeta } _ { d - 1 } = 1 , } \\ { \dots , } \\ { p _ { d - n + 2 } + \hat { \zeta } _ { d - n + 2 } = 1 , } \\ { p _ { d - n + 1 } + \hat { \zeta } _ { d - n + 1 } < 1 . } \end{array}
+$$
+
+These adjustments are applied to $n$ positions as follows:
+
+$$
+P _ { i } ^ { \prime } = \left\{ \begin{array} { l l } { p _ { i } + \hat { \zeta } _ { i } , } & { i = d - n + 1 , \cdots , d } \\ { p _ { i } - \zeta , } & { i = d + 1 } \\ { p _ { i } , } & { o t h e r w i s e , } \end{array} \right. \quad s . t . \ \zeta = \sum _ { i = 1 } ^ { d } \zeta _ { i } = \sum _ { i = d + 1 } ^ { D } \zeta _ { i } = \sum _ { i = d - n + 1 } ^ { d } \hat { \zeta } _ { i } .
+$$
+
+Next, we construct the following inequality chain:
+
+$$
+\mathbb { E } ( p _ { 1 } , \mathrm { ~ } p _ { 2 } , \mathrm { ~ } p _ { 3 } , \mathrm { ~ } \cdots , \mathrm { ~ } p _ { d } + \hat { \zeta } _ { d } , \mathrm { ~ } p _ { d + 1 } - \hat { \zeta } _ { d } , \mathrm { ~ } p _ { d + 2 } , \mathrm { ~ } \cdots , \mathrm { ~ } p _ { D } )
+$$
+
+Each step in the inequality chain (59) can be proven using the same method as proving $\mathbb { E } [ L ] _ { \mathrm { c o n c e n t r a t e } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$ . Next, we construct another inequality:
+
+The inequality (60) can be proven using the same method as proving inequality (36).
+
+Finally, we construct a third inequality:
+
+<table><tr><td>E(p1* Pd-n+1+d-n+1 Pd-1+Sd-1 Pa+d Pd+1-Sd+1 Pd+2-Sd+2PD-(D)</td></tr></table>
+
+To prove inequality (61), we can follow the same method as proving the first to the second-to-last line in inequality (29). In that proof, we observed that each step involved moving a non-negative value $\zeta _ { i }$ from an earlier position to a later position, and the validity of the inequality was independent of the specific value of $\zeta _ { i }$ or the positions involved. In the case of inequality (61), we have $\zeta _ { 1 } , \cdots , \zeta _ { d } \geq 0$ and $\zeta _ { d - n + 1 } , \cdot \cdot \cdot , \zeta _ { d } \leq \hat { \zeta } _ { d - n + 1 } , \cdot \cdot \cdot , \hat { \zeta } _ { d } .$ . This means that we are essentially moving positive values from earlier positions to later positions, similar to the process in inequality (29). Therefore, we can use the same method to establish inequality (61).
+
+With inequalities (61), (60) and the inequality chain (59) established, we conclude that $\mathbb { E } [ L ] _ { \mathrm { i m p r o v e d } } \geq \mathbb { E } [ L ] _ { \mathrm { o r i g i n a l } }$ . Therefore, we finish the proof of Theorem 3.1 in the main paper.
+
+# B. Experiment Results on A100
+
+This section gives the inference results on a single NVIDIA A100 GPU. It is reasonable that there are different final speedups on MI250 and A100. However, note that the mean average tokens $( \tau )$ are also slightly different. This is because of the use of FP16 precision during inference, and different FP16 processing mechanisms in MI250 and A100 GPUs. Using FP32 precision yields identical $\tau$ values across both platforms, yet significantly increases the inference latency.
+
+Table 4. Speedup ratios and mean accepted tokens $( \tau )$ of different methods on NVIDIA A100. V represents Vicuna, L2 represents LLaMA2-Chat, and L3 represents LLaMA3-Instruct. We present the results of different methods across six datasets. Mean represents the average performance across these six datasets.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Method</td><td colspan="2">MT-Bench</td><td colspan="2">HumanEval</td><td colspan="2">GSM8K</td><td colspan="2">Alpaca</td><td colspan="2">CNN/DM</td><td colspan="2">Natural Ques.</td><td colspan="2">Mean</td></tr><tr><td>Speedup</td><td>T</td><td>Speedup</td><td>T</td><td>Speedup</td><td>T</td><td>Speedup</td><td></td><td>Speedup</td><td>T</td><td>Speedup</td><td>T</td><td>Speedup</td><td>T</td></tr><tr><td colspan="10">Temperature=0</td><td colspan="7"></td></tr><tr><td rowspan="3">V 7B</td><td></td><td></td><td></td><td>3.2 </td><td>5.36</td><td></td><td></td><td></td><td></td><td></td><td></td><td>2.48 </td><td>3.82</td><td>3.03 </td><td>4.8%</td></tr><tr><td>Gumaglo-2urs)</td><td>3.30</td><td>5.03</td><td></td><td></td><td>3.21x</td><td>4.04</td><td>3.0</td><td>4.82</td><td>2.57x</td><td>4.4%</td><td></td><td></td><td></td><td></td></tr><tr><td>GuEangle-2 rs)</td><td>3.03 x</td><td>4.93</td><td>3.60 </td><td></td><td>3.34x</td><td>$.08</td><td>3.09 x</td><td>4.99</td><td>2.47 x</td><td>4.40</td><td>2.48 </td><td>3:81</td><td></td><td>4.92</td></tr><tr><td rowspan="3">V 13B L2 7B</td><td></td><td></td><td></td><td></td><td>5.42</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3.98 x</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>3.28 </td><td>4.83</td><td>3.17</td><td>4.34</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gumagle-2s)</td><td>3.28</td><td>4.72</td><td>3.72 x</td><td>5.39</td><td></td><td></td><td></td><td></td><td>2.61x</td><td>4.90</td><td>2.731</td><td>4.16</td><td>3.14x</td><td>4.64</td></tr><tr><td rowspan="2">L2 13B</td><td></td><td></td><td></td><td></td><td>5.33</td><td></td><td>$.83</td><td></td><td></td><td></td><td></td><td></td><td></td><td>3.23 </td><td></td></tr><tr><td>Gulaglo-urs</td><td>3.02 x</td><td>4:37</td><td>3.08</td><td></td><td>3.60</td><td></td><td>3.10</td><td>4.63</td><td>2.72x</td><td>4.38</td><td>3.77</td><td>4.19</td><td></td><td>4.84</td></tr><tr><td rowspan="2">L3 8B</td><td></td><td></td><td></td><td></td><td>5.92</td><td>3.70</td><td>4.42</td><td>2.78</td><td>4.87</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gumagle-2s)</td><td>2.71</td><td>4.48</td><td>3.20</td><td></td><td></td><td></td><td></td><td></td><td>2.34x</td><td>3.80</td><td>2.36 x</td><td>3.63</td><td>2.97x</td><td>4.45</td></tr><tr><td colspan="10">Temperature=1</td><td colspan="3"></td><td colspan="3"></td></tr><tr><td rowspan="2">V 7B</td><td></td><td></td><td></td><td>3.9 x</td><td>4.82</td><td>2.59 x</td><td></td><td>3.5 </td><td></td><td></td><td></td><td>2.29 x</td><td></td><td></td><td></td></tr><tr><td>Gumagle-urs)</td><td>3.73</td><td>4:33</td><td></td><td></td><td></td><td>4.44</td><td></td><td>4.25</td><td>2.28x</td><td>3.87</td><td></td><td>3.63</td><td>2.78</td><td>4.17</td></tr><tr><td rowspan="2">V 13B</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gumanle-2 s</td><td>2.73 </td><td>4.36</td><td>3.14</td><td>4.85</td><td>2.81x</td><td>4.63</td><td>2.75 x</td><td>4.67</td><td>2.3 x</td><td>4.05</td><td>2.23x</td><td>3.53</td><td></td><td>4.45</td></tr><tr><td rowspan="2">L2 7B</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>2.68 x</td><td></td></tr><tr><td>Gumaglo-2us)</td><td>3.94x</td><td>4.63</td><td>3.42</td><td>5.:39</td><td>3.01 </td><td>4.94</td><td>2.89x</td><td>4.43</td><td>2.60 x</td><td>4.87</td><td>2.60 </td><td>4.02</td><td>3.86</td><td>4.48</td></tr><tr><td rowspan="2">L2 13B</td><td></td><td></td><td></td><td></td><td>5:37</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>3.900</td><td></td></tr><tr><td>Gumaglo-2urs)</td><td>2.80</td><td>4.85</td><td>3.11x</td><td></td><td>3.3</td><td>4.78</td><td>2.93 </td><td>4.50</td><td>2.4x</td><td>4.35</td><td>2.9</td><td>4.12</td><td></td><td>4.53</td></tr><tr><td rowspan="2">L3 8B</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Gumagle-2us)</td><td>2.34 </td><td>3.97</td><td>3.8 </td><td>4.81</td><td>2.81 </td><td>4.62</td><td>2.3 </td><td>4.54</td><td>2.04 x</td><td>3.56</td><td>2.02</td><td>3.49</td><td>2.39</td><td>4.08</td></tr></table>
+
+# C. Training Details and Hyper-parameters
+
+Similar to Eagle-2 (Li et al., 2024b), we employ both the regression loss $\mathcal { L } _ { r e g }$ and the classification loss $\mathcal { L } _ { c l s }$ to train the draft model. The total loss $\mathcal { L }$ is defined as a weighted combination of these two components:
+
+$$
+\mathcal { L } = w _ { r e g } \cdot \mathcal { L } _ { r e g } + w _ { c l s } \cdot \mathcal { L } _ { c l s } ,
+$$
+
+where $w _ { r e g }$ and $w _ { c l s }$ denote the weighting coefficients for the regression loss $\mathcal { L } _ { r e g }$ and the classification loss $\mathcal { L } _ { c l s }$ , respectively.
+
+Similar to EAGLE-2’s tree attention, we select the top 10 output tokens from each Transformer head as input for the subsequent head, i.e., topk $_ { = 1 0 }$ . For FTA, we extract the top 35 output tokens from each MLP head , i.e., $\mathord { \mathrm { s } } = 3 5$ . Hyperparameters can be found in Tab. 5.
+
+Table 5. Hyper-parameter configurations of Gumiho.   
+
+<table><tr><td rowspan="3">Hyper-parameters</td><td>Vicuna 7B/13B</td><td>LLaMA2 70B</td></tr><tr><td>LLaMA2 7B/13B</td><td>LLaMA3 70B</td></tr><tr><td>LLaMA3 8B</td><td></td></tr><tr><td>Learning rate</td><td>2e-4</td><td>1e-4</td></tr><tr><td>Transformer layer number</td><td></td><td>2</td></tr><tr><td>MLP head number</td><td></td><td>5</td></tr><tr><td>Batch size</td><td></td><td>4</td></tr><tr><td>Wcls</td><td></td><td>0.1</td></tr><tr><td>Wreg</td><td></td><td>1</td></tr><tr><td>Training epoch</td><td></td><td>10</td></tr><tr><td>Optimizer</td><td></td><td>AdamW</td></tr><tr><td>(B1, 2)</td><td></td><td>(0.9, 0.95)</td></tr><tr><td>Per MLP structure</td><td></td><td>[2*hidden state, 1*hidden_state]*1 -&gt; [1*hidden_state, 1*hidden state]*5</td></tr><tr><td>topk</td><td></td><td>10</td></tr><tr><td>s</td><td></td><td>35</td></tr></table>
+
+# D. Ablation Study on Head Accuracy
+
+![](images/705ce011e9678848cb4c309b00e6f373471484ece753d1416e13fa2b23d289c0.jpg)  
+Figure 6. Comparison of draft head accuracy on two datasets (MT-Bench and GSM8K). Both results are based on Vicuna 7B with the temperature set to 0.
+
+We conducted a comparative analysis of head-wise accuracy between our method and EAGLE-2 based on Vicuna 7B with temperature set to 0 on MT-Bench and GSM8K datasets. It should be noted that we have a total of seven draft heads, while EAGLE-2 only has six heads. Therefore, to facilitate comparison, we have only conducted an accuracy comparison between the first six heads of ours and the six heads of EAGLE-2. As illustrated in Fig. 6, our approach enhances the accuracy of front heads, which are responsible for generating the initial tokens in the draft sequence. The precision of these early tokens substantially impacts the final mean accepted tokens $( \tau )$ . Our back heads employ a parallel MLP architecture, resulting in lower accuracy compared to EAGLE-2. This accuracy distribution aligns with our theoretical findings. Our theorem demonstrates that optimizing the accuracy distribution across heads, specifically through enhancing precision in front heads while proportionally reducing accuracy in back heads, leads to better overall mean accepted tokens $( \tau )$ .

data/sampled_jsons/0WQJ6DFSKp_Socialized Coevolution- Advancing a Better World through Cross-Task Collaboration/auto/0WQJ6DFSKp_Socialized Coevolution- Advancing a Better World through Cross-Task Collaboration.md

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+# Socialized Coevolution: Advancing a Better World through Cross-Task Collaboration
+
+Xinjie Yao 1 2 3 Yu Wang 1 2 3 Pengfei Zhu 1 2 3 Wanyu Lin 4 Ruipu Zhao 1 Zhoupeng Guo 5 Weihao Li 6 Qinghua Hu 1 2 3
+
+# Abstract
+
+# 1. Introduction
+
+Traditional machine societies rely on data-driven learning, overlooking interactions and limiting knowledge acquisition from model interplay. To address these issues, we revisit the development of machine societies by drawing inspiration from the evolutionary processes of human societies. Motivated by Social Learning (SL), this paper introduces a practical paradigm of Socialized Coevolution (SC). Compared to most existing methods focused on knowledge distillation and multitask learning, our work addresses a more challenging problem: not only enhancing the capacity to solve new downstream tasks but also improving the performance of existing tasks through inter-model interactions. Inspired by cognitive science, we propose Dynamic Information Socialized Collaboration (DISC), which achieves SC through interactions between models specialized in different downstream tasks. Specifically, we introduce the dynamic hierarchical collaboration and dynamic selective collaboration modules to enable dynamic and effective interactions among models, allowing them to acquire knowledge from these interactions. Finally, we explore potential future applications of combining SL and SC, discuss open questions, and propose directions for future research, aiming to spark interest in this emerging and exciting interdisciplinary field. Our code will be publicly available at https://github.com/yxjdarren/SC.
+
+![](images/5a13fc7c1e0dcb1dae5f49dfdfe0099653db4f5850815cf88f8787ffd2c7b021.jpg)  
+Figure 1: Social learning in human society versus socialized coevolution in machine society.
+
+Similar to human society (Humphrey, 1976; Dunbar, 1998), each model in machine society evolves through knowledge acquisition, as illustrated in Figure 1. Traditional learning paradigms largely depend on data-driven, single-mode learning. While some paradigms (Caruana, 1997; Hinton, 2015; Zhu et al., 2024) allow knowledge transfer from other models, they are limited by blind imitation and difficulty in retaining specialized expertise. In human history, cultural evolution has been driven by continuous capability enhancement, surpassing existing critical skills for survival (Henrich, 2016; Laland, 2017). This enhancement is achieved through ongoing interaction with the environment, rather than extensive trial and data collection. This effective learning paradigm, known as socialized coevolution (SC), has been studied in cognitive science (Thompson et al., 2022).
+
+In machine learning, Knowledge Distillation (KD) (Hinton, 2015) is closely related to SC, as it overcomes the limitation of relying solely on data for knowledge acquisition. However, KD often blindly mimics the teacher model’s predictions, which may not always represent valuable knowledge. Unlike humans, KD lacks mutual teaching and learning. An alternative is Multi-Task Learning (MTL) (Caruana, 1997), a shared learning paradigm. While MTL improves performance by leveraging task correlations, it often sacrifices task-specific expertise for generalization, making it hard to maintain performance on individual tasks, particularly when tasks are highly heterogeneous. Thus, current paradigms are limited in achieving effective coevolution.
+
+Existing paradigms struggle to learn new knowledge and improve capabilities through interaction, as humans do. Due to task and data heterogeneity, models find it difficult to interact effectively. For instance, KD causes blind imitation, hindering the learning of complementary task information and degrading original performance. Similarly, MTL sacrifices task-specific expertise for generalization. In intelligent autonomous systems, the challenge is to acquire more knowledge while maintaining or enhancing original task expertise. By revisiting existing paradigms, we find that two problems are still open: (1) dynamic interactive learning in heterogeneous task scenarios, and (2) expanding general capabilities without compromising task specialization.
+
+To address these problems, we analyze two paradigms: KD and MTL. As shown in Figure 2, both KD and MTL fail to balance the enhancement of original task capabilities and the expansion of general task capabilities due to blind imitation and an overemphasis on generalization. Moreover, KD and MTL rely heavily on large datasets for knowledge transfer, whereas SC enables dynamic cross-task interaction at low cost, facilitating coevolution. Therefore, two significant issues should be explored to realize SC:
+
+1: How to achieve dynamic interactive learning?
+
+# 2: How to balance specialization and generalization?
+
+In line with cognitive science (Tomasello et al., 1993; Mesoudi, 2021; Yao et al., 2024), new knowledge should be learned from experts in different tasks and integrated with individual needs for coevolution. To achieve this, we adopt dynamic collaboration, allowing models to selectively interact and acquire valuable knowledge while avoiding blind imitation. Most cultural evolution, driven by population genetics (Mesoudi, 2021) and cognitive science (Tomasello, 2016), inspires our approach, which combines populationgenetic-style modeling with directional bias transformation. By organizing experts into a society, maintaining their independence, and enabling interactions, we expand capabilities while enhancing original performance, achieving SC.
+
+To validate this premise, we design an SC-based framework. We first establish that sociability is fundamental to SC, enabling models with complementary strengths to learn mutually. Based on this, the framework is designed around organizational structures, interaction modes, and communication mechanisms. SC is implemented using hierarchical organizational structures, progressive interaction modes, and strong-guided communication mechanisms. Specifically, we create a society of independent classification and detection models, enabling selective collaboration for logitsbased mutual learning and hierarchical feature interaction.
+
+These contributions are detailed as follows:
+
+• We introduce a practical learning paradigm, Socialized Coevolution (SC), in which models achieve coevolution through mutual interaction. • We discuss SC using an information-theoretic framework, where sociability enables models with complementary strengths to mutually and dynamically learn. • We propose a novel insight into SC methodology, where organizational structures, interaction modes, and communication mechanisms ensure a trade-off between specialization and generalization.
+
+![](images/6854920c5ee1b6f0b5798ee8b09097a4139d61240b8cef351d3b9dc67e68f89a.jpg)  
+Figure 2: Comparison of different learning paradigms.
+
+# 2. Related Work
+
+# 2.1. Knowledge Distillation
+
+Knowledge Distillation (KD), as a paradigm for knowledge transfer, focuses on enabling student models to efficiently inherit and apply the knowledge and capabilities of teacher models. Based on the type of information emphasized during the distillation process and the levels at which distillation occurs, existing KD approaches (Gou et al., 2021) can be classified into three categories: (1) Response-based distillation emphasizes the outputs or logits of the teacher model, aiming to align the logits of the student model as closely as possible with those of the teacher model to enhance performance (Jin et al., 2023); (Sun et al., 2024); (Wei et al., 2024). (2) Feature-based distillation focuses on knowledge transfer within the intermediate layers of the model, using the feature representations from the teacher model’s intermediate layers to guide the corresponding layers of the student model (Hao et al., 2024); (Lu et al., 2024b); (Huang et al., 2024). (3) Relation-based distillation targets the relative relationships between different samples in the feature space. By capturing the teacher model’s rich representations of these relationships, the student model learns to capture the inherent structure of the data (Zhang et al., 2024); (Yang et al., 2022); (Xiao & Yamasaki, 2024).
+
+Existing KD methods primarily rely on the knowledge obtained by the teacher model, often overlooking the reliability of the knowledge for each individual sample, as illustrated in Figure 2a. The key challenge in KD lies in how to effectively transfer knowledge by considering both the teacher’s capabilities and the student’s needs. To address this, SC employs dynamic interactive learning to avoid blind reliance, providing insights across different samples.
+
+# 2.2. Multi-Task Learning
+
+Multi-Task Learning (MTL) is a paradigm in machine learning that aims to learn multiple related tasks simultaneously, enabling the knowledge from one task to benefit others. This approach seeks to enhance the generalization performance of all tasks involved. Existing methods (Vandenhende et al., 2022) can be broadly categorized into two main groups: (1) Encoder-focused architectures share information within the encoder, using either hard- or softparameter sharing, before decoding each task with an independent, task-specific head (Agiza et al., 2024); (Xu et al., 2024); (Yang et al., 2024b). (2) Decoder-focused architectures first employ a multi-task network to make initial task predictions, and then use features from these initial predictions to further share or exchange information during the decoding stage (Bhattacharjee et al., 2022); (Chen et al., 2024); (Lu et al., 2024a). In addition to the aforementioned approaches, other methods exist outside these categories. For instance, multilinear relationship networks (Long et al., 2017) leverage tensor priors to capture task interactions, while soft layer ordering (Meyerson & Miikkulainen, 2017) enables flexible sharing of network layers across tasks.
+
+Existing MTL research primarily focuses on the sharing of general knowledge across different tasks, often neglecting the preservation of task-specific expertise, as illustrated in Figure 2b. The key challenge in MTL lies in the trade-off between task specialization and generalization. To address this, SC leverages social structures to categorize the model into main and auxiliary decision roles, offering insights into this dilemma.
+
+# 3. Sociability in Coevolution
+
+In this section, we address the issues outlined in Section 1 by providing precise definitions of the problem setup, followed by a concise theoretical analysis. The full proofs of the theorems are provided in the supplementary.
+
+Problem setups: Let $\mathcal { M } = \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } , . . . , \mathcal { M } _ { N } \}$ denote a set of $N$ models. Let $X$ represent the input space, and $Y$ denote the label space. The data associated with the $n$ - th model is represented as $\mathcal { D } _ { M _ { n } } = \{ x _ { i } , y _ { i } \} _ { i = 1 } ^ { K _ { \mathcal { M } _ { n } } }$ }KMn , where $K _ { \mathcal { M } _ { n } }$ denotes the total number of samples for $\mathcal { M } _ { n }$ . Each sample $x _ { i } \in \mathbb { R } ^ { D }$ corresponds to a label $y _ { i } \in Y _ { M _ { n } }$ . To illustrate this setup more clearly, consider two models, $\mathcal { M } _ { 1 }$ and $\mathcal { M } _ { 2 }$ , as examples. The expert task sets for these two models, $\mathcal { M } _ { 1 }$ and $\mathcal { M } _ { 2 }$ , are distinct, i.e., $\mathcal { T } _ { 1 }$ and $\mathcal { T } _ { 2 }$ represent the sets of tasks at which each model excels. Specifically, the performance of the models on these tasks is described as $\dot { \mathcal { M } } _ { 1 } ^ { \mathcal { T } _ { 1 } } > \mathcal { M } _ { 2 } ^ { \mathcal { T } _ { 1 } }$ and $\mathcal { M } _ { 1 } ^ { \mathcal { T } _ { 2 } } < \mathcal { M } _ { 2 } ^ { \mathcal { T } _ { 2 } }$ , indicating that $\mathcal { M } _ { 1 }$ outperforms $\mathcal { M } _ { 2 }$ on task $\mathcal { T } _ { 1 }$ , while $\mathcal { M } _ { 2 }$ excels on task $\mathcal { T } _ { 2 }$ After coevolution, the society $\pmb { S } = \{ \mathcal { M } _ { 1 } , \mathcal { M } _ { 2 } \}$ , formed by the two models, can acquire superior expertise compared to each individual model, i.e., $\mathbf { \mathcal { S } } ^ { T _ { 1 } } > \mathbf { \bar { \mathcal { M } } } _ { 1 } ^ { T _ { 1 } }$ and $\hat { S ^ { \mathcal { T } _ { 2 } } } >$ $\mathcal { M } _ { 2 } ^ { \mathcal { T } _ { 2 } }$ . This collaboration enables broader task coverage and enhances performance on original tasks.
+
+Definition 3.1. (Specialization and Generalization) The society $s$ learns more tasks while improving performance on original tasks, i.e., $\mathcal { T } _ { S } = \mathcal { T } _ { 1 } \cup \mathcal { T } _ { 2 }$ , with $\tilde { S ^ { \tilde { \tau _ { 1 } } } } > \mathcal M _ { 1 } ^ { \mathcal T _ { 1 } }$ and $\mathbf { \mathcal { S } } ^ { \mathcal { T } _ { 2 } } > \mathcal { M } _ { 2 } ^ { \mathcal { T } _ { 2 } }$ .
+
+To clearly elucidate the significance of coevolution, we define sociability information and subsequently conduct an analysis based on this premise:
+
+Definition 3.2. (Sociability Information) Given the input variables $X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } }$ , and the target $Y$ , the sociability information provided by the models is defined as:
+
+$$
+\begin{array} { r l } & { \Phi _ { \mathcal { M } _ { 1 } } = I ( X _ { \mathcal { M } _ { 1 } } ; Y | X _ { \mathcal { M } _ { 2 } } ) , } \\ & { } \\ & { \Phi _ { \mathcal { M } _ { 2 } } = I ( X _ { \mathcal { M } _ { 2 } } ; Y | X _ { \mathcal { M } _ { 1 } } ) . } \end{array}
+$$
+
+The metrics $\Phi _ { \mathcal { M } _ { 1 } }$ and $\Phi _ { { \mathcal { M } } _ { 2 } }$ quantify the sociability information of $X _ { \mathcal { M } _ { 1 } }$ and $X _ { \mathcal { M } _ { 2 } }$ in SC. Higher values reflect greater sociability and coevolution potential, representing collaboration effectiveness and information exchange between $X _ { \mathcal { M } _ { 1 } }$ and $X _ { \mathcal { M } _ { 2 } }$ . The following relation can be derived from information theory:
+
+$$
+I ( X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ; Y ) = \Phi _ { \mathcal { M } _ { 1 } } + \Phi _ { \mathcal { M } _ { 2 } } + I ( X _ { \mathcal { M } _ { 1 } } ; X _ { \mathcal { M } _ { 2 } } ; Y ) .
+$$
+
+Bayes error rate: The Bayes error rate (Fukunaga & Hummels, 1987) represents the minimum error, with $P _ { e _ { c } } ^ { m u l }$ and $P _ { e _ { c } } ^ { s i n }$ denoting the errors for multi-model and single-model scenarios, respectively, for $X _ { \mathcal { M } _ { 1 } }$ and $X _ { \mathcal { M } _ { 2 } }$ .
+
+$$
+P _ { e _ { c } } ^ { m u l } = \mathbb { E } _ { x _ { M _ { 1 } } , x _ { M _ { 2 } } \sim P _ { X _ { M _ { 1 } } , X _ { M _ { 2 } } } } [ 1 - \operatorname* { m a x } _ { y \in Y } P ( Y = y | x _ { M _ { 1 } } , x _ { M _ { 2 } } ) ] ,
+$$
+
+$$
+P _ { e _ { c } } ^ { s i n } = \mathbb { E } _ { x _ { \mathcal { M } _ { 1 } } \sim P _ { X _ { \mathcal { M } _ { 1 } } } } [ 1 - \operatorname* { m a x } _ { y \in Y } P ( Y = y | x _ { \mathcal { M } _ { 1 } } ) ] .
+$$
+
+Theorem 3.3. (Key Factor in Coevolution) Building on prior work (Cover, 1999; Feder & Merhav, 1994; $L i$ et al., 2023a), we focus on $P _ { e _ { c } } ^ { m u l }$ and $P _ { e _ { c } } ^ { s i n }$ , as follows:
+
+$$
+\begin{array} { r } { \frac { H ( Y | X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) - \log 2 } { \log | Y | } \leq P _ { e _ { c } } ^ { m u l } \leq 1 - \exp ( - H ( Y \mid X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) ) , } \end{array}
+$$
+
+$$
+\begin{array} { r } { \frac { H ( Y | X _ { \mathcal { M } _ { 1 } } ) - \log 2 } { \log | Y | } \leq P _ { e _ { c } } ^ { s i n } \leq 1 - \exp ( - H ( Y \mid X _ { \mathcal { M } _ { 1 } } ) ) . } \end{array}
+$$
+
+Since
+
+$$
+\Phi _ { \mathcal { M } _ { 2 } } = H ( Y | X _ { \mathcal { M } _ { 1 } } ) - H ( Y | X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) .
+$$
+
+We can derive
+
+$$
+\frac { H ( Y | X _ { { \cal M } _ { 1 } } ) - \Phi _ { { \cal M } _ { 2 } } - \log 2 } { \log | Y | } \leq P _ { e _ { c } } ^ { m u l } \leq 1 - \exp ( - H ( Y \mid X _ { { \cal M } _ { 1 } } ) + \Phi _ { { \cal M } _ { 2 } } ) .
+$$
+
+Remark 3.4. The difference between $P _ { e _ { c } } ^ { m u l }$ and $P _ { e _ { c } } ^ { s i n }$ reflects the effect of SC, with $\Phi _ { { \mathcal { M } } _ { 2 } }$ being the key factor.
+
+Theorem 3.5. (Sociability Information in Data) Building on prior work (Zuo et al., 2024), given the optimal parameter $w ^ { * }$ for the data $\mathcal { D }$ and the normalizing constant $p ( \mathcal { D } )$ , we have:
+
+$$
+\begin{array} { r } { D ) = \int p ( \mathcal { D } \mid w ) p ( w ) d w < \int p \left( \mathcal { D } \mid w ^ { * } \right) p ( w ) d w = p \left( \mathcal { D } \mid w ^ { * } \right) . } \end{array}
+$$
+
+Since
+
+$$
+\mathcal { D } _ { S } = \mathcal { D } _ { M _ { 1 } } \cup \mathcal { D } _ { \Phi _ { M _ { 2 } } } .
+$$
+
+We can derive
+
+$$
+\log p ( w ^ { * } \mid \mathcal { D } _ { S } ) > \log p ( w ^ { * } \mid \mathcal { D } _ { M _ { 1 } } ) .
+$$
+
+Remark 3.6. Training on sociability data ${ \mathcal { D } } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ enhances $w ^ { * }$ , proving the effect of SC in optimization.
+
+Theorem 3.7. (Sociability Information in Model) Building on prior work (Zuo et al., 2024), given the optimal parameter $w ^ { * }$ consists of the backbone parameter $\boldsymbol { w } _ { f } ^ { * }$ and the head parameter $w _ { h } ^ { * }$ , we have:
+
+$$
+\log p \left( \boldsymbol { w } ^ { * } \mid \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) = \log p \left( \boldsymbol { w } _ { f } ^ { * } \mid \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f \left( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) \right) .
+$$
+
+# Since
+
+$$
+\log p \left( w _ { h } ^ { * } \mid f \left( \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) \right) < \log p \left( w _ { h } ^ { * } \mid f ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) ^ { \Phi _ { \mathcal { M } _ { 2 } } } \right) .
+$$
+
+We can derive
+
+$$
+\log p \left( \boldsymbol { w } ^ { * } \mid \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) < \log p \left( \boldsymbol { w } _ { f } ^ { * } \mid \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f ( \mathcal { D } _ { M _ { 1 } } ) ^ { \Phi _ { \mathcal { M } _ { 2 } } } \right) .
+$$
+
+Remark 3.8. Training on sociability features $f ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) ^ { \Phi _ { \mathcal { M } _ { 2 } } }$ enhances $w ^ { * }$ , proving the effect of SC in optimization.
+
+# 4. Methodology
+
+In this section, to address the two issues highlighted in Section 1, we introduce dynamic interactive learning as a means to balance specialization and generalization. This approach is realized in two ways. First, we facilitate dynamic interactions between teacher and student models, allowing them to adjust based on their respective strengths and needs. Second, we organize the models into a society where, depending on the task, some models take on the role of decision-makers while others assist in the main process. The concepts of mutual growth through teaching and leveraging individual strengths are considered forms of SC.
+
+We first analyze the driving factors of SC, introduce its core framework, and then detail the dynamic interactive learning mechanism along with our approach to balancing specialization and generalization.
+
+# 4.1. Key Factors of SC
+
+In the face of the challenge to balance specialization and generalization within SC, a natural question arises: what are the key factors influencing SC?
+
+![](images/5b931aecb5575da86c96d7c232888b45651d6090d3fda5f6e82394d3f3db637b.jpg)  
+Figure 3: Comparison of different weights on datasets.
+
+To address the aforementioned question, we consider two downstream tasks: classification and detection. Specifically, features provide patch-wise information, while logits offer image-wise information. Consequently, we conduct exploratory experiments on the interaction between features and logits under varying weights. From Figure 3, it can be inferred that neither extremely low nor high weights are suitable for effective interaction. When the weight is too low, the knowledge interaction provides limited assistance, failing to improve the performance. Conversely, when the weight is too high, the knowledge becomes overly complex for the model to learn effectively. Notably, the model exhibits superior performance under a variety of weight combinations. Based on this observation, we hypothesize that a dynamic weighting mechanism for interaction may effectively balance specialization and generalization.
+
+Theorem 4.1. (Generalization Bound of Collaboration) Building on prior work (Zhang et al., 2023), we focus on the dynamic weighting mechanism. Let $D _ { t r a i n } = \{ x _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ be a training dataset of $N$ samples, and errˆ $( f ^ { m } )$ be the empirical error of the $m$ -th model $f ^ { m }$ on $D _ { t r a i n }$ . For any hypothesis $f \in \mathcal H$ (i.e., $\mathcal { H } : \mathcal { X }  \{ - 1 , 1 \} ,$ ), with probability at least $1 - \delta$ , the generalization error is bounded by:
+
+$$
+\begin{array} { r l r } { G E r r o r ( f ) \le \underbrace { \displaystyle \sum _ { m = 1 } ^ { M } \mathbb { E } ( w ^ { m } ) e \hat { r } r ( f ^ { m } ) } _ { T e r m - L ( e m p i r i c a l l o s s } + \underbrace { \displaystyle \sum _ { m = 1 } ^ { M } \mathbb { E } ( w ^ { m } ) \Re _ { m } ( f ^ { m } ) } _ { T e r m - C ( c o m p l e x i t y ) } } & \\ { \quad \quad } & { \quad \quad + \underbrace { \displaystyle \sum _ { m = 1 } ^ { M } C o v ( w ^ { m } , \ell ^ { m } ) } _ { T e r m - C o v ( c o v a r i a n c e ) } + M \sqrt { \frac { \ln ( 1 / \delta ) } { 2 N } } , } & \end{array}
+$$
+
+where $\mathbb { E } ( w ^ { m } )$ is the expected collaboration weights, $\Re _ { m } ( f ^ { m } )$ is the Rademacher complexity of model $f ^ { m }$ , and $C o v ( w ^ { m } , \ell ^ { m } )$ is the covariance between the collaborative weight and the loss. In static collaboration, the weights $w _ { \mathrm { s t a t i c } } ^ { m }$ remain constant, so $C o v ( w _ { \mathrm { s t a t i c } } ^ { m } , \ell ^ { m } ) = 0$ . By contrast, in dynamic collaboration, the weight $w _ { \mathrm { d v n a m i c } } ^ { m }$ increases as the loss $\ell ^ { m }$ decreases, yielding $C o v ( w _ { \mathrm { d y n a m i c } } ^ { m } , \ell ^ { m } ) \leq 0$ This negative covariance lowers Term-Cov, thereby tightening the upper bound of generalization error of collaboration, i.e., $\mathcal { O } ( \mathrm { G E r r o r } ( f _ { \mathrm { d y n a m i c } } ) ) \leq \mathcal { O } ( \mathrm { G E r r o r } ( f _ { \mathrm { s t a t i c } } ) )$ .
+
+Remark 4.2. Dynamic collaboration yields a tighter generalization error bound compared to static collaboration.
+
+# 4.2. Socialized Coevolution Framework
+
+We take a closer look at SC to identify the key factors forming its framework. Inspired by (Zuo et al., 2024; Li et al., 2023a), we perform an information-theoretic analysis of SC in the previous section and explore the impact of sociability information. Based on this, we focus on three aspects: organizational structures, interaction modes, and communication mechanisms, aiming to design a unified SC framework $U ( \cdot )$ and implement dynamic interactive learning to balance specialization and generalization.
+
+$$
+U ( { \mathcal { M } } _ { 1 } , { \mathcal { M } } _ { 2 } ) = \psi ( \varphi ( { \mathcal { M } } _ { 1 } , { \mathcal { M } } _ { 2 } ) , \beta ) ,
+$$
+
+where $\mathcal { M } _ { 1 }$ and $\mathcal { M } _ { 2 }$ are two models, $\varphi ( \cdot )$ represents the organizational structures, $\beta$ denotes the communication mechanisms, and $\psi ( \cdot )$ represents the interaction modes.
+
+# 4.3. Dynamic Information Socialized Collaboration
+
+Driven by the above analysis and (Li et al., 2023b; Shao et al., 2021; Cao et al., 2024), our goal is to address SC from three aspects: hierarchical organizational structures, progressive interaction modes, and strong-guided communication mechanisms. We introduce a new approach based on SC, called Dynamic Information Socialized Collaboration (DISC), as shown in Figure 4. For clarity, we provide a detailed explanation of the three aspects included in DISC.
+
+Hierarchical organizational structures: In DISC, we design the Dynamic Hierarchical Collaboration (DHC) module to effectively leverage hierarchical auxiliary information, aiding the main model in learning. Specifically, for different tasks, we treat the specialized model as the main entity and the other models as auxiliary information. For instance, in classification tasks, the model specialized in classification focuses on global-wise information, while the model specialized in detection emphasizes patch-wise information. The key to achieving SC lies in fully utilizing the patch-wise knowledge embedded in the detection model. The hierarchical organizational structure facilitates the progressive abstraction and capture of both low-level and high-level image features, enabling effective handling of complex visual tasks. Therefore, we define patch-wise information in the form of a hierarchical structure, as follows:
+
+$$
+i n p u t _ { m a i n } ^ { l } = o u t p u t _ { m a i n } ^ { l - 1 } + D H C \left( l - 1 \right) ,
+$$
+
+where inputlmain denotes the input of the $l$ -th layer of the main model, outputl−1main denotes the output of the $( l - 1 )$ - th layer of the main model, and $D H C ( l - 1 )$ denotes the hierarchical output of the auxiliary model. Next, we will provide a detailed explanation of $D H C ( l - 1 )$ in progressive interaction modes.
+
+Progressive interaction modes: In DISC, DHC mitigates excessive reliance on auxiliary information by employing progressive interaction, effectively utilizing hierarchical auxiliary data to support the main model’s learning process. Initially, the model depends heavily on data, but as optimization advances, the information provided by the data becomes limited. At this stage, gradually incorporating auxiliary information from other dimensions becomes essential for further learning. Overuse of auxiliary information early in the process can impede the model’s ability to learn from data. Thus, the key to SC lies in the gradual integration of auxiliary information, enabling the model to overcome the limitations of data-driven learning. This progressive interaction mode is defined through DHC as follows:
+
+![](images/2dfd8d21a8f79a612faf87aebc29d1f4d4c6f38709caca8ad82374e5095b2289.jpg)  
+Figure 4: An overview of our proposed Dynamic Information Socialized Collaboration (DISC). We design the Dynamic Hierarchical Collaboration (DHC) and Dynamic Selective Collaboration (DSC) modules, integrating organizational structures, interaction modes, and communication mechanisms as the core elements of SC.
+
+$$
+\begin{array} { r } { D H C \left( l - 1 \right) = \left| \sin \left( \frac { \operatorname* { m i n } \left( E _ { n } , E _ { t } \right) } { E _ { l o o p s } } \pi \right) \right| \sum _ { i = 1 } ^ { l - 1 } o u t p u t _ { a u x } ^ { i } , } \end{array}
+$$
+
+where $E _ { n }$ represents the current training epoch, $E _ { t }$ denotes the threshold epoch, $E _ { l o o p s }$ indicates the total number of training epochs, and $o u t p u t _ { a u x } ^ { i }$ denotes the output of the $i$ -th layer of the auxiliary model.
+
+Strong-guided communication mechanisms: In DISC, we designed the Dynamic Selective Collaboration (DSC) module to leverage strong-guided communication mechanisms to enhance the learning process of the main model. Specifically, the weights assigned to the main and auxiliary models vary dynamically depending on the sample, meaning the weight of the main model is not necessarily greater than that of the auxiliary model. Fixing the weights as constants prevents the models from adapting flexibly to their performance. The key to achieving SC is to assign appropriate weights dynamically to each model. A model’s weight should negatively correlate with its own loss and positively correlate with the losses of other models. Strong-guided communication mechanisms dynamically assign weights based on how well each model handles different samples, enabling effective learning for every sample. This dynamic weighting is defined through strong-guided communication mechanisms as follows:
+
+$$
+\begin{array} { r } { D S C \left( m \right) = \left( p _ { m } + \frac { \log \prod _ { j \neq m } ^ { | { \mathcal { M } } | } p _ { j } } { \log \prod _ { i = 1 } ^ { | { \mathcal { M } } | } p _ { i } } \right) \cdot o u t p u t _ { m } ^ { l o g i t s } , } \end{array}
+$$
+
+where $p _ { m }$ is the prediction of $m$ -th model.
+
+The DHC and DSC modules, featuring hierarchical organizational structures, progressive interaction modes, and strong-guided communication mechanisms, are integrated into the overall loss function as follows:
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { o v e r a l l } = \mathcal { L } _ { c l s } \left( y , f ^ { D S C } ( f ^ { D H C } ( x ) ) \right) } \\ & { \qquad + \mathcal { L } _ { r e g } \left( y , f ^ { D S C } ( f ^ { D H C } ( x ) ) \right) , } \end{array}
+$$
+
+where $x$ is the input, $y$ is the label, $\mathcal { L } _ { c l s }$ represents the classification loss, $\mathcal { L } _ { \boldsymbol { r } \boldsymbol { e } \boldsymbol { g } }$ represents the regression loss, $f ^ { D H C } ( \cdot )$ denotes the collaboration with DHC, and $f ^ { D S C } ( \cdot )$ denotes the collaboration with DSC. $\mathcal { L } _ { c l s }$ is employed for classification tasks, while both $\mathcal { L } _ { c l s }$ and $\mathcal { L } _ { r e g }$ are utilized for detection tasks. For a clearer understanding of training and inference, we have described the detailed algorithms in Algorithms 1 and 2.
+
+# Algorithm 1 Training for DISC.
+
+Input: Datasets $\mathcal { D } _ { \mathcal { M } _ { n } }$ , model $\mathcal { M } _ { n }$ , main task $\mathcal { T } _ { \mathrm { m a i n } }$ ; Output: Embedding with DHC $f _ { \mathcal { M } _ { n } } ^ { D H C } ( \cdot )$ , embedding with DSC $f _ { \mathcal { M } _ { n } } ^ { D S C } ( \cdot )$ , downstream task head $f _ { \mathcal { M } _ { n } } ^ { \mathrm { h e a d } } ( \cdot )$ ;
+
+1: if $\mathcal { T } _ { \mathrm { m a i n } } =$ classification then   
+2: while not converged do   
+3: Fix detection model ${ \mathcal { M } } _ { \mathrm { d e t } }$ ;   
+4: Get a mini-batch of training data from $\mathcal { D } _ { \mathcal { M } _ { c l s } }$ ;   
+5: Calculate the overall loss $\mathcal { L } _ { \mathrm { o v e r a l l } }$ using Eq. (21);   
+6: Update the classification model, i.e., $f _ { \mathcal { M } _ { c l s } } ^ { D H C } ( \cdot )$   
+$f _ { \mathcal { M } _ { c l s } } ^ { D S C } ( \cdot )$ , and $f _ { \mathcal { M } _ { c l s } } ^ { \mathrm { h e a d } } \left( \cdot \right)$ ;   
+7: end while   
+8: end if   
+9: if $\mathcal { T } _ { \mathrm { m a i n } } =$ detection then   
+10: while not converged do   
+11: Fix classification model $\mathcal { M } _ { \mathrm { c l s } }$ ;   
+12: Get a mini-batch of training data from DM ;   
+13: Calculate the overall loss $\mathcal { L } _ { \mathrm { o v e r a l l } }$ using Eq. (21);   
+14: Update the detection model, i.e., $f _ { \mathcal { M } _ { d e t } } ^ { D H C } ( \cdot )$   
+$f _ { \mathcal { M } _ { d e t } } ^ { D S C } ( \cdot )$ , and $f _ { \mathcal { M } _ { d e t } } ^ { \mathrm { h e a d } } ( \cdot )$ ;   
+15: end while   
+16: end if
+
+# Algorithm 2 Inference for DISC.
+
+Given components: Backbone $f _ { \mathcal { M } _ { n } } ^ { b } ( \cdot )$ , downstream task head $f _ { \mathcal { M } _ { n } } ^ { \mathrm { h e a d } } ( \cdot )$ ;
+
+Input: Test sample $x$ ;
+
+Output: Final prediction $y ^ { * }$ ;
+
+1: Calculate image feature $f _ { \mathcal { M } _ { c l s } } ^ { b } ( x )$ (x) and f bM ;   
+2: Calculate weights of the feature $\mathcal { W } _ { c l s } ^ { D H C }$ and $\mathcal { W } _ { d o t } ^ { D H C }$ ;   
+3: Calculate weights of the logits $\mathcal { W } _ { c l s } ^ { D S C }$ and $\mathcal { W } _ { d e t } ^ { D S C }$ ;   
+4: Return final prediction $y ^ { * }$ .
+
+# 5. Experiments
+
+In this section, we compare DISC with state-of-the-art methods on the CIFAR100 and $\mathrm { \ V O C { 0 7 + 1 2 } }$ datasets. We discuss task-driven knowledge transfer methods, and the ablation studies demonstrate the effectiveness of the DHC and DSC modules. Our model is implemented in PyTorch (Paszke et al., 2019) and deployed on an NVIDIA RTX 3090 GPU.
+
+# 5.1. Implementation Details
+
+Datasets: We evaluate the proposed method on CIFAR100 (Krizhevsky et al., 2009) and $\mathrm { \Delta V O C 0 7 } { + } 1 2$ (Everingham et al., 2010; 2015) datasets using a data-efficient setting. Specifically, we employ $10 \%$ of the training set and the complete test set, with CIFAR100 used for evaluating classification and $\mathrm { \ V O C { 0 7 + 1 2 } }$ for assessing detection.
+
+Compared methods: To enable comparison with KD and MTL in the SC setting, we adopt the hard-sharing mechanism from MTL, based on KD methods such as LSKD (Sun et al., 2024), CrossKD (Wang et al., 2024), and PPAL (Yang et al., 2024a). In this approach, the hidden layers trained under the guidance of the teacher model are shared across all tasks, while separate output layers for specific tasks are retained. Additionally, we compare several classic methods, including SwAV (Caron et al., 2020), DeepClusterV2 (Caron et al., 2018), MoCo v2 (Chen et al., 2020), and CLIP (Radford et al., 2021).
+
+# 5.2. Specialization and Generalization are All You Need
+
+We analyze the Fine-Tune (FT), KD, and MTL knowledge transfer methods based on expert models for classification and detection, as shown in Table 1. Then, we compare DISC with state-of-the-art methods, as shown in Table 2.
+
+Table 1: Comparison of performance across different tasks before and after evolution on CIFAR100 and $\mathrm { \ V O C { 0 7 + 1 2 } }$ datasets. The $1 ^ { s t } / 2 ^ { n d }$ best results are indicated in red/blue.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3"> Before evolution (%) </td><td colspan="3">After evolution (%) </td></tr><tr><td>CLS</td><td>DET AVG</td><td>CLS</td><td></td><td>DET</td><td>AVG</td></tr><tr><td>Expert(CLS)+FT</td><td>70.68 0.00</td><td>35.34</td><td>0.68</td><td></td><td>65.00</td><td>32.84</td></tr><tr><td>Expert(CLS)+KD</td><td>70.68 0.00</td><td>35.34</td><td>0.61</td><td></td><td>66.35</td><td>33.48</td></tr><tr><td>Expert(CLS)+MTL</td><td>70.68 0.00</td><td>35.34</td><td></td><td>70.68</td><td>59.92</td><td>65.30</td></tr><tr><td>Expert(DET)+FT</td><td>0.47</td><td>83.89 42.18</td><td>64.08</td><td>0.00</td><td></td><td>32.04</td></tr><tr><td>Expert(DET)+KD</td><td>0.47</td><td>83.89 42.18</td><td></td><td>66.82</td><td>0.00</td><td>33.41</td></tr><tr><td>Expert(DET)+MTL 0.47</td><td></td><td>83.89 42.18</td><td></td><td>43.05</td><td>83.89</td><td>63.47</td></tr><tr><td>DISC</td><td>70.68 0.00</td><td>35.34</td><td></td><td></td><td>72.28(+1.60) 84.92(+1.03) 78.60(+13.30)</td><td></td></tr></table>
+
+Table 2: Comparison of performance across different tasks on CIFAR100 and $\mathrm { \ V O C { 0 7 + 1 2 } }$ datasets. The $1 ^ { s t } / 2 ^ { n d }$ best results are indicated in red/blue.   
+
+<table><tr><td>Method</td><td>CLS (%) </td><td>DET (%) </td><td>AVG (%) </td></tr><tr><td>SwAV (Caron et al., 2020)</td><td>65.90</td><td>69.20</td><td>67.55</td></tr><tr><td>DeepClusterV2 (Caron et al., 2018)</td><td>66.10</td><td>70.00</td><td>68.05</td></tr><tr><td>MoCo v2 (Chen et al., 2020)</td><td>66.00</td><td>70.20</td><td>68.10</td></tr><tr><td>CLIP (Radford et al., 2021)</td><td>58.70</td><td>68.60</td><td>63.65</td></tr><tr><td>LSKD(CLS)+MTL (Sun et al., 2024)</td><td>71.05</td><td>63.37</td><td>67.21</td></tr><tr><td>LSKD(DET)+MTL (Sun et al., 2024)</td><td>50.13</td><td>83.91</td><td>67.02</td></tr><tr><td>CrossKD(CLS)+MTL (Wang et al., 2024) 71.13</td><td></td><td>63.85</td><td>67.49</td></tr><tr><td>CrossKD(DET)+MTL (Wang et al., 2024) 52.96</td><td></td><td>84.01</td><td>68.49</td></tr><tr><td>PPAL(CLS)+MTL (Yang et al., 2024a)</td><td>70.95</td><td>62.31</td><td>66.63</td></tr><tr><td>PPAL(DET)+MTL (Yang et al., 2024a)</td><td>50.68</td><td>83.95</td><td>67.32</td></tr><tr><td>DISC</td><td></td><td>72.28(+1.15) 84.92(+0.91) 78.60(+10.11)</td><td></td></tr></table>
+
+FT and KD lose task generalization: As shown in Table 1, both FT and KD obtain new downstream task capabilities at the cost of sacrificing performance on existing tasks. This raises the question of whether acquiring new task abilities alone constitutes evolution. Our goal is for the model to retain or even improve performance on old tasks while learning new ones. Additionally, even focusing on new tasks, FT and KD underperform compared to DISC, as they fail to utilize cross-task knowledge. Both FT and KD can cause models to blindly follow the evolutionary process, degrading original task capabilities and losing generalization.
+
+MTL overlooks specialization for new tasks: As shown in Table 1, we observe that MTL preserves performance on existing tasks by freezing the model for those tasks. While MTL trains new task-specific parameters for learning new downstream tasks, the model cannot fully optimize for the new tasks due to the need to maintain performance on existing ones. MTL achieves basic evolution but still falls short of our goal. MTL struggles to fully learn the capabilities for new downstream tasks while failing to enhance performance on existing tasks. In the evolutionary process, MTL prioritizes maintaining generalization for existing tasks, overlooking specialization for new tasks.
+
+DISC balances specialization and generalization: Based on the SC paradigm, DISC trains task-specific parameters while dynamically leveraging auxiliary information from other models for collaborative learning. DISC’s hierarchical organizational structures utilize both task-specific data and guidance from other tasks, acquiring intra-task and crosstask knowledge. Its progressive interaction modes enable curriculum-like learning, while strong-guided communication mechanisms allow dynamic weighting of learning and inference. DISC not only enhances new task capabilities but also improves existing task performance through model interactions, as shown in Tables 1 and 2.
+
+Analysis of specialization and generalization: Specialization and generalization are critical in SC. Specialization means the model can effectively learn new tasks or improve performance on existing ones, while generalization refers to the model’s ability to perform well across multiple tasks. As shown in Table 2, classical models struggle to maintain their original performance in data-efficient settings and on ResNet50, indicating an over-reliance on large datasets and model parameters. Although the combination of KD and MTL comes close to achieving the SC objective, there is still a noticeable gap compared to the DISC model we propose.
+
+# 5.3. Ablation Study
+
+To further verify the significance of each module, i.e., DHC and DSC, in DISC based on SC, we conduct the ablation study as shown in Table 3. The results reveal that the model with DHC and DSC attains the best performance.
+
+Table 3: Ablation study on CIFAR100 and $\mathrm { \ V O C { 0 7 + 1 2 } }$ . In the table, $" \ j ^ { , , }$ denotes DISC with the module.   
+
+<table><tr><td>Method</td><td>DHC</td><td>DSC</td><td>CLS (%) </td><td>DET (%) </td><td>AVG (%) </td></tr><tr><td rowspan="4">DISC</td><td rowspan="2">V</td><td></td><td>70.74</td><td>83.28</td><td>77.01</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td>71.26</td><td>84.24</td><td>77.76</td></tr><tr><td>V</td><td></td><td>72.28(+1.54)</td><td>84.92(+1.64)</td><td>78.60(+1.59)</td></tr></table>
+
+Without DHC and DSC: We observe that DHC and DSC are essential to DISC, as they effectively utilize the sociability information of cross-task models. In contrast, direct interaction between different downstream task models follows a static interaction mechanism that aggregates features from the same layer with uniform weights, which restricts improvements in both specialization and generalization.
+
+Using only DHC: We observe significant performance improvement. This indicates that hierarchical cross-task features interacting dynamically and progressively are highly effective in enhancing representational capability. DHC provides insight into cross-task knowledge transfer.
+
+Using only DSC: We observe that the model’s performance improves, but not as significantly as with DHC. While DHC focuses on enhancing representational capability, DSC focuses on improving decision-making capability. This suggests that both representational and decision-making capabilities are crucial for model performance. DSC provides insight into dynamic decision-making.
+
+Combining DHC and DSC: We find that the model achieves the expected results, not only adding new downstream task capabilities but also improving the performance of existing tasks through model interactions. This shows that representational ability forms the foundation for achieving good performance, while decision-making ability helps push the performance beyond its limits.
+
+# 5.4. Further Analysis
+
+The DISC, designed with SC, incorporates task-specific characteristics through the DHC and DSC modules. Classification focuses on image-wise logits interaction, while detection emphasizes patch-wise feature interaction. Despite their different focuses, the complementary information from both tasks enhances representational and decision-making capabilities, as illustrated in Figure 5.
+
+Adversarial competition: As shown in the red dashed box in Figure 5a, DHC and DSC do not consistently improve performance across all classes and may even experience performance degradation due to adversarial competition. DHC focuses on low-level intra-class information, while DSC emphasizes high-level inter-class information. Relying on only one of these modules is affected by large intraclass variation and high inter-class similarity, leading to adversarial competition.
+
+Dynamic coevolution: As illustrated in Figure 5b, the synergistic integration of DHC and DSC significantly improves class-wise performance, culminating in enhanced overall model efficacy. This underscores the critical role of both low-level and high-level information in augmenting the model’s perceptual capabilities. Mastering the effective collaboration of this complementary information is fundamental to realizing SC, with DISC offering valuable insights into this complex interplay.
+
+![](images/79ac07fb92cba65e910645fa2be24098cee2124fe9c4559aeb3890c14a0d32b4.jpg)  
+(a) Comparison of various modules on $\mathrm { \ V O C { 0 7 + 1 2 } }$ dataset
+
+![](images/efcdad576e46b215766afdd4ac535069289458aceea2a7538d9c65442ad08092.jpg)  
+Figure 5: The performance differences for each class.
+
+# 5.5. Discussion
+
+SC is a cross-task collaborative paradigm that facilitates knowledge interaction, allowing models to co-evolve while improving both specialization and generalization. Alongside advancing learning paradigms, SC is applicable to defect detection, autonomous driving, and emergency rescue. Table 4 presents a comparative analysis of learning paradigms in addressing diverse challenges.
+
+Table 4: Comparison of different learning paradigms.   
+
+<table><tr><td>Paradigm Specialization</td><td></td><td>Generalization Dynamism Hierarchy</td><td></td><td></td></tr><tr><td>KD</td><td></td><td>x</td><td>X</td><td>x</td></tr><tr><td>MTL</td><td>x</td><td></td><td>x</td><td>x</td></tr><tr><td> SC</td><td></td><td></td><td></td><td></td></tr></table>
+
+i) Specialization: The model needs to preserve or improve the performance of original tasks while learning new tasks.
+
+ii) Generalization: As new tasks arise, the model needs to adapt without compromising existing performance.
+
+iii) Dynamism: Model interaction should be weighted by each model’s proficiency with each sample.
+
+iv) Hierarchy: Model interaction should be diverse and comprehensive to maximize knowledge utilization.
+
+Existing KD and MTL paradigms face challenges in balancing specialization and generalization in dynamic environments. SC overcomes this by enabling knowledge sharing and coevolution via DHC and DSC modules across different downstream tasks.
+
+# 6. Conclusion
+
+In this paper, we introduce a practical SC paradigm with a rigorous mathematical foundation and an informationtheoretic explanation. Additionally, we carefully design DISC based on SC to not only incorporate new downstream task capabilities but also enhance the performance of existing tasks through model interactions. Essentially, we view specialization and generalization as two sides of the same coin, not independent issues. The dynamic interaction between features and logits provides new insights into SC, highlighting the importance of tailored approaches to maximize strengths. In future work, we plan to further explore the potential of combining multiple modalities and tasks.
+
+# Acknowledgements
+
+This work was supported in part by the National Science and Technology Major Project under Grant 2022ZD0116500, in part by the National Natural Science Foundation of China under Grants 62436002, 62476195, U23B2049, and 62222608, in part by Tianjin Natural Science Funds for Distinguished Young Scholar under Grant 23JCJQJC00270, in part by Zhejiang Provincial Natural Science Foundation of China under Grant LD24F020004, in part by Tianjin Young Scientific and Technological Talents Project under grant QN20230305, in part by Tianjin Science and Technology Plan Project under grant 24YDTPJC00150, in part by Natural Science Foundation of Tianjin under grant 24JCYBJC00950, and in part by the Huawei Ascend Computing Platform.
+
+# Impact Statement
+
+This paper presents work aimed at advancing the field of coevolution in machine learning. Our goal is to propose a socialized coevolution learning paradigm to enhance the performance of existing downstream tasks and facilitate the learning of new downstream tasks, making full use of the complementarities between different tasks. However, due to the presence of model heterogeneity in dynamic open environments, the application of our approach to real-world scenarios may inevitably encounter heterogeneity challenges.
+
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+
+# A. Supplementary Material
+
+The supplementary material contains comprehensive details on the implementations and experimental results referenced in the main paper, along with supplementary theoretical analysis and in-depth discussions. It is organized as follows:
+
+• In Section B, we offer a thorough overview of the methods compared in the main paper, accompanied by a detailed description of the datasets utilized. • In Section C, we present the full set of experimental results, accompanied by an in-depth analysis that thoroughly evaluates the model’s performance. • In Section D, we provide a comprehensive theoretical analysis of sociability information, exploring its implications within the context of the study. • In Section E, we explore SC’s insights across realworld applications and learning paradigms, aiming to spark interest in this dynamic interdisciplinary field.
+
+# B. Implementation Details
+
+In this section, we offer a detailed description of the methods compared in the main paper, as well as the datasets used. For classification as the primary task, the model is trained for 1500 epochs using SGD with a batch size of 64. The threshold is set to 300, with a learning rate of 0.01, momentum of 0.9, and weight decay of 0.001. For detection as the primary task, the model is trained for 600 epochs using SGD with a batch size of 32. The threshold is also set to 300, with a learning rate of 0.03, momentum of 0.9, and weight decay of 0.001. All reported results represent the mean values obtained from 3 independent trials.
+
+# B.1. Compared Methods
+
+In this subsection, we provide an overview of the methods compared in the main paper, outlining their key characteristics. The methods considered are as follows:
+
+• SwAV (Caron et al., 2020): state-of-the-art unsupervised learning method, which improves representation by using online clustering with swapped prediction.   
+• DeepClusterV2 (Caron et al., 2018): state-of-theart unsupervised learning method, which jointly trains convolutional networks and uses $\mathbf { k }$ -means clustering for end-to-end optimization.   
+• MoCov2 (Chen et al., 2020): state-of-the-art contrastive learning method, which improves image representation by incorporating a projection head and stronger data augmentation.   
+• CLIP (Radford et al., 2021): state-of-the-art contrastive learning method, which pre-trains on a large dataset of pairs and enables zero-shot transfer across various tasks without task-specific training.   
+• LSKD (Sun et al., 2024): state-of-the-art knowledge distillation method, which standardizes logits using Z-score before softmax.   
+• CrossKD (Wang et al., 2024): state-of-the-art knowledge distillation method, which aligns student and teacher outputs through cross-head mimicking.   
+• PPAL (Yang et al., 2024a): state-of-the-art active learning method, which combines uncertainty-based and diversity-based sampling.
+
+# B.2. Datasets
+
+The $\mathrm { \ V O C { 0 7 + 1 2 } }$ training dataset contains 47,221 objects, while the testing dataset includes 14,976 objects. The CIFAR100 training dataset consists of 50,000 images, with 10,000 images in the testing set. In the data-efficient setting, we utilize $10 \%$ of the training data: for $\mathrm { \ V O C { 0 7 + 1 2 } }$ $( 1 0 \% )$ , the training set contains 4,696 objects, and the testing set remains at 14,976 objects. For CIFAR100 $( 1 0 \% )$ , the training set includes 5,000 images, with the testing set consisting of 10,000 images. We list the details of the datasets in Table 5.
+
+# C. Full Experimental Results
+
+In this section, we present additional experimental results, including analyses of parameters and visualizations.
+
+# C.1. Analysis of Parameters
+
+Dynamic progressive interaction: Inspired by curriculum learning (Bengio et al., 2009), we apply sine and cosine in Dynamic Information Socialized Collaboration (DISC) to achieve dynamic progressive interaction. DISC incorporates two key hyper-parameters: training loops $E _ { \mathrm { l o o p s } }$ and threshold $E _ { t }$ , both of which influence the dynamic progressive interaction of the model. As shown in Figure 6, the sine function achieves superior performance on CIFAR100 when $E _ { \mathrm { l o o p s } } = 1 5 0 0$ and $E _ { t } = 3 0 0$ , while $E _ { \mathrm { l o o p s } } = 6 0 0$ and $E _ { t } = 3 0 0$ yield better results on $\mathrm { \ V O C { 0 7 + 1 2 } }$ .
+
+# C.2. Analysis of Visualizations
+
+Competition and coevolution: In the main paper, we analyze Figure 5 to demonstrate that using Dynamic Hierarchical Collaboration (DHC) or Dynamic Selective Collaboration (DSC) alone cannot fully realize coevolution for certain categories, due to high intra-class variance and inter-class similarity. Only their combined application effectively overcomes these challenges. To further investigate, we perform
+
+Table 5: Detailed datasets.   
+
+<table><tr><td>Datasets</td><td>Training dataset</td><td>Testing dataset</td><td>Detailed classes</td></tr><tr><td>VOC07+12</td><td>47,221</td><td>14,976</td><td>aeroplane, bicycle, bird, boat, bus, car, cat, chair, cow, dining table, dog, horse, motorbike, person, potted plant, sheep, sofa, train, TV monitor</td></tr><tr><td>VOC07+12(10%)</td><td>4,696</td><td>14,976</td><td>aeroplane, bicycle, bird, boat, bus, car, cat, chair, cow, dining table, dog, horse, motorbike, person, potted plant, sheep, sofa, train, TV monitor</td></tr><tr><td>CIFAR100</td><td>50,000</td><td>10,000</td><td>apple, aquarium fish, baby, bear, beaver, bed, bee, beetle, bicycle, bottle, bowl, boy, bridge, bus, butterfly, camel, can, castle, caterpillar, cattle, chair, chimpanzee, clock, cloud, cockroach, couch, crab, crocodile, cup, dinosaur, dolphin, elephant, flatfish, forest, fox, girl, hamster, house, kangaroo, keyboard, lamp, lawn mower, leopard, lion, lizard, lobster, man, maple tree, motorcycle, mountain, mouse, mushroom, oak tree, orange, orchid, otter, palm tree, pear, pickup truck, pine tree, plain, plate, poppy, porcupine, possum, rabbit, raccoon, ray, road, rocket,rose, sea, seal, shark, shrew, skunk, skyscraper, snail, snake, spider, squirrel, streetcar, sunflower, sweet pepper, table, tank, telephone, television, tiger, tractor, train, trout, tulip, turtle, wardrobe, whale, willow tree, wolf, woman, worm</td></tr><tr><td>CIFAR100(10%)</td><td>5,000</td><td>10,000</td><td>apple, aquarium fish, baby, bear, beaver, bed, bee, beetle, bicycle, bottle, bowl, boy, bridge, bus, butterfly, camel, can, castle, caterpillar, cattle, chair, chimpanzee, clock, cloud, cockroach, couch, crab, crocodile, cup, dinosaur, dolphin, elephant, flatfish, forest, fox, girl, hamster, house, kangaroo, keyboard, lamp, lawn mower, leopard, lion, lizard, lobster, man, maple tree, motorcycle, mountain, mouse, mushroom, oak tree, orange, orchid, otter, palm tree, pear, pickup truck, pine tree, plain, plate, poppy, porcupine, possum, rabbit, raccoon, ray, road, rocket,rose, sea, seal, shark, shrew, skunk, skyscraper, snail, snake, spider, squirrel, streetcar, sunflower, sweet pepper, table, tank, telephone, television, tiger, tractor, train, trout, tulip, turtle, wardrobe, whale, willow tree, wolf, woman, worm</td></tr></table>
+
+![](images/18a4346acfabcfb6f7ed2267a48797dee7bc6d00a454e2e39804494b201feefd.jpg)  
+Figure 6: Comparison of different weights.
+
+Class Activation Maps (CAMs) analysis (Zhou et al., 2016), as shown in Figure 7. DHC emphasizes low-level intra-class features, which can cause confusion for categories with similar textures, while DSC focuses on high-level inter-class features, which may confuse categories with high inter-class similarity. To address these issues, we integrate DHC and DSC into the DISC model, eliminating adversarial competition and promoting coevolution.
+
+# D. Theoretical Analysis
+
+Below, we provide the omitted proofs from Sections 3 and 4 (refer to the main paper).
+
+# D.1. Proof of Theorem 3.3
+
+Proof. We leverage previous results (Cover, 1999; Feder & Merhav, 1994; Li et al., 2023a):
+
+$$
+- \log { \left( 1 - P _ { e _ { c } } ^ { m u l } \right) } \leq H ( Y \mid X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) ,
+$$
+
+$$
+H ( Y \mid X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) \leq \log 2 + P _ { e _ { c } } ^ { m u l } \log | Y | .
+$$
+
+Combine the two inequalities and put $P _ { e _ { c } } ^ { m u l }$ in the middle:
+
+$$
+\begin{array} { r } { \frac { H ( Y | X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) - \log 2 } { \log | Y | } \leq P _ { e _ { c } } ^ { m u l } \leq 1 - \exp ( - H ( Y \mid X _ { \mathcal { M } _ { 1 } } , X _ { \mathcal { M } _ { 2 } } ) ) , } \end{array}
+$$
+
+which is the first result in the theorem. Then we apply the results to P sin:
+
+$$
+\begin{array} { r } { \frac { H ( Y \mid X _ { \mathcal { M } _ { 1 } } ) - \log 2 } { \log \mid Y \mid } \leq P _ { e _ { c } } ^ { s i n } \leq 1 - \exp ( - H ( Y \mid X _ { \mathcal { M } _ { 1 } } ) ) . } \end{array}
+$$
+
+![](images/9e2ce61f35d342345cfae916a22b84c21247580d6e0a6b5b3cbec86168022b65.jpg)  
+Figure 7: The visualization with different modules.
+
+Since
+
+$$
+\begin{array} { r l } {  { \Phi _ { X _ { \mathcal M _ { 2 } } } = I ( X _ { \mathcal M _ { 2 } } ; Y \mid X _ { \mathcal M _ { 1 } } ) } } \\ & { = I ( Y ; X _ { \mathcal M _ { 1 } } , X _ { \mathcal M _ { 2 } } ) - I ( Y ; X _ { \mathcal M _ { 1 } } ) } \\ & { = [ H ( Y ) - H ( Y \mid X _ { \mathcal M _ { 1 } } , X _ { \mathcal M _ { 2 } } ) ] - [ H ( Y ) - H ( Y \mid X _ { \mathcal M _ { 1 } } ) ] } \\ & { = H ( Y | X _ { \mathcal M _ { 1 } } ) - H ( Y | X _ { \mathcal M _ { 1 } } , X _ { \mathcal M _ { 2 } } ) . } \end{array}
+$$
+
+# We can derive
+
+$$
+\frac { H ( Y | X _ { M _ { 1 } } ) - \Phi _ { X _ { M _ { 2 } } } - \log 2 } { \log | Y | } \leq P _ { e _ { c } } ^ { m u l } \leq 1 - \exp ( - H ( Y \mid X _ { M _ { 1 } } ) + \Phi _ { X _ { M _ { 2 } } } ) .
+$$
+
+# D.2. Proof of Theorem 3.5
+
+Proof. We leverage previous results (Zuo et al., 2024; Li et al., 2023a) and assume that $w ^ { * }$ is the optimal parameter for the given dataset $\mathcal { D } _ { \mathcal { M } _ { 1 } }$ . Hence, we have:
+
+$$
+\begin{array} { r l } & { \log p \left( \boldsymbol { w ^ { * } } \mid \mathcal { D } _ { \mathcal { M } _ { 1 } } \right) } \\ & { = \log p \left( \mathcal { D } _ { \mathcal { M } _ { 1 } } \mid \boldsymbol { w } ^ { * } \right) + \log p \left( \boldsymbol { w } ^ { * } \right) - \log p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) . } \end{array}
+$$
+
+For $p ( \mathcal { D } _ { \mathbf { \mathcal { M } } _ { 1 } } )$ , we have
+
+$$
+p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) = \int p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } \mid w ) p ( w ) d w .
+$$
+
+Since
+
+$$
+\begin{array} { r } { \int p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } \mid w ) p ( w ) d w < \int p \left( \mathcal { D } _ { \mathcal { M } _ { 1 } } \mid w ^ { * } \right) p ( w ) d w . } \end{array}
+$$
+
+We can derive
+
+$$
+p ( \mathcal { D _ { M _ { 1 } } } ) < p \left( \mathcal { D _ { M _ { 1 } } } \mid w ^ { * } \right) .
+$$
+
+After integrating $\mathcal { D } _ { \mathcal { M } _ { 1 } }$ and ${ \mathcal { D } } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ , the combined dataset is $\mathcal { D } _ { S } = \mathcal { D } _ { \mathcal { M } _ { 1 } } \cup \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ , yielding
+
+$$
+\begin{array} { r l } & { \log p ( w ^ { * } \mid \mathcal { D } _ { S } ) } \\ & { = \log p ( \mathcal { D } _ { S } \mid w ^ { * } ) + \log p ( w ^ { * } ) - \log p ( \mathcal { D } _ { S } ) } \\ & { = \log p ( \mathcal { D } _ { M _ { 1 } } , \mathcal { D } _ { \Phi _ { M _ { 2 } } } \mid w ^ { * } ) + \log p ( w ^ { * } ) } \\ & { \quad - \log p ( \mathcal { D } _ { M _ { 1 } } , \mathcal { D } _ { \Phi _ { M _ { 2 } } } ) } \\ & { = \log p ( \mathcal { D } _ { M _ { 1 } } \mid w ^ { * } ) + \log p ( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \mid w ^ { * } ) + \log p ( w ^ { * } ) } \\ & { \quad - \log p ( \mathcal { D } _ { M _ { 1 } } ) - \log p ( \mathcal { D } _ { \Phi _ { M _ { 2 } } } ) } \\ & { = \log p ( \mathcal { D } _ { M _ { 1 } } \mid w ^ { * } ) + \log p ( w ^ { * } ) - \log p ( \mathcal { D } _ { M _ { 1 } } ) } \\ & { \quad + \log p ( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \mid w ^ { * } ) - \log p ( \mathcal { D } _ { \Phi _ { M _ { 2 } } } ) . } \end{array}
+$$
+
+Assuming the supplementary augmented dataset ${ \mathcal { D } } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ closely resembles the data distribution of $\mathcal { D } _ { \mathcal { M } _ { 1 } }$ , we have
+
+$$
+\begin{array} { r } { p \left( \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \mid \boldsymbol { w } ^ { * } \right) > p \left( \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) . } \end{array}
+$$
+
+We can derive
+
+$$
+\begin{array} { r l } & { \log p ( w ^ { * } \mid \mathcal { D } _ { S } ) } \\ & { > \log p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } \mid w ^ { * } ) + \log p ( w ^ { * } ) - \log p ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) } \\ & { = \log p ( w ^ { * } \mid \mathcal { D } _ { \mathcal { M } _ { 1 } } ) . } \end{array}
+$$
+
+have
+
+$$
+\begin{array} { r l } & { \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f \left( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) \right) } \\ & { = \log p \left( f \left( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) \mid \boldsymbol { w } _ { h } ^ { * } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \right) - \log p \left( f \left( \mathcal { D } _ { \Phi _ { M _ { 2 } } } \right) \right) } \\ & { < \log p \left( f \left( \mathcal { D } _ { M _ { 1 } } \right) ^ { \Phi _ { M _ { 2 } } } \mid \boldsymbol { w } _ { h } ^ { * } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \right) } \\ & { - \log p \left( f \left( \mathcal { D } _ { M _ { 1 } } \right) ^ { \Phi _ { M _ { 2 } } } \right) } \\ & { = \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f ( \mathcal { D } _ { M _ { 1 } } ) ^ { \Phi _ { M _ { 2 } } } \right) . } \end{array}
+$$
+
+We can derive
+
+$$
+\begin{array} { r l } & { \log p \left( \boldsymbol { w ^ { * } } \mid \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) } \\ & { = \log p \left( \boldsymbol { w } _ { f } ^ { * } \mid \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f \left( \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) \right) } \\ & { < \log p \left( \boldsymbol { w } _ { f } ^ { * } \mid \mathcal { D } _ { \Phi _ { \mathcal { M } _ { 2 } } } \right) + \log p \left( \boldsymbol { w } _ { h } ^ { * } \mid f ( \mathcal { D } _ { \mathcal { M } _ { 1 } } ) ^ { \Phi _ { \mathcal { M } _ { 2 } } } \right) . } \end{array}
+$$
+
+# D.4. Proof of Theorem 4.1
+
+Proof. Let $D _ { \mathrm { t r a i n } } ~ = ~ \{ x _ { i } , y _ { i } \} _ { i = 1 } ^ { N }$ be a training dataset of $N$ samples, and $e ^ { { \hat { r } } r } ( f ^ { m } )$ be the empirical error of the $m$ - th model $f ^ { m }$ on $D _ { \mathrm { t r a i n } }$ . For any hypothesis $f \in \mathcal { H }$ (i.e., $\mathcal { H } : \mathcal { X }  \{ - 1 , 1 \} )$ , with probability at least $1 - \delta$ , the generalization error is bounded by:
+
+$$
+\begin{array} { r l r } {  { \mathrm { G E r r o r } ( f ) \le \underbrace { \displaystyle \sum _ { m = 1 } ^ { M } \mathbb { E } ( w ^ { m } ) e \hat { r } r ( f ^ { m } ) } _ { \mathrm { T e m . ~ L ( \hat { \mathrm { e m p i r i c a l l o s } } ) } } + \underbrace { \sum _ { m = 1 } ^ { M } \mathbb { E } ( w ^ { m } ) \Re _ { m } ( f ^ { m } ) } _ { \mathrm { T e m . ~ C ( \hat { \mathrm { c o m p l e x i t y } } ) } } } } \\ & { } & { + \underbrace { \displaystyle \sum _ { m = 1 } ^ { M } C o v ( w ^ { m } , \ell ^ { m } ) } _ { \mathrm { T e m . ~ C o v ( \mathrm { c o v a r i a n c e } ) } } + M \sqrt { \frac { \ln ( 1 / \delta ) } { 2 N } } , \quad \quad } \end{array}
+$$
+
+# D.3. Proof of Theorem 3.7
+
+Proof. We decompose the DISC into two components: the backbone with parameters $\boldsymbol { w } _ { f } ^ { * }$ and the head with parameters $w _ { h } ^ { * }$ . Using the supplementary augmented dataset ${ \mathcal { D } } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ , we obtain
+
+$$
+\begin{array} { r l } & { \log p \left( w ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) } \\ & { = \log p \left( w _ { f } ^ { * } , w _ { h } ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) } \\ & { = \log p \left( w _ { f } ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) + \log p \left( w _ { h } ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) } \\ & { = \log p \left( w _ { f } ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) + \log p \left( w _ { h } ^ { * } \mid f \left( \mathcal D _ { \Phi _ { M _ { 2 } } } \right) \right) } \\ & { = \log p \left( w _ { f } ^ { * } \mid \mathcal D _ { \Phi _ { M _ { 2 } } } \right) + \log p \left( f \left( \mathcal D _ { \Phi _ { M _ { 2 } } } \right) \mid w _ { h } ^ { * } \right) } \\ & { + \log p \left( w _ { h } ^ { * } \right) - \log p \left( f \left( \mathcal D _ { \Phi _ { M _ { 2 } } } \right) \right) . } \end{array}
+$$
+
+Since ${ \mathcal { D } } _ { \Phi _ { \mathcal { M } _ { 2 } } }$ is the supplementary augmentation of the dataset $\mathcal { D } _ { \mathcal { M } _ { 1 } }$ , it has a similar distribution with $\mathcal { D } _ { \mathcal { M } _ { 1 } }$ , we where $\mathbb { E } ( w ^ { m } )$ is the expected collaboration weights, $\Re _ { m } ( f ^ { m } )$ is the Rademacher complexity of model $f ^ { m }$ , and $C o v ( w ^ { m } , \ell ^ { m } )$ is the covariance between the weight and the loss.
+
+In static collaboration, the weights wmstatic are constant, hence
+
+$$
+C o v ( w _ { \mathrm { s t a t i c } } ^ { m } , \ell ^ { m } ) = 0 .
+$$
+
+In dynamic collaboration, the collaboration weight $w _ { \mathbf { d y } } ^ { m }$ namic increases as the model loss $\ell ^ { m }$ decreases. Thus,
+
+$$
+C o v ( w _ { \mathrm { d y n a m i c } } ^ { m } , \ell ^ { m } ) \leq 0 ,
+$$
+
+which effectively reduces the Term-Cov and thereby lowers the generalization bound.
+
+Based on the principle of convexity, it can be concluded that:
+
+$$
+\sum \mathbb { E } ( w _ { \mathrm { d y n a m i c } } ^ { m } ) e \hat { r } r ( f ^ { m } ) \leq \sum w _ { \mathrm { s t a t i c } } ^ { m } e \hat { r } r ( f ^ { m } ) .
+$$
+
+$$
+\sum \mathbb { E } ( w _ { \mathrm { d y n a m i c } } ^ { m } ) \mathfrak { R } _ { m } ( f ^ { m } ) \leq \sum w _ { \mathrm { s t a t i c } } ^ { m } \mathfrak { R } _ { m } ( f ^ { m } ) .
+$$
+
+Since the confidence term $M { \sqrt { \frac { \ln ( 1 / \delta ) } { 2 N } } }$ ln(1/δ) is independent of the collaboration strategy, it remains the same. Therefore, suppose the hypothesis space is $\mathcal { H } : \mathcal { X }  \{ - 1 , 1 \}$ . Then for any fdynamic, $f _ { \mathrm { s t a t i c } } \in \mathcal { H }$ , and for $1 > \delta > 0$ , it holds that
+
+$$
+\mathcal { O } ( \mathrm { G E r r o r } ( f _ { \mathrm { d y n a m i c } } ) ) \leq \mathcal { O } ( \mathrm { G E r r o r } ( f _ { \mathrm { s t a t i c } } ) ) .
+$$
+
+# E. Future Applications and Future Research
+
+In this section, we provide an in-depth exploration of SC’s insights across diverse real-world applications and learning paradigms, highlighting its potential to bridge disciplinary boundaries. By shedding light on its foundational principles and practical implications, we aim to spark broader interest in this rapidly evolving and interdisciplinary domain.
+
+# E.1. Future Applications of SC
+
+This section delves into the future applications of SC in emerging fields. We have studied various potential application scenarios, analyzed their unique requirements, challenges, and opportunities, explored the different uses of SC across different domains, and provided practical guidance on how to implement it.
+
+Defect detection: Defect detection and segmentation tasks for industrial products typically involve processing highdimensional, noisy image data. Moreover, the manifestations of faults are diverse, involving different materials, structures, and processes, and may appear as small cracks, corrosion, wear, etc. This requires models to handle complex and heterogeneous data in a diversified manner, which poses a significant challenge for the application of SC. Future research should focus on ensuring that the coevolution between different models effectively enhances their individual performance without causing a decline in performance. Additionally, the interaction between detection and segmentation models requires precise design, enabling them to mutually promote each other’s learning, rather than simply optimizing their own objectives independently.
+
+Autonomous driving: In autonomous driving, road environment segmentation and object detection tasks need to handle complex and dynamic scenarios, including varying weather conditions, visibility, and road situations. Ensuring that SC between different models can effectively enhance each other’s performance in such dynamically changing environments is a significant challenge. By applying SC to autonomous driving, the detection model can assist the segmentation model in accurately identifying different traffic signs, pedestrians, vehicles, and other objects, while the segmentation model can provide more precise background information and spatial distribution for object detection. Future research should focus on designing adaptive feedback mechanisms that allow models to mutually promote each other during joint training and flexibly adjust according to the demands of different scenarios and tasks.
+
+Emergency rescue: In emergency rescue scenarios, Augmented Reality (AR) technology can play a crucial role. In complex rescue environments, AR can help rescue personnel quickly find the best route by real-time planning and displaying safe paths and obstacle avoidance information. For lost item classification, AR can overlay real-time classification data and item features onto the real-world scene, assisting rescue personnel in quickly identifying and categorizing items. The application of SC in AR for emergency rescue scenarios presents the following challenges. Emergency rescue scenes are typically highly dynamic and uncertain, requiring models to not only have high accuracy in detection and classification but also to possess strong robustness. Additionally, real-time detection and classification tasks have different requirements: object detection focuses on quickly and accurately locating and identifying key targets at the rescue site, while item classification focuses on the rapid classification and archiving of items. In this context, research should focus on designing SC mechanisms that enable the two models to mutually enhance each other, creating a synergistic effect.
+
+# E.2. Future Research and Expansion of SC
+
+In addition to the future applications of SC mentioned above, in future research, SC can also be integrated with various existing learning paradigms to be further extended, achieving better performance across a range of tasks.
+
+Federated Learning (FL): FL is a form of distributed learning that allows models on different devices to be trained locally and only share updates with a central server. This approach ensures data privacy, making it particularly suitable for collaborative learning in multi-device environments. Future research should focus on ensuring collaborative learning between models, optimizing communication efficiency, and accelerating the evolution process in multi-device environments, while adhering to data privacy protection requirements.
+
+In the context of FL, models on different devices can autonomously evolve locally and share the results of their local updates through a socialization mechanism to enhance collaborative effects. For example, in industrial defect detection tasks, due to the involvement of proprietary research and development secrets in industrial products, full opensource sharing is not feasible. However, companies can contribute their defect detection models, which can then be shared via the socialization mechanism on a central server, driving advancements in industrial defect detection and segmentation. The integration of SC with FL enables models distributed across different locations to collaborate in evolution, safeguarding data privacy while enhancing the overall intelligence level of the system.
+
+Continual Learning (CL): CL aims to enable models to gradually expand their knowledge while continuously re ceiving new tasks and data, without forgetting previously learned information. This learning paradigm is particularly suitable for intelligent systems that operate over long periods. Future research should focus on combining continual learning with SC, ensuring that models can share data and knowledge while avoiding catastrophic forgetting.
+
+In the framework of CL, models can autonomously evolve when faced with constantly changing environments, and share local updates through a socialization mechanism to enhance collaborative effects and reduce forgetting. For example, in autonomous driving tasks, as road environments and traffic conditions continuously change, new objects or scenarios may emerge, such as suddenly appearing wild animals, different forms of accident vehicles, or even unseen obstacles. Although these new targets or situations might represent entirely new categories, through CL, the autonomous driving system can gradually adapt and expand its knowledge base, while SC helps prevent catastrophic forgetting, ensuring that the model does not lose the ability to handle previous tasks while adapting to new ones. The integration of SC and CL enables autonomous driving models in different environments to evolve collaboratively, not only enhancing model adaptability but also ensuring the system as a whole maintains stronger robustness and performance over the long term.
+
+Online Learning (OL): OL is a process in which models continuously learn from streaming data, typically updating as data arrives gradually, thus avoiding the computational burden associated with batch training. SC can leverage the real-time feedback mechanism of OL, allowing different models to update collaboratively based on the latest data, rather than waiting for batch data accumulation.
+
+In a constantly changing environment, the coevolution between models can accelerate their learning process and ensure that collaborative optimization does not cause performance fluctuations due to the dynamic changes in the data stream. For example, in emergency rescue scenarios, due to the complexity of on-site conditions, the collected data often contains missing values, noise, and biases, leading to potential shifts in sample distribution. This presents a major challenge for OL. However, within the SC framework, different models can collaborate with each other, where the detection model can help the segmentation model identify missing or partially damaged objects, and the segmentation model can provide more precise contextual information to improve the detection model’s adaptation to the environment. In this way, SC can enhance the model’s adaptability, ensuring that it maintains high performance and robustness even when faced with distribution shifts and drift in the data.

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+{"idx": 1, "title": "Unraveling coevolutionary dynamics using ecological genomics", "date": "", "ddg_snippet": "We discuss the recent advances in coevolutionary theory and genomics as well as shortcomings, to identify coevolving genes that take into account this spatial and temporal variability of coevolution , and propose a practical guide to understand the dynamic of coevolution using an ecological genomics lens.", "subpage_snippet": "", "source": "www.sciencedirect.com", "link": "https://www.sciencedirect.com/science/article/pii/S0168952522001160", "content": "We discuss the recent advances in coevolutionary theory and genomics as well as shortcomings, to identify coevolving genes that take into account this spatial and temporal variability of coevolution , and propose a practical guide to understand the dynamic of coevolution using an ecological genomics lens."}
+{"idx": 2, "title": "synthesis of coevolution across levels of biological organization ...", "date": "", "ddg_snippet": "Examples of coevolution across different levels of biological organization, and whether such coevolution usually occurs between parties with shared interests (mutualistic coevolution ) or parties with opposed interests (antagonistic coevolution ).", "subpage_snippet": "", "source": "academic.oup.com", "link": "https://academic.oup.com/evolut/article/78/2/211/7471347", "content": "Examples of coevolution across different levels of biological organization, and whether such coevolution usually occurs between parties with shared interests (mutualistic coevolution ) or parties with opposed interests (antagonistic coevolution )."}
+{"idx": 3, "title": "Dynamic Cooperative Coevolution for Large Scale Optimization", "date": "", "ddg_snippet": "The cooperative coevolution (CC) framework achieves a promising performance in solving large scale global optimization problems. The framework encounters difficulties on nonseparable problems, where variables interact with each other. Using the static grouping methods, variables will be theoretically grouped into one big subcomponent, whereas the random grouping strategy endures low efficiency ...", "subpage_snippet": "", "source": "ieeexplore.ieee.org", "link": "https://ieeexplore.ieee.org/document/8628260", "content": "The cooperative coevolution (CC) framework achieves a promising performance in solving large scale global optimization problems. The framework encounters difficulties on nonseparable problems, where variables interact with each other. Using the static grouping methods, variables will be theoretically grouped into one big subcomponent, whereas the random grouping strategy endures low efficiency ..."}
+{"idx": 4, "title": "Socialized Coevolution: Advancing a Better World through Cross-Task ...", "date": "", "ddg_snippet": "Specifically, we introduce the dynamic hierarchical collabo-ration and dynamic selective collaboration mod-ules to enable dynamic and effective interactions among models, allowing them to acquire knowl-edge from these interactions.", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/pdf?id=0WQJ6DFSKp", "content": "Specifically, we introduce the dynamic hierarchical collabo-ration and dynamic selective collaboration mod-ules to enable dynamic and effective interactions among models, allowing them to acquire knowl-edge from these interactions."}
+{"idx": 5, "title": "The Hierarchical Coevolutionary Units of Ecological Networks", "date": "", "ddg_snippet": "Cohesive groups displayed hierarchical organisation, and potential coevolutionary effects overflowing lower-scale groups were contained by higher-scale groups, underscoring the hierarchy's impact. However, indirect coevolutionary effects blurred group boundaries and hierarchy, particularly under strong selection from ecological interactions.", "subpage_snippet": "", "source": "pubmed.ncbi.nlm.nih.gov", "link": "https://pubmed.ncbi.nlm.nih.gov/39354909/", "content": "Cohesive groups displayed hierarchical organisation, and potential coevolutionary effects overflowing lower-scale groups were contained by higher-scale groups, underscoring the hierarchy's impact. However, indirect coevolutionary effects blurred group boundaries and hierarchy, particularly under strong selection from ecological interactions."}
+{"idx": 6, "title": "GitHub - yxjdarren/SC: The paper has been accepted to ICML 2025.", "date": "", "ddg_snippet": "The code repository for \" Socialized Coevolution : Advancing a Better World through Cross-Task Collaboration \" (the paper has been accepted by ICML 2025) in PyTorch.", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/yxjdarren/SC", "content": "The code repository for \" Socialized Coevolution : Advancing a Better World through Cross-Task Collaboration \" (the paper has been accepted by ICML 2025) in PyTorch."}
+{"idx": 7, "title": "两篇论文被icml 2025录用 - Visdrone", "date": "", "ddg_snippet": "论文题目： Socialized Coevolution : Advancing a Better World through Cross-Task Collaboration 作者：姚鑫杰（博士研究生），王煜，朱鹏飞，林婉瑜，赵睿朴，郭周鹏，李维浩，胡清华 论文概述： 近期，团队提出了一个多智能体跨任务协同感知的社会化学习框架，已被2025国际机器学习大会（ICML）录用。在动态开放 ...", "subpage_snippet": "", "source": "aiskyeye.com", "link": "https://aiskyeye.com/论文《socialized-coevolution-advancing-a-better-world-through-cross-task-collaboration》和《task-gated-multi-expert-collaboration-network-for-degraded-multi/", "content": "论文题目： Socialized Coevolution : Advancing a Better World through Cross-Task Collaboration 作者：姚鑫杰（博士研究生），王煜，朱鹏飞，林婉瑜，赵睿朴，郭周鹏，李维浩，胡清华 论文概述： 近期，团队提出了一个多智能体跨任务协同感知的社会化学习框架，已被2025国际机器学习大会（ICML）录用。在动态开放 ..."}
+{"idx": 8, "title": "The Concept of 'Co-evolution' and Its Application in the Social ...", "date": "", "ddg_snippet": "Kauffman [ 19 ] argues that co-evolution is the central dynamic of all self-organizing behaviour. It allows dynamic system change to occur, and allows innovative structures to emerge. Mitleton-Kelly was one of the first complexity theorists to apply the concept of co-evolution within the social sciences.", "subpage_snippet": "", "source": "link.springer.com", "link": "https://link.springer.com/chapter/10.1007/978-3-642-36614-7_3", "content": "Kauffman [ 19 ] argues that co-evolution is the central dynamic of all self-organizing behaviour. It allows dynamic system change to occur, and allows innovative structures to emerge. Mitleton-Kelly was one of the first complexity theorists to apply the concept of co-evolution within the social sciences."}
+{"idx": 9, "title": "From employee socialization to co‐evolution: A lifespan ...", "date": "", "ddg_snippet": "Various theories have highlighted how employees evolve in their organization and how organizations influence this process, but only portray part of the complex relations among these interacting social entities. We thus propose a meta-theory to unify these multiple theories, including symbolic interactionism, employee/organizational socialization theory, human resource management (HRM) systems ...", "subpage_snippet": "", "source": "iaap-journals.onlinelibrary.wiley.com", "link": "https://iaap-journals.onlinelibrary.wiley.com/doi/full/10.1111/apps.12572", "content": "Various theories have highlighted how employees evolve in their organization and how organizations influence this process, but only portray part of the complex relations among these interacting social entities. We thus propose a meta-theory to unify these multiple theories, including symbolic interactionism, employee/organizational socialization theory, human resource management (HRM) systems ..."}

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+{"idx": 1, "title": "AI-Powered Paper Summarization about the arXiv paper 2212 . 09251 v1", "date": "", "ddg_snippet": "Easy-to-read summary of the arXiv paper 2212 . 09251 v1 entitled Discovering Language Model Behaviors with Model-Written Evaluations.", "subpage_snippet": "", "source": "summarizepaper.com", "link": "https://summarizepaper.com/en/arxiv-id/2212.09251v1/", "content": "Easy-to-read summary of the arXiv paper 2212 . 09251 v1 entitled Discovering Language Model Behaviors with Model-Written Evaluations."}
+{"idx": 2, "title": "[ 2212 . 09251 ] Discovering Language Model Behaviors with... | Notion", "date": "", "ddg_snippet": "https://arxiv.org/abs/ 2212 . 09251 ?utm_source=chatgpt.com&utm_medium=email. Ethan Perez, Sam Ringer, Kamilė Lukošiūtė, Karina Nguyen, Edwin Chen, Scott Heiner, Craig Pettit...", "subpage_snippet": "", "source": "tmasada.notion.site", "link": "https://tmasada.notion.site/2212-09251-Discovering-Language-Model-Behaviors-with-Model-Written-Evaluations-1ea0e0d51e4b818f9a88d0a90068fd6c", "content": "https://arxiv.org/abs/ 2212 . 09251 ?utm_source=chatgpt.com&utm_medium=email. Ethan Perez, Sam Ringer, Kamilė Lukošiūtė, Karina Nguyen, Edwin Chen, Scott Heiner, Craig Pettit..."}
+{"idx": 3, "title": "Paper page - Discovering Language Model Behaviors with...", "date": "", "ddg_snippet": "Abstract . Automatically generated evaluations from language models reveal novel behaviors, including cases of inverse scaling with model size and reinforcement learning from human feedback.", "subpage_snippet": "", "source": "huggingface.co", "link": "https://huggingface.co/papers/2212.09251", "content": "Abstract . Automatically generated evaluations from language models reveal novel behaviors, including cases of inverse scaling with model size and reinforcement learning from human feedback."}
+{"idx": 4, "title": "Discovering Language Model Behaviors with Model-Written Evaluations", "date": "", "ddg_snippet": "DOI:10.48550/arXiv. 2212 . 09251 . Authors Abstract . As language models (LMs) scale, they develop many novel behaviors, good and bad, exacerbating the need to evaluate how they behave.", "subpage_snippet": "", "source": "www.researchgate.net", "link": "https://www.researchgate.net/publication/366423471_Discovering_Language_Model_Behaviors_with_Model-Written_Evaluations", "content": "DOI:10.48550/arXiv. 2212 . 09251 . Authors Abstract . As language models (LMs) scale, they develop many novel behaviors, good and bad, exacerbating the need to evaluate how they behave."}
+{"idx": 5, "title": "[PDF] Discovering Language Model Behaviors with... | Semantic Scholar", "date": "", "ddg_snippet": "Sign In. Create Free Account. DOI:10.48550/arXiv. 2212 . 09251 .", "subpage_snippet": "", "source": "www.semanticscholar.org", "link": "https://www.semanticscholar.org/paper/Discovering-Language-Model-Behaviors-with-Perez-Ringer/cef330bacf014d60daabbd489647b2006af130ca", "content": "Sign In. Create Free Account. DOI:10.48550/arXiv. 2212 . 09251 ."}
+{"idx": 6, "title": "A Unifying Lens on Language Model Evaluation", "date": "", "ddg_snippet": "Anonymous EMNLP submission. Abstract . Micali, 1984; Yao, 1982), which measures a distriabs/ 2212 . 09251 . 491. 434 shia, Zhiyi Ma, Tristan Thrush, Sebastian Riedel", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/pdf?id=EvTnm0vASV", "content": "Anonymous EMNLP submission. Abstract . Micali, 1984; Yao, 1982), which measures a distriabs/ 2212 . 09251 . 491. 434 shia, Zhiyi Ma, Tristan Thrush, Sebastian Riedel"}
+{"idx": 7, "title": "GitHub - anthropics/evals", "date": "", "ddg_snippet": "@misc{perez2022discovering, doi = {10.48550/ARXIV. 2212 . 09251 }, url = {https://arxiv.org/abs/ 2212 . 09251 }, author = {Perez, Ethan and Ringer, Sam and...", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/anthropics/evals", "content": "@misc{perez2022discovering, doi = {10.48550/ARXIV. 2212 . 09251 }, url = {https://arxiv.org/abs/ 2212 . 09251 }, author = {Perez, Ethan and Ringer, Sam and..."}
+{"idx": 8, "title": "Measuring Progress on Scalable Oversight for Large Language Models...", "date": "", "ddg_snippet": "read more. Abstract : Developing safe and useful general-purpose AI systems will require us to make progress on scalable oversight: the problem of supervising systems that potentially...", "subpage_snippet": "", "source": "scispace.com", "link": "https://scispace.com/papers/measuring-progress-on-scalable-oversight-for-large-language-39zgpech", "content": "read more. Abstract : Developing safe and useful general-purpose AI systems will require us to make progress on scalable oversight: the problem of supervising systems that potentially..."}
+{"idx": 9, "title": "Sycophancy in Generative-AI Chatbots - NN/G", "date": "", "ddg_snippet": "Ethan Perez et al., 2022. Discovering Language Model Behaviors with Model-Written Evaluations. arXiv: 2212 . 09251 .", "subpage_snippet": "", "source": "www.nngroup.com", "link": "https://www.nngroup.com/articles/sycophancy-generative-ai-chatbots/", "content": "Ethan Perez et al., 2022. Discovering Language Model Behaviors with Model-Written Evaluations. arXiv: 2212 . 09251 ."}

data/sampled_jsons/23zxLtvder_SPD-_Sync-Point_Drop_for_Efficient_Tensor_Parallelism_of_Large_Language_Models.jsonl

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+{"idx": 0, "title": "SPD: Sync-Point Drop for Efficient Tensor Parallelism of Large Language ...", "date": "", "ddg_snippet": "With the rapid expansion in the scale of large language models (LLMs), enabling efficient distributed inference across multiple computing units has become increasingly critical. However, communication overheads from popular distributed inference techniques such as Tensor Parallelism pose a significant challenge to achieve scalability and low latency. Therefore, we introduce a novel ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/abs/2502.20727", "content": "With the rapid expansion in the scale of large language models (LLMs), enabling efficient distributed inference across multiple computing units has become increasingly critical. However, communication overheads from popular distributed inference techniques such as Tensor Parallelism pose a significant challenge to achieve scalability and low latency. Therefore, we introduce a novel ..."}
+{"idx": 1, "title": "Spd: Sync-point Drop for Efficient Tensor Par Allelism of Large ...", "date": "", "ddg_snippet": "ability and low latency. Therefore, we introduce a novel optimization technique, Sync-Point Drop (SPD) to reduce communication overheads in tensor parallelism by dropping synchronization on attention outputs. In detail, we first propose a block design that allows execution to proceed without communication through SPD. Second, we identify regions of communication redundancy, where dropping ...", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/pdf?id=uoU4ypjAmN", "content": "ability and low latency. Therefore, we introduce a novel optimization technique, Sync-Point Drop (SPD) to reduce communication overheads in tensor parallelism by dropping synchronization on attention outputs. In detail, we first propose a block design that allows execution to proceed without communication through SPD. Second, we identify regions of communication redundancy, where dropping ..."}
+{"idx": 2, "title": "\"SPD: Sync-Point Drop for efficient tensor parallelism of Large ... - dblp", "date": "", "ddg_snippet": "Bibliographic details on SPD: Sync-Point Drop for efficient tensor parallelism of Large Language Models .", "subpage_snippet": "", "source": "dblp.org", "link": "https://dblp.org/rec/journals/corr/abs-2502-20727", "content": "Bibliographic details on SPD: Sync-Point Drop for efficient tensor parallelism of Large Language Models ."}
+{"idx": 3, "title": "Abdullah K. on LinkedIn: SPD: Sync-Point Drop for efficient tensor ...", "date": "", "ddg_snippet": "🚀 Exciting advancements in Large Language Models (LLMs) are on the horizon! The recent study introduces a groundbreaking optimization technique known as Sync-Point Drop (SPD), aimed at ...", "subpage_snippet": "", "source": "www.linkedin.com", "link": "https://www.linkedin.com/posts/abdullah-kasri_spd-sync-point-drop-for-efficient-tensor-activity-7302389847258710017-S4Q8", "content": "🚀 Exciting advancements in Large Language Models (LLMs) are on the horizon! The recent study introduces a groundbreaking optimization technique known as Sync-Point Drop (SPD), aimed at ..."}
+{"idx": 4, "title": "SPD: Sync-Point Drop for Efficient Tensor Parallelism of Large Language ...", "date": "", "ddg_snippet": "This paper introduces a method called Sync-Point Drop (SPD) to help large language models (LLMs) work faster on multiple computers. By skipping some communication steps in the process, SPD reduces...", "subpage_snippet": "", "source": "bytez.com", "link": "https://bytez.com/docs/icml/46606/paper", "content": "This paper introduces a method called Sync-Point Drop (SPD) to help large language models (LLMs) work faster on multiple computers. By skipping some communication steps in the process, SPD reduces..."}
+{"idx": 5, "title": "Decoder block structure with sync-point drop (in 2-GPUs distributed ...", "date": "", "ddg_snippet": "'X', 'Yi', 'Zi' and 'Pi' denotes a hidden representation of each device (i) on ' • ' in the figure. from publication: SPD: Sync-Point Drop for efficient tensor parallelism of Large Language ...", "subpage_snippet": "", "source": "www.researchgate.net", "link": "https://www.researchgate.net/figure/Decoder-block-structure-with-sync-point-drop-in-2-GPUs-distributed-inference-case-Wi_fig2_389510233", "content": "'X', 'Yi', 'Zi' and 'Pi' denotes a hidden representation of each device (i) on ' • ' in the figure. from publication: SPD: Sync-Point Drop for efficient tensor parallelism of Large Language ..."}
+{"idx": 6, "title": "SPD: Sync-Point Drop for Efficient Tensor Parallelism of Large Language ...", "date": "", "ddg_snippet": "Abstract With the rapid expansion in the scale of large language models (LLMs), enabling eficient dis-tributed inference across multiple computing units has become increasingly critical. However, com-munication overheads from popular distributed inference techniques such as Tensor Parallelism pose a significant challenge to achieve scalability and low latency. Therefore, we introduce a novel ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2502.20727", "content": "Abstract With the rapid expansion in the scale of large language models (LLMs), enabling eficient dis-tributed inference across multiple computing units has become increasingly critical. However, com-munication overheads from popular distributed inference techniques such as Tensor Parallelism pose a significant challenge to achieve scalability and low latency. Therefore, we introduce a novel ..."}
+{"idx": 7, "title": "SPD: Sync-Point Drop for Efficient Tensor Parallelism of Large...", "date": "", "ddg_snippet": "Serving Large Language Models (LLMs) requires distributing computation across multiple GPUs to handle their size and complexity efficiently. However, this distribution introduces delays due to frequent synchronization between devices during model execution. We developed a technique called Sync-Point Drop (SPD) that selectively removes unnecessary synchronization steps while running the model ...", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/forum?id=23zxLtvder", "content": "Serving Large Language Models (LLMs) requires distributing computation across multiple GPUs to handle their size and complexity efficiently. However, this distribution introduces delays due to frequent synchronization between devices during model execution. We developed a technique called Sync-Point Drop (SPD) that selectively removes unnecessary synchronization steps while running the model ..."}
+{"idx": 8, "title": "Han-Byul Kim", "date": "", "ddg_snippet": "Research SPD: Sync-Point Drop for efficient tensor parallelism of Large Language Models Han-Byul Kim, Duc Hoang, Arnav Kundu, Mohammad Samragh, Minsik Cho International Conference on Machine Learning (ICML), 2025 #Distributed_Inference, # Tensor_Parallelism MetaMix: Meta-state Precision Searcher for Mixed-precision Activation Quantization", "subpage_snippet": "", "source": "hanbyulkim.github.io", "link": "https://hanbyulkim.github.io/", "content": "Research SPD: Sync-Point Drop for efficient tensor parallelism of Large Language Models Han-Byul Kim, Duc Hoang, Arnav Kundu, Mohammad Samragh, Minsik Cho International Conference on Machine Learning (ICML), 2025 #Distributed_Inference, # Tensor_Parallelism MetaMix: Meta-state Precision Searcher for Mixed-precision Activation Quantization"}
+{"idx": 9, "title": "Optimizing LLM Inference Across Multiple GPUs - zerna.io", "date": "", "ddg_snippet": "Reducing Communication Bottlenecks in Tensor Parallelism Sync-Point Drop (SPD) is a novel optimization technique that selectively eliminates synchronization points in tensor -parallel LLM inference, significantly reducing communication overhead.", "subpage_snippet": "", "source": "zerna.io", "link": "http://zerna.io/en/page/engineering/presentation_set/engineering-llm-research/presentation/engineering-model-optimization/slide/engineering-paper-2502_20727", "content": "Reducing Communication Bottlenecks in Tensor Parallelism Sync-Point Drop (SPD) is a novel optimization technique that selectively eliminates synchronization points in tensor -parallel LLM inference, significantly reducing communication overhead."}

data/sampled_jsons/2405.17618_pdf_equation_7_RA2C_loss_formula.jsonl

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+{"idx": 0, "title": "PDF Determining Inductor Power Losses - Mouser Electronics", "date": "", "ddg_snippet": "This paper has introduced the basic formulas used in determining inductor power loss . It also discussed the importance of knowing where inductor power losses come from in order to reduce heat creation and thus improve overall eficiency.", "subpage_snippet": "", "source": "www.mouser.com", "link": "https://www.mouser.com/pdfDocs/Coilcraft_inductorlosses.pdf", "content": "This paper has introduced the basic formulas used in determining inductor power loss . It also discussed the importance of knowing where inductor power losses come from in order to reduce heat creation and thus improve overall eficiency."}
+{"idx": 1, "title": "PDF Accurate Calculation of AC Loss of an Inductor in Power ... - PSMA", "date": "", "ddg_snippet": "This process is repeated over wide range of parameters to produce our own empirical data Fig 10: AC loss plotted against Switching frequency This empirical data is then used to plot a AC loss graph & create an equation to calculate AC loss as shown in figure 10", "subpage_snippet": "", "source": "www.psma.com", "link": "https://www.psma.com/sites/default/files/uploads/tech-forums-magnetics/presentations/is94-accurate-estimation-losses-power-inductor-power-electronics-applications.pdf", "content": "This process is repeated over wide range of parameters to produce our own empirical data Fig 10: AC loss plotted against Switching frequency This empirical data is then used to plot a AC loss graph & create an equation to calculate AC loss as shown in figure 10"}
+{"idx": 2, "title": "Symmetric Reinforcement Learning Loss for Robust Learning on Diverse ...", "date": "", "ddg_snippet": "We define a symmetric RL loss , whose fundamental working mechanism aligns with the robust loss function of supervised learning (Wang et al., 2019), to make the RL learning procedure more robust for A2C and PPO.", "subpage_snippet": "", "source": "ar5iv.labs.arxiv.org", "link": "https://ar5iv.labs.arxiv.org/html/2405.17618", "content": "We define a symmetric RL loss , whose fundamental working mechanism aligns with the robust loss function of supervised learning (Wang et al., 2019), to make the RL learning procedure more robust for A2C and PPO."}
+{"idx": 3, "title": "Darcy-Weisbach Equation: Flow Resistance & Pressure Loss Calculator", "date": "", "ddg_snippet": "The Darcy-Weisbach equation can be used to calculate the major pressure and head loss due to friction in ducts, pipes or tubes.", "subpage_snippet": "", "source": "www.engineeringtoolbox.com", "link": "https://www.engineeringtoolbox.com/darcy-weisbach-equation-d_646.html", "content": "The Darcy-Weisbach equation can be used to calculate the major pressure and head loss due to friction in ducts, pipes or tubes."}
+{"idx": 4, "title": "PDF How to Perform a Heat-Loss Calculation — Part 2 Simple ways to ...", "date": "", "ddg_snippet": "The heat loss formula for determining transmission losses through floors, roofs, and walls is Q = A • U • ∆T. In other words, the rate of heat flow through a building assembly (in Btu/h) is equal to the area of the assembly (in ft2) times the U-factor (in Btu/ft2 • hr • F°) of the assembly times the ∆T (in F°).", "subpage_snippet": "", "source": "lorisweb.com", "link": "https://lorisweb.com/CMGT235/DIS02/Heat+Loss+Calculations+Part+2.pdf", "content": "The heat loss formula for determining transmission losses through floors, roofs, and walls is Q = A • U • ∆T. In other words, the rate of heat flow through a building assembly (in Btu/h) is equal to the area of the assembly (in ft2) times the U-factor (in Btu/ft2 • hr • F°) of the assembly times the ∆T (in F°)."}
+{"idx": 5, "title": "PDF Home - Nason Mechanical Systems", "date": "", "ddg_snippet": "Steam Tra Ener Losses Modified Napier Formula , can be used to estimate steam loss through a trap blowing to atmosphere. The following is the equation used in calculating the approximate loss : Modified Napier Formula Steam loss in lbs./hr = (24.24) * (aPa) * (D*D) Where. Pa = Pressure in Absolute ex. Gauge Pressure + 14.7 D = Orifice Diameter in Trap in Inches. Example: Inverted Bucket Trap on ...", "subpage_snippet": "", "source": "nasonmechanical.com", "link": "https://nasonmechanical.com/wp-content/uploads/2018/04/Steam-Trap-Energy-Losses.pdf", "content": "Steam Tra Ener Losses Modified Napier Formula , can be used to estimate steam loss through a trap blowing to atmosphere. The following is the equation used in calculating the approximate loss : Modified Napier Formula Steam loss in lbs./hr = (24.24) * (aPa) * (D*D) Where. Pa = Pressure in Absolute ex. Gauge Pressure + 14.7 D = Orifice Diameter in Trap in Inches. Example: Inverted Bucket Trap on ..."}
+{"idx": 6, "title": "PDF Analog | Embedded processing | Semiconductor company | TI.com", "date": "", "ddg_snippet": "Analog | Embedded processing | Semiconductor company | TI.com", "subpage_snippet": "", "source": "www.ti.com", "link": "https://www.ti.com/lit/ml/slup125/slup125.pdf", "content": "Analog | Embedded processing | Semiconductor company | TI.com"}
+{"idx": 7, "title": "Abstract - arXiv.org", "date": "", "ddg_snippet": "3.3 Symmetric cross entropy atasets. Cross Entropy (CE) loss ( Equation 4) performs effectively when the data is clean; however, it encounters challenges in the presence f noise. Given a true distribution q and a predicted distribution p, p is learned based on the information derived from q according to informatio", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2405.17618", "content": "3.3 Symmetric cross entropy atasets. Cross Entropy (CE) loss ( Equation 4) performs effectively when the data is clean; however, it encounters challenges in the presence f noise. Given a true distribution q and a predicted distribution p, p is learned based on the information derived from q according to informatio"}
+{"idx": 8, "title": "Power Inductor Basic Course - Chapter 3 - Murata Manufacturing Co., Ltd.", "date": "", "ddg_snippet": "We talk here about the important DC-DC converter characteristics of efficiency, ripple voltage, and load response, and explain the kinds of characteristics required of power inductors. We explain in detail the impact these three important characteristics have on circuit design and performance. We provide information to help select the optimal inductor.", "subpage_snippet": "", "source": "www.murata.com", "link": "https://www.murata.com/en-global/products/inductor/power/overview/learn/basic_03", "content": "We talk here about the important DC-DC converter characteristics of efficiency, ripple voltage, and load response, and explain the kinds of characteristics required of power inductors. We explain in detail the impact these three important characteristics have on circuit design and performance. We provide information to help select the optimal inductor."}
+{"idx": 9, "title": "Refrigeration And Air Conditioning - C P Arora - Google Drive", "date": "", "ddg_snippet": "This book provides comprehensive insights into refrigeration and air conditioning, covering fundamental principles and practical applications for students and professionals.", "subpage_snippet": "", "source": "drive.google.com", "link": "https://drive.google.com/file/d/1Ig00qWzgp8i2eHo9TuGMdJd7g2vvqYZu/view?usp=sharing", "content": "This book provides comprehensive insights into refrigeration and air conditioning, covering fundamental principles and practical applications for students and professionals."}

data/sampled_jsons/2408.10672_Ts-Attn_tensor_shape_before_Attn_intra.jsonl

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+{"idx": 0, "title": "A arXiv:2408.10672v3 [cs.LG] 27 Mar 2025", "date": "", "ddg_snippet": "olds identical shape with the input Xin. In our Ts-Attn block, we employ an Attn block Attninter for the first cross-solution information sharing stage, and the other Attn block Attnintra for the second cross-dimension information sharing stage (illustrated i", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2408.10672", "content": "olds identical shape with the input Xin. In our Ts-Attn block, we employ an Attn block Attninter for the first cross-solution information sharing stage, and the other Attn block Attnintra for the second cross-dimension information sharing stage (illustrated i"}
+{"idx": 1, "title": "RuntimeError: Mask shape should match input shape · Issue ...", "date": "", "ddg_snippet": "Oct 3, 2024 · hello，author，i have a question want to ask you. when i use clip.model.text_encode , the text. tensor . shape must is [77,77]?,meaning that the text must is consist of 77 sequences? the attn _mask could...", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/mlfoundations/open_clip/issues/954", "content": "Oct 3, 2024 · hello，author，i have a question want to ask you. when i use clip.model.text_encode , the text. tensor . shape must is [77,77]?,meaning that the text must is consist of 77 sequences? the attn _mask could..."}
+{"idx": 2, "title": "Transformer encoder layer with pytorch : The shape of the 2D ...", "date": "", "ddg_snippet": "Mar 9, 2023 · RuntimeError: The shape of the 2D attn _mask is torch.Size([16, 512]), but should be (16, 16). I don't understand why that doesn't work because the mask length will be equal to the number of the tokens ?", "subpage_snippet": "", "source": "stackoverflow.com", "link": "https://stackoverflow.com/questions/75686820/transformer-encoder-layer-with-pytorch-the-shape-of-the-2d-attn-mask-is-torch", "content": "Mar 9, 2023 · RuntimeError: The shape of the 2D attn _mask is torch.Size([16, 512]), but should be (16, 16). I don't understand why that doesn't work because the mask length will be equal to the number of the tokens ?"}
+{"idx": 3, "title": "Custom 4D tensor caused shape mismatch error · Issue #35290 ...", "date": "", "ddg_snippet": "Try to add a dimension with torch.unsqueeze () for broadcasting the expanded_ attn _mask.", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/huggingface/transformers/issues/35290", "content": "Try to add a dimension with torch.unsqueeze () for broadcasting the expanded_ attn _mask."}
+{"idx": 4, "title": "Qwen3 Fails w/4D Attn Mask when using FA2 #39608 - GitHub", "date": "", "ddg_snippet": "Jul 23, 2025 · System Info transformers: 4.53.3 torch 2.6.0 flash- attn 2.7.4.post1 Who can help? @ArthurZucker Information The official example scripts My own modified scripts Tasks An officially supported task i...", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/huggingface/transformers/issues/39608", "content": "Jul 23, 2025 · System Info transformers: 4.53.3 torch 2.6.0 flash- attn 2.7.4.post1 Who can help? @ArthurZucker Information The official example scripts My own modified scripts Tasks An officially supported task i..."}
+{"idx": 5, "title": "arXiv:2408.10672v2 [cs.LG] 26 Sep 2024", "date": "", "ddg_snippet": "ds identical shape with the input Xin. In our Ts-Attn block, we employ an Attn block Attninter for the first cross-solution information sharing stage, and the other Attn block Attnintra for the second cross-dimension information sharing stage", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2408.10672v2", "content": "ds identical shape with the input Xin. In our Ts-Attn block, we employ an Attn block Attninter for the first cross-solution information sharing stage, and the other Attn block Attnintra for the second cross-dimension information sharing stage"}
+{"idx": 6, "title": "Installing flash- attn without compiling it | Simon Willison’s TILs", "date": "", "ddg_snippet": "pip install flash- attn --no-build-isolation.flash_ attn -2.6.3+cu123torch2.4cxx11abiFALSE-cp310-cp310-linux_x86_64.whl. This seemed to work (and installed in just a couple of seconds): import flash_ attn flash_ attn .__version__.", "subpage_snippet": "", "source": "til.simonwillison.net", "link": "https://til.simonwillison.net/python/installing-flash-attention", "content": "pip install flash- attn --no-build-isolation.flash_ attn -2.6.3+cu123torch2.4cxx11abiFALSE-cp310-cp310-linux_x86_64.whl. This seemed to work (and installed in just a couple of seconds): import flash_ attn flash_ attn .__version__."}
+{"idx": 7, "title": "[Bug] Fix tensor shape mismatch in fused_qkv_norm_rottary for...", "date": "", "ddg_snippet": "Without padding, the output tensor 's size mismatches the expected view, leading to reshape failures. Proposed Fix: Pad 'batch_size * seq_len' to the nearest multiple of 256 before quantization and output creation. After processing, slice back to the original size.", "subpage_snippet": "", "source": "github.com", "link": "https://github.com/nunchaku-tech/nunchaku/issues/642", "content": "Without padding, the output tensor 's size mismatches the expected view, leading to reshape failures. Proposed Fix: Pad 'batch_size * seq_len' to the nearest multiple of 256 before quantization and output creation. After processing, slice back to the original size."}
+{"idx": 8, "title": "tensorflow: Keras application - Tensor is not an element of this graph...", "date": "", "ddg_snippet": "ValueError: Tensor Tensor (\"vgg_base/Placeholder:0\", shape =(3, 3, 3, 64), dtype=float32) is not an element of this graph. During handling of the above exception, another exception occurred", "subpage_snippet": "", "source": "errorism.dev", "link": "https://errorism.dev/issues/tensorflow-tensorflow-keras-application---tensor-is-not-an-element-of-this-graph-on-eval-after-train", "content": "ValueError: Tensor Tensor (\"vgg_base/Placeholder:0\", shape =(3, 3, 3, 64), dtype=float32) is not an element of this graph. During handling of the above exception, another exception occurred"}
+{"idx": 9, "title": "How to resolve the error regarding the 2D ` attn _mask` shape ... - Glari...", "date": "", "ddg_snippet": "- **Adjust Mask Shape **: You can reshape or create your ` attn _mask` to match the expected dimensions.- **Debug Input Sizes**: Check the dimensions of the inputs being fed into the attention layer. Ensure that the input tensor shapes are consistent with what the model expects.", "subpage_snippet": "", "source": "askai.glarity.app", "link": "https://askai.glarity.app/search/How-to-resolve-the-error-regarding-the-2D--attn-mask--shape-in-PyTorch", "content": "- **Adjust Mask Shape **: You can reshape or create your ` attn _mask` to match the expected dimensions.- **Debug Input Sizes**: Check the dimensions of the inputs being fed into the attention layer. Ensure that the input tensor shapes are consistent with what the model expects."}

data/sampled_jsons/2410.10562_climate_activism_reddit_subreddits_activation_percentage_distribution_table.jsonl

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+{"idx": 0, "title": "[2410.10562] Causal Modeling of Climate Activism on Reddit", "date": "", "ddg_snippet": "Oct 14, 2024 · Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/abs/2410.10562", "content": "Oct 14, 2024 · Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ..."}
+{"idx": 1, "title": "Causal Modeling of Climate Activism on Reddit - arXiv.org", "date": "", "ddg_snippet": "We developed a rich and comprehensive causal model to study the interplay between diferent determinants of climate activism on Reddit . This work represents a first attempt to apply a multi-causal model to social media data.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2410.10562", "content": "We developed a rich and comprehensive causal model to study the interplay between diferent determinants of climate activism on Reddit . This work represents a first attempt to apply a multi-causal model to social media data."}
+{"idx": 2, "title": "Causal Modeling of Climate Activism on Reddit", "date": "", "ddg_snippet": "14 Oct 2024 — In this work, we develop a comprehensive causal model of how and why Reddit users engage with activist communities driving mass climate protests.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/html/2410.10562v1", "content": "14 Oct 2024 — In this work, we develop a comprehensive causal model of how and why Reddit users engage with activist communities driving mass climate protests."}
+{"idx": 3, "title": "On the Inference of Sociodemographics on Reddit", "date": "", "ddg_snippet": "7 Feb 2025 — Reddit offers a rich platform for sociodemographic inference due to its diverse user base and activity patterns. Existing methods often leverage ...", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/html/2502.05049v1", "content": "7 Feb 2025 — Reddit offers a rich platform for sociodemographic inference due to its diverse user base and activity patterns. Existing methods often leverage ..."}
+{"idx": 4, "title": "Causal Modeling of Climate Activism on Reddit", "date": "", "ddg_snippet": "Oct 14, 2024 · Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ...", "subpage_snippet": "", "source": "researchtrend.ai", "link": "https://researchtrend.ai/papers/2410.10562", "content": "Oct 14, 2024 · Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ..."}
+{"idx": 5, "title": "activists - Reddit", "date": "", "ddg_snippet": "This is a sub-Reddit for activists to come together and collaborate in their efforts.", "subpage_snippet": "", "source": "www.reddit.com", "link": "https://www.reddit.com/r/activists/", "content": "This is a sub-Reddit for activists to come together and collaborate in their efforts."}
+{"idx": 6, "title": "\"Causal Modeling of Climate Activism on Reddit.\" - dblp", "date": "", "ddg_snippet": "Bibliographic details on Causal Modeling of Climate Activism on Reddit .", "subpage_snippet": "", "source": "dblp.org", "link": "https://dblp.org/rec/journals/corr/abs-2410-10562", "content": "Bibliographic details on Causal Modeling of Climate Activism on Reddit ."}
+{"idx": 7, "title": "Causal Modeling of Climate Activism on Reddit | Cool Papers ...", "date": "", "ddg_snippet": "Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ...", "subpage_snippet": "", "source": "papers.cool", "link": "https://papers.cool/arxiv/2410.10562", "content": "Climate activism is crucial in stimulating collective societal and behavioral change towards sustainable practices through political pressure. Although multiple factors contribute to the participation in activism , their complex relationships and the scarcity of data on their interactions have restricted most prior research to studying them in isolation, thus preventing the development of a ..."}
+{"idx": 8, "title": "Climate Skepticism on Reddit | redditClimate", "date": "", "ddg_snippet": "In the interest of making a novel contribution, we decided to study climate skepticism on Reddit . Sample Bias A taste of Reddit demographics [ref]: 67-69% male 54% US-based users Similar racial distribution to US pop. 58% age 18-29, 33% age 30-49 So Reddit appears to be skewed towards males and a younger demographic.", "subpage_snippet": "", "source": "redditclimate.github.io", "link": "https://redditclimate.github.io/redditClimate/", "content": "In the interest of making a novel contribution, we decided to study climate skepticism on Reddit . Sample Bias A taste of Reddit demographics [ref]: 67-69% male 54% US-based users Similar racial distribution to US pop. 58% age 18-29, 33% age 30-49 So Reddit appears to be skewed towards males and a younger demographic."}
+{"idx": 9, "title": "(PDF) On the Inference of Sociodemographics on Reddit", "date": "", "ddg_snippet": "2024. Causal Modeling of Climate Activism on Reddit . arXiv preprint arXiv: 2410 . 10562 . Lokala, U.; Srivastava, A.; Dastidar, T. G.; Chakraborty, T.• probability to observe an activation over subreddit j. given the class y; • the probability to observe xindependent activations in.", "subpage_snippet": "", "source": "www.researchgate.net", "link": "https://www.researchgate.net/publication/388847848_On_the_Inference_of_Sociodemographics_on_Reddit", "content": "2024. Causal Modeling of Climate Activism on Reddit . arXiv preprint arXiv: 2410 . 10562 . Lokala, U.; Srivastava, A.; Dastidar, T. G.; Chakraborty, T.• probability to observe an activation over subreddit j. given the class y; • the probability to observe xindependent activations in."}

data/sampled_jsons/2502.00921_arxiv_MATH_dataset_Table_1_CW_correct_CW_incorrect_ΔCW.jsonl

@@ -0,0 +1,10 @@
+{"idx": 0, "title": "[ 2502 . 00921 ] Blink of an eye: a simple theory for feature localization in...", "date": "", "ddg_snippet": "arXiv : 2502 . 00921 (cs). [Submitted on 2 Feb 2025 (v 1 ), last revised 5 Jun 2025 (this version, v2)].Finally, we validate our predictions empirically for LLMs and find that critical windows often coincide with failures in problem solving for various math and reasoning benchmarks.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/abs/2502.00921", "content": "arXiv : 2502 . 00921 (cs). [Submitted on 2 Feb 2025 (v 1 ), last revised 5 Jun 2025 (this version, v2)].Finally, we validate our predictions empirically for LLMs and find that critical windows often coincide with failures in problem solving for various math and reasoning benchmarks."}
+{"idx": 1, "title": "deepmind/ math _ dataset · Datasets at Hugging Face", "date": "", "ddg_snippet": "Dataset Card for \" math _ dataset \".This dataset code generates mathematical question and answer pairs, from a range of question types at roughly school-level difficulty.", "subpage_snippet": "", "source": "huggingface.co", "link": "https://huggingface.co/datasets/deepmind/math_dataset", "content": "Dataset Card for \" math _ dataset \".This dataset code generates mathematical question and answer pairs, from a range of question types at roughly school-level difficulty."}
+{"idx": 2, "title": "[SOLVED] 'InvalidArgumentError: Input is empty' error on dataset ...", "date": "", "ddg_snippet": "I also cannot use my training dataset to train my model as I get a similar error. I am creating the folder directories using the Split Folders API, and I am oversampling my training dataset because of label imbalance.", "subpage_snippet": "", "source": "www.developerload.com", "link": "https://www.developerload.com/39invalidargumenterror-input-is-empty39-error-on-dataset-created-with-39tfkerasutilsimagedatasetfromdirectory39", "content": "I also cannot use my training dataset to train my model as I get a similar error. I am creating the folder directories using the Split Folders API, and I am oversampling my training dataset because of label imbalance."}
+{"idx": 3, "title": "3D models database | Printables.com", "date": "", "ddg_snippet": "Haptic Models. Math . Other 3D Objects for Learning.", "subpage_snippet": "", "source": "www.printables.com", "link": "https://www.printables.com/model?ordering=downloads", "content": "Haptic Models. Math . Other 3D Objects for Learning."}
+{"idx": 4, "title": "Тренировочные варианты ОГЭ 2024-2025-2026 по... — math 100.ru", "date": "", "ddg_snippet": "Тренировочный вариант № 181 ОГЭ УСЛОЖНЁННЫЙ Тренировочный вариант № 180 ОГЭ из заданий банка ФИПИ Тренировочный вариант № 179 ОГЭ УСЛОЖНЁННЫЙ Тренировочный вариант...", "subpage_snippet": "", "source": "math100.ru", "link": "https://math100.ru/trenirovochnie-varianti-oge-new/", "content": "Тренировочный вариант № 181 ОГЭ УСЛОЖНЁННЫЙ Тренировочный вариант № 180 ОГЭ из заданий банка ФИПИ Тренировочный вариант № 179 ОГЭ УСЛОЖНЁННЫЙ Тренировочный вариант..."}
+{"idx": 5, "title": "How to find the 20th and 80th percentile of a data set - YouTube", "date": "", "ddg_snippet": "The kth percentile of a data set is the data value that appeared in the kth position after the dataset has...", "subpage_snippet": "", "source": "www.youtube.com", "link": "https://www.youtube.com/watch?v=MSQpvuPL2cw", "content": "The kth percentile of a data set is the data value that appeared in the kth position after the dataset has..."}
+{"idx": 6, "title": "Fisch - Secret Cave: How to Unlock the Luminescent and Crimson...", "date": "", "ddg_snippet": "Table of Contents. 1 . Step 1 : Unsealing the First Gate – The Luminescent Cavern.Be warned: this process can be incredibly expensive. The cost is random, and it’s not uncommon for players to spend millions of in-game cash to get the correct mutation.", "subpage_snippet": "", "source": "trioner.com", "link": "https://trioner.com/fisch-secret-cave-how-to-unlock-the-luminescent-and-crimson-caverns/", "content": "Table of Contents. 1 . Step 1 : Unsealing the First Gate – The Luminescent Cavern.Be warned: this process can be incredibly expensive. The cost is random, and it’s not uncommon for players to spend millions of in-game cash to get the correct mutation."}
+{"idx": 7, "title": "Number Names 1 to 100 | Spelling | 1 to 100 in Words", "date": "", "ddg_snippet": "The following table has 1 to 100 spellings in English and learning these is beneficial for students in higher grades. List of Number Names from 1 to 100. Observe the following table which shows numbers one to hundred in English.", "subpage_snippet": "", "source": "www.cuemath.com", "link": "https://www.cuemath.com/numbers/number-names-1-to-100/", "content": "The following table has 1 to 100 spellings in English and learning these is beneficial for students in higher grades. List of Number Names from 1 to 100. Observe the following table which shows numbers one to hundred in English."}
+{"idx": 8, "title": "Kaggle: Your Machine Learning and Data Science Community", "date": "", "ddg_snippet": "arXiv Dataset . Usability 8.8 · 2 GB. arXiv dataset and metadata of 1 .7M+ scholarly papers across STEM.", "subpage_snippet": "", "source": "www.kaggle.com", "link": "https://www.kaggle.com/", "content": "arXiv Dataset . Usability 8.8 · 2 GB. arXiv dataset and metadata of 1 .7M+ scholarly papers across STEM."}
+{"idx": 9, "title": "Canon i -SENSYS MF651 Cw Printer and Scanner Drivers Download...", "date": "", "ddg_snippet": "Printer Model: Canon i -SENSYS MF651 Cw . How to install the printer driver correctly : Do not connect the printer to the computer. Start installing the software first.If you installed the driver incorrectly , uninstall the driver, restart your computer, and reinstall the software.", "subpage_snippet": "", "source": "drivernew.com", "link": "https://drivernew.com/?p=41007", "content": "Printer Model: Canon i -SENSYS MF651 Cw . How to install the printer driver correctly : Do not connect the printer to the computer. Start installing the software first.If you installed the driver incorrectly , uninstall the driver, restart your computer, and reinstall the software."}

data/sampled_jsons/2502.10875_Table_1_dataset_statistics_train_density_user-item_interactions.jsonl

@@ -0,0 +1,10 @@
+{"idx": 0, "title": "Find Open Datasets and Machine Learning Projects | Kaggle", "date": "", "ddg_snippet": "Download Open Datasets on 1000s of Projects + Share Projects on One Platform. Explore Popular Topics Like Government, Sports, Medicine, Fintech, Food, More.", "subpage_snippet": "", "source": "www.kaggle.com", "link": "https://www.kaggle.com/datasets", "content": "Download Open Datasets on 1000s of Projects + Share Projects on One Platform. Explore Popular Topics Like Government, Sports, Medicine, Fintech, Food, More."}
+{"idx": 1, "title": "yandex/yambda · Datasets at Hugging Face", "date": "", "ddg_snippet": "Key Features. About Dataset . Statistics . User History Length Distribution. Item Interaction Count. Data Format. File Descriptions.", "subpage_snippet": "", "source": "huggingface.1319lm.top", "link": "https://huggingface.1319lm.top/datasets/yandex/yambda", "content": "Key Features. About Dataset . Statistics . User History Length Distribution. Item Interaction Count. Data Format. File Descriptions."}
+{"idx": 2, "title": "Free Public Datasets for Data Science Projects", "date": "", "ddg_snippet": "In this post we can find free public datasets for Data Science projects. There is a big number of datasets which cover different areas - machine learning, presentation, data analysis and visualization.", "subpage_snippet": "", "source": "datascientyst.com", "link": "https://datascientyst.com/datasets/", "content": "In this post we can find free public datasets for Data Science projects. There is a big number of datasets which cover different areas - machine learning, presentation, data analysis and visualization."}
+{"idx": 3, "title": "UCI Machine Learning Repository | Discover datasets around the world!", "date": "", "ddg_snippet": "The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. One class is linearly separable from the other 2; the latter are not linearly separable from each other.Variables Table . Variable Name.", "subpage_snippet": "", "source": "archive.ics.uci.edu", "link": "https://archive.ics.uci.edu/dataset/53/iris", "content": "The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. One class is linearly separable from the other 2; the latter are not linearly separable from each other.Variables Table . Variable Name."}
+{"idx": 4, "title": "Hotel2vec: Learning Hotel Embeddings from User Click Sessions with...", "date": "", "ddg_snippet": "Data are summarized in Table 1 . We randomly split the sessions into training , validation, and test with a ratio of 8:1:1. Table 1 . Dataset statistics . Number of user click sessions Number of unique hotels.", "subpage_snippet": "", "source": "ceur-ws.org", "link": "https://ceur-ws.org/Vol-2974/paper5.pdf", "content": "Data are summarized in Table 1 . We randomly split the sessions into training , validation, and test with a ratio of 8:1:1. Table 1 . Dataset statistics . Number of user click sessions Number of unique hotels."}
+{"idx": 5, "title": "Dataset Search", "date": "", "ddg_snippet": "Feedback. Sign in. Dataset Search.Learn more about Dataset Search.", "subpage_snippet": "", "source": "datasetsearch.research.google.com", "link": "https://datasetsearch.research.google.com/", "content": "Feedback. Sign in. Dataset Search.Learn more about Dataset Search."}
+{"idx": 6, "title": "PNCF: neural collaborative filtering based on pre- trained embedding", "date": "", "ddg_snippet": "We construct two user - item interaction matrices with users' explicit SP S value and users' view activities as implicit feedback.Content-based collaborative filtering (CCF) predicts user - item interactions based on both users' interaction history and items' content information.", "subpage_snippet": "", "source": "www.academia.edu", "link": "https://www.academia.edu/119343832/PNCF_neural_collaborative_filtering_based_on_pre_trained_embedding", "content": "We construct two user - item interaction matrices with users' explicit SP S value and users' view activities as implicit feedback.Content-based collaborative filtering (CCF) predicts user - item interactions based on both users' interaction history and items' content information."}
+{"idx": 7, "title": "What is the influence of the user and items on a recommendation...", "date": "", "ddg_snippet": "User - Item . In the intricate world of recommendation systems, the foundation lies in how users and items are represented.By mapping users and items into this latent space, recommendation algorithms can effectively learn and leverage complex interactions .", "subpage_snippet": "", "source": "ac-programming.com", "link": "https://ac-programming.com/content/Reco/Key/user-item.html", "content": "User - Item . In the intricate world of recommendation systems, the foundation lies in how users and items are represented.By mapping users and items into this latent space, recommendation algorithms can effectively learn and leverage complex interactions ."}
+{"idx": 8, "title": "A Geometric Approach to Personalized Recommendation with...", "date": "", "ddg_snippet": "Table 1 : Dataset Statistics , the Item - User interaction .Table 2: Compositional Query Statistics . Dataset . Personalized.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/html/2502.10875v1", "content": "Table 1 : Dataset Statistics , the Item - User interaction .Table 2: Compositional Query Statistics . Dataset . Personalized."}
+{"idx": 9, "title": "(PDF) A Geometric Approach to Personalized Recommendation with...", "date": "", "ddg_snippet": "Table 1 : Dataset Statistics , the Item - User interaction DU& the Item -Attribute interaction DA. The Train /Test split is created using algorithm 1to test set -theoretic generalization.", "subpage_snippet": "", "source": "www.researchgate.net", "link": "https://www.researchgate.net/publication/389091382_A_Geometric_Approach_to_Personalized_Recommendation_with_Set-Theoretic_Constraints_Using_Box_Embeddings", "content": "Table 1 : Dataset Statistics , the Item - User interaction DU& the Item -Attribute interaction DA. The Train /Test split is created using algorithm 1to test set -theoretic generalization."}

data/sampled_jsons/2503.16979_Equation_6_z_i_=_motion_feature_interpolation.jsonl

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+{"idx": 0, "title": "arXiv:2503.16979v1 [cs.CV] 21 Mar 2025", "date": "", "ddg_snippet": "Interpolate and Motion Decode: Using the 3D motion features stored at anchor points, we can assign each Gaus-sian point a motion feature by interpolating from its K near-est anchors in the neighborhood:", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2503.16979", "content": "Interpolate and Motion Decode: Using the 3D motion features stored at anchor points, we can assign each Gaus-sian point a motion feature by interpolating from its K near-est anchors in the neighborhood:"}
+{"idx": 1, "title": "Correlated Skin Surface and Tumor Motion Modeling for ...", "date": "", "ddg_snippet": "In order to verify the correlation between each variable and tumor movement, we also calculated the correlation coefficient between tumor motion, dimension-reduced data, and NDI markers, respectively, by using Pearson method as Equation ( 6 ). The results are in Table 2.", "subpage_snippet": "", "source": "pmc.ncbi.nlm.nih.gov", "link": "https://pmc.ncbi.nlm.nih.gov/articles/PMC7688455/", "content": "In order to verify the correlation between each variable and tumor movement, we also calculated the correlation coefficient between tumor motion, dimension-reduced data, and NDI markers, respectively, by using Pearson method as Equation ( 6 ). The results are in Table 2."}
+{"idx": 2, "title": "Learning aggressive animal locomotion skills for quadrupedal ...", "date": "", "ddg_snippet": "6 days ago · Recent advancements in deep reinforcement learning have significantly improved the locomotion capabilities of quadrupedal robots. Quadrupedal robots have demonstrated impressive locomotion skills ...", "subpage_snippet": "", "source": "www.nature.com", "link": "https://www.nature.com/articles/s44182-025-00048-x", "content": "6 days ago · Recent advancements in deep reinforcement learning have significantly improved the locomotion capabilities of quadrupedal robots. Quadrupedal robots have demonstrated impressive locomotion skills ..."}
+{"idx": 3, "title": "True Amplitude Seismic Imaging With Wave Equation-Based ...", "date": "", "ddg_snippet": "Nowadays, the amplitude fidelity of seismic imaging becomes more important than ever. Although reverse time migration (RTM) adopts the full wave equation as true amplitude seismic wave propagator, it is still not sufficient for true amplitude seismic imaging since migration is only the adjoint operator corresponding to the forward modeling process.", "subpage_snippet": "", "source": "www.tib.eu", "link": "https://www.tib.eu/en/search/id/ieee:ed9a88631415699a71673c1de39a1c2dbb6a61a7/True-Amplitude-Seismic-Imaging-With-Wave-Equation", "content": "Nowadays, the amplitude fidelity of seismic imaging becomes more important than ever. Although reverse time migration (RTM) adopts the full wave equation as true amplitude seismic wave propagator, it is still not sufficient for true amplitude seismic imaging since migration is only the adjoint operator corresponding to the forward modeling process."}
+{"idx": 4, "title": "An online method for ship trajectory compression using AIS data", "date": "", "ddg_snippet": "In conclusion, when applied to vessel AIS trajectory data compression, it enhances the operational efficiency of computers. Furthermore, it establishes a theoretical foundation for future processing of vessel AIS trajectory big data, ship motion pattern recognition, ship motion feature extraction and research on the behaviour of single vessels.", "subpage_snippet": "", "source": "www.cambridge.org", "link": "https://www.cambridge.org/core/services/aop-cambridge-core/content/view/7A3C7C520C7F2DA5DB3F9B6450A22B7D/S0373463324000171a.pdf/div-class-title-an-online-method-for-ship-trajectory-compression-using-ais-data-div.pdf", "content": "In conclusion, when applied to vessel AIS trajectory data compression, it enhances the operational efficiency of computers. Furthermore, it establishes a theoretical foundation for future processing of vessel AIS trajectory big data, ship motion pattern recognition, ship motion feature extraction and research on the behaviour of single vessels."}
+{"idx": 5, "title": "(PDF) Analyse Automatique des Macro et Micro Expressions ...", "date": "", "ddg_snippet": "In this manner, first, we propose a deep Recurrent Convolutional Auto-Encoder to capture spatial and motion feature changes of natural facial behaviours. Then, a statistical based model for estimating the probability density function of normal facial behaviours while associating a discriminating score to spot micro-expressions is learned based ...", "subpage_snippet": "", "source": "www.academia.edu", "link": "https://www.academia.edu/144057854/Analyse_Automatique_des_Macro_et_Micro_Expressions_Faciales_Détection_et_Reconnaissance_par_Machine_Learning", "content": "In this manner, first, we propose a deep Recurrent Convolutional Auto-Encoder to capture spatial and motion feature changes of natural facial behaviours. Then, a statistical based model for estimating the probability density function of normal facial behaviours while associating a discriminating score to spot micro-expressions is learned based ..."}
+{"idx": 6, "title": "Computer Vision and Pattern Recognition Mar 2025", "date": "", "ddg_snippet": "Title: Do Vision Models Develop Human-Like Progressive Difficulty Understanding? Zeyi Huang, Utkarsh Ojha, Yuyang Ji, Donghyun Lee, Yong Jae Lee. Subjects: ...", "subpage_snippet": "", "source": "www.arxiv.org", "link": "https://www.arxiv.org/list/cs.CV/2025-03?skip=1200&show=2000", "content": "Title: Do Vision Models Develop Human-Like Progressive Difficulty Understanding? Zeyi Huang, Utkarsh Ojha, Yuyang Ji, Donghyun Lee, Yong Jae Lee. Subjects: ..."}

data/sampled_jsons/26JsumCG0z_The Value of Prediction in Identifying the Worst-Off/auto/26JsumCG0z_The Value of Prediction in Identifying the Worst-Off.md

@@ -0,0 +1,735 @@
+# The Value of Prediction in Identifying the Worst-Off
+
+Unai Fischer-Abaigar 1 2 Christoph Kern 1 2 Juan Carlos Perdomo 3
+
+# Abstract
+
+Machine learning is increasingly used in government programs to identify and support the most vulnerable individuals, prioritizing assistance for those at greatest risk over optimizing aggregate outcomes. This paper examines the welfare impacts of prediction in equity-driven contexts, and how they compare to other policy levers, such as expanding bureaucratic capacity. Through mathematical models and a real-world case study on long-term unemployment amongst German residents, we develop a comprehensive understanding of the relative effectiveness of prediction in surfacing the worst-off. Our findings provide clear analytical frameworks and practical, data-driven tools that empower policymakers to make principled decisions when designing these systems.
+
+# 1. Introduction
+
+Faced with pressure to modernize, large bureaucracies are increasingly adopting risk prediction tools to improve efficiency and better serve their populations. Beyond optimizing aggregate outcomes, investments in these programs often aim to address historical inequities and prioritize the needs of the worst-off. For instance, in 2012, Wisconsin launched a risk prediction system to explicitly address deep racial disparities in academic achievement and improve high school graduation rates amongst underserved students. More broadly, such systems are particularly relevant in settings where normative considerations demand prioritizing those at the greatest risk of adverse outcomes, and where well-established downstream interventions can meaningfully benefit these vulnerable individuals.
+
+From a design perspective, these risk predictors are challenging to evaluate because their value cannot be assessed without reference to the broader social context. The value of a risk predictor is ultimately determined by its impact on bottom-line welfare (e.g., graduation rates) and how these welfare impacts compare to those of other bureaucratic alternatives (Johnson & Zhang, 2022). For example, to understand whether investments in prediction are truly valuable in Wisconsin, we need to assess how much better the risk predictor is at identifying at-risk students relative to existing policies. We also need to understand whether sophisticated prediction systems yield higher graduation rates amongst the underserved than structural investments in teacher training or better facilities.
+
+Equity-driven programs are pervasive in applications like social housing, poverty targeting, and unemployment assistance. In these contexts, many government agencies are exploring how algorithmic prediction systems may be an improvement over their current profiling processes (Kortner ¨ & Bonoli, 2023). Yet, due to the absence of an overarching framework that allows the systematic assessment of the relative impacts of different design decisions, efforts to improve predictive accuracy are rarely studied in concert with other policy levers such as expanding screening capacity.
+
+Building on recent work in a budding area of learning in resource allocation contexts, we develop tools to evaluate the design and broader impact of prediction systems that aim to identify the worst-off members of a population. We develop a holistic understanding of the value of statistical prediction in these contexts through theoretical insights into foundational statistical models and a real-world case study on identifying long-term unemployment. Our results establish clear theoretical and empirical criteria characterizing the relative value of core design decisions within these problems. Specifically, we identify when improving prediction provides a higher marginal benefit in helping an institution identify the worst-off. This is compared to alternative strategies, such as keeping prediction accuracy fixed, expanding bureaucratic capacity and screening a larger population.
+
+Interestingly, we show that prediction is a first and last-mile effort. The impacts of improving prediction are always outweighed by those of expanded screening capacity, except for when the system explains either none or almost all of the variance in outcomes. While this relationship is moderated by costs, it still largely holds when prediction improvements are more cost-efficient than measures that expand access.
+
+These results are counternarrative to current efforts in empirical public policy where agencies focus on incremental improvements within complex prediction systems, starting from the solid baseline performances of their current processes (Desiere et al., 2019; Desiere & Struyven, 2021). Furthermore, implementing more complex profiling systems at scale comes with operational costs (such as staff training and data collection) which need to be contextualized by the cost-benefit ratio of expanding access. Our empirical case study explicates how to systematically assess the relative gains of these design components in a real-world application setting, translating formal insights into critical guidance for designers of these systems.
+
+Our results provide theoretically principled and empirically grounded tools for policymakers to make informed decisions when designing prediction systems to identify the worst-off. They also offer a practical framework to help determine how much should be invested in prediction relative to other interventions and how to decide when prediction systems are “good enough” for deployment.
+
+# 1.1. Overview of Framework and Contributions
+
+Setup. We consider a scenario where a decision-maker seeks to identify worst-off members of a population, as determined by a real-valued welfare metric $Y \in \mathbb { R }$ , with the goal of prioritizing them for further screening and support. The population is represented by a distribution $\mathcal { D }$ over features $X$ and outcomes $Y$ . The planner aims to identify all individuals whose outcomes $Y$ fall below some threshold $t ( \beta )$ , $Y \leqslant t ( \beta )$ . Here, $\beta \in [ 0 , 1 ]$ is a parameter (quantile) that determines the size of the population that is at risk, $\operatorname* { P r } [ Y \leqslant t ( \beta ) ] = \beta$ . For instance, in poverty prediction, $Y$ is income, and the goal is to identify everyone whose income is below some value.
+
+To solve this problem, the social planner has access to data $( X , Y ) \sim { \mathcal { D } }$ and builds a screening policy $\pi : \mathcal { X }  \{ 0 , 1 \}$ that determines whether an individual with features $x$ is screened from the broader population to see if they belong to the worst-off group. Learning plays a fundamental role since the optimal policy is to predict each person’s expected outcome, $\dot { \boldsymbol { f } } ( \boldsymbol { x } ) = \dot { \boldsymbol { Y } } \approx \mathbb { E } \left[ \boldsymbol { Y } \right] \boldsymbol { X } = \boldsymbol { x } \boldsymbol { ] }$ and screen those in the bottom fraction, $\pi _ { f } ( x ) = 1 \{ f ( x ) \leqslant t ( \alpha ) \}$ .
+
+Unpacking this further, $\alpha \in [ 0 , 1 ]$ , is a design parameter that determines how many people the planner can screen, $\operatorname* { P r } [ f ( x ) \leqslant t ( \alpha ) ] = \alpha$ . The amount of resources $\alpha$ need not be equal to the size of the target population $\beta$ . For instance, an organization might have normative goal of identifying the poorest $5 \%$ of individuals, but only have the resource to screen $1 \%$ of the population. Conversely, they might realize that predictions are not perfect, and that to identify the bottom $5 \%$ , they might have to screen $1 0 \%$ of the population.
+
+Given a predictor $f$ , a screening budget of $\alpha$ , and a target parameter $\beta$ , the value of a prediction system is equal to the fraction of the at-risk population that it identifies,
+
+$$
+V ( \alpha , f ; \beta ) = \operatorname* { P r } _ { \mathcal { D } } [ f ( x ) \leqslant t ( \alpha ) \ | \ Y \leqslant t ( \beta ) ] ,
+$$
+
+where again $t ( \alpha ) , t ( \beta )$ are chosen to respect the design constraints. We focus on this notion of value since our driving motivation is to analyze domains like unemployment assistance, or poverty prediction, where there is no harm in the prediction system raising a false positive $\pi ( x ) =$ $1 , Y > t ( \beta ) )$ . By and large, the true value of the system is equal to the extent that it helps an institution efficiently identify the needy amongst a large, diverse population.
+
+The focus of our work is to build a holistic understanding of prediction in these contexts by evaluating the relative impacts of different design parameter, such as expanding screening capacity or improving prediction, on this notion of bottom-line value $V ( \alpha , f ; \beta )$ . We develop these insights through theoretical investigations as well as in-depth empir ical case study.
+
+Mathematical Results. Following Perdomo (2024), we formalize the relative value of prediction for the worst-off by studying the prediction-access ratio or PAR. Intuitively, the PAR measures the relative change in value achieved by optimizing different policy levers.
+
+$$
+\mathrm { P A R } = \frac { \mathrm { M a r g i n a l ~ V a l u e ~ o f ~ E x p a n d i n g ~ A c c e s s } } { \mathrm { M a r g i n a l ~ V a l u e ~ o f ~ B e t t e r ~ P r e d i c t i o n } } .
+$$
+
+We formally define this quantity in Equation 3. While initially developed to specifically study the value of prediction in allocation problems where allocating goods to individuals had heterogeneous effects, here we extend this concept to analyze the value of prediction in a related, but distinct, setting where we aim to identify the worst-off.
+
+Small values of the PAR (i.e. $\mathrm { P A R } < 1$ ) indicate that small improvements in prediction yield a much larger (relative) impact in the ability to target the worst-off than a small expansion in screening capacity. The opposite is true if the PAR is greater than 1. Calculating this quantity is a fundamental step in deciding which policy lever makes economic sense.
+
+Costs. A full cost-benefit analysis requires combining the prediction-access ratio with the (marginal) costs of improvements in capacity $\boldsymbol { C } _ { \mathrm { A c c e s s } }$ and prediction $C _ { \mathrm { P r e d } }$ . Once we factor in costs, it is easy to decide what to focus on. A social planner should expand access whenever
+
+$$
+\frac { C _ { \mathrm { A c c e s s } } } { C _ { \mathrm { P r e d } } } < \mathrm { P A R }
+$$
+
+and invest in better prediction otherwise. The (marginal) costs that competing policy levers carry are inherently context-dependent and will vary across application domains. In many applied settings the cost ratio is comparatively well understood, for example, the salary of an additional caseworker or the cost of a household survey. By presenting PAR separately from costs, we isolate the welfare side of the equation; domain experts can then plug in their own cost estimates to reach a policy decision. In particular, the PAR tells us how much we should be willing to pay for improvements in prediction versus expanding access.
+
+We encourage future work to explore scenarios with more complex or less clearly defined cost structures. For instance, many practical applications involve recurring costs, such as ongoing staff salaries or periodic data collection, and fixed costs, such as initial investments in infrastructure or predictive model development. Analyzing how these cost structures affect welfare decisions over time, including amortization of fixed investments or identifying the point at which specific improvements become cost-effective, would significantly enhance our understanding of the relative value of prediction.
+
+To build intuition for the value of prediction in identifying the worst-off, we examine the prediction access ratio in one of the most basic statistical models. The outcomes $Y$ are Gaussian, and the learner has access to a predictor $f ( x ) = { \hat { Y } }$ such that the errors $Y - { \hat { Y } }$ are also Gaussian and independent of $\hat { Y }$ . While extremely simple, the model yields surprisingly precise numerical insights that exactly match up in our real-world case study, where, of course, none of these assumptions hold. In this setting the quality of $\hat { Y }$ is fully summarized by the coefficient of determination $R ^ { 2 } = \dot { \operatorname { c o r r } } ( Y , \hat { Y } ) ^ { 2 }$ .
+
+Our first result identifies when local improvements in prediction have the highest impact:
+
+Theorem 1.1 (Informal, see Theorem 3.2). If $\alpha$ is at least a constant, the local improvements in $V$ with respect to $R ^ { 2 }$ diverge in two regimes: (1) $R ^ { 2 } \to 1$ and $\alpha = \beta$ , or (2) $R ^ { 2 } \to 0$ . In both cases, the prediction-access ratio satisfies $\mathrm { P A R } ( \alpha , \beta ) = 0$ .
+
+Predictions have the highest marginal impact at low and high $R ^ { 2 }$ -values, making them a first- and last-mile effort. Our second result characterizes when the opposite is true. We prove that whenever screening capacities are severely limited relative to the size of the population one aims to identify $\alpha \ll \beta$ , the benefits of increasing $\alpha$ are overwhelming. Furthermore, it shows that the impacts of improving access are still relatively larger exactly in the regime where most current systems operate: $f$ explains $\approx 2 0 \%$ of the variance and $\alpha$ is equal to, or even slightly larger, than $\beta$ .
+
+Theorem 1.2 (Informal, see Theorem 3.1, Proposition 3.3). If the predictor $f$ explains an $R ^ { 2 }$ fraction of the variance, where $R ^ { 2 }$ is at least a constant, then the prediction access ratio is at least $\Omega ( \alpha ^ { - 1 / ( 1 - R ^ { 2 } ) } )$ . Furthermore, $i f 0 . 1 5 \leqslant$ $R ^ { 2 } \leqslant 0 . 8 5$ and $\alpha \leqslant \beta$ or $0 . 2 \leqslant R ^ { 2 } \leqslant 0 . 5$ , $\beta \geqslant 0 . 1 5$ , and $\alpha \leqslant 0 . 5$ then the local prediction-access ratio is at least $^ { l }$ .
+
+Empirical Results. We complement our theoretical discussion by presenting a methodology for policymakers to evaluate the prediction-access ratio in practice. Using a real-world administrative dataset on hundreds of thousands of jobseekers in Germany, we show that our theoretical findings generalize to a more complex, real-world context that closely resembles algorithmic profiling systems widely implemented in many countries. Notably, our results reveal that when considering non-local improvements, expanding screening capacity has an even greater impact compared to enhancing prediction accuracy.
+
+# 1.2. Related Work
+
+Machine learning is increasingly used in the public sector to allocate support by predicting individuals at risk of adverse outcomes (Fischer-Abaigar et al., 2024), with applications spanning a wide range of problem domains (Desiere et al., 2019; Blumenstock, 2016; Perdomo et al., 2023; Chan et al., 2012; Potash et al., 2015; Chouldechova et al., 2018). A large methodological literature draws on decision theory, operations research, economics, and machine learning to learn allocation rules from data (Elmachtoub & Grigas, 2022; Kitagawa & Tetenov, 2018; Manski, 2004; Fernandez-Lor ´ ´ıa & Provost, 2022), with recent work in causal inference focusing on learning treatment policies from observational data (Athey & Wager, 2021; Kallus, 2021). However, many decision-makers rely on separately trained predictive risk scoring-systems to solve “prediction policy problems” (Kleinberg et al., 2015). Recently, this work has been extended using causal inference to train and evaluate these systems (Coston et al., 2023; Guerdan et al., 2023; Boehmer et al., 2024).
+
+The widespread use of risk-scoring systems has raised concerns regarding their tradeoffs, pitfalls, and validity (Wang et al., 2024; Coston et al., 2023; Fischer-Abaigar et al., 2024). These concerns include not only questions of empirical performance but also of fairness and equity in how predictive systems shape access to public services (Barocas et al., 2023). Recent work explores alternative design choices—such as employing aggregate rather than individual-level predictions (Shirali et al., 2024), balancing immediate needs with information-gathering (Wilder & Welle, 2024), and introducing randomization (Jain et al., 2024)—to improve downstream outcomes.
+
+Perdomo (2024) studies the prediction-access ratio under both linear and probit models, with the latter closely related to our work. While they focus on binary welfare outcomes, we adopt a continuous welfare metric and a distinct policy objective: rather than evaluating changes in overall expected welfare, we measure the fraction of truly worst-off individuals who are identified. This captures a mathematically and conceptually distinct setting frequently encountered in the public sector. For instance, employment agencies often prioritize identifying and assisting individuals in greatest need, rather than optimizing average employment outcomes across all jobseekers. In addition, we introduce a set of empirical tools to analyze these tradeoffs in practice, while the work of Perdomo (2024) is purely theoretical.
+
+# 2. Formal Framework
+
+We start by formally defining our screening problem.
+
+Definition 2.1 (Screening Problem). The screening problem seeks to identify a decision rule $\pi \colon \mathbb { R }  \{ 0 , 1 \}$ that fraction of the worst-off population that is identified while adhering to resource constraints $\alpha \in ( 0 , 1 )$ that bound the percentage of the population that can be screened by the social planner:
+
+$$
+\operatorname* { m a x } _ { \pi : \mathbb { R } \to \{ 0 , 1 \} } \mathbb { E } \left[ \pi ( { \hat { Y } } ) = 1 \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \right] { \mathrm { s . t . } } \mathbb { E } \left[ \pi ( { \hat { Y } } ) \right] \leqslant \alpha
+$$
+
+The quantile $F _ { Y } ^ { - 1 } ( \beta )$ denotes the welfare cutoff that identifies the worst-off $\beta \in ( 0 , 1 )$ fraction of the population.
+
+Given perfect knowledge of the welfare outcomes ${ \hat { Y } } = Y$ , the optimal decision policy is simple: rank individuals based on their outcomes $Y$ and intervene in the bottom $\alpha$ -fraction of the population. In the general case, we have:
+
+Proposition 2.2. The optimal policy $\pi ^ { * } \colon  { \mathbb { R } } \to \{ 0 , 1 \}$ to solve the screening problem (Definition 2.1) is equal to $\pi ^ { * } ( \hat { Y } _ { i } ) = 1 \{ s ( \hat { Y } _ { i } ) \stackrel {  } { \geqslant } F _ { s } ^ { - 1 } ( 1 - \alpha ) \}$ where $F _ { s } ^ { - 1 } ( 1 - \alpha )$ is the $( 1 - \alpha )$ -quantile of $s ( \hat { Y } ) = \operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } ]$ .
+
+Policy Value in Gaussian Setting. For the theoretical investigation, we assume independent, identically distributed errors $\varepsilon = Y { - } \hat { Y } \stackrel { i i d } { \sim } { \mathcal { N } } ( 0 , \gamma ^ { 2 } )$ that are independent of $\hat { Y }$ . In this setting, the screening problem can be solved by ranking individuals in ascending order of their predicted outcomes $\hat { Y }$ and screening the bottom $\alpha$ -fraction (see Proposition C.1), achieving the policy value:
+
+$$
+V ( \pi ^ { * } ) = \operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \ | \ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ]
+$$
+
+In addition, we assume welfare outcomes $Y \sim { \mathcal { N } } ( \mu , \eta ^ { 2 } )$ Because $\varepsilon$ is independent of $\hat { Y }$ , this implies that $Y$ and $\dot { Y }$ follow a bivariate normal distribution.
+
+Proposition 2.3. (Policy Value in Gaussian Setting) Let Y − Yˆ iid∼ N (0, γ2) and Y ∼ N (µ, η2), then the value $V ( \pi ^ { * } )$ of the optimal screening policy $\pi ^ { * }$ is given by
+
+$$
+\begin{array} { r } { V ( \pi ^ { * } ) = V ( \alpha , \beta , R ^ { 2 } ) = \frac { \Phi _ { 2 } \left( \Phi ^ { - 1 } ( \alpha ) , \Phi ^ { - 1 } ( \beta ) ; \rho \right) } { \beta } } \end{array}
+$$
+
+where $\Phi _ { 2 } \left( \cdot \right)$ denotes the bivariate standard normal $C D F$ with correlation $\rho = \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } / \eta$ and $\Phi ^ { - 1 } \left( \cdot \right)$ is the quantile function of the standard normal distribution.
+
+In this model, the goodness of the predictions $\hat { Y }$ are entirely captured by the coefficient of determination $R ^ { 2 }$ , which equals the squared correlation $\rho ^ { 2 }$ between $Y$ and $\hat { Y }$ .
+
+Our analysis extends to the log-normal distribution $\log Y \sim$ $\mathcal { N } ( \mu , \eta ^ { 2 } )$ under a a multiplicative error model $Y = \overset { \cdot } { Y } \cdot u$ with $\log u \ \sim \ \mathcal N ( 0 , \gamma ^ { 2 } )$ . Taking logarithms, leads to $\log Y = \log { \hat { Y } } + \log u$ . Since the logarithm is strictly increasing, the ordering of $Y$ and $\hat { Y }$ is preserved under transformation. This allows us to apply the same framework to the log-transformed variables $\log Y$ and $\log { \hat { Y } }$ . This extension is particularly useful because many welfare outcomes, such as income distributions (Clementi & Gallegati, 2005), can be approximated by a log-normal distribution.
+
+Visualization. For a given screening capacity $\alpha$ and $R ^ { 2 }$ value, we can illustrate the corresponding screening policy by plotting the probability $\operatorname* { P r } \{ \hat { Y } \overset { \cdot } { \leqslant } F _ { \hat { Y } } ^ { - 1 } \overset { \cdot } { ( \alpha ) } | Y = \overset { \cdot } { y } \}$ that an individual with welfare outcome $Y ^ { ' } = y$ is screened. As shown in Figure 1, lower values of $Y$ correspond to higher probabilities of being screened. We focus on evaluating how effectively the screening policy identifies individuals in the worst-off segment of the population (i.e., on the left side of the $\beta$ cutoff).
+
+# 3. Theoretical Results
+
+The decision-maker has (at least) two pathways to raise the policy value, which we refer to as policy levers:
+
+• Expanding Access Increasing the screening threshold from $\alpha$ to $\alpha + \Delta _ { \alpha }$ . If full screening were possible $( \alpha = 1 )$ ), the $\beta$ -fraction would be fully identified, as $\begin{array} { r } { V \big ( \pi ^ { * } \big ) = \frac { \Phi _ { 2 } \big ( \Phi ^ { - 1 } ( \alpha ) , \Phi ^ { - 1 } ( \beta ) ; \rho \big ) } { \rho _ { . } } = \frac { \Phi \big ( \Phi ^ { - 1 } ( \beta ) \big ) } { \rho _ { . } \beta _ { . } } = 1 } \end{array}$ • Improving Predictions Investing in better predictive models, modeled as increasing $R ^ { 2 }$ to $R ^ { 2 } +$ $\Delta _ { R ^ { 2 } }$ . Perfect predictions $R ^ { 2 } \ = \ 1 )$ leads to optimal allocation of available capacities: $V ( \pi ^ { * } ) =$ $\scriptstyle { \frac { 1 } { \beta } } \Phi \left( \operatorname* { m i n } ( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ) \right)$ .
+
+Figure 1 showcases improvements in access and prediction. Increasing capacity expands the fraction of the population screened, while improving $R ^ { 2 }$ shifts probability mass across the $\beta$ threshold, enhancing targeting accuracy.
+
+Following Perdomo (2024), a key quantity of interest is the prediction-access ratio (PAR), which quantifies the relative improvements in policy value from enhancing predictions versus improving access to screening. Specifically, the PAR is defined as:
+
+$$
+\begin{array} { r } { \mathrm { P A R } = \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) } } \end{array}
+$$
+
+In other words, the PAR can inform a social planner how much more they should be willing to pay for improvements in screening capacity relative to prediction. For example, a $\mathrm { P A R } > 2$ implies that expanding the screening capacity by $\Delta _ { \alpha }$ yields at least twice the increase in policy value compared to investing in improved predictions by $\Delta _ { R ^ { 2 } }$ . Consequently, the social planner should prioritize investments in screening capacity, provided the costs of doing so are not more than double those of improving predictions.
+
+![](images/7638b95add28dc8fff4a4bcdbdbae2c50a8daf9c66ffd14fd25a90116d2dd91c.jpg)  
+Figure 1. Screening Policy in Gaussian Setting. (Left) Probability of being screened for an individual with a specific welfare outcome $Y$ , given $R ^ { 2 } = 0 . 2 5$ , $\alpha = 0 . 2$ , and $\beta = 0 . 2$ . The dashed line represents the unconstrained oracle policy, which perfectly screens those in need. (Middle) Policy with expanded screening capacity, where $\alpha$ increases by $\Delta _ { \alpha } = 0 . 2$ . (Right) Policy under an improved prediction model with $R ^ { 2 } + \Delta _ { R ^ { 2 } }$ , where $\Delta _ { R ^ { 2 } } = 0 . 2$ . The shaded area under $\operatorname* { P r } [ \hat { Y } \leqslant \dot { F } _ { \hat { Y } } ^ { - 1 } ( \alpha ) \ | \ Y = y ]$ , weighted by $f _ { Y } ( y )$ and normalized by $\operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ]$ , corresponds to the policy value.
+
+# 3.1. Theoretical Bounds for the Prediction-Access Ratio
+
+In our setting, direct calculation of the PAR is challenging due to the policy value being analytically intractable and the problem featuring strong non-linearities. We derive bounds for specific cases and regimes that we consider particularly insightful, with a focus on marginal local improvements. In our empirical investigation, we find that the main results generalize well to a more complex, real-world setting.
+
+# What should priorities be if screening is very limited?
+
+Theorem 3.1 (PAR for Small Screening Capacities). For any $0 < R ^ { 2 } < 1$ , $\Delta _ { R ^ { 2 } } , \Delta _ { \alpha } > 0$ and $0 < \beta \leqslant 0 . 5$ there exists a threshold $t ( \beta , R ^ { 2 } , \Delta _ { R ^ { 2 } } )$ such that for any $\alpha + \Delta _ { \alpha } \leqslant$ $t _ { : }$ , $\mathrm { P A R } ( \alpha , R ^ { 2 } , \Delta _ { \alpha } , \Delta _ { R ^ { 2 } } )$ is at least
+
+$$
+\textstyle { \frac { \Delta _ { \alpha } } { \Delta _ { R ^ { 2 } } } } { \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } } \left( 5 . 1 \cdot \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { - { \frac { 1 } { 1 - R ^ { 2 } } } + o ( 1 ) }
+$$
+
+where $o ( 1 )$ goes to zero as α approaches zero.
+
+Suppose the available screening capacity $\alpha + \Delta _ { \alpha }$ is very small $( \alpha + \Delta _ { \alpha } \ll \beta )$ , and assume there is a baseline level of predictability (i.e., $R ^ { 2 }$ is bounded away from 0). Then Theorem 3.1 implies that the PAR can become very large. Specifically, for small $\alpha$ , $\Phi ^ { - 1 } \left( 1 - \alpha \right)$ grows asymptotically like $\sqrt { \log { ( 1 / \alpha ) } }$ . Consequently, the polynomial growth of $\alpha ^ { - 1 / ( 1 - R ^ { 2 } ) }$ drives the PAR to increase rapidly as $\alpha$ decreases. It follows that in the scarce capacity regime, expanding the screening capacity has a far greater impact than improvements in prediction accuracy.
+
+# When does prediction have the highest impact?
+
+Theorem 3.2 (Maximally Effective (Local) Prediction Improvements). Let $0 < \beta < 1$ be fixed and $0 \textless \alpha \textless 1$ Consider the points that maximize the local rate of change
+
+in policy value $V$ with respect to improvements in $R ^ { 2 }$ :
+
+$$
+( \alpha _ { * } , R _ { * } ^ { 2 } ) = \operatorname * { a r g m a x } _ { ( \alpha , R ^ { 2 } ) \in ( 0 , 1 ) \times ( 0 , 1 ) } \operatorname * { l i m } _ { \Delta \to 0 } \frac { V ( \alpha , \beta , R ^ { 2 } + \Delta ) - V ( \alpha , \beta , R ^ { 2 } ) } { \Delta }
+$$
+
+The local improvements in $V$ diverge—and are maximized— in two regimes: (1) $R _ { * } ^ { 2 } \to 1$ , $\alpha _ { * } = \beta$ , and (2) $R _ { * } ^ { 2 } \to 0$ . For both regimes, setting $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = \Delta$ , the local predictionaccess ratio satisfies $\begin{array} { r } { \operatorname* { l i m } _ { \Delta \to 0 } \operatorname { P A R } ( \alpha , \beta , \Delta ) \to 0 } \end{array}$ .
+
+According to Theorem 3.2, marginal improvements in prediction are most impactful in two distinct regimes. First, when predictive capacity is very low, even a small initial investment can lead to disproportionately large improvements, provided that a minimal baseline of screening capacity is present. Second, as $R ^ { 2 }$ approaches one, further marginal improvements can also have a significant relative impact, specifically around the point where the screening capacity $\alpha$ matches the requirements for screening the entire $\beta$ -segment of the population. See Figure 2.
+
+# When are small increases in screening capacity more impactful than improving predictions?
+
+Proposition 3.3 (PAR for Local Improvements). Let $R ^ { 2 }$ , $\beta$ , and $\alpha$ satisfy either $R ^ { 2 } \in ( 0 . 1 5 , 0 . 8 5 )$ , $\beta \in ( 0 . 0 3 , 0 . 5 )$ , and $\alpha \leqslant \beta$ , or $R ^ { 2 } \in ( 0 . 2 , 0 . 5 )$ , $\beta \geqslant 0 . 1 5$ , and $\alpha \leqslant 0 . 5 .$ . If $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = \Delta$ , then $\begin{array} { r } { \operatorname* { l i m } _ { \Delta \to 0 } \operatorname { P A R } ( \alpha , \beta , \Delta ) \geqslant 1 } \end{array}$ .
+
+We find that the PAR remains above one as long as $\alpha \leqslant \beta$ and $R ^ { 2 }$ is not too extreme. For larger $\beta$ values (i.e., $\beta \geqslant$ 0.15) the PAR stays above one even for large $\alpha$ provided $R ^ { 2 }$ remains in a moderate range. Crucially, this represents the standard parameter regime in which most allocation programs operate, characterized by a moderate baseline of predictions and resource levels comparable to $\beta$ .
+
+Numerical Simulations. We complement our theoretical investigation with numerical simulations of the PAR for different $\alpha$ , $\beta$ and $R ^ { 2 }$ values (see Figure 2). Consistent with our theoretical results, the PAR becomes large for small screening capacities $( \alpha \ll \beta )$ and remains above one for $\alpha \leqslant \beta$ , provided a small baseline level of predictive performance has been established. The bounds in Proposition 3.3 are conservative, with $\mathrm { P A R } > 1$ observed for a broad range of $R ^ { 2 }$ values. Prediction improvements are particularly impactful when $R ^ { 2 }$ is small. Although the PAR falls below one in the high- $R ^ { 2 }$ and high- $\alpha$ regime, allocation is nearly perfect, making further improvements a “last mile” effort.
+
+Discussion. We found several insights relevant to policymakers aiming to iteratively improve a screening system. First, establishing a baseline level of predictive performance is usually a good starting point. Once this is achieved, expanding the screening capacity becomes the next priority. For very small capacities, Theorem 3.1 tell us that the PAR can increase significantly, making investments in screening capacity highly impactful.
+
+Generally, expanding capacity to at least the level where everyone in need could hypothetically be screened $( \alpha \geqslant$ $\beta$ ) is likely cost-efficient. Once both screening capacity and predictive accuracy are high and the allocation system is close to optimal, improvements in prediction become relatively more valuable again for perfecting the system. However, this regime may rarely be reached in practice.
+
+In Figure 2, we display the PAR for a cost ratio of $1 / 4$ As expected, the regions where investing in $R ^ { 2 }$ is more efficient expand, and some of the earlier nonquantitative bounds no longer apply. Nevertheless, the key insights remain consistent: when screening capacities are small, investments in expanding them are very effective, while improvements in $R ^ { 2 }$ are more important when predictive accuracy is low.
+
+# 4. Empirically Evaluating the PAR
+
+While our theory offers broad intuition when expanding screening capacity or improving predictions is most effective, policymakers need practical tools for their own systems. To support this, we develop a methodology to compute and interpret the prediction-access ratio using empirical data, helping social planners identify the most efficient policy levers for their unique problem context.
+
+Policy Value. As before, we define the allocation policy’s value as the probability that the worst-off individuals are successfully identified:, i.e. $V ( \alpha , \beta ) = \mathrm { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \ |$ $Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ]$ . In practice, this can be measured using a recall-like metric, capturing the proportion of truly at-risk individuals screened by the policy.
+
+$$
+\begin{array} { r } { V ( \alpha , \beta ) \approx \frac { \sum _ { i = 1 } ^ { n } 1 \{ \hat { Y } _ { i } \leqslant F _ { \hat { Y } , n } ^ { - 1 } ( \alpha ) \} 1 \{ Y _ { i } \leqslant F _ { Y , n } ^ { - 1 } ( \beta ) \} } { \sum _ { i = 1 } ^ { n } 1 \{ Y _ { i } \leqslant F _ { Y , n } ^ { - 1 } ( \beta ) \} } } \end{array}
+$$
+
+Increasing Screening Capacity. Given a chosen $\Delta _ { \alpha }$ the policy improvement can be directly computed $V ( \alpha +$ $\Delta _ { \alpha } , \beta ) \mathrm { - } V ( \alpha , \beta )$ by recalculating the empirical policy value at the new threshold. For example, in cash transfer programs (Blumenstock, 2016), a key question is how many resources $\alpha ^ { * }$ are required to reach a specified fraction $p$ of poor households, i.e. $\begin{array} { r } { \alpha ^ { * } = \operatorname* { i n f } _ { \alpha \in ( 0 , 1 ) } \{ \alpha \colon V ( \alpha , \beta ) \geqslant p \} } \end{array}$ .
+
+Improving Predictions. A decision-maker can improve a model’s predictions through various pathways:
+
+a) Data Collection Collect additional samples and increase the frequency of data collection. Social prediction systems are often vulnerable to distribution shifts over time in dynamic and evolving environments (Fischer-Abaigar et al., 2024; Aiken et al., 2023).   
+b) Data Quality Improve data quality (i.e., reduce errors and missing data) by means such as standardizing data collection processes, implementing centralized data management systems, and offering targeted training programs for staff.   
+c) Collect Additional Features In government, this may involve integrating separate data sources across institutions (Sun & Medaglia, 2019; Wirtz et al., 2019).   
+d) Advanced Modeling Techniques Utilize more sophisticated modeling techniques, which might capture more complex patterns in the data but are often more costly to operationalize.
+
+In resource-constrained settings, planners often focus on incremental improvements rather than rebuilding entire systems. For instance, collecting a small amount of additional data may boost $R ^ { 2 }$ by a few points, uniformly reducing errors. To simulate such minor gains, we scale the model’s residuals $\hat { Y } _ { + } = \hat { Y } + \delta ( Y - \hat { Y } )$ , choosing $\delta \in ( 0 , 1 )$ so that $R ^ { 2 }$ increases by a target $\Delta _ { R ^ { 2 } }$ (see Appendix B.3). This preserves the overall error structure, allowing us to gauge how a “similar but slightly better” model affects policy outcomes.
+
+This approach can be extended in several ways. For example, residuals could be adjusted for specific subgroups to account for uneven prediction improvements (e.g., targeted data collection for rural or underrepresented populations). Alternatively, planners could retrain models under different conditions—such as sample size, feature set, or architecture—and compare the resulting policy value.
+
+# 5. Case Study: Identifying Long-Term Unemployment in Germany
+
+Public employment services (PES) across the globe make use of profiling approaches to identify jobseekers at risk of long-term unemployment to target preventative measures (Loxha & Morgandi, 2014). Starting from traditional rulebased approaches, many PES either test or already deploy algorithmic profiling to identify jobseekers in need of support (Desiere et al., 2019; Kortner & Bonoli ¨ , 2023). While these profiling tools assist in allocating programs that account for large shares of PES spending — making design choices critical (Kern et al., 2024) — systematic assessments of their relative value compared to other measures for improving jobseekers’ outcomes remain absent.
+
+![](images/b8a6c97ec9b625037fc1b3acf0f617ce29e727e636909ccb2d6e31b7384b9210.jpg)  
+Figure 2. Numerical Simulation of the Prediction-Access Ratio (PAR), Equation 3, for $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = 0 . 0 1$ and $\beta = 0 . 2$ . (Left) The PAR values. (Right) $1 / 4 \times \mathrm { P A R }$ , representing a cost ratio of $1 / 4$ . Each point represents a screening capacity $\alpha$ ( $\mathbf { \hat { x } }$ -axis) and $R ^ { 2 }$ value $\mathbf { \widetilde { y } }$ -axis), with the color bar showing the PAR clipped to the range [0.5, 2.0]. Dotted black lines represent $\mathrm { P A R } = 1$ , where improvements in $\alpha$ and $R ^ { 2 }$ are equally effective. The purple line marks the region in the $( \alpha , R ^ { 2 } )$ space where the policy value $V ( \alpha , \beta , R ^ { 2 } )$ exceeds 0.9.
+
+![](images/9b2f21e6c3848a4021f808cba481dd31621b56d4fe5c6b0d5ccb53247b9ada8e.jpg)  
+Figure 3. Prediction-Access Ratio for $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = 0 . 1$ across three regimes: (left) constant prediction $( R ^ { 2 } = 0$ ), (middle) trained model $R ^ { 2 } = 0 . 1 5 )$ , and (right) near-perfect prediction $R ^ { 2 } = 0 . 9$ ). As expected from our theoretical intuition, the PAR is large for small $\alpha$ and in the middle plot, which represents the typical regime for allocation systems.
+
+We secured access to a dataset1 on German jobseekers derived from German administrative labor market records that cover a large portion of the German labor force. It merges multiple administrative data sources, containing a wide spectrum of individual labor market information — including records on employment histories, received benefits, unemployment periods, participation in job training programs and demographic information. Such administrative records are the primary data source used by PES to build algorithmic profiling models (Bach et al., 2023).
+
+Experimental Setup. We train a model to predict how long a newly registered jobseeker remains unemployed, defining the target $Y$ as unemployment duration in days (capped at 24 months). Following Bach et al. (2023), we use a set of covariates capturing demographic information, labor market history, and most recent job details. To ensure full 24-month observations and mimic a realistic deployment scenario, we focus on unemployment spells beginning between 2010 and
+
+2015, resulting in data on 274,515 different jobseekers and 553,980 unemployment spells. We refer to Appendix B.1 for additional information on the experimental setup and data.
+
+Our focus is the $\beta$ -fraction of jobseekers with the longest expected unemployment durations, representing those most at risk. In Germany, being unemployed for over one year (about $1 5 \%$ of cases in our data; Figure 8) meets the legal definition of long-term unemployment (Bach et al., 2023), but some countries adopt different cutoffs (Desiere et al., 2019).
+
+# 5.1. Results
+
+We train a CatBoost model (see Appendix B.2 for details), achieving an $R ^ { 2 }$ of 0.15 on the test set. This level of predictive power aligns well with what is typically observed in social prediction tasks (Salganik et al., 2020) and similar applied settings (Desiere et al., 2019).
+
+How much does the screening capacity need to increase to target a significant fraction of high-risk jobseekers? As expected, larger screening capacities increase both the policy value and the number of high-risk jobseekers screened (see Figure 4(a)). Focusing on the (German) LTU cutoff $( \beta \approx 0 . 1 5 )$ , our policy value aligns well with findings of previous studies2 (Bach et al., 2023).
+
+A planner might begin by setting $\alpha = \beta$ , ensuring that, in theory, enough capacity is provided to screen and support every high-risk jobseeker. A natural question then arises: how much additional capacity $\Delta _ { \alpha }$ would be required to screen at least a specified percentage of high-risk individuals? This additional capacity represents the overhead that must be invested to account for imperfect predictions. We observe that the $\Delta _ { \alpha }$ required to ensure at least $7 5 \%$ of high-risk jobseekers are screened remains consistently around 0.25 across different $\beta$ values. While the policy value increases as $\alpha = \beta$ rises, the marginal improvements gained from increasing access decrease for higher $\alpha$ , resulting in a somewhat stable $\Delta _ { \alpha }$ across $\beta$ . In practice, this means we need to screen $2 5 \%$ more of the population to ensure adequate coverage.
+
+What is the impact of improving screening capacity versus prediction errors? We simulate small improvements in the $R ^ { 2 }$ value by uniformly scaling the residuals by a multiplicative factor. To ensure that this approach approximates a realistic pathway of (marginally) improving the model, we train various models at different sample sizes. We then verify that as $R ^ { 2 }$ increases with the amount of training data, the variance of the residuals decreases, while the distribution remains largely unchanged in shape (see Figure 12). We then evaluate the prediction-access ratio for $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = 0 . 1$ in three scenarios : (1) the trained CatBoost model with $R ^ { 2 } = 0 . 1 5$ , (2) near-perfect predictions with $R ^ { 2 } = 1 - \Delta _ { R ^ { 2 } }$ and (3) constant predictions $R ^ { 2 } = 0$ ), effectively randomizing screening decisions.
+
+We observe a rise in the PAR for small screening capacities $\alpha$ (see Figure 3), consistent with Theorem 3.1. Under random allocation $R ^ { 2 } = 0 _ { , }$ ), the PAR stays below one for $\alpha \geqslant 0 . 1$ . This result aligns somewhat with Theorem 3.2, where we found that the (local) PAR approaches zero as $R ^ { 2 } \to 0$ . Because we consider $\Delta = 0 . 1$ (rather than an infinitesimal improvement, see Figure 13 for $\Delta = 0 . 0 1$ ), the PAR remains large at small $\alpha$ . For the CatBoost model $R ^ { 2 } = 0 . 1 5 )$ ), capacity improvements stay relatively more effective (i.e., $\mathrm { P A R } > 1$ ) for larger $\alpha$ , matching Proposition 3.3, where we found that for moderate $R ^ { 2 }$ and $\alpha \leqslant \beta$ , the local PAR remains above one. Meanwhile, near-perfect predictions $( R ^ { 2 } = 0 . 9$ ) make capacity investments highly efficient, causing the PAR to diverge for $\alpha < \beta$ , then drop sharply near $\alpha = \beta$ because the allocation becomes nearly optimal. When $\alpha \geqslant \beta$ , the PAR stabilizes at one as numerator and denominator both approach zero.
+
+These observations broadly match our theoretical findings, despite the non-local improvements and more complex residual structure. Notably, the theory’s focus on local improvements offers a conservative perspective on capacity investments: even under random allocation $R ^ { 2 } = 0$ ), securing a modest screening capacity $( 5 - 1 0 \% )$ is often the first priority, while at very high $R ^ { 2 }$ , gains in policy value diminish so rapidly once $\alpha \geqslant \beta$ that the relative advantage of further prediction investments becomes negligible.
+
+When do small improvements in prediction error have the largest impact? From theory (Theorem 3.2), we expect local policy value improvements from better predictions to diverge as $R ^ { 2 } \to 0$ and $R ^ { 2 } \to 1$ when $\alpha = \beta$ . This aligns with our results in Figure 4: for small $\Delta _ { R ^ { 2 } }$ , the rate of local improvements in $V ( R ^ { 2 } )$ with respect to $R ^ { 2 }$ diverges. The location of the maximum in $\alpha$ also follows from the theory: as $R ^ { 2 } \to 1$ , the rate only diverges for $\alpha = \beta$ , while for small $R ^ { 2 }$ the maximum is at $\alpha \approx 0 . 5$ .
+
+What are the relative benefits and tradeoffs of using a simpler vs more complex prediction model? We compare a shallow 4-depth decision tree with the CatBoost model. As expected, the simpler tree shows a small drop in predictive power $5 \%$ decrease in $R ^ { 2 }$ ) which translates into a $1 { - } 8 \%$ reduction in policy value (see Figure 15). Compared to a uniform $5 \%$ increase in $R ^ { 2 }$ achieved by scaling the residuals (see Figure 14), the differences in policy value are only partially similar across $\alpha$ . The CatBoost model does not provide a uniform improvement over the decision tree; for instance, it performs better at distinguishing longer unemployment spells.
+
+Despite this performance gap, the simpler model offers potential advantages: it fits on a single sheet of paper, demands minimal computational infrastructure, can be easily explained to frontline case workers and resembles the categorical prioritization rules common in public institutions. (Johnson & Zhang, 2022). Because more complex models incur higher costs, a planner might instead increase screening capacity. Formally, we define
+
+$$
+\begin{array} { r } { \Delta _ { \alpha } ^ { * } = \underset { \Delta _ { \alpha } \in ( 0 , 1 - \beta ) } { \operatorname* { i n f } } \left. \Delta _ { \alpha } : \frac { V _ { \mathrm { T R E E } } ( \alpha + \Delta _ { \alpha } , \beta ) - V _ { \mathrm { T R E E } } ( \alpha , \beta ) } { V _ { \mathrm { C A T } } ( \alpha , \beta ) - V _ { \mathrm { T R E E } } ( \alpha , \beta ) } \gtrsim 1 \right. } \end{array}
+$$
+
+the smallest $\Delta _ { \alpha } ^ { * }$ that matches the policy-value gains of the CatBoost model. Empirically, $\Delta _ { \alpha } ^ { * }$ mostly rises with $\alpha$ (see Figure 15), consistent with our finding that the PAR decreases with $\alpha$ . By framing the difference between models in terms of additional screenings, planners can directly compare the cost of increased capacity to that of deploying a more complex model.
+
+# 6. Conclusion
+
+This paper develops a framework for quantifying the relative value of prediction in identifying the worst-off. We formalize tradeoffs between expanding screening capacity and improving predictive models, and show through both mathematical analysis and a real-world case study that prediction is not always the most important piece of the puzzle in so cial allocation systems. Future work could examine more specific application settings and cost structures, including distinctions between fixed and recurring costs, and explore policy levers that improve prediction unevenly, for example, by reducing errors in high-risk subgroups or by increasing robustness to distributional shifts. More broadly, we see a need for clearer theoretical foundations to understand the role of prediction in public-sector allocation, particularly in relation to the institutional and administrative systems in which it is embedded.
+
+![](images/f29d48fe78883edb30abe67c1303422ed10bd4168dfe15a44c6681329107c798.jpg)  
+Figure 4. (a) Policy value across different screening capacities $( \alpha )$ and worst-off fractions $\beta$ evaluated on the test set using the CatBoost regression model. A $\beta$ value of 0.15 corresponds to the 12-month cutoff used to define long-term unemployment in Germany. (b, c) The rate of local improvements in $V ( R ^ { 2 } )$ with respect to small changes in $R ^ { 2 }$ . Panel (b) shows that the local improvements become increasingly large as $\Delta _ { R ^ { 2 } }$ approaches zero. Panel (c) illustrates that improvements in prediction have the greatest impact when the capacity precisely matches the targeted fraction of the population $\alpha = \beta$ ). Note that these are on a logarithmic scale.
+
+![](images/e64041f8966c6720a8c2f9eeab2508bc5fdbac6176e88db2c438b1cbf3b9d1ed.jpg)  
+Figure 5. The minimum additional screening capacity that would need to be invested for the decision tree to achieve a policy value comparable to that of the CatBoost model.
+
+# Acknowledgements
+
+This work is supported by the DAAD programme Konrad Zuse Schools of Excellence in Artificial Intelligence, sponsored by the Federal Ministry of Education and Research and by the Volkswagen Foundation, grant “Consequences of Artificial Intelligence for Urban Societies (CAIUS)”. Juan Carlos Perdomo is supported by the Center for Research on Computation and Society (CRCS) at Harvard University and by the Alfred P. Sloan Foundation grant G-2020-13941. We would like to thank the anonymous reviewers for their insightful comments, as well as Frauke Kreuter, Patrick Schenk and Moritz Hardt for their valuable feedback.
+
+# Impact Statement
+
+Our work offers a principled framework for evaluating the relative benefits of using predictive models to target the most vulnerable populations, helping public agencies allocate limited resources more effectively. However, formalizing complex institutional processes inevitably omits important real-world details, risking biases or misalignments if assumptions are not carefully examined. We encourage policymakers and researchers to incorporate fairness, transparency, and accountability measures when implementing these methods, particularly in resource-constrained contexts where small design changes can disproportionately affect marginalized communities.
+
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+
+# A. Theoretical Investigation
+
+![](images/b92bb0bafcca9d9f7e9e88790331440f1a5400b907e8e6b13c252d3b7080f8db.jpg)  
+(a) Normal Welfare Distribution
+
+![](images/fc8c63cff226cda2827699ba36f7db23e1fa1adaddbb23fa6fe7dfee5a67612d.jpg)  
+Figure 6. Normal welfare distribution, with vertical lines marking the quantile cutoff $\beta = 0 . 2$ ). The shaded region to the left of the vertical line represents the worst-off segment of the population.   
+Figure 7. Numerical Simulation of the Prediction-Access Ratio (PAR), Equation 3, for $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = 0 . 0 1$ and $\beta = 0 . 0 5$ . Each point represents a screening capacity $\alpha$ ( $\mathbf { X }$ -axis) and $R ^ { 2 }$ value $\mathbf { \widetilde { y } }$ -axis), with the color bar showing the PAR clipped to the range [0.5, 2.0]. The vertical black line marks $\beta$ , indicating the threshold above which sufficient resources are available to screen everyone under perfect prediction. Dotted black lines represent $\mathrm { P A R } = 1$ , where improvements in $\alpha$ and $R ^ { 2 }$ are equally effective. The purple line marks the region in the $( \alpha , R ^ { 2 } )$ space where the policy value $V ( \alpha , \beta , R ^ { 2 } )$ exceeds 0.9. Values above 0.9 are located in the upper-right region beyond the purple line.
+
+# B. Experiments
+
+# B.1. Experimental Setup and Labor Market Data
+
+The dataset is provided via a Scientific Use File by the Research Data Centre (FDZ) of the German Federal Employment Agency (BA) at the Institute for Employment Research (IAB) (Schmucker & vom Berge, 2023a;b). It is a $2 \%$ weakly anonymized random sample of the complete German labor market records from 1975 to 2017 and contains information on 1,827,903 individuals across 62,340,521 observations (Schmucker & vom Berge, 2023b).
+
+We follow the same set of covariates and aggregation procedure for individual unemployment spells as described in Bach et al. (2023), incorporating demographic characteristics, labor market histories, and information about the most recent job. This results in 56 numerical variables and 24 categorical variables, which are one-hot encoded for model training. Figure 8 shows a histogram of individual unemployment durations, which we use as the basis for constructing the outcome variables. The distribution is right-skewed, with a concentration on short durations near zero and a long tail. Such a pattern is commonly observed in other welfare-related outcomes, such as health or income metrics. We define as prediction target the duration of the unemployment period in days $Y$ , capped at 24 months3. Differentiating tail values is less important for decision-making, and capping also allows training across years with varying observation windows.
+
+![](images/02a357af0d751dd19ae81e8b0a237cf7c4d9b403e3084c0b780e204761a96eba.jpg)  
+Figure 8. Unemployment duration The red line marks the 12 month threshold used to classify a jobseeking episode as long-term unemployment (LTU).
+
+To avoid the impact of significant labor market reforms in Germany and to ensure full observation of unemployment durations up to 24 months, we restrict our analysis to unemployment episodes that began between 2010 and 2015. We use records from 2010 and 2011 to build the training dataset, records from 2012 for validation, and evaluate test performance on data from 2015 (see Figure 9). We left a gap between the training and test data periods to allow enough time for the outcomes in the training data to have been fully observed at test time, in order to mimic a realistic deployment scenario starting at the beginning of 2015.
+
+![](images/d5c75d8f3879d13946e83581ef63393dc0e54df289af7280a25070105fc55919.jpg)  
+Figure 9. Stacked timeline diagram illustrating training (2010–2013), validation (2012–2014), and test (2015–2017) data periods. Red dashed boundaries within each colored box indicate the possible start dates of unemployment episodes, while the full colored boxes represent the entire observation phases for each dataset.
+
+# B.2. Training Details
+
+We use CatBoost (https://catboost.ai) for model training. The model was trained for a maximum 5,000 iterations with an early stopping criterion (early stopping rounds $= 2 0$ ) based on validation performance. Additionally, we train a shallow Decision Tree (max depth $= 4$ ) using the scikit-learn package. All hyperparameters are kept at their default settings unless otherwise specified.
+
+# B.3. Prediction Improvements
+
+To simulate an increase in predictive power by a specified amount $\Delta _ { R ^ { 2 } }$ , we adjust the model’s predictions $\hat { Y }$ using the residuals $Y - { \hat { Y } }$ . Starting with the original predictions $\hat { Y }$ and true outcomes $Y$ , we define the adjusted predictions as
+
+$$
+\hat { Y } _ { + } = \hat { Y } + \delta ( Y - \hat { Y } )
+$$
+
+We can then determine the $\delta$ corresponding to an increase of $\Delta _ { R ^ { 2 } }$ in the model’s $R ^ { 2 }$ :
+
+$$
+\delta = 1 - \sqrt { 1 - \Delta _ { R ^ { 2 } } \frac { \sum _ { i = 1 } ^ { n } ( Y _ { i } - \bar { Y } ) ^ { 2 } } { \sum _ { i = 1 } ^ { n } ( Y _ { i } - \hat { Y } _ { i } ) ^ { 2 } } }
+$$
+
+For a specified $\delta$ , the new residuals are
+
+$$
+Y - \hat { Y } _ { + } = ( 1 - \delta ) ( Y - \hat { Y } )
+$$
+
+Consequently, the variance decreases by a multiplicative factor: $\operatorname { V a r } ( Y - \hat { Y } _ { + } ) = ( 1 - \delta ) ^ { 2 } \operatorname { V a r } ( Y - \hat { Y } )$ .
+
+![](images/ba93581f23e5ea641bc7c19782cd2699d502baf5d9b725bf4a43b11396905a21.jpg)  
+Figure 10. Residual Distribution Before and After Adjustment Figure (a) shows the residual distribution for the original predictions $\Delta _ { R ^ { 2 } } = 0 )$ ), while Figure (b) shows the residual distribution after increasing the $R ^ { 2 }$ -value $\Delta _ { R ^ { 2 } } = 0 . 1 \AA$ ) for the CatBoost model. The adjustment preserves the overall structure of the residuals.
+
+![](images/5aae351bab0bc0d8606f6563d817b8edf8134953dda6114222cad5dcbf38d377.jpg)  
+Figure $1 I$ . The $R ^ { 2 }$ value on the test set for varying training set size (CatBoost Regression).
+
+![](images/c251b2f3e8d995ff7879ec51d8a8868bd95d7f8da285c1626862f03f229d85b1.jpg)  
+Figure 12. Residual distributions on the test set for models trained with varying training set sizes.
+
+# B.4. Additional Figures
+
+![](images/fa398d122245fa3d7a51a8430a8464f33a5f4c9f5cead8503963b8dc141d37e5.jpg)  
+Figure 13. Prediction-Access Ratio for ${ R } ^ { 2 } = 0$ and $\Delta _ { R ^ { 2 } } = \Delta _ { \alpha } = 0 . 0 1$ .
+
+![](images/8071d3a456dd565f4ffba6ee0da8af40a3a785478feb9da3cb1abdeb4c7ff064.jpg)  
+Figure 14. $V ( R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( R ^ { 2 } )$ for CatBoost model and $\Delta _ { R ^ { 2 } } = 0 . 0 5$ .
+
+![](images/672676278485340d8cbcc324c56ae86944c9f8b3d6804df4131d74784c45e401.jpg)  
+Figure 15. The difference in policy value between a 4-depth decision tree and CatBoost model.
+
+# B.5. Binary Classification
+
+Instead of predicting the exact duration of unemployment, the problem can be reframed as a binary classification task. For a fixed $\beta$ , we can define a binary outcome: $Y = \bar { 1 } \bar { \{ Y \geqslant F _ { Y , n } ^ { - 1 } ( \bar { 1 } - \beta ) \} }$ . This approach more directly encodes the target of interest: identifying individuals who may require further screening or assistance. If the chosen classifier provides estimates of class probabilities ${ \hat { p } } ( x )$ , it can be used to formulate a decision policy $1 \{ \hat { p } ( x ) \geqslant F _ { n , \hat { p } } ^ { - 1 } ( 1 - \alpha ) \}$ . However, this forces us to specify $\beta$ and the resulting decision threshold prior to model training. This requirement reduces flexibility compared to a continuous prediction setup, making classification more appropriate when the model is not intended for use in other tasks and when $\beta$ remains constant across the deployment context. Additionally, directly converting durations to labels discards information on the precise unemployment durations that could be valuable for the modeling process.
+
+As can be seen in Figure 16, the resulting policy values and true positive counts remain very similar compared to the regression case.
+
+![](images/0d791a12acd6e0cb2193afe91a45023b57b7a415f1d1afa5af3b65058515ab13.jpg)  
+Figure 16. Policy Value and True Positive Count on Test Set (Classification).
+
+# C. Additional Propositions
+
+Proposition C.1. (Optimal Policy with Gaussian Error) I $f \varepsilon = Y - \hat { Y } \sim \mathcal { N } ( 0 , \gamma ^ { 2 } )$ , then the optimal policy $\pi ^ { * } : \mathbb { R }  \{ 0 , 1 \}$ to solve the screening problem (Definition 2.1) is equal to:
+
+$$
+\pi ^ { * } ( \hat { Y } _ { i } ) = 1 \{ \hat { Y } _ { i } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \}
+$$
+
+where $F _ { \hat { Y } } ^ { - 1 } ( \alpha )$ is the $\alpha$ -quantile of $\hat { Y }$ . The value of the policy is $V ( \pi ^ { * } ) = \operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \ | \ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ] .$
+
+Proof. Since $Y = { \hat { Y } } + \varepsilon$ where $\varepsilon \sim \mathcal { N } ( 0 , \gamma ^ { 2 } )$ , it follows for the conditional distribution $Y \mid \hat { Y } \sim \mathcal { N } ( \hat { Y } , \gamma ^ { 2 } )$ . Since $\boldsymbol { Y } \mid \hat { \boldsymbol { Y } }$ is Gaussian, we can express the conditional probability from Proposition 2.2 in terms of the CDF of the standard normal distribution,
+
+$$
+\operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } ] = \Phi \left( \frac { F _ { Y } ^ { - 1 } ( \beta ) - \hat { Y } } { \gamma } \right)
+$$
+
+To reproduce the ranking induced by $\operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } ]$ , individuals can be ranked in ascending order of $\hat { Y }$ . Thus, we can express the optimal policy (Proposition 2.2) in terms of a ranking of $\hat { Y }$ ,
+
+$$
+\pi ^ { * } ( \hat { Y } _ { i } ) = 1 \{ \hat { Y } _ { i } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \}
+$$
+
+where $F _ { \hat { Y } } ^ { - 1 } ( \alpha )$ is the $\alpha$ -quantile of $\hat { Y }$ . The value $V ( \pi ^ { * } )$ that can by achieved by the optimal screening policy $\pi ^ { * }$ can then be expressed as:
+
+$$
+\begin{array} { r l } & { V ( \pi ^ { * } ) = \mathbb { E } \left[ \pi ^ { * } ( \hat { Y } ) = 1 \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \right] = \mathbb { E } \left[ 1 \{ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \} \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \right] } \\ & { \qquad = \operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ] } \end{array}
+$$
+
+# D. Proofs
+
+# D.1. Optimal Policy for Screening Problem: Proof of Proposition 2.2
+
+Proof. We rewrite the policy value,
+
+$$
+\begin{array} { r l } & { \mathbb { E } \left[ \pi ( \hat { Y } _ { i } ) = 1 \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \right] = \frac { \mathbb { E } \left[ \pi ( \hat { Y } _ { i } ) 1 \{ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \} \right] } { \operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ] } } \\ & { = \frac { 1 } { \beta } \mathbb { E } \left[ \pi ( \hat { Y } _ { i } ) \mathbb { E } \left[ 1 \{ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \} \mid \hat { Y } _ { i } \right] \right] } \\ & { = \frac { 1 } { \beta } \mathbb { E } \left[ \pi ( \hat { Y } _ { i } ) \mathrm { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } _ { i } ] \right] } \end{array}
+$$
+
+To maximize the objective, individuals $\hat { Y _ { i } }$ with the largest scores $s ( \hat { Y } _ { i } ) = \operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } _ { i } ]$ should be prioritized. Thus, the optimal policy is to intervene $( \pi ( \hat { Y } _ { i } ) = 1 \rangle$ ) for the top $\alpha$ -fraction of the population ranked by $\operatorname* { P r } [ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \mid \hat { Y } ]$ . ■
+
+# D.2. Optimal Policy Value in Gaussian Setting: Proof of Proposition 2.3
+
+Following Proposition C.1, the value of the optimal screening policy $\pi ^ { * }$ can then be expressed as:
+
+$$
+V ( \pi ^ { * } ) = \operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \ | \ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ]
+$$
+
+We can rewrite the conditional probability in terms of the joint distribution of $Y$ and $\hat { Y }$ , and note that ${ \operatorname* { P r } } \{ Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \} =$ $F _ { Y } ( F _ { Y } ^ { - 1 } ( \beta ) ) = \beta$ ,
+
+$$
+\operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ] = \frac { 1 } { \beta } \operatorname* { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) , Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ]
+$$
+
+We then standardize $Y \sim { \mathcal { N } } ( \mu , \eta ^ { 2 } )$ and $\hat { Y } \sim \mathcal N ( \mu , \eta ^ { 2 } - \gamma ^ { 2 } )$ and make use that for a normal random variable with mean $\mu$ and variance $\sigma ^ { 2 }$ the quantile function is $F ^ { - 1 } ( p ) = \mu + \sigma \Phi ^ { - 1 } \left( p \right)$ .
+
+$$
+\begin{array} { r l r } & { } & { \frac { 1 } { \beta } \mathrm { P r } [ \hat { Y } \leqslant F _ { \hat { Y } } ^ { - 1 } ( \alpha ) , Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) ] = \frac { \mathrm { P r } \left\{ Z _ { 1 } \leqslant \frac { F _ { \hat { Y } } ^ { - 1 } ( \alpha ) - \mu } { \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } } , Z _ { 2 } \leqslant \frac { F _ { Y } ^ { - 1 } ( \beta ) - \mu } { \eta } \right\} } { \beta } } \\ & { } & { = \frac { \mathrm { P r } \left\{ Z _ { 1 } \leqslant \Phi ^ { - 1 } \left( \alpha \right) , Z _ { 2 } \leqslant \Phi ^ { - 1 } \left( \beta \right) \right\} } { \beta } } \end{array}
+$$
+
+$Z _ { 1 }$ and $Z _ { 2 }$ are standard Gaussian with $\begin{array} { r } { \operatorname { C o v } ( Z _ { 1 } , Z _ { 2 } ) = \mathbb { E } \left[ Z _ { 1 } Z _ { 2 } \right] = \frac { 1 } { \eta \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } } \mathrm { C o v } ( \hat { Y } , \hat { Y } + \varepsilon ) = \frac { \operatorname { C o v } ( \hat { Y } , \hat { Y } ) } { \eta \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } } = \frac { \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } } { \eta } } \end{array}$ Thus, they are distributed according to a standard bivariate normal distribution with correlation $\begin{array} { r } { \rho = \mathrm { C o v } ( Z _ { 1 } , Z _ { 2 } ) = \frac { \sqrt { \eta ^ { 2 } - \gamma ^ { 2 } } } { \eta } } \end{array}$ Thus,
+
+$$
+V ( \pi ^ { * } ) = \mathbb { E } \left[ \pi ^ { * } ( \hat { Y } ) = 1 \mid Y \leqslant F _ { Y } ^ { - 1 } ( \beta ) \right] = \frac { 1 } { \beta } \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right)
+$$
+
+where
+
+$$
+\Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) \right) = \int _ { - \infty } ^ { \Phi ^ { - 1 } \left( \alpha \right) } \int _ { - \infty } ^ { \Phi ^ { - 1 } \left( \beta \right) } \phi _ { 2 } \left( z _ { 1 } , z _ { 2 } ; \rho \right) \mathrm { d } z _ { 1 } \mathrm { d } z _ { 2 }
+$$
+
+and
+
+$$
+\phi _ { 2 } \left( z _ { 1 } , z _ { 2 } \right) = \frac { 1 } { 2 \pi \sqrt { 1 - \rho ^ { 2 } } } e ^ { - 1 / 2 ( z _ { 1 } ^ { 2 } - 2 \rho z _ { 1 } z _ { 2 } + z _ { 2 } ^ { 2 } ) / ( 1 - \rho ^ { 2 } ) }
+$$
+
+# D.3. Prediction-Access Ratio for Small Screening Capacities: Proof of Theorem 3.1
+
+Using Taylor’s theorem,
+
+$$
+V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) = \Delta _ { R ^ { 2 } } \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } + p _ { R ^ { 2 } } \Delta _ { R ^ { 2 } } )
+$$
+
+where $p _ { R ^ { 2 } } \in ( 0 , 1 )$ . We know from Lemma D.3,
+
+$$
+\frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R _ { * } ^ { 2 } ) \leqslant \frac { 1 } { \beta \sqrt { 8 \pi R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \phi \left( \frac { \Phi ^ { - 1 } \left( \alpha \right) - \sqrt { R _ { * } ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } } \right)
+$$
+
+where $R _ { * } ^ { 2 } : = R ^ { 2 } + p _ { R ^ { 2 } } \Delta _ { R ^ { 2 } }$ . For $\alpha < 0 . 5$ and $\beta \leqslant 0 . 5$ , we know $\Phi ^ { - 1 } \left( \alpha \right) < 0$ and $\Phi ^ { - 1 } \left( \beta \right) \leqslant 0$ . It follows, that for any $\varepsilon _ { 1 } > 0 , 0 < R _ { * } ^ { 2 }$ and $0 < \beta$ , there exists a threshold value $t _ { 1 } > 0$ , such that for all $\alpha \leqslant t _ { 1 }$ , we have
+
+$$
+( 1 + \varepsilon _ { 1 } ) \frac { \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } } \leqslant \frac { \Phi ^ { - 1 } \left( \alpha \right) - \sqrt { R _ { * } ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } } \leqslant ( 1 - \varepsilon _ { 1 } ) \frac { \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } }
+$$
+
+If $\alpha < \beta$ we find $\Phi ^ { - 1 } \left( \alpha \right) - \sqrt { R _ { * } ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) < 0$ . Since $\phi ( x ) \leqslant \phi ( x ^ { \prime } )$ for $x \leqslant x ^ { \prime } < 0$ ,
+
+$$
+\begin{array} { r } { \frac { 1 } { \beta \sqrt { 8 \pi R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \phi \left( \frac { \Phi ^ { - 1 } \left( \alpha \right) - \sqrt { R _ { * } ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } } \right) \leqslant \frac { 1 } { \beta \sqrt { 8 \pi R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \phi \left( ( 1 - \varepsilon _ { 1 } ) \frac { \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R _ { * } ^ { 2 } } } \right) } \\ { = \frac { 1 } { \beta \sqrt { 8 \pi R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \phi \left( \kappa \Phi ^ { - 1 } \left( \alpha \right) \right) } \\ { = \frac { 1 } { \beta \sqrt { 8 \pi R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \phi \left( \kappa \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) } \end{array}
+$$
+
+where $\begin{array} { r } { \kappa : = \frac { ( 1 - \varepsilon _ { 1 } ) } { \sqrt { 1 - R _ { * } ^ { 2 } } } } \end{array}$ . For any $\varepsilon _ { 2 } > 0$ , there exists a threshold $t _ { 2 } > 0$ , such that for all $\alpha \leqslant t _ { 2 }$ , we can apply Lemma B.5. from Perdomo (2024) to arrive at the following inequality:
+
+$$
+\phi \left( \kappa \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) \leqslant \frac { 1 } { \sqrt { 2 \pi } } \left( \left( 1 + \varepsilon _ { 2 } \right) \sqrt { 2 \pi } \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { \kappa ^ { 2 } }
+$$
+
+Thus,
+
+$$
+V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) \leqslant \Delta _ { R ^ { 2 } } \frac { 1 } { \beta 4 \pi \sqrt { R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } } \left( ( 1 + \varepsilon _ { 2 } ) \sqrt { 2 \pi } \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { \kappa ^ { 2 } }
+$$
+
+We can use Taylor’s theorem again and from Lemma D.1 we know that
+
+$$
+\begin{array} { l } { { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) = \Delta _ { \alpha } \displaystyle \frac { \partial } { \partial \alpha } V ( \alpha + p _ { \alpha } \Delta _ { \alpha } , \beta , R ^ { 2 } ) } } \\ { { { } } } \\ { { = \Delta _ { \alpha } \displaystyle \frac { 1 } { \beta } \Phi \left( \displaystyle \frac { \Phi ^ { - 1 } ( \beta ) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha + p _ { \alpha } \Delta _ { \alpha } ) } { \sqrt { 1 - R ^ { 2 } } } \right) } } \end{array}
+$$
+
+where $p _ { \alpha } \in ( 0 , 1 )$ . Since $0 < \beta$ and $0 < R ^ { 2 }$ there will always be a small enough $\alpha + \Delta _ { \alpha }$ such that
+
+$$
+\Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha + p _ { \alpha } \Delta _ { \alpha } \right) \geqslant 0
+$$
+
+Since $\Phi \left( x \right) \geqslant 1 / 2$ for $x \geqslant 0$ , it follows
+
+$$
+\frac { \Delta _ { \alpha } } { 2 \beta } \leqslant V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } )
+$$
+
+It follows for the prediction-access ratio,
+
+$$
+\frac { \Delta _ { \alpha } } { \Delta _ { R ^ { 2 } } } 2 \pi \sqrt { R _ { * } ^ { 2 } ( 1 - R _ { * } ^ { 2 } ) } \left( ( 1 + \varepsilon _ { 2 } ) \sqrt { 2 \pi } \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { - ( 1 - \varepsilon _ { 1 } ) ^ { 2 } \frac { 1 } { 1 - R _ { * } ^ { 2 } } } \leqslant \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta ) } ,
+$$
+
+For small $\alpha$ , $\Phi ^ { - 1 } \left( 1 - \alpha \right)$ grows asymptotically like $\sqrt { \log { ( 1 / \alpha ) } }$ . Consequently, the polynomial growth of $\alpha ^ { - 1 / ( 1 - R ^ { 2 } ) }$ drives the PAR to increase rapidly as $\alpha$ decreases. Since $\frac { 1 } { 1 - R _ { * } ^ { 2 } }$ increases with $R _ { * } ^ { 2 }$ and $R ^ { 2 } \leqslant R _ { * } ^ { 2 }$ , we can lower bound the PAR by inserting $R ^ { 2 }$ instead of $R _ { * } ^ { 2 }$ :
+
+$$
+\frac { \Delta _ { \alpha } } { \Delta _ { R ^ { 2 } } } 2 \pi \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } \left( ( 1 + \varepsilon _ { 2 } ) \sqrt { 2 \pi } \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { - ( 1 - \varepsilon _ { 1 } ) ^ { 2 } \frac { 1 } { 1 - R ^ { 2 } } } \leqslant \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) } ,
+$$
+
+We can simplify the lower-bound by noting that $0 < \varepsilon _ { 1 }$ and $0 < \varepsilon _ { 2 }$ can be made arbitrarily small by selecting a sufficiently small threshold $t$ for $\alpha + \Delta _ { \alpha }$ . Specifically, $\varepsilon _ { 2 } < 1$ holds for $\alpha \leqslant 0 . 0 5$ (see Lemma A.6 in Perdomo (2024)).
+
+$$
+\frac { \Delta _ { \alpha } } { \Delta _ { R ^ { 2 } } } \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } \left( 5 . 1 \cdot \alpha \Phi ^ { - 1 } \left( 1 - \alpha \right) \right) ^ { - \frac { 1 } { 1 - R ^ { 2 } } + o ( 1 ) } \leqslant \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) }
+$$
+
+# D.4. Maximally Effective (Local) Prediction Improvements: Proof of Theorem 3.2
+
+We know from Lemma D.2,
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { l i m } _ { \Delta \to 0 } \frac { V ( \alpha , \beta , R ^ { 2 } + \Delta ) - V ( \alpha , \beta , R ^ { 2 } ) } { \Delta } = \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) } \\ { = \displaystyle \frac { 1 } { 2 \beta \sqrt { R ^ { 2 } } } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } \end{array}
+$$
+
+We insert $\phi _ { 2 } \left( \cdot \right)$ and arrive at
+
+$$
+\begin{array} { c } { { { \displaystyle { \frac { \partial } { \partial R ^ { 2 } } } V ( \alpha , \beta , R ^ { 2 } ) = \underbrace { \frac { 1 } { 4 \pi \beta \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } } } } _ { T _ { 1 } } } } \\ { { { \displaystyle { \times \underbrace { \exp \bigg ( - \frac { 1 } { 2 ( 1 - R ^ { 2 } ) } ( \Phi ^ { - 1 } ( \alpha ) { } ^ { 2 } + \Phi ^ { - 1 } ( \beta ) { } ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha ) \Phi ^ { - 1 } ( \beta ) ) \bigg ) } } } } } \end{array}
+$$
+
+The prefactor $T _ { 1 }$ diverges as $R ^ { 2 } \to 1$ or $R ^ { 2 } \to 0$ .
+
+If $R ^ { 2 } \to 1$ , the exponential term will generally suppress the polynomial growth of the prefactor. However for $\alpha = \beta$ , we find for the exponent
+
+$$
+\begin{array} { c c } { { \displaystyle - \frac { 1 } { 2 ( 1 - R ^ { 2 } ) } ( \Phi ^ { - 1 } ( \alpha ) ^ { 2 } + \Phi ^ { - 1 } ( \beta ) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) ) = - \displaystyle \frac { 1 - \sqrt { R ^ { 2 } } } { 1 - R ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } } } \\ { { \displaystyle = - \frac { 1 } { ( 1 + \sqrt { R ^ { 2 } } ) } \Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } } } \\ { { \displaystyle R _ { \equiv } ^ { 2 } = ^ { 1 } - \frac { 1 } { 2 } \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } } } \end{array}
+$$
+
+which is finite. Therefore, $\textstyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } )$ becomes unboundedly large if $\alpha = \beta$ and $R ^ { 2 } \to 1$
+
+If $R ^ { 2 } \to 0$ , the prefector $T _ { 1 }$ diverges again to $+ \infty$ . The exponent then simplifies to
+
+$$
+- \frac { 1 } { 2 ( 1 - R ^ { 2 } ) } ( \Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) ) = - \frac { 1 } { 2 } ( \Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } )
+$$
+
+If $\alpha$ and $\beta$ are not set arbitrarily small or large $\textstyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } )$ will diverge. The local PAR (Lemma D.1)
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { l i m } _ { \Delta \to 0 } \frac { V ( \alpha + \Delta , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta ) - V ( \alpha , \beta , R ^ { 2 } ) } = \frac { \frac { \partial } { \partial \alpha } V ( \alpha , \beta , R ^ { 2 } ) } { \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) } } \\ { \displaystyle \quad = \frac { \Phi \left( \frac { \Phi ^ { - 1 } ( \beta ) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha ) } { \sqrt { 1 - R ^ { 2 } } } \right) } { \frac { 1 } { 2 \sqrt { R ^ { 2 } } } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } } \end{array}
+$$
+
+approaches zero in both regimes.
+
+# D.5. Prediction-Access Ratio for Local Improvements: Proof of Proposition 3.3
+
+We know
+
+$$
+\operatorname* { l i m } _ { \Delta  0 } \frac { V ( \alpha + \Delta , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta ) - V ( \alpha , \beta , R ^ { 2 } ) } = \frac { \frac { \partial } { \partial \alpha } V ( \alpha , \beta , R ^ { 2 } ) } { \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) }
+$$
+
+Using Lemma D.1 and Lemma D.3 we find a lower bound for the PAR:
+
+$$
+\underbrace { \sqrt { 8 \pi R ^ { 2 } ( 1 - R ^ { 2 } ) } } _ { T _ { 1 } } \underbrace { \frac { \Phi \left( \frac { \Phi ^ { - 1 } ( \beta ) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha ) } { \sqrt { 1 - R ^ { 2 } } } \right) } { \phi \left( \frac { \Phi ^ { - 1 } ( \beta ) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha ) } { \sqrt { 1 - R ^ { 2 } } } \right) } } _ { T _ { 2 } } \leqslant \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) }
+$$
+
+We then denote $\begin{array} { r } { z : = \frac { \Phi ^ { - 1 } ( \beta ) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } ( \alpha ) } { \sqrt { 1 - R ^ { 2 } } } } \end{array}$ and $\begin{array} { r } { T _ { 2 } = \frac { \Phi ( z ) } { \phi ( z ) } } \end{array}$ . We know from Lemma D.4 that $\frac { \Phi ( z ) } { \phi ( z ) }$ increases with $z$ . It follows that we need to find the smallest possible $z$ to find a lower bound for $T _ { 2 }$ . Generally, $z$ decreases with $\alpha$ and increases with $\beta$ . We treat both cases separately:
+
+1. For $\alpha \leqslant \beta$ we find $\begin{array} { r } { \frac { \Phi ^ { - 1 } ( \beta ) ( 1 - \sqrt { R ^ { 2 } } ) } { \sqrt { 1 - R ^ { 2 } } } \leqslant z } \end{array}$ . Since $\textstyle \frac { 1 - { \sqrt { R ^ { 2 } } } } { \sqrt { 1 - R ^ { 2 } } }$ decreases with $R ^ { 2 }$ and $\beta \leqslant 0 . 5$ we can lower bound the expression by setting $R ^ { 2 } = 0 . 1 5$ and $\beta = 0 . 0 3$ . Thus, $- 1 . 2 5 \leqslant z$ and $0 . 5 9 \leqslant T _ { 2 }$ . Since $0 . 1 5 \leqslant R ^ { 2 } \leqslant 0 . 8 5$ we can lower bound the prefactor $1 . 7 9 \leqslant T _ { 1 }$ .
+
+2. For $\alpha \leqslant 0 . 5$ , it follows $\frac { \Phi ^ { - 1 } ( \beta ) } { \sqrt { 1 - R ^ { 2 } } } \leqslant z$ by setting $\Phi ^ { - 1 } \left( \alpha = 0 . 5 \right) = 0$ . Since $0 . 1 5 \leqslant \beta$ and $0 . 2 \leqslant R ^ { 2 } \leqslant 0 . 5$ , it follows $0 . 5 2 \leqslant T _ { 2 }$ and $2 \leqslant T _ { 1 }$
+
+In both cases, we can combine the lower bounds of $T _ { 1 }$ and $T _ { 2 }$ to find
+
+$$
+1 \leqslant \frac { V ( \alpha + \Delta _ { \alpha } , \beta , R ^ { 2 } ) - V ( \alpha , \beta , R ^ { 2 } ) } { V ( \alpha , \beta , R ^ { 2 } + \Delta _ { R ^ { 2 } } ) - V ( \alpha , \beta , R ^ { 2 } ) }
+$$
+
+# D.6. Technical Lemmas
+
+Lemma D.1 (Derivative w.r.t. $\alpha$ ).
+
+$$
+\frac { \partial } { \partial \alpha } V ( \alpha , \beta , R ^ { 2 } ) = \frac { 1 } { \beta } \Phi \left( \frac { \Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R ^ { 2 } } } \right)
+$$
+
+Proof. In the Gaussian setting we find for the policy value (Proposition 2.3),
+
+$$
+V ( \alpha , \beta , R ^ { 2 } ) = \frac { \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } { \beta }
+$$
+
+We first apply Leibniz integral rule,
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { \partial } { \partial \alpha } V ( \alpha , \beta , R ^ { 2 } ) = \frac { \partial } { \partial \alpha } \frac { \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } { \beta } } \\ & { \quad \quad \quad \quad = \frac { 1 } { \beta } \displaystyle \int _ { - \infty } ^ { \Phi ^ { - 1 } \left( \beta \right) } \phi _ { 2 } \left( z _ { 1 } , \Phi ^ { - 1 } \left( \alpha \right) ; \rho \right) \mathrm { d } z _ { 1 } \frac { \partial } { \partial \alpha } \Phi ^ { - 1 } \left( \alpha \right) } \\ & { \quad \quad \quad = \frac { 1 } { \beta \phi \left( \Phi ^ { - 1 } \left( \alpha \right) \right) } \displaystyle \int _ { - \infty } ^ { \Phi ^ { - 1 } \left( \beta \right) } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , z _ { 2 } ; \rho \right) \mathrm { d } z _ { 2 } } \end{array}
+$$
+
+We insert the bivariate density $\phi _ { 2 } \left( \cdot \right)$ and substitute $z _ { 2 } - \rho \Phi ^ { - 1 } \left( \alpha \right) = u \sqrt { 1 - \rho ^ { 2 } }$
+
+$$
+\begin{array} { l } { \displaystyle \frac { 1 } { \beta \phi ( \Phi ^ { - 1 } ( \alpha ) ) } \int _ { - \infty } ^ { \Phi ^ { - 1 } ( \beta ) } \phi _ { 2 } ( \Phi ^ { - 1 } ( \alpha ) , z _ { 2 } ; \rho ) \mathrm { d } z _ { 2 } } \\ { = \displaystyle \frac { 1 } { \beta \phi ( \Phi ^ { - 1 } ( \alpha ) ) } \frac { 1 } { 2 \pi \sqrt { 1 - \rho ^ { 2 } } } \int _ { - \infty } ^ { + \infty } e ^ { - 1 / ( z _ { 2 } ^ { - 2 } - 2 \rho z _ { 2 } \Phi ^ { - 1 } ( \alpha ) + \Phi ^ { - 1 } ( \alpha ) ^ { 2 } ) / ( 1 - \rho ^ { 2 } ) } \mathrm { d } z _ { 2 } } \\ { = \displaystyle \frac { 1 } { 2 \pi \beta \phi ( \Phi ^ { - 1 } ( \alpha ) ) } \int _ { - \infty } ^ { \Phi ^ { - 1 } ( \beta ) - \rho \phi ^ { - 1 } ( \alpha ) ) / \sqrt { 1 - \rho ^ { 2 } } } e ^ { - 1 / 2 ( \alpha ^ { 2 } ( 1 - \rho ^ { 2 } ) + \rho ^ { 2 } + 1 ( \alpha ) ^ { 2 } + ( 1 - \rho ^ { 2 } ) \Phi ^ { - 1 } ( \alpha ) ^ { 2 } ) / ( 1 - \rho ^ { 2 } ) } \mathrm { d } \theta } \\ { = \displaystyle \frac { 1 } { 2 \pi \beta \phi ( \Phi ^ { - 1 } ( \alpha ) ) } e ^ { - 1 / 2 \Phi ^ { - 1 } ( \alpha ) ^ { 2 } } \int _ { - \infty } ^ { \Phi ^ { - 1 } ( \beta ) - \rho \phi ^ { - 1 } ( \alpha ) ) / \sqrt { 1 - \rho ^ { 2 } } } e ^ { - 1 / 2 \pi ^ { 2 } } \mathrm { d } u } \\  = \displaystyle \frac { 1 } { \beta } \Phi ( \frac { \Phi ^ { - 1 } ( \beta ) - \rho \Phi ^ { - 1 } ( \alpha ) } { \sqrt { 1 - \rho ^ { 2 } } } ) = \frac { 1 } { \beta } \Phi ( \frac  \Phi ^ { - 1 } ( \beta ) - \ \end{array}
+$$
+
+Lemma D.2 (Derivative w.r.t. $R ^ { 2 }$ ).
+
+$$
+\frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) = \frac 1 { 2 \beta \sqrt { R ^ { 2 } } } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) \right)
+$$
+
+where $\phi _ { 2 } \left( \cdot \right)$ is the standard bivariate density.
+
+Proof.
+
+$$
+\begin{array} { l } { \displaystyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) = \frac { \partial } { \partial R ^ { 2 } } \frac { \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } { \beta } } \\ { \displaystyle \qquad = \frac { 1 } { \beta } \frac { \partial \rho } { \partial R ^ { 2 } } \frac { \partial } { \partial \rho } \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } \\ { \displaystyle \qquad = \frac { 1 } { 2 \beta \sqrt { R ^ { 2 } } } \frac { \partial } { \partial \rho } \Phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } \\ { \displaystyle \qquad = \frac { 1 } { 2 \beta \sqrt { R ^ { 2 } } } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) } \end{array}
+$$
+
+where $\phi _ { 2 } \left( \cdot \right)$ is the standard bivariate density. We utilized $R ^ { 2 } = \rho ^ { 2 }$ , and in the final step applied the partial derivative of the standard bivariate cumulative distribution with respect to its correlation $\rho$ (Drezner & Wesolowsky, 1990).
+
+Lemma D.3 (Upper bound of $R ^ { 2 }$ derivative). Let $0 \leqslant \sqrt { R ^ { 2 } } \leqslant 1$ . Then,
+
+$$
+\begin{array} { r } { \displaystyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) \leqslant \frac { 1 } { \beta \sqrt { 8 \pi R ^ { 2 } ( 1 - R ^ { 2 } ) } } \phi \left( \frac { \Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R ^ { 2 } } } \right) } \\ { \displaystyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) \leqslant \frac { 1 } { \beta \sqrt { 8 \pi R ^ { 2 } ( 1 - R ^ { 2 } ) } } \phi \left( \frac { \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) - \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R ^ { 2 } } } \right) } \end{array}
+$$
+
+Proof. We know from Lemma D.2,
+
+$$
+\begin{array} { l } { { \displaystyle { \frac { \partial } { \partial R ^ { 2 } } } V ( \alpha , \beta , R ^ { 2 } ) = \frac { 1 } { 2 \beta \sqrt { R ^ { 2 } } } \phi _ { 2 } \left( \Phi ^ { - 1 } \left( \alpha \right) , \Phi ^ { - 1 } \left( \beta \right) ; \rho \right) \ ~ } } \\ { { \displaystyle ~ = \frac { 1 } { 4 \pi \beta \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } } } } \\ { { \displaystyle ~ \times \exp \left( - \frac { 1 } { 2 ( 1 - R ^ { 2 } ) } ( \Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) ) \right) \ } } \end{array}
+$$
+
+Since $0 \leqslant \sqrt { R ^ { 2 } } \leqslant 1$ ,
+
+$$
+\begin{array} { c } { { ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) \geqslant R ^ { 2 } \Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) } } \\ { { { } } } \\ { { = \left( \Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \right) ^ { 2 } \geqslant 0 } } \end{array}
+$$
+
+Similarly,
+
+$$
+\Phi ^ { - 1 } \left( \alpha \right) ^ { 2 } + \Phi ^ { - 1 } \left( \beta \right) ^ { 2 } - 2 \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) \Phi ^ { - 1 } \left( \beta \right) \geqslant \left( \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) - \Phi ^ { - 1 } \left( \alpha \right) \right) ^ { 2 } \geqslant 0
+$$
+
+Thus,
+
+$$
+\begin{array} { c } { { \displaystyle \frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) \leqslant \frac { 1 } { 4 \pi \beta \sqrt { R ^ { 2 } ( 1 - R ^ { 2 } ) } } \exp \left( - \frac { 1 } { 2 ( 1 - R ^ { 2 } ) } ( \Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) ) ^ { 2 } \right) } } \\ { { \displaystyle = \frac { 1 } { \beta \sqrt { 8 \pi R ^ { 2 } ( 1 - R ^ { 2 } ) } } \phi \left( \frac { \Phi ^ { - 1 } \left( \beta \right) - \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R ^ { 2 } } } \right) } } \end{array}
+$$
+
+and
+
+$$
+\frac { \partial } { \partial R ^ { 2 } } V ( \alpha , \beta , R ^ { 2 } ) \leqslant \frac { 1 } { \beta \sqrt { 8 \pi R ^ { 2 } ( 1 - R ^ { 2 } ) } } \phi \left( \frac { \sqrt { R ^ { 2 } } \Phi ^ { - 1 } \left( \beta \right) - \Phi ^ { - 1 } \left( \alpha \right) } { \sqrt { 1 - R ^ { 2 } } } \right)
+$$
+
+# Lemma D.4. The ratio
+
+$$
+\frac { \Phi \left( z \right) } { \phi \left( z \right) }
+$$
+
+is increasing in $z$ .
+
+Proof. We compute the derivative of the ratio Φ(z)ϕ(z) ,
+
+$$
+\frac { \partial } { \partial z } \frac { \Phi \left( z \right) } { \phi \left( z \right) } = \frac { \phi ^ { 2 } ( z ) + z \phi \left( z \right) \Phi \left( z \right) } { \phi ^ { 2 } ( z ) } = 1 + z \frac { \Phi \left( z \right) } { \phi \left( z \right) }
+$$
+
+For $z \geqslant 0$ the derivative is clearly positive. For $z < 0$ we start by rewriting,
+
+$$
+1 + z { \frac { \Phi \left( z \right) } { \phi \left( z \right) } } = { \frac { z } { \phi \left( z \right) } } \left( { \frac { \phi \left( z \right) } { z } } + \Phi \left( z \right) \right)
+$$
+
+Since ∂∂z  ϕ(z)z + Φ (z) $\begin{array} { r } { \frac { \partial } { \partial z } \left( \frac { \phi ( z ) } { z } + \Phi \left( z \right) \right) = \frac { - z ^ { 2 } \phi \left( z \right) - \phi \left( z \right) } { z ^ { 2 } } + \phi \left( z \right) = \frac { - \phi \left( z \right) } { z ^ { 2 } } < 0 } \end{array}$ and $\begin{array} { r } { \frac { \phi ( z ) } { z } + \Phi ( z ) \stackrel { z  - \infty } {  } 0 } \end{array}$ , it follows for $z < 0$ that $\begin{array} { r } { \left( \frac { \phi \left( z \right) } { z } + \Phi \left( z \right) \right) < 0 . } \end{array}$ . Thus for any $z$ ,
+
+$$
+\frac { \partial } { \partial z } \frac { \Phi \left( z \right) } { \phi \left( z \right) } \geqslant 0
+$$

data/sampled_jsons/27tMzmzDjO_A_Geometric_Approach_to_Personalized_Recommendation_Generalization_Spectrum_Gap_Section_5.jsonl

@@ -0,0 +1,10 @@
+{"idx": 0, "title": "A Geometric Approach to Personalized Recommendation with Set-Theoretic ...", "date": "", "ddg_snippet": "In this work, we formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence we represent each user and attribute by a hyper-rectangle or box (i.e. a Cartesian product of intervals).", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/abs/2502.10875", "content": "In this work, we formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent. To capture this set-theoretic dependence we represent each user and attribute by a hyper-rectangle or box (i.e. a Cartesian product of intervals)."}
+{"idx": 1, "title": "A Geometric Approach to Personalized Recommendation with Set-Theoretic ...", "date": "", "ddg_snippet": "This paper aims to advance the field of Machine Learn-ing by introducing a geometric approach to personalized recommendation under set-theoretic constraints. Our pri-mary contribution is methodological, focusing on improving representation learning for structured preference modeling.", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/pdf?id=27tMzmzDjO", "content": "This paper aims to advance the field of Machine Learn-ing by introducing a geometric approach to personalized recommendation under set-theoretic constraints. Our pri-mary contribution is methodological, focusing on improving representation learning for structured preference modeling."}
+{"idx": 2, "title": "A Geometric Approach to Personalized Recommendation with...", "date": "", "ddg_snippet": "Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall.", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/forum?id=0HWAbWgI3T", "content": "Queries involving set-theoretic constraints can be efficiently computed directly on the embedding space by performing geometric operations on the representations. We empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30% overall."}
+{"idx": 3, "title": "Personalized recommendation: From clothing to academic", "date": "", "ddg_snippet": "A personalized recommendation system can assist specific shoppers, students, and researchers based on their interaction, demonstrating an enormous capacity for improvement by generating intelligent recommendations , such as clothing, research articles, books, and other related information.", "subpage_snippet": "", "source": "link.springer.com", "link": "https://link.springer.com/article/10.1007/s11042-022-12259-7", "content": "A personalized recommendation system can assist specific shoppers, students, and researchers based on their interaction, demonstrating an enormous capacity for improvement by generating intelligent recommendations , such as clothing, research articles, books, and other related information."}
+{"idx": 4, "title": "Geometric Approach to Personalized Recommendation with Set-Theoretic ...", "date": "", "ddg_snippet": "Abstract Personalized item recommendation typically suf-fers from data sparsity, which is most often ad-dressed by learning vector representations of users and items via low-rank matrix factorization.", "subpage_snippet": "", "source": "arxiv.org", "link": "https://arxiv.org/pdf/2502.10875", "content": "Abstract Personalized item recommendation typically suf-fers from data sparsity, which is most often ad-dressed by learning vector representations of users and items via low-rank matrix factorization."}
+{"idx": 5, "title": "PDF A Data-Driven Approach to Personalized Bundle Pricing and Recommendation", "date": "", "ddg_snippet": "As a direct example of this, strategic recommendation services that utilize such data e ectively are currently undergoing massive and rapid expansion. StitchFix, which o ers personalized clothing styling for its customers and only became cash- ow positive in 2014, achieved $250M in revenue in 2015 that nearly tripled to $730M by the end of 2016.", "subpage_snippet": "", "source": "ide.mit.edu", "link": "https://ide.mit.edu/wp-content/uploads/2019/08/Georgia-paper.pdf?x34977", "content": "As a direct example of this, strategic recommendation services that utilize such data e ectively are currently undergoing massive and rapid expansion. StitchFix, which o ers personalized clothing styling for its customers and only became cash- ow positive in 2014, achieved $250M in revenue in 2015 that nearly tripled to $730M by the end of 2016."}
+{"idx": 6, "title": "A Geometric Approach to Personalized Recommendation With Set T C U B ...", "date": "", "ddg_snippet": "Personalized item recommendation typically suffers from data sparsity, which is most often addressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by assum-ing users and movies can be represented by linearly dependent latent features, it does not capture more complicated interactions. For example, vector ...", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/pdf?id=0HWAbWgI3T", "content": "Personalized item recommendation typically suffers from data sparsity, which is most often addressed by learning vector representations of users and items via low-rank matrix factorization. While this effectively densifies the matrix by assum-ing users and movies can be represented by linearly dependent latent features, it does not capture more complicated interactions. For example, vector ..."}
+{"idx": 7, "title": "A Geometric Approach to Personalized Recommendation with...", "date": "", "ddg_snippet": "This paper proposes a geometric approach to personalized item recommendation using box embeddings. Unlike traditional vector-based methods that struggle with set-theoretic operations, the authors introduce axis-aligned hyperrectangles (boxes) to model users, items, and attributes.", "subpage_snippet": "", "source": "openreview.net", "link": "https://openreview.net/forum?id=27tMzmzDjO", "content": "This paper proposes a geometric approach to personalized item recommendation using box embeddings. Unlike traditional vector-based methods that struggle with set-theoretic operations, the authors introduce axis-aligned hyperrectangles (boxes) to model users, items, and attributes."}
+{"idx": 8, "title": "Personalization in personalized marketing: Trends and ways forward ...", "date": "", "ddg_snippet": "The science mapping of personalized marketing research was conducted using bibliographic coupling and co-word analysis, resulting in six themes that underpin the field's intellectual structure, namely personalized recommendation , personalized relationship, personalization-privacy paradox, personalized advertising, personalization concept and ...", "subpage_snippet": "", "source": "onlinelibrary.wiley.com", "link": "https://onlinelibrary.wiley.com/doi/full/10.1002/mar.21670", "content": "The science mapping of personalized marketing research was conducted using bibliographic coupling and co-word analysis, resulting in six themes that underpin the field's intellectual structure, namely personalized recommendation , personalized relationship, personalization-privacy paradox, personalized advertising, personalization concept and ..."}
+{"idx": 9, "title": "[PDF] A Geometric Approach to Personalized Recommendation with Set ...", "date": "", "ddg_snippet": "This work formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent, and empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30 \\\\% overall. Personalized item recommendation typically suffers from data sparsity, which is most ...", "subpage_snippet": "", "source": "www.semanticscholar.org", "link": "https://www.semanticscholar.org/paper/A-Geometric-Approach-to-Personalized-Recommendation-Dasgupta-Boratko/07df63a8f1aab135768c58da100c740a3fa574f1", "content": "This work formulate the problem of personalized item recommendation as matrix completion where rows are set-theoretically dependent, and empirically demonstrate the superiority of box embeddings over vector-based neural methods on both simple and complex item recommendation queries by up to 30 \\\\% overall. Personalized item recommendation typically suffers from data sparsity, which is most ..."}

data/sampled_jsons/2aKHuXdr7Q_Going Deeper into Locally Differentially Private Graph Neural Networks/auto/2aKHuXdr7Q_Going Deeper into Locally Differentially Private Graph Neural Networks.md

@@ -0,0 +1,626 @@
+# Going Deeper into Locally Differentially Private Graph Neural Networks
+
+Longzhu He 1 Chaozhuo Li 1 Peng Tang 2 Sen Su 1
+
+# Abstract
+
+Graph Neural Networks (GNNs) have demonstrated superior performance in a variety of graph mining and learning tasks. However, when node representations involve sensitive personal information or variables related to individuals, learning from graph data can raise significant privacy concerns. Although recent studies have explored local differential privacy (LDP) to address these concerns, they often introduce significant distortions to graph data, severely degrading private learning utility (e.g., node classification accuracy). In this paper, we present UPGNET, an LDP-based privacy-preserving graph learning framework that enhances utility while protecting user data privacy. Specifically, we propose a three-stage pipeline that generalizes the LDP protocols for node features, targeting privacy-sensitive scenarios. Our analysis identifies two key factors that affect the utility of privacy-preserving graph learning: feature dimension and neighborhood size. Based on the above analysis, UPGNET enhances utility by introducing two core layers: High-Order Aggregator (HOA) layer and the Node Feature Regularization (NFR) layer. Extensive experiments on real-world datasets indicate that UPGNET significantly outperforms existing methods in terms of both privacy protection and learning utility.
+
+# 1. Introduction
+
+In recent years, Graph Neural Networks (GNNs) have shown superior performance in various domains, including social sciences (Hamilton et al., 2017), graph mining (Li et al., 2019), and bioinformatics (Fout et al., 2017). GNNs have also achieved state-of-the-art performance on a range of downstream graph learning tasks, such as node classification (Kipf & Welling, 2017), link prediction (Zhang & Chen,
+
+![](images/a309311e9edd6ea878e95efe7653e5dc5fe08112307813f78ea94a24250be34e.jpg)  
+Figure 1. Comparison of (a) prior works and (b) ours in the locally private graph learning scenario. The scenario comprises a cloud server and multiple users situated across different clients. Users’ sensitive node features $\mathbf { x }$ are perturbed to $\mathbf { x } ^ { \prime }$ using LDP before uploading to the cloud server for graph learning. Our approach achieves higher utility by integrating $\nsucc$ than prior works.
+
+2018), and community detection (Chen et al., 2019). In real-world scenarios, graphs frequently contain significant amounts of sensitive personal information, such as user profiles on social networks. However, recent studies (Wang & Wang, 2022; Shen et al., 2022; Zhang et al., 2022; Meng et al., 2023) have proposed various privacy attack methods targeting GNN models, which pose serious security and privacy challenges during their training. Therefore, designing an efficient privacy-preserving GNN framework to protect users’ private information is of paramount importance.
+
+To collect and analyze private data from decentralized data owners, local differential privacy (LDP) (Kasiviswanathan et al., 2011) has been increasingly accepted as the de facto standard for data privacy in the research community (Erlingsson et al., 2014; Ding et al., 2017; Wang et al., 2019a;b). In the LDP protocol, multiple users must interact with an untrustworthy server that may exploit their private data. Each user perturbs their data locally, typically through noise injection (Dwork et al., 2006), to ensure privacy. The perturbed data is then transmitted to the server, which conducts data analysis and learning based on this information. Locally private graph learning have recently gained significant attention from researchers (Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022; Jin & Chen, 2022; Pei et al., 2023). Fig. 1 illustrates this scenario, where a cloud server interacts with decentralized users. Each user perturbs their sensitive node features $\mathbf { x }$ to $\mathbf { x } ^ { \prime }$ under LDP before submitting them to the server for private graph learning (e.g., node classification).
+
+However, in locally private graph learning, existing LDP protocols for perturbing node features cause significant performance degradation in GNNs (Sajadmanesh & GaticaPerez, 2021; Lin et al., 2022). Specifically, LDP regulates noise injection through the privacy budget ϵ, where a smaller $\epsilon$ introduces more noise. In practice, $\epsilon$ is highly limited and node features are typically multidimensional with multiple attributes. As a result, each attribute receives only a minimal fraction of the budget, leading to substantial information loss. While server-side aggregation mitigates noise to some extent, it also introduces excessive estimation errors, further reducing utility (e.g., node classification accuracy). Thus, the key challenge is how to maximize the utility of privacy preserving graph learning while ensuring user privacy.
+
+Contributions. To address this challenge, we present a threestage pipeline that generalizes the current LDP protocols for perturbing node features. Our analysis of the pipeline reveals two key factors that influence the estimation error in feature aggregation: feature dimension and neighborhood size. We conclude that reducing the effective feature dimension and expanding the effective neighborhood size help minimize the estimation error and thus enhance the utility.
+
+Based on these findings, we propose UPGNET, a utilityenhanced framework for locally privacy-preserving graph learning that minimizes estimation error and maximizes utility from two key perspectives (Comparison with prior works is presented in Fig. 1). First, to reduce the effective feature dimensions, we introduce a Node Feature Regularization (NFR) layer based on $L _ { 1 }$ -regularization (Buhlmann ¨ & Van De Geer, 2011), a classical optimization technique that promotes sparsity in solutions. We conduct a theoretical analysis using proximal gradient descent (Nitanda, 2014; Li & Lin, 2015) to derive a one-off, non-iterative solution. This enables the NFR to obtain sparse embeddings through feature selection, thereby reducing effective feature dimensions during aggregation. Second, to expand the effective neighborhood size, we propose a multi-hop aggregation method, the High-Order Aggregator (HOA) layer. Although our theoretical analysis indicates that increasing neighborhood size improves utility, real-world graphs often feature small neighborhoods. A straightforward solution, such as aggregating multi-layer node features (Abu-El-Haija et al., 2019; Gasteiger et al., 2019; Chen et al., 2020), suffers from over-smoothing (Rusch et al., 2023). As the number of layers increases, node embeddings tend to converge, diminishing the effectiveness of higher-order neighbors in correcting errors. To address this, our HOA layer leverages personalized aggregation and Dirichlet energy (Zhou et al., 2021; Rusch et al., 2023) analysis, effectively mitigating oversmoothing and reducing noise bias injection. UPGNET comprises two architectures: H-N (HOA followed by NFR) and N-H (NFR followed by HOA), both of which can be independently integrated with any GNN architecture. We evaluate overall performance of UPGNET and the contributions of each component under different parameters through theoretical analysis and extensive experiments.
+
+Our contributions are summarized as follows. $\textcircled{1}$ We propose a three-stage pipeline to systematically generalize the LDP protocols for perturbing node features. By analyzing the pipeline, we identify two key factors influencing the estimation error of feature aggregation. $\textcircled{2}$ Based on the above analysis, we propose NFR and HOA layers to reduce the estimation error and integrate them with LDP protocols, introducing UPGNET, a utility-enhanced privacy-preserving graph learning framework. $\textcircled{3}$ Extensive experiments on real datasets demonstrate that UPGNET excels in achieving privacy preservation and superior graph learning utility.
+
+# 2. Preliminaries
+
+This section first defines the problem (Sec. 2.1), then provides the essential background on LDP (Sec. 2.2) and GNN (Sec. 2.3), and finally describes the threat model (Sec. 2.4).
+
+# 2.1. Problem Definition
+
+Consider a graph $\mathcal { G } = ( \nu , \mathcal { E } )$ , where $\mathcal { V } = \{ v _ { 1 } , v _ { 2 } , \ldots , v _ { | \mathcal { V } | } \}$ represents the set of nodes and $\mathcal { E }$ is the set of edges. Each decentralized user $v \in \mathcal V$ locally possesses a $d$ -dimensional feature vector $\mathbf { x } _ { v } \in \mathbb { R } ^ { d }$ , and the feature matrix is defined as $\mathbf { X } \in \mathbb { R } ^ { | \nu | \times d }$ . Following previous work (Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022; Pei et al., 2023), we assume that the server has access to $\nu$ and $\mathcal { E }$ , but $\mathbf { X }$ remains private and inaccessible to the server.1 Consequently, we face the challenge of performing graph learning on $\mathcal { G }$ while protecting the privacy of node features. Consistent with previous work (Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022), this paper focuses on the node classification task in graph learning. Specifically, the node set $\mathcal { V } = \mathcal { V } _ { l } \cup \mathcal { V } _ { u }$ is the union of the set of labeled nodes $\mathcal { V } _ { l }$ and unlabeled ones $\nu _ { u }$ . The label $\mathbf { y } _ { v }$ for each node $v$ is derived from a set of possible labels, denoted $\mathcal { Y } = \{ y _ { 1 } , y _ { 2 } , \dots , y _ { c } \}$ . The objective of node classification (Kipf & Welling, 2017) is to learn a function $f : \mathcal { V } \to \mathcal { V }$ that assigns labels to unlabeled nodes based on the graph structure and available node features.
+
+# 2.2. Local Differential Privacy
+
+LDP (Kasiviswanathan et al., 2011; Yang et al., 2024; He et al., 2025) has been extensively studied and widely deployed in decentralized data collection and analysis scenarios. In particular, major companies such as Apple (Thakurta et al., 2017), Google (Erlingsson et al., 2014), and Microsoft (Ding et al., 2017) have adopted LDP. In an LDP setting, there exists a server and multiple users, each possessing sensitive data. Users are not required to transmit their private data to an untrustworthy server. Instead, each user initially perturbs their data using a perturbation mechanism $\mathcal { M }$ and then transmits the perturbed data to the server. Following the collection of perturbed data from each user, the server performs data analysis and learning based on these data, ensuring that user privacy remains uncompromised. The formal definition of $\epsilon$ -LDP is provided below.
+
+Definition 1 (ϵ-LDP). A local perturbation mechanism $\mathcal { M }$ satisfies $\epsilon$ -local differential privacy $\epsilon$ -LDP), where $\epsilon > 0$ , if and only if for any user’s private data $x$ and $x ^ { \prime }$ , we have:
+
+$$
+\forall y \in \operatorname { R a n g e } ( \mathcal { M } ) : \operatorname* { P r } [ \mathcal { M } ( x ) = y ] \leq e ^ { \epsilon } \cdot \operatorname* { P r } [ \mathcal { M } ( x ^ { \prime } ) = y ] ,
+$$
+
+where $\mathrm { R a n g e } ( { \mathcal { M } } )$ denotes the set of all possible outputs of the perturbation mechanism $\mathcal { M }$ . In essence, LDP guarantees that the data aggregator on the server side can’t reconstruct the data source, regardless of any prior knowledge. The parameter $\epsilon$ , called the privacy budget, plays a pivotal role in balancing privacy and utility. A smaller (resp. larger) $\epsilon$ provides stronger (resp. weaker) privacy preservation but also results in lower (resp. higher) utility. LDP has several important properties, such as immunity to post-processing and sequential composition (Dwork, 2008).
+
+# 2.3. Graph Nerual Networks
+
+In recent years, GNNs have gained popularity for graph mining and learning. The primary goal of GNNs is to learn embeddings for each node in a graph by combining initial node features with the graph’s topology. These learned node embeddings can then be applied to various downstream tasks such as node classification (Kipf & Welling, 2017; Sun et al., 2024a;b). A typical $K$ -layer GNN consists of $K$ graph convolutional layers. Each layer aggregates information from neighboring nodes and updates the node’s embedding. Following $K$ aggregation iterations, the embedding of a node captures information from its neighbors within $K$ hops. The formal definition of the $k$ -th layer is as follows:
+
+$$
+\begin{array} { r l } & { \mathbf { h } _ { \mathcal { N } ( v ) } ^ { k } = \mathrm { A G G R E G A T E } _ { k } ( \{ \mathbf { h } _ { u } ^ { k - 1 } , \forall u \in \mathcal { N } ( v ) \} ) , } \\ & { \quad \quad \mathbf { h } _ { v } ^ { k } = \mathrm { U P D A T E } _ { k } ( \mathbf { h } _ { \mathcal { N } ( v ) } ^ { k } ) , } \end{array}
+$$
+
+where $\mathcal { N } ( v )$ represents the set of neighbors of node $v$ (which could include $v$ itself). For any node $u \in \mathcal { N } ( v )$ , $\mathbf { h } _ { u } ^ { k - 1 }$ denotes the embedding of node $u$ at layer $k - 1$ . The aggregation functions amax, are denoted as $\mathrm { A G G R E G A T E } _ { k } ( \cdot )$ $k$ s mean, su functions. $\mathbf { h } _ { \mathcal { N } ( v ) } ^ { k }$ represents the output of the aggregation function on $\mathcal { N } ( v )$ $\mathrm { U P D A T E } _ { k } ( \cdot )$ denotes a learnable non-linear function at layer $k$ , such as a neural network. Initially, $\mathbf { h } _ { v } ^ { 0 } = \mathbf { x } _ { v }$ , indicating that the initial embedding of node $v$ is its feature vector $\mathbf { x } _ { v }$ .
+
+# 2.4. Threat Model
+
+As shown in Fig. 1, different users upload graph data to a third-party untrustworthy server. On the one hand, under a semi-honest adversary setup, although the server follows our LDP protocol honestly, it may attempt to individually learn the private information of the data owner. On the other hand, an attacker (Wang & Wang, 2022; Shen et al., 2022; Zhang et al., 2022; Meng et al., 2023) can target the GNN model to extract private information from the victim node, leading to disclosure of the user’s privacy.
+
+# 3. Methodology
+
+In this section, we begin with a theoretical analysis of prior work on locally differentially private graph neural networks (LDPGNN) in Sec. 3.1, identifying the key factors limiting their utility. Following this, we introduce UPGNET, a utility-enhanced private graph learning model, in Sec. 3.2.
+
+# 3.1. Theoretical Analysis of LDPGNN
+
+A series of studies (Sajadmanesh & Gatica-Perez, 2021; Du et al., 2021; Lin et al., 2022; Jin & Chen, 2022; Qi et al., 2024; Pei et al., 2023) have been conducted on private graph learning based on LDP. However, no effort has been made to establish a unified framework for revisiting existing work to further enhance the utility of private graph learning. In this subsection, our objective is to identify the key factors influencing the aggregation estimation error by constructing and analyzing a unified node feature LDP pipeline.
+
+# 3.1.1. NODE FEATURE LDP PIPELINE
+
+Currently, two state-of-the-art LDP mechanisms are applied to node features: the piecewise mechanism (PM) (Wang et al., $2 0 1 9 \mathrm { a }$ ; Pei et al., 2023) and the multi-bit mechanism (MBM) (Du et al., 2021; Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022; Jin & Chen, 2022) (for more details on PM and MBM see App. A). We propose a unified node-feature LDP pipeline that generalizes these two approaches. Assuming that the node-feature LDP mechanism is denoted as $\mathcal { M }$ and the total privacy budget employed is $\epsilon$ , the node-feature LDP mechanism can be outlined in three steps:
+
+Perturbation. In total, there are $| \nu |$ users, and each user $v$ possesses a $d$ -dimensional node feature $\mathbf { x } _ { v }$ containing their sensitive information. Users employ an LDP mechanism $\mathcal { M }$ to protect their privacy. Initially, the mechanism $\mathcal { M }$ randomly selects $m$ dimensions from $d$ dimensions without replacement, with $m$ being a configurable parameter controlling the number of perturbed dimensions. Subsequently, each sampled dimension undergoes random perturbation with a privacy budget denoted as $\epsilon / m$ , while the remaining $d - m$ dimensions are set to 0. Ultimately, $\mathbf { x } _ { v }$ undergoes the $\mathcal { M }$ mechanism to yield $\mathbf { x } _ { v } ^ { \prime }$ , denoted as $\mathbf { x } _ { v } ^ { \prime } = \mathcal { M } ( \mathbf { x } _ { v } )$ .
+
+Calibration. Following perturbation, $\mathbf { x } _ { v } ^ { \prime }$ remains biased, i.e., $\mathbb { E } \left[ \mathbf { x } _ { v } ^ { \prime } \right] \neq \mathbf { x } _ { v }$ . Let $\pmb { \sigma } = \mathbb { E } \left[ \mathbf { x } _ { v } ^ { \prime } \right] - \mathbf { x } _ { v }$ , representing the expected bias shift. The server calibrates the perturbed values with $\sigma$ . Note that $\pmb { \sigma } = 0$ indicates an unbiased estimate.
+
+Aggregation. After receiving the feature vectors $\mathbf { x } _ { v } ^ { \prime }$ for all users $v \in \mathcal V$ , the server aggregates these vectors as follows:
+
+$$
+\widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } = \mathop { \mathrm { A G G R E G A T E } } \left( \{ \mathbf { x } _ { u } ^ { \prime } , \forall u \in \mathcal { N } ( v ) \} \right) ,
+$$
+
+where $\widehat { \mathbf { h } } _ { \mathcal { N } ( v ) }$ represents the estimated embedding for any given node $v$ after undergoing the AGGREGATE $( \cdot )$ . This estimate is derived by aggregating the perturbed node feature vectors $\mathbf { x } _ { u } ^ { \prime }$ from all nodes $u$ adjacent to the target node $v$ .
+
+Theorem 2. Assuming $\pmb { \sigma } = 0$ , the aggregator function defined by Eq. (4) is an unbiased estimate, i.e., for any $v$ ,
+
+$$
+\mathbb { E } [ \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ] = \mathbf { h } _ { \mathcal { N } ( v ) } .
+$$
+
+As demonstrated in Thm. 2, the aggregation process is an unbiased estimate when $\sigma = 0$ and the AGGREGATE function is linear, meaning the output is a weighted summation of the inputs. For the proof, please refer to App. B.1.
+
+# 3.1.2. KEY FACTOR ANALYSIS
+
+In Sec. 3.1.1, we establish an LDP analytical pipeline for the node features, consisting of three stages: perturbation, calibration, and aggregation. In the context of privacypreserving graph learning, the aggregation stage significantly impacts the overall utility of graph learning. Highquality aggregation helps mitigate the injected noise to a greater extent. Therefore, our objective is to explore the key factors that directly influence the estimation error in the aggregation stage. The estimation error is defined as the discrepancy between $\widehat { \mathbf { h } } _ { \mathcal { N } ( v ) }$ , obtained by aggregating the perturbed node features $\mathbf { x } ^ { \prime }$ , and $\mathbf { h } _ { \mathcal { N } ( v ) }$ , obtained by aggregating the original node features $\mathbf { x }$ . This discrepancy is represented as $\xi _ { i } = \lvert ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \rvert$ , $i \in \{ 1 , \ldots , d \}$ . Based on Bernstein’s inequality (Giroux et al., 1979), Thm. 3 provides an analysis of the various factors influencing the discrepancy.
+
+Theorem 3. Given the aggregator for the first layer and $\delta > 0$ , with probability at least $1 - \delta$ , for any node $v$ , we have:
+
+$$
+\begin{array} { r } { \operatorname* { m a x } \xi _ { i } = \mathcal { O } ( \sqrt { d \log ( d / \delta ) } / ( \epsilon \sqrt { | \mathcal { N } ( v ) | } ) ) , i \in \{ 1 , \dots , d \} . } \end{array}
+$$
+
+According to Thm. 3, after eliminating the known privacy budget parameter $\epsilon$ and the analysis parameter $\delta$ , two key factors that impact the estimation error are the feature dimension $d$ and the neighborhood size $\left| \mathcal { N } ( v ) \right|$ . From Eq. (6), we infer that a smaller effective $d$ is more conducive to reducing the estimation error, while a larger effective $\vert \mathcal { N } ( v ) \vert$ is also advantageous to minimizing the estimation error. Therefore, in Sec. 3.2, our objective is enhance the utility of privacy-preserving graph learning by influencing $d$ and $\left| \mathcal { N } ( v ) \right|$ . Please see App. B.2 for the proof of Thm. 3.
+
+# 3.2. UPGNET: Utility-Enhanced Private GNNs
+
+In this section, we propose a utility-enhanced private graph learning framework called UPGNET. Designed for various node feature LDP protocols, UPGNET incorporates plugand-play NFR and HOA layers that significantly boost utility. Fig. 2 (a) provides an overview of UPGNET, and Fig. 2 (b) illustrates the two architectures of UPGNET.
+
+# 3.2.1. OVERVIEW
+
+This subsection introduces the foundational design principles of UPGNET. According to Thm. 3, the key factors influencing the estimation error during the aggregation process are the feature dimension and the neighborhood size. Naturally, our objective is to minimize the estimation error during aggregation by targeting these two crucial factors, thereby enhancing the practicality of private graph learning. To this end, we design from the following two perspectives:
+
+Expanding the Effective Neighborhood Size. In order to extend the effective neighborhood size, direct multi-layer aggregation is a potential approach. However, our analysis reveals that this method is significantly limited by oversmoothing (Rusch et al., 2023), which adversely affected the denoising performance. To address this, we propose a Higher-Order Aggregator (HOA) layer, leveraging personalized aggregation and Dirichlet energy analysis to effectively mitigate over-smoothing and reduce noise injection.
+
+Reducing the Effective Feature Dimensions. In order to minimize estimation error by reducing the effective feature dimensions, we focus on the aggregation stage2. Based on $L _ { 1 }$ -regularization and proximal gradient descent (PGD) (Nitanda, 2014; Li & Lin, 2015; Duan et al., 2022), we introduce the Node Feature Regularizer (NFR) layer.
+
+Through theoretical analysis and experimental validation, UPGNET, in both the H-N (see Sec. 3.2.2) and the N-H architecture (see Sec. 3.2.3), effectively integrates the HOA layer and the NFR layer to reduce estimation error and enhance learning utility across various LDP mechanisms.
+
+# 3.2.2. UPGNET IN H-N ARCHITECTURE
+
+In this part, we introduce the H-N architecture of UPGNET, where the perturbed node features $\mathbf { x } ^ { \prime }$ are successively enhanced by the HOA layer followed by the NFR layer.
+
+Higher-Order Aggregator. According to Thm. 3, as the size of $\mathcal { N } ( v )$ (i.e., $| \mathcal { N } ( v ) | )$ increases, the estimation error decreases at a rate proportional to the square root of the node degree. This indicates that larger $\left| \mathcal { N } ( v ) \right|$ results in smaller estimation errors. However, in practical scenarios, $\left| \mathcal { N } ( v ) \right|$ is usually quite small. One approach to address this issue is to expand $\mathcal { N } ( v )$ by directly aggregating multiple layers of node features across $K$ hops (Abu-El-Haija et al., 2019; Gasteiger et al., 2019; Chen et al., 2020; Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022) (defined as the SKA method). However, the SKA scheme encounters the oversmoothing (Rusch et al., 2023), where increasing the value of $K$ causes node embeddings to converge, reducing the effectiveness of information aggregation from higher-order neighbors and skewing the calibration of aggregation errors. To better understand the over-smoothing issue, we examine the Dirichlet energy (Rusch et al., 2023) of the estimated node embeddings on the graph, a primary measure of over-smoothing in deep GNNs. Specifically, the estimated embedding $\widehat { \mathbf { h } }$ is modeled as a combination of the original embedding $\mathbf { h }$ and the noise signal $\eta$ , i.e., $\widehat { \mathbf { h } } = \mathbf { h } + \boldsymbol { \eta }$ . The Dirichlet energy $\Upsilon ( \cdot )$ of $\widehat { \mathbf { h } }$ is then defined as:
+
+![](images/8b4b1a97ecdbd5879437da731e9d879342b0c1c7c4573d62620c2afc436dd2c9.jpg)  
+Figure 2. (a) Overview of our proposed UPGNET. On the client side, local node features x are perturbed using LDP to obtain $\mathbf { x } ^ { \prime }$ . On the server side, $\mathbf { x } ^ { \prime }$ undergoes processing through an Node Feature Regularization (NFR) layer and an High-Order Aggregator (HOA) layer to enhance utility before being input into the GNN for graph learning, enabling downstream tasks such as node classification. (b) UPGNET features two distinct architectures: H-N (HOA layer followed by NFR layer) and N-H (NFR layer followed by HOA layer).
+
+$$
+\Upsilon ( \widehat { \mathbf { h } } ) = \frac { 1 } { | \mathcal { V } | } \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathcal { N } ( v _ { i } ) } \big ( \Big \| \mathbf { h } _ { i } ^ { k } - \mathbf { h } _ { j } ^ { k } \Big \| _ { 2 } ^ { 2 } + \Big \| \underbrace { \eta _ { i } ^ { k } - \eta _ { j } ^ { k } } _ { \mathrm { n o i s e ~ s i g n a l } } \Big | _ { 2 } ^ { 2 } ,
+$$
+
+where $k \in \{ 1 , 2 , \cdots , K \}$ represents the step parameter. Eq. (7) highlights two key observations: $\textcircled{1}$ Over-smoothing exists. As $k$ increases, the difference between the embedding $\mathbf { h } _ { i }$ and its neighbor embedding $\mathbf { h } _ { j }$ diminishes. After several rounds of propagation, all node features converge, causing a sharp decline in the first term of $\Upsilon ( \widehat { \mathbf { h } } )$ , resulting in the typical oversmoothing phenomenon. $\textcircled{2}$ Noise exacerbates over-smoothing. Under the LDP, when $\epsilon$ approaches 0, the noise term $\| \eta _ { i } - \eta _ { j } \| _ { 2 } ^ { 2 }$ dominates the Dirichlet energy. This accelerates the homogenization of node features, leading to faster energy decay and thus intensifying the oversmoothing effect. Therefore, mitigating the over-smoothing problem to extend more effective neighborhood sizes is crucial.
+
+To address the aforementioned issues, we propose a utilityenhanced graph convolution layer called the High-Order Aggregator (HOA), as shown in Alg. 1. Compared to SKA, it has two advantages in reducing estimation error for noisy data: $\textcircled{1}$ mitigating over-smoothing and $\textcircled{2}$ reducing noise
+
+# Algorithm 1 High-Order Aggregator (HOA) Layer
+
+$$
+\begin{array} { r l } & { \mathbf { \tilde { a } } _ { \mathcal { N } ( v ) } ^ { \mathrm { ~ ~ } } = 0 , \mathbf { x } _ { \mathcal { N } ( v ) } ^ { 0 } = \mathbf { x } _ { v } } \\ & { \mathbf { f o r } \ k = 1 \mathrm { \textbf { t o } } K \mathbf { d o } } \\ & { \quad \mathbf { x } _ { \mathcal { N } ( v ) } ^ { k } = \mathrm { A G G R E G A T E } \left( \{ \mathbf { x } _ { \mathcal { N } ( u ) } ^ { k - 1 } , \forall u \in \mathcal { N } ( v ) \backslash \{ v \} \} \right) } \\ & { \quad \mathbf { h } _ { \mathcal { N } ( v ) } ^ { k } = \mathbf { h } _ { \mathcal { N } ( u ) } ^ { k - 1 } + \mathbf { x } _ { \mathcal { N } ( v ) } ^ { k } } \end{array}
+$$
+
+$$
+\begin{array} { r } { \mathbf { h } _ { v } = \frac { 1 } { K } \mathbf { h } _ { \mathcal { N } ( v ) } ^ { K } } \\ { \mathbf { \tilde { \Sigma } } _ { \mathbf { \tilde { \Sigma } } } \mathbf { \Sigma } _ { \mathbf { \tilde { \Sigma } } } } \end{array}
+$$
+
+bias injection. First, as shown in Thm. 4, the energy ratio $\Phi _ { K }$ between $\mathrm { H O A } ( \cdot )$ and $\operatorname { S K A } ( { \mathord { \cdot } } )$ approaches 0 as $K  \infty$ This indicates that HOA is less influenced by information from infinite-hop receptive fields, allowing it to mitigate over-smoothing and expand into larger neighborhoods while still aggregating data from smaller ones. Second, HOA employs personalized weightings during neighbor information aggregation, assigning the highest weight to the nearest neighbor. The insight behind our approach is that information from the closest neighbor is more effective in calibrating noise for that specific node, justifying its higher weight, while more distant neighbors receive lower weights. This design further alleviates noise bias injection. See App. B.3 for the proof of Thm. 4 and more details.
+
+Theorem 4. Let $\Upsilon _ { H O A } ^ { k }$ and $\Upsilon _ { S K A } ^ { k }$ kSKA represent the Dirichlet energies of $H O A ( \cdot )$ and $S K A ( \cdot )$ at layer $k$ , respectively. The energy ratio of $H O A ( \cdot )$ to $S K A ( \cdot )$ across $K$ layers satisfies:
+
+$$
+\begin{array} { r } { \Phi _ { K } = \operatorname* { l i m } _ { K \to \infty } \left( \sum _ { k = 1 } ^ { K } \Upsilon _ { H O A } ^ { k } / \sum _ { k = 1 } ^ { K } \Upsilon _ { S K A } ^ { k } \right) = 0 . } \end{array}
+$$
+
+As in (Sajadmanesh & Gatica-Perez, 2021), we also employ the GCN aggregator function (Kipf & Welling, 2017) and perform the aggregations without including the self-loop, which will facilitate the reduction of the total noise.
+
+Node Feature Regularization. Regularization is a classical method for optimizing minimization tasks (Hoyer, 2004; Buhlmann & Van De Geer¨ , 2011; Negahban et al., 2017; Duan et al., 2022). Among various regularization techniques, $L _ { 1 }$ -regularization tends to produce sparse solutions. In other words, parameters obtained through $L _ { 1 }$ - regularization are more likely to have fewer non-zero components. This property facilitates the implementation of embedded feature selection, aligning with our objective of reducing the effective feature dimension. Next, we formalize the $L _ { 1 }$ -regularization problem within UPGNET under the H-N architecture. We employ the mean aggregation function and define the embedding of node $v$ after the server aggregates the perturbed feature vectors as follows:
+
+$$
+\widehat { \mathbf { h } } _ { v } \mathrm { = A G G R E G A T E } \left( \big \{ \mathbf { x } _ { u } ^ { \prime } , \forall u \in \mathcal { N } ( v ) \big \} \right) = \frac { 1 } { | \mathcal { N } ( v ) | } \sum _ { u \in \mathcal { N } ( v ) } \mathbf { x } _ { u } ^ { \prime } ,
+$$
+
+where $\mathbf { x } _ { u } ^ { \prime }$ represents the noisy node features of node $u$ . We define the loss function $\mathcal { L } _ { 1 } ( w )$ as follows:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { 1 } ( \boldsymbol { w } ) = \displaystyle \frac { 1 } { 2 \left| \mathcal { N } ( \boldsymbol { v } ) \right| } \cdot \sum _ { \boldsymbol { u } \in \mathcal { N } ( \boldsymbol { v } ) } \left\| \mathbf { x } _ { \boldsymbol { u } } ^ { \prime } - \boldsymbol { w } \right\| _ { 2 } ^ { 2 } . } \end{array}
+$$
+
+Based on this, we add the $L _ { 1 }$ -regularization terms $\lVert \pmb { w } \rVert _ { 1 }$ to $\mathcal { L } _ { 1 } ( w )$ to obtain the enhanced node embedding $\widetilde { \mathbf { h } } _ { v }$ for any node $v$ as $\begin{array} { r } { \widetilde { \mathbf { h } } _ { v } = \arg \operatorname* { m i n } _ { { \pmb w } \in \mathbb { R } ^ { d } } \mathcal { L } _ { 1 } ( { \pmb w } ) + \mu _ { 1 } \| { \pmb w } \| _ { 1 } } \end{array}$ . Thm. 5 derives the above $L _ { 1 }$ -regularization problem by employing the proximal gradient descent (PGD) (Nitanda, 2014; Li & Lin, 2015; Duan et al., 2022) method.
+
+Theorem 5. For any node $v \in \mathcal V$ and any feature dimension $i \in \{ 1 , \ldots , d \}$ , $( \widetilde { \mathbf { h } } _ { v } ) _ { i }$ in the following equation can efficiently achieve feature selection for $( \widehat { \mathbf { h } } _ { v } ) _ { i }$ :
+
+$$
+( \widetilde { \mathbf { h } } _ { v } ) _ { i } = \mathrm { s i g n } \left( ( \widehat { \mathbf { h } } _ { v } ) _ { i } \right) \cdot \operatorname* { m a x } \left( | ( \widehat { \mathbf { h } } _ { v } ) _ { i } | - \mu _ { 1 } , 0 \right) ,
+$$
+
+where $\mathrm { s i g n } ( \cdot )$ denotes the sign function, which takes 1 if $( \widehat { \mathbf { h } } _ { v } ) _ { i } > 0$ , 0 if $( \widehat { \mathbf { h } } _ { v } ) _ { i } = 0$ , and $^ { - 1 }$ if $( \widehat { \mathbf { h } } _ { v } ) _ { i } < 0$ . The optimal value for $\mu _ { 1 }$ is $\tau _ { 1 } B / \bar { d } ^ { K }$ , where $\tau _ { 1 } \in ( 0 , 1 )$ , with $B$ as the boundary of the perturbed node features, $\bar { d }$ as the approximate average degree of the graph, and $K$ as the step parameter of the HOA layer. By Thm. 5, we conclude that Eq. (11) can efficiently achieve feature selection for $\widehat { \mathbf { h } } _ { v }$ , thereby contributing to enhancing the utility of private graph learning. See App. B.4 for the proof.
+
+# 3.2.3. UPGNET IN N-H ARCHITECTURE
+
+Under the N-H architecture of UPGNET, the HOA is consistent with Alg. 1. The NFR specifically aims to enhance utility through efficient feature selection of the perturbed node features $\mathbf { x } ^ { \prime }$ directly using $L _ { 1 }$ -regularization. The objective function $\mathcal { L } _ { 2 }$ of HOA is formalized as follows:
+
+$$
+\mathcal { L } _ { 2 } ( \mathbf { x } ) = \frac { 1 } { 2 } \left\| \mathbf { x } ^ { \prime } - \mathbf { x } \right\| _ { 2 } ^ { 2 } + \mu _ { 2 } \left\| \mathbf { x } \right\| _ { 1 } .
+$$
+
+Table 1. Statistics of graph datasets.   
+
+<table><tr><td>Dataset</td><td>Classes</td><td>Nodes</td><td>Edges</td><td>Features</td><td>Deg.</td></tr><tr><td>Cora</td><td>7</td><td>2,708</td><td>5,278</td><td>1,433</td><td>3.90</td></tr><tr><td>Citeseer</td><td>6</td><td>3,327</td><td>4,552</td><td>3,703</td><td>2.74</td></tr><tr><td>LastFM</td><td>18</td><td>7,624</td><td>27,806</td><td>7,842</td><td>7.29.</td></tr><tr><td>Facebook</td><td>4</td><td>22,470</td><td>170,912</td><td>4,714</td><td>15.21</td></tr></table>
+
+Deg. in the table denotes the average node degree.
+
+Theorem 6. For any node $v$ and any $i \in \{ 1 , \ldots , d \}$ , $( \widetilde { \mathbf { x } } _ { v } ) _ { i }$ in Eq. (13) can efficiently achieve feature selection:
+
+$$
+( \widetilde { \mathbf { x } } _ { v } ) _ { i } = \mathrm { s i g n } \left( ( \mathbf { x } _ { v } ^ { \prime } ) _ { i } \right) \cdot \operatorname* { m a x } \left( | ( \mathbf { x } _ { v } ^ { \prime } ) _ { i } | - \mu _ { 2 } , 0 \right) ,
+$$
+
+where the optimal value for $\mu _ { 1 }$ is $\tau _ { 2 } B$ , where $\tau _ { 2 } \in ( 0 , 1 )$ , with $B$ as the boundary of the perturbed node features. According to Thm. 6, we conclude that Eq. (13) can efficiently achieve feature selection for $\mathbf { x } _ { v } ^ { \prime }$ , thus enhancing the utility of graph learning. See Sec. C and App. B.5 for more details.
+
+# 3.2.4. PRIVACY AND COMPLEXITY ANALYSIS
+
+Privacy Analysis. The PM and MBM satisfy $\epsilon$ -LDP for each node. The entire training process remains LDP compliant due to the robustness of DP against the post-processing theorem (Dwork et al., 2014). Moreover, any subsequent prediction is bounded by the post-processing theorem (Dwork et al., 2014), since the LDP protocol is applied only once to the private data. This ensures that LDP holds for all nodes throughout the process. For more details, see App. D.
+
+Complexity Analysis. The computational complexity of UPGNET mainly arises from its two key components: the HOA and the NFR. Through analysis, the overall complexity of UPGNET is $\mathcal { O } ( K \cdot | \mathcal { E } | \cdot d + | \mathcal { V } | \cdot d )$ , scaling linearly with graph size and feature dimensionality. This ensures that UPGNET remains both practical and scalable for largescale graphs with high-dimensional data. See App. E for more detailed analysis and comparisons with baselines.
+
+# 4. Experiments
+
+In this section, we conduct a series of experiments to validate the performance of UPGNET and its core components. More experimental results can be found in App. F.
+
+# 4.1. Experimental Setting
+
+Datasets. We conduct experiments on four representative graph datasets: Cora (Yang et al., 2016), Citeseer (Yang et al., 2016), LastFM (Rozemberczki & Sarkar, 2020), and Facebook (Rozemberczki et al., 2021). These datasets are commonly used in graph machine learning (Wu et al., 2020; Zhang et al., 2020). Table 1 provides the statistics for these datasets, with specific descriptions as follows:
+
+• Cora and CiteSeer. They are well-known citation networks, where each node represents a scientific paper, and the edges denote citation links. Each node contains a bag-of-words feature vector and a label for each category.
+
+![](images/dad892a55e626dfa29101a309285e7f9f2138de7e8d5a8cd5f07ffcdc83e255a.jpg)  
+Figure 3. Performance of UPGNET and other baselines. X-axis represents $\epsilon$ and y-axis represents test accuracy $( \% )$ .
+
+• Facebook. This social network consists of nodes as official Facebook pages, with edges representing mutual liking relationships. Each node has a feature extracted from the site description and a label indicating the category.
+
+• LastFM. Nodes in this dataset represent users of the music streaming service LastFM and links represent friendships between them. The classification task is to predict the users’ home country given the artists liked them.
+
+Baselines. To comprehensively assess the performance of UPGNET, we compare it with the following baselines: The NonPriv sets $\epsilon = \infty$ and inputs clean (non-perturbed) node features directly into the GNN for graph learning. In contrast, $B A S E$ utilizes the GNN for graph learning directly after using node feature LDP protocols to perturb feature vectors, without incorporating additional utility enhancement strategies. LPGNN (Sajadmanesh & Gatica-Perez,
+
+2021) and Solitude (Lin et al., 2022) apply different strategies to achieve locally differentially private graph learning. In addition, we consider the multi-bit mechanism (MBM) (Sajadmanesh & Gatica-Perez, 2021) and the piecewise mechanism (PM) (Wang et al., 2019a) independently.
+
+Parameter Settings. All datasets are randomly divided into $5 0 / 2 5 / 2 5 \%$ for training, validation, and test sets, respectively. To evaluate the performance of UPGNET, we use three representative GNN architectures, graph convolutional networks (GCN) (Kipf & Welling, 2017), GraphSAGE (Hamilton et al., 2017), and graph attention networks (GAT) (Velickovic et al., 2018) as backbone models. By default, the dataset used is Cora, the LDP protocol applied is the MBM, the GNN model is the GCN, and UPGNET adopts the N-H architecture. For more details, see App. F.1 and F.2.
+
+Evaluation Metrics. Consistent with prior work (Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022), we conduct experiments on the node classification task, using classification accuracy as the primary metric to evaluate the performance of UPGNET. All models undergo 500 training iterations and the best model is chosen for testing based on validation loss. Accuracy is measured over 10 consecutive runs, and we report the average along with $9 5 \%$ confidence intervals calculated by bootstrapping over 1000 samples.
+
+![](images/aa2c19d8babdfa2639b8b3a48d99ab2d87186f11050fdf5f2b6e3c974e9eb67d.jpg)  
+Figure 4. (a) Comparison of the performance of UPGNET under different GNN models. (b) Comparison of the performance of UPGNET in H-N vs. N-H architectures.
+
+# 4.2. Evaluating the Performance of UPGNET
+
+In this experiment, we vary $\epsilon$ in $\{ 0 . 0 1 , 0 . 1 , 1 . 0 , 2 . 0 , 3 . 0 \}$ to thoroughly validate the performance under different noise scales. The experimental results across four datasets and various GNN backbone models are presented in Fig. 3. It shows that, in all cases, UPGNET consistently achieves higher accuracy than BASE, LPGNN and Solitude, and in some instances, it even approaches the accuracy of NonPriv. For example, in the case of Fig. 3 (f), UPGNET increases accuracy by about $5 \%$ over LPGNN and Solitude when $\epsilon =$ 3.0. Moreover, for Facebook (Fig. 3 (d), (h)), the accuracy of UPGNET is closely aligned with that of NonPriv. All these observations underscore the superior utility of UPGNET.
+
+# 4.3. Comparison of Different GNN Models
+
+Fig. 4(a) intuitively compares the accuracy of UPGNET under different privacy budgets $\epsilon$ and across various GNN models. The results reveal that the GAT slightly worse than GCN and GraphSAGE. Specifically, under high privacy settings (e.g., when $\epsilon = 0 . 0 1$ ), the accuracy gap between GAT and the other two models becomes more pronounced. The GAT model introduces an attention mechanism that learns attention coefficients to weight the neighbors in neighborhood aggregation. Consequently, this makes GAT more sensitive to feature perturbations, resulting in a greater degradation of utility in high-noise settings. On the other hand, the utility of GAT remains comparable when $\epsilon \geq 0 . 1$ .
+
+# 4.4. Ablation Study on the Performance of NFR
+
+In this experiment, we investigate the utility enhancement of Node Feature Regularizer (NFR) layer for two node feature LDP mechanisms: piecewise mechanism (PM) and multi-bit mechanism (MBM). Table 2 shows the accuracy comparison between MBM and $\mathrm { M B M ^ { \star } }$ $( \mathrm { M B M } + \mathrm { N F R }$ layer) as well as PM and $\mathrm { { \mathbb { P } M ^ { \star } } }$ $\mathrm { P M } + \mathrm { N F R }$ layer) across different datasets and ϵ. The results in Table 2 clearly demonstrate that applying the NFR layer improves graph learning accuracy in all cases. Moreover, Table 2 and Fig. 5 indicates that NFR layer is more effective in improving graph learning accuracy when the privacy budget is small compared to when the privacy budget is large. For instance, in the case of the Cora dataset with MBM perturbing the node features, when $\epsilon = 0 . 0 1$ , $\mathrm { M B M ^ { \star } }$ improves accuracy by approximately $7 \%$ over MBM. When $\epsilon = 1 . 0$ , $\mathrm { M B M ^ { \star } }$ enhances accuracy by $2 . 6 \%$ over MBM. This is attributed to the fact that when $\epsilon$ is small, more noise is injected into the node features, so our NFR layer calibrates the noise and improves the accuracy more significantly.
+
+![](images/20ba70879db95eec8b8cfb3e409042a4734bf0f76d2ebeabf289c18331509491.jpg)  
+Figure 5. Comparison of the utility enhancement of $\mathrm { M B M } ^ { \star }$ and $\mathrm { { P M } ^ { \star } }$ under different $\epsilon \in \{ 0 . 0 1 , 0 . 1 , 1 . 0 , 2 . 0 , 3 . 0 \}$ . X-axis represents privacy budget $\epsilon$ and $\mathbf { y } .$ -axis represents test accuracy enhancement.
+
+Table 2. Applying NFR layer to node feature LDP mechanisms PM and MBM to achieve utility enhancement. $\mathrm { M B M } ^ { \star }$ and $\mathrm { { \mathbb { P } M } } ^ { \star }$ represent the integration of the NFR layer with MBM and PM, respectively.   
+
+<table><tr><td>Dataset</td><td>Mech.</td><td>e = 0.01</td><td>e = 0.1</td><td>e = 1.0</td><td>Overall</td></tr><tr><td rowspan="4">Cora</td><td>MBM</td><td>64.5</td><td>77.9</td><td>78.9</td><td></td></tr><tr><td>MBM*</td><td>71.36.8</td><td>80.62.7</td><td>81.52.6</td><td> 4.0</td></tr><tr><td>PM</td><td>65.3</td><td>78.5</td><td>79.4</td><td></td></tr><tr><td>PM*</td><td>72.0 6.7</td><td>80.5 2.0</td><td>81.2  1.8</td><td> 3.5</td></tr><tr><td rowspan="4">CiteSeer</td><td>MBM</td><td>52.5</td><td>61.7</td><td>65.4</td><td></td></tr><tr><td>MBM*</td><td>.57.2  4.7</td><td>65.0  3.3</td><td>67.8 2.4</td><td> 3.5</td></tr><tr><td>PM</td><td>53.1</td><td>62.8</td><td>66.1</td><td></td></tr><tr><td>PM*</td><td>57.4  4.3</td><td>64.7  1.9</td><td>67.41.3</td><td>2.5</td></tr><tr><td rowspan="4">LastFM</td><td>MBM</td><td>61.4</td><td>75.8</td><td>79.5</td><td></td></tr><tr><td>MBM*</td><td>67.15.7</td><td>.79.4  3.6</td><td>81.7 2.2</td><td>3.8</td></tr><tr><td>PM</td><td>66.1</td><td>77.1</td><td>80.1</td><td></td></tr><tr><td>PM*</td><td>68.7 2.6</td><td>79.52.4</td><td>81.9  1.8</td><td> 2.3</td></tr><tr><td rowspan="4">Facebook</td><td>MBM</td><td>78.6</td><td>86.6</td><td>87.3</td><td></td></tr><tr><td> MBM*</td><td>84.9 6.3</td><td>90.9  4.3</td><td>91.4  4.1</td><td> 4.9</td></tr><tr><td>PM</td><td>78.4</td><td>88.4</td><td>88.5</td><td></td></tr><tr><td>PM*</td><td>85.1  6.7</td><td>90.8  3.7</td><td>91.8 3.3</td><td> 4.6</td></tr></table>
+
+$\uparrow$ indicates the increase in utility after incorporating the NFR.
+
+# 4.5. Ablation Study on the Performance of HOA
+
+In this experiment, we investigate two aspects: first, whether Higher-Order Aggregator (HOA) layer can mitigate the oversmoothing in multilayer aggregation; and second, whether the HOA can effectively enhance the performance of private graph learning. To achieve these objectives, we compare
+
+![](images/1e61a508d1524d71ff4d9f69eff85f043547dc30914be21dab0b11063bb24b8c.jpg)  
+Figure 6. Effect of HOA vs. SKA on graph learning performance across various steps $K \in \{ 0 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \}$ .
+
+HOA with SKA and set $K = \{ 0 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \}$ for both HOA and SKA, consider the privacy budget $\epsilon = 0 . 0 1$ . Fig. 6 shows the performance of HOA compared to SKA on accuracy for different values of $K$ . As illustrated in Fig. 6, for all datasets, the accuracy trend under SKA initially rises with increasing $K$ , but after a certain point, it sharply declines, while the accuracy under HOA continues to improve steadily. For example, in the LastFM dataset (Fig. 6 (c)), SKA’s accuracy peaks at $K = 4$ but then drops rapidly, falling below $40 \%$ accuracy at $K = 6 4$ . This sharp decline is due to the oversmoothing effect in SKA, where larger $K$ values cause node embeddings to become overly similar, diminishing graph learning performance. In contrast, the HOA algorithm shows a steady increase in accuracy as $K$ grows, indicating that the proposed HOA layer successfully mitigates oversmoothing and effectively aggregates useful information from expanded neighborhoods. Furthermore, for $K \in \{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \}$ , HOA consistently outperforms SKA, demonstrating that the HOA can significantly enhance the learning utility. See App. F.3 and F.5 for more details.
+
+# 4.6. Comparison of Different Architectures
+
+We evaluate the effect of both the N-H and H-N architectures on utility by varying $\epsilon \in \{ 0 . 0 1 , 0 . 1 , 1 . 0 , 2 . 0 , 3 . 0 \}$ and conducting comparisons using the GCN. As shown in Fig. 4(b), for the Cora, the N-H architecture outperforms the H-N slightly in terms of accuracy. Notably, with smaller ϵ, the N-H architecture excels in feature dimension optimization, which allows it to better handle the information loss from noise injection. However, as $\epsilon$ increases, the performance gap between the N-H and H-N architectures narrows, indicating that the early application of the NFR layer in the N-H architecture is more effective in expanding the neighborhood range and enhancing utility when the noise is higher.
+
+# 5. Related work
+
+Recently, a series of works related to locally differentially private GNNs have been proposed. Sajadmanesh & GaticaPerez (2021) propose a privacy-preserving graph learning framework called LPGNN, which assumes that node features are private and the server has access to the graph topology, aligning with the scenario of this paper. In LPGNN, each user perturbs their features using the multi-bit mechanism (MBM). However, this paper demonstrates that MBM introduces excessive noise to the feature vector, thereby reducing the utility of the final private graph learning process. Similarly, Du et al. (2021), Lin et al. (2022) and Jin & Chen (2022) utilize the MBM to perturb node features or employ the SKA (Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022) to calibrate noisy features. Besides MBM, the PM (Wang et al., 2019a; Pei et al., 2023) has also been explored for privacy-preserving graph learning. Yet, it faces similar utility challenges. In contrast to these approaches, our paper introduces a utility-enhanced locally private graph learning framework applicable to various node feature perturbation mechanisms, including MBM and PM, to further enhance the utility of private graph learning.
+
+In addition to node feature perturbation, other efforts have addressed link privacy under LDP. Lin et al. (2022) adopted a naive randomized response mechanism (Qin et al., 2017) to protect adjacency lists. Hidano & Murakami (2024) propose a link LDP mechanism called DPRR, which injects noise into adjacency lists and node degrees respectively and calibrates the noisy links by a degree-sampling method. Zhu et al. (2023) propose a Bayesian estimation-based link LDP mechanism called BLINK. Similar to DPRR, BLINK injects noise into adjacency lists and node degrees, and then estimates the ground truth graph using the adjacency lists as prior and the node degrees as evidence. This work, in contrast, focuses on protecting node features, and is orthogonal to these approaches. It can be seamlessly integrated with existing methods designed to protect neighbor lists.
+
+# 6. Conclusion
+
+In this paper, we initially establish a pipeline to generalize the LDP protocols for perturbing node features. Through our analysis, we identify two key factors that affect the estimation error. Building on these insights, we propose UPGNET, which incorporates NFR and HOA layers. The generalization and effectiveness of UPGNET and its components are validated through theoretical analysis and extensive experiments in various datasets and parameter settings.
+
+# Acknowledgements
+
+This work is supported in part by the National Natural Science Foundation of China (62072052), in part by the National Natural Science Foundation of China (62372051).
+
+# Impact Statement
+
+This paper presents work aiming to advance the field of privacy-preserving graph learning. It focuses on enhancing the utility of private graph learning while preserving the privacy of graph data using local differential privacy. Our approach introduces a more efficient framework for privacypreserving graph learning, enhancing the overall utility of learning tasks while ensuring that user data remains protected from potential privacy breaches. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here. We believe this work will contribute to the development of more efficient and scalable privacy-preserving graph learning frameworks.
+
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+
+# A. Node Feature LDP Mechanisms
+
+# A.1. Piecewise Mechanism
+
+The one-dimensional piecewise mechanism (PM) (Wang et al., 2019a) takes $x \in [ - 1 , 1 ]$ as input3 and outputs the perturbed value $x ^ { \prime } \in [ - Q , Q ]$ , where $\begin{array} { r } { Q = \frac { e ^ { \epsilon / 2 } + 1 } { e ^ { \epsilon / 2 } - 1 } } \end{array}$ . The perturbed value is sampled from the following distribution:
+
+$$
+\begin{array} { r } { \operatorname* { P r } [ x ^ { \prime } = c | x ] = \left\{ \begin{array} { l l } { p , } & { \mathrm { i f } c \in [ l ( x ) , r ( x ) ] } \\ { \frac { p } { e ^ { \epsilon } } , } & { \mathrm { i f } c \in [ - Q , l ( x ) ) \cup ( r ( x ) , Q ] } \end{array} \right. , } \end{array}
+$$
+
+where p = eϵ−eϵ/2ϵ/2 , $\begin{array} { r } { l ( x ) = \frac { Q + 1 } { 2 } \cdot x - \frac { Q - 1 } { 2 } } \end{array}$ , and $r ( x ) = l ( x ) + Q - 1$ . In its high-dimensional form, the piecewise mechanism initially uniformly samples $m$ dimensions from the $d$ dimensions without replacement. Subsequently, it perturbs the data in each of the selected dimensions using the $\epsilon / m$ -LDP. According to the sequential composition property (Dwork, 2008) of the LDP mechanism, it adheres to $\epsilon$ -LDP. After receiving all the perturbed data, the server performs de-biasing on the reporting data $x ^ { * }$ using the following equation: $\begin{array} { r } { x ^ { \prime } = \frac { d } { m } \cdot x ^ { * } } \end{array}$ . The variance of $x ^ { \prime }$ is as follows:
+
+$$
+V a r [ x ^ { \prime } ] = \frac { d ( e ^ { \epsilon / 2 m } + 3 ) } { 3 m ( e ^ { \epsilon / 2 m } - 1 ) ^ { 2 } } + \biggl [ \frac { d \cdot e ^ { \epsilon / 2 m } } { m ( e ^ { \epsilon / 2 m } - 1 ) } - 1 \biggr ] \cdot x ^ { 2 } .
+$$
+
+The range of $x ^ { \prime }$ is as follows:
+
+$$
+x ^ { \prime } \in \left[ - \frac { d } { m } \cdot \frac { e ^ { \epsilon / 2 } + 1 } { e ^ { \epsilon / 2 } - 1 } , \frac { d } { m } \cdot \frac { e ^ { \epsilon / 2 } + 1 } { e ^ { \epsilon / 2 } - 1 } \right] .
+$$
+
+# A.2. Multi-bit Mechanism
+
+The multi-bit mechanism (MBM) (Sajadmanesh & Gatica-Perez, 2021) is an extension of the 1-bit mechanism (Ding et al., 2017). In its one-dimensional form, for any original data $x \in [ - 1 , 1 ]$ , the distribution followed by the perturbed value $x ^ { \prime } \in \{ - 1 , 1 \}$ is as follows:
+
+$$
+\operatorname* { P r } [ x ^ { \prime } = c | x ] = { \left\{ \begin{array} { l l } { { \frac { 1 } { e ^ { \epsilon } + 1 } } + { \frac { x + 1 } { 2 } } \cdot { \frac { e ^ { \epsilon } - 1 } { e ^ { \epsilon } + 1 } } , } & { { \mathrm { i f } } c = 1 } \\ { { \frac { e ^ { \epsilon } } { e ^ { \epsilon } + 1 } } - { \frac { x + 1 } { 2 } } \cdot { \frac { e ^ { \epsilon } - 1 } { e ^ { \epsilon } + 1 } } , } & { { \mathrm { i f } } c = - 1 } \end{array} \right. } .
+$$
+
+In its high-dimensional form, similar to the piecewise mechanism, each sampled dimension carries out $\epsilon / m$ -LDP perturbation. After receiving all the perturbed data, the server transforms the reporting data $x ^ { * }$ to its unbiased estimate $\begin{array} { r } { x ^ { \prime } = \frac { d } { m } \cdot \frac { e ^ { \epsilon / m } + 1 } { e ^ { \epsilon / m } - 1 } \cdot x ^ { * } } \end{array}$ . The variance of $x ^ { \prime }$ is as follows:
+
+$$
+V a r [ x ^ { \prime } ] = \frac { d } { m } \cdot \left( \frac { e ^ { \epsilon / m } + 1 } { e ^ { \epsilon / m } - 1 } \right) ^ { 2 } - x ^ { 2 } .
+$$
+
+The range of $x ^ { \prime }$ is as follows:
+
+$$
+x ^ { \prime } \in \left\{ - \frac { d } { m } \cdot \frac { e ^ { \epsilon / m } + 1 } { e ^ { \epsilon / m } - 1 } , 0 , \frac { d } { m } \cdot \frac { e ^ { \epsilon / m } + 1 } { e ^ { \epsilon / m } - 1 } \right\} .
+$$
+
+# B. Theoretical Proof
+
+# B.1. Proof for Theorem 2
+
+Proof. According to Eq. (4), we have: $\begin{array} { r } { \mathbb { E } \left[ \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } \right] = \mathbb { E } \left[ \Delta \mathrm { G G R E G A T E } \left( \{ \mathbf { x } _ { u } ^ { \prime } , \forall u \in \mathcal { N } ( v ) \} \right) \right] . } \end{array}$
+
+Since AGGREGATE $( \cdot )$ is linear, we have:
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } \right] = \mathrm { A G G R E G A T E } \left( \left\{ \mathbb { E } \left[ \mathbf { x } _ { u } ^ { \prime } \right] , \forall u \in \mathcal { N } ( v ) \right\} \right) . } \end{array}
+$$
+
+Considering $\sigma = 0$ , we have $\mathbb { E } [ \mathbf { x } _ { u } ^ { \prime } ] = \mathbf { x } _ { u }$ and hence have:
+
+$$
+\mathbb { E } \left[ \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } \right] = \mathrm { A G G R E G A T E } \left( \{ \mathbf { x } _ { u } , \forall u \in \mathcal { N } ( v ) \} \right) = \mathbf { h } _ { \mathcal { N } ( v ) } .
+$$
+
+3Without loss of generality, this paper assumes that each dimension of the input data is normalized to the range $[ - 1 , 1 ]$ .
+
+# B.2. Proof for Theorem 3
+
+Proof. For any node $u \in \mathcal V$ and dimension $i \in \{ 1 , 2 , \ldots , d \}$ , following the perturbation and calibration stages, we determine that $x _ { u , i } ^ { \prime } \in [ - B , B ]$ , where $B$ is a finite boundary. We define $X = x _ { u , i } ^ { \prime } - x _ { u , i }$ . Given $x _ { u , i } \in [ - 1 , 1 ]$ , we have: $| X | \leq B + 1$ , and through the calibration stage, if $\sigma = 0$ , we know that $\mathbb { E } \left[ X \right] = \mathbb { E } \left[ x _ { u , i } ^ { \prime } \right] - x _ { u , i } = 0$ .
+
+Assuming the mean aggregator is employed, for any node $v \in \mathcal V$ and any dimension $i \in \{ 1 , 2 , \ldots , d \}$ , we have:
+
+$$
+( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } = \frac { 1 } { | \mathcal { N } ( v ) | } \sum _ { u \in \mathcal { N } ( v ) } x _ { u , i } , \quad ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } = \frac { 1 } { | \mathcal { N } ( v ) | } \sum _ { u \in \mathcal { N } ( v ) } x _ { u , i } ^ { \prime } .
+$$
+
+Thus, based on the Bernstein inequality, we have:
+
+$$
+\begin{array} { r l } & { \operatorname* { P r } \left[ \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| \geq \lambda \right] = \operatorname* { P r } \left[ \left| \displaystyle \sum _ { u \in \mathcal { N } ( v ) } ( x _ { u , i } ^ { \prime } - x _ { u , i } ) \right| \geq \lambda \left| \mathcal { N } ( v ) \right| \right] } \\ & { \quad \leq 2 \cdot \exp \left\{ - \frac { \lambda ^ { 2 } \left| \mathcal { N } ( v ) \right| } { \left| \mathcal { N } ( v ) \right| \sum _ { u \in \mathcal { N } ( v ) } V a r [ X ] + \frac { 2 } { 3 } \lambda ( B + 1 ) } \right\} = 2 \cdot \exp \left\{ - \frac { \lambda ^ { 2 } \left| \mathcal { N } ( v ) \right| } { 2 V a r [ x _ { u , i } ^ { \prime } ] + \frac { 2 } { 3 } \lambda ( B + 1 ) } \right\} } \end{array}
+$$
+
+Considering Eq. (15) and Eq. (18), we have:
+
+$$
+V a r [ x _ { u , i } ^ { \prime } ] = \mathcal { O } ( \frac { m d } { \epsilon ^ { 2 } } ) .
+$$
+
+Considering Eq. (16) and Eq. (19), we have:
+
+$$
+B = \mathcal { O } ( \frac { d } { \epsilon } ) .
+$$
+
+Substituting the Eq. (23) and Eq. (24) into Eq. (22), we obtain:
+
+$$
+\operatorname* { P r } \left[ \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| \geq \lambda \right] \leq 2 \cdot \exp \left\{ - \frac { \lambda ^ { 2 } \left| \mathcal { N } ( v ) \right| } { \mathcal { O } ( \frac { m d } { \epsilon ^ { 2 } } ) + \lambda \mathcal { O } ( \frac { d } { \epsilon } ) } \right\} .
+$$
+
+By applying the union bound, we have:
+
+$$
+\begin{array} { r l } & { \operatorname* { P r } \left[ \underset { i \in \{ 1 , \ldots , d \} } { \operatorname* { m a x } } \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| \geq \lambda \right] = \underset { i = 1 } { \overset { d } { \bigcup } } \operatorname* { P r } \left[ \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| \geq \lambda \right] } \\ & { \quad \leq \underset { i = 1 } { \overset { d } { \sum } } \operatorname* { P r } \left[ \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| \geq \lambda \right] = 2 d \cdot \exp \left\{ - \frac { \lambda ^ { 2 } \left| \mathcal { N } ( v ) \right| } { \mathcal { O } \left( \frac { m d } { \epsilon ^ { 2 } } \right) + \lambda \mathcal { O } \left( \frac { d } { \epsilon } \right) } \right\} . } \end{array}
+$$
+
+To ensure that $\begin{array} { r } { \operatorname* { m a x } _ { i \in \{ 1 , . . . , d \} } \left| ( \widehat { \mathbf { h } } _ { \mathcal { N } ( v ) } ) _ { i } - ( \mathbf { h } _ { \mathcal { N } ( v ) } ) _ { i } \right| < \lambda } \end{array}$ holds with at least $1 - \delta$ probability, it is sufficient to set $\begin{array} { r } { \delta = 2 d \cdot \exp \left\{ - \frac { \lambda ^ { 2 } | N ( v ) | } { \mathcal { O } ( \frac { m d } { \epsilon ^ { 2 } } ) + \lambda \mathcal { O } ( \frac { d } { \epsilon } ) } \right\} } \end{array}$ . Solving the above for $\lambda$ , we get: $\lambda = \mathcal { O } ( \sqrt { d \log ( d / \delta ) } / ( \epsilon \sqrt { | \mathcal { N } ( v ) | } ) ) .$
+
+# B.3. Proof for Theorem 4
+
+Proof. Considering that
+
+$$
+\Upsilon ( \widehat { \mathbf { h } } ) = \frac { 1 } { | \mathcal { V } | } \sum _ { \substack { i \in \mathcal { V } } } \sum _ { j \in \mathcal { N } ( v _ { i } ) } \left\| \widehat { \mathbf { h } } _ { i } ^ { k } - \widehat { \mathbf { h } } _ { j } ^ { k } \right\| _ { 2 } ^ { 2 } = \frac { 1 } { | \mathcal { V } | } \sum _ { \substack { i \in \mathcal { V } } } \sum _ { j \in \mathcal { N } ( v _ { i } ) } \left( \left\| \mathbf { h } _ { i } ^ { k } - \mathbf { h } _ { j } ^ { k } \right\| _ { 2 } ^ { 2 } + \underbrace { \left\| \eta _ { i } ^ { k } - \eta _ { j } ^ { k } \right\| _ { 2 } ^ { 2 } } _ { \mathrm { n o i s e ~ s i g n a l } } \right)
+$$
+
+and
+
+$$
+\Phi _ { K } = \operatorname* { l i m } _ { K  \infty } \frac { \sum _ { k = 1 } ^ { K } \mathcal { E } _ { \mathtt { H O A } } ^ { k } } { \sum _ { k = 1 } ^ { K } \mathcal { E } _ { \mathtt { S K A } } ^ { k } } = \operatorname* { l i m } _ { K  \infty } \frac { \sum _ { k = 1 } ^ { K } ( \frac { 1 } { | \mathcal { V } | } \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathcal { N } ( v _ { i } ) } | \widehat { \mathbf { h } } _ { 1 i } ^ { k } - \widehat { \mathbf { h } } _ { 1 j } ^ { k } | _ { 2 } ^ { 2 } ) } { \sum _ { k = 1 } ^ { K } ( \frac { 1 } { | \mathcal { V } | } \sum _ { i \in \mathcal { V } } \sum _ { j \in \mathcal { N } ( v _ { i } ) } | \widehat { \mathbf { h } } _ { 2 i } ^ { k } - \widehat { \mathbf { h } } _ { 2 j } ^ { k } | _ { 2 } ^ { 2 } ) } ,
+$$
+
+although both HOA and SKA introduce the noise term $\eta ^ { k }$ in each iteration, the key difference lies in how they handle the propagation and amplification of noise. As explained in the main paper, HOA employs a personalized aggregation strategy, allowing it to better control the noise amplification effect in subsequent layers. Specifically, by assigning smaller aggregation weights to the neighbors in more distant levels, HOA can mitigate the cumulative impact of noise. Consequently, as $K$ (the number of aggregation steps) approaches infinity, the effect of noise diminishes more effectively in HOA, leading to the condition $\Phi _ { K } = 0$ .
+
+# B.4. Proof for Theorem 5
+
+Proof. For the $L _ { 1 }$ -regularization problem addressed in this paper, we employ the proximal gradient descent (PGD) (Nitanda, 2014; Li & Lin, 2015; Duan et al., 2022) method for its solution. First, we obtain the derivative of $\mathcal { L } _ { 1 } ( w )$ :
+
+$$
+\nabla \mathcal { L } _ { 1 } ( w ) = \frac { 1 } { | \mathcal { N } ( v ) | } \sum _ { u \in \mathcal { N } ( v ) } ( w - \mathbf { x } _ { u } ^ { \prime } ) = w - \frac { 1 } { | \mathcal { N } ( v ) | } \sum _ { u \in \mathcal { N } ( v ) } \mathbf { x } _ { u } ^ { \prime } = w - \widehat { \mathbf { h } } _ { v } .
+$$
+
+Then, the derivative of $\nabla \mathcal { L } _ { 1 } ( \boldsymbol { w } )$ is $\begin{array} { r } { \frac { \mathrm { d } \nabla { \mathcal { L } } _ { 1 } ( w ) } { \mathrm { d } w } = 1 } \end{array}$ . According to Cauchy mean value theorem, we have:
+
+$$
+\begin{array} { r } { \| \nabla \mathcal { L } _ { 1 } ( \boldsymbol { w } ) - \nabla \mathcal { L } _ { 1 } ( \boldsymbol { w } _ { k } ) \| _ { 2 } ^ { 2 } \leq \| \boldsymbol { w } - \boldsymbol { w } _ { k } \| _ { 2 } ^ { 2 } , } \end{array}
+$$
+
+where ${ \pmb w } _ { k }$ represents the result of the $k$ -th iteration.
+
+We can approximate $\mathcal { L } _ { 1 } ( w )$ around ${ \pmb w } _ { k }$ using second-order Taylor expansion, given by:
+
+$$
+\begin{array} { l } { \displaystyle \mathcal { L } _ { 1 } ( \pmb { w } ) \simeq \mathcal { L } _ { 1 } ( \pmb { w } _ { k } ) + \langle \nabla \mathcal { L } _ { 1 } ( \pmb { w } _ { k } ) , \pmb { w } - \pmb { w } _ { k } \rangle + \frac { 1 } { 2 } \left. \pmb { w } - \pmb { w } _ { k } \right. ^ { 2 } } \\ { \displaystyle = \frac { 1 } { 2 } \left. \pmb { w } - ( \pmb { w } _ { k } - \nabla \mathcal { L } _ { 1 } ( \pmb { w } _ { k } ) ) \right. _ { 2 } ^ { 2 } + \mathrm { c o n s t } , } \end{array}
+$$
+
+where const is a constant independent of $\pmb { w }$ and $\langle \cdot , \cdot \rangle$ denotes the inner product. The minimum of the above equation is obtained at ${ \pmb w } _ { k + 1 }$ as follows:
+
+$$
+\pmb { w } _ { k + 1 } = \pmb { w } _ { k } - \nabla \mathcal { L } _ { 1 } ( \pmb { w } _ { k } ) .
+$$
+
+We then introduce the $L _ { 1 }$ -regularization term in the iteration:
+
+$$
+\pmb { w } _ { k + 1 } = \arg \operatorname* { m i n } _ { \pmb { w } } \frac { 1 } { 2 } \left\| \pmb { w } - \left( \pmb { w } _ { k } - \nabla \mathcal { L } _ { 1 } ( \pmb { w } _ { k } ) \right) \right\| _ { 2 } ^ { 2 } + \mu _ { 1 } \left\| \pmb { w } \right\| _ { 1 } .
+$$
+
+Let ${ \pmb w } _ { i }$ denote the $i$ -th dimension of $\textbf { \em w }$ . Since each dimension is independent, we have:
+
+$$
+( \pmb { w } _ { k + 1 } ) _ { i } = \arg \operatorname* { m i n } _ { \pmb { w } _ { i } } \frac { 1 } { 2 } \left\| \pmb { w } _ { i } - ( ( \pmb { w } _ { k } ) _ { i } - \nabla \mathcal { L } ( ( \pmb { w } _ { k } ) _ { i } ) ) \right\| _ { 2 } ^ { 2 } + \mu _ { 1 } \left\| ( \pmb { w } ) _ { i } \right\| _ { 1 } .
+$$
+
+Let $z _ { i } = ( \pmb { w } _ { k } ) _ { i } - \nabla \mathcal { L } _ { 1 } ( ( \pmb { w } _ { k } ) _ { i } ) = ( \pmb { w } _ { k } ) _ { i } - \left( ( \pmb { w } _ { k } ) _ { i } - ( \widehat { \mathbf { h } } _ { v } ) _ { i } \right) = ( \widehat { \mathbf { h } } _ { v } ) _ { i }$ . Computing $( { \pmb w } _ { k + 1 } ) _ { i }$ depends on whether ${ \pmb w } _ { i }$ is positive, negative or 0. Below, we compute it in the following cases:
+
+• If $w _ { i } > 0$ , by setting the gradient ${ \pmb w } _ { i } - z _ { i } + \mu _ { 1 }$ of Eq. (34) equal to zero, we obtain $( { \pmb w } _ { k + 1 } ) _ { i } = z _ { i } - \mu _ { 1 } > 0$ , hence $z _ { i } > \mu _ { 1 }$ .   
+• If $w ^ { i } < 0$ , similarly, we obtain $( { \pmb w } _ { k + 1 } ) _ { i } = z _ { i } + \mu _ { 1 } < 0$ , hence $z _ { i } < - \mu _ { 1 }$ .   
+• If ${ \pmb w } _ { i } = 0$ , we obtain $( { \pmb w } _ { k + 1 } ) _ { i } = 0$ , in which case $| z _ { i } | \leq \mu _ { 1 }$ .
+
+Consequently, we have:
+
+$$
+( \pmb { w } _ { k + 1 } ) _ { i } = \mathrm { s i g n } ( z _ { i } ) \cdot \operatorname* { m a x } \left( | z _ { i } | - \mu _ { 1 } , 0 \right) ,
+$$
+
+where sign denotes the sign function, which takes 1 if $( \widehat { \mathbf { h } } _ { v } ) _ { i } > 0 , 0$ if $( \widehat { \mathbf { h } } _ { v } ) _ { i } = 0$ , and $^ { - 1 }$ if $( \widehat { \mathbf { h } } _ { v } ) _ { i } < 0$ . Applying $( \pmb { w } _ { k + 1 } ) _ { i } = ( \widetilde { \mathbf { h } } _ { v } ) _ { i }$ and $z _ { i } = ( \widehat { \mathbf { h } } _ { v } ) _ { i }$ .
+
+$$
+( \widetilde { \mathbf { h } } _ { v } ) _ { i } = \mathrm { s i g n } \left( ( \widehat { \mathbf { h } } _ { v } ) _ { i } \right) \cdot \operatorname* { m a x } \left( | ( \widehat { \mathbf { h } } _ { v } ) _ { i } | - \mu _ { 1 } , 0 \right) ,
+$$
+
+The optimal value for $\mu _ { 1 }$ is $\tau _ { 1 } B / \bar { d } ^ { K }$ , where $\tau _ { 1 } \in ( 0 , 1 )$ , with $B$ as the boundary of the perturbed node features, $\bar { d }$ as the approximate average degree of the graph, and $K$ as the step parameter of the HOA layer. Here, $\bar { d } ^ { K }$ approximates the number of neighbors, as a larger number of neighbors tends to smooth out the feature values, allowing for a relatively smaller regularization parameter $\mu _ { 1 }$ . This formulation reflects how the regularization strength can adjust according to the feature magnitude and the neighborhood structure.
+
+# B.5. Proof for Theorem 6
+
+Proof. Under the N-H architecture, the NFR layer aims to enhance utility through efficient feature selection of the perturbed node features $\mathbf { x } ^ { \prime }$ using $L _ { 1 }$ -regularization. The objective function $\mathcal { L } _ { 2 }$ is defined as follows:
+
+$$
+\mathcal { L } _ { 2 } ( \mathbf { x } ) = \frac { 1 } { 2 } \left\| \mathbf { x } ^ { \prime } - \mathbf { x } \right\| _ { 2 } ^ { 2 } + \mu _ { 2 } \left\| \mathbf { x } \right\| _ { 1 } .
+$$
+
+Our goal is to minimize the above loss function. The proof procedure is similar to that of Theorem 5 and is not repeated.
+
+# C. More Details of N-H Architecture
+
+In this section, We provide more details on the N-H architecture of UPGNET, where the perturbed node features $\mathbf { x } ^ { \prime }$ are sequentially enhanced by first passing through the NFR layer followed by the HOA layer.
+
+Node Feature Regularization. Under the N-H architecture, the NFR layer specifically aims to enhance utility through efficient feature selection of the perturbed node features $\mathbf { x } ^ { \prime }$ directly using $L _ { 1 }$ -regularization. The objective function $\mathcal { L } _ { 2 }$ is formalized as follows:
+
+$$
+\mathcal { L } _ { 2 } ( \mathbf { x } ) = \frac { 1 } { 2 } \left\| \mathbf { x } ^ { \prime } - \mathbf { x } \right\| _ { 2 } ^ { 2 } + \mu _ { 2 } \left\| \mathbf { x } \right\| _ { 1 } .
+$$
+
+Our goal is to minimize the above loss function. Thm. 7 derives this regularization problem. According to Thm. 7, we conclude that Eq. (39) can efficiently achieve feature selection for $\mathbf { x } _ { v } ^ { \prime }$ , thereby enhancing the utility of private graph learning.
+
+Theorem 7. For any node $v$ and any $i \in \{ 1 , \ldots , d \}$ , $( \widetilde { \mathbf { x } } _ { v } ) _ { i }$ in the following equation can efficiently achieve feature selection:
+
+$$
+( \widetilde { \mathbf { x } } _ { v } ) _ { i } = \mathrm { s i g n } \left( ( \mathbf { x } _ { v } ^ { \prime } ) _ { i } \right) \cdot \operatorname* { m a x } \left( | ( \mathbf { x } _ { v } ^ { \prime } ) _ { i } | - \mu _ { 2 } , 0 \right) ,
+$$
+
+where the optimal value for $\mu _ { 1 }$ is $\tau _ { 2 } B$ , where $\tau _ { 2 } \in ( 0 , 1 )$ , with $B$ as the boundary of the perturbed node features.
+
+Higher-Order Aggregator. After utility enhancement through the NFR layer, additional utility enhancement is achieved by executing Alg. 1 on the enhanced feature vector $\widetilde { \mathbf { x } } _ { v }$ through the HOA layer.
+
+# D. Privacy Analysis
+
+The piecewise mechanism (PM) (Wang et al., 2019a; Pei et al., 2023) and multi-bit mechanism (MBM) (Du et al., 2021; Sajadmanesh & Gatica-Perez, 2021; Lin et al., 2022; Jin & Chen, 2022) satisfy $\epsilon$ -local differential privacy for each node. The entire training process remains LDP-compliant due to the robustness of differential privacy against post-processing (Dwork et al., 2014) (Thm. 8). Moreover, any subsequent prediction is bounded by the post-processing theorem (Dwork et al., 2014), since the LDP is applied only once to the private data. This ensures that LDP holds for all nodes throughout the process.
+
+Definition 8 (Post-processing). If an algorithm $\boldsymbol { \mathcal { A } } ( \cdot )$ satisfies $\epsilon$ -local differential privacy, then any further processing of its output by another algorithm $B ( \cdot )$ (i.e., process $B ( \mathcal { A } ( \cdot ) ) )$ also maintains $\epsilon$ -local differential privacy.
+
+# E. Complexity Analysis
+
+The computational complexity of UPGNET mainly arises from its two key components: the HOA layer and the NFR layer. The HOA layer performs $K$ steps of high-order neighborhood aggregation, where each step aggregating feature vectors from neighboring nodes. This process has a time complexity of $\mathcal { O } ( \vert \mathcal { E } \vert \cdot d )$ , where $| \mathcal { E } |$ is the number of edges and $d$ is the feature dimension, resulting in a total complexity of $O ( K \cdot | { \boldsymbol { \mathcal { E } } } | \cdot d )$ . The NFR layer applies lightweight element-wise transformations to each node’s feature vector, contributing a complexity of $\mathcal { O } ( | \mathcal { V } | \cdot d )$ , where $| \nu |$ is the number of nodes. Combining these, the overall computational complexity of UPGNET is $\mathcal { O } ( K \cdot | \mathcal { E } | \cdot d + | \mathcal { V } | \cdot d )$ , scaling linearly with graph size and feature dimensionality. This ensures that UPGNET remains both practical and scalable for large-scale graphs with high-dimensional data. Furthermore, compared to other methods (Solitude (Lin et al., 2022) and LPGNN (Sajadmanesh & Gatica-Perez, 2021)), the computational complexity of UPGNET introduces only an additional factor $| \nu | \cdot d$ . This factor is linear with respect to the number of nodes and the feature dimension, making it highly efficient in practice. Moreover, graph pruning (Yu et al., 2022; Liu et al., 2023) or parallelization techniques, such as GPU-based sparse matrix operations (Lee et al., 2020), can be applied to further reduce computational overhead.
+
+# F. Additional Details of Experiments
+
+# F.1. More Parameter Design
+
+To evaluate the performance of UPGNET, we use three state-of-the-art GNN architectures, graph convolutional networks (GCN) (Kipf & Welling, 2017) GraphSAGE (Hamilton et al., 2017) and graph attention networks (GAT) (Velickovic et al., 2018), as the backbone models. All GNN models have two graph convolutional layers, each with a hidden dimension of size 16 and a SeLU activation function (Klambauer et al., 2017) followed by dropout. The GAT model has four attention heads. To obtain the best hyperparameters, we use grid search for selection: both learning rate and weight decay are chosen from $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } \}$ , and dropout is chosen from $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 0 ^ { - 1 } \}$ . The HOA’s step parameter is denoted by $K$ . Based on the selected best hyperparameters, the best $K$ within $\{ 0 , 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \}$ for all $\epsilon \in \{ 0 . 0 1 , 0 . 1 , 1 , 2 , 3 \}$ . The parameters $\tau _ { 1 }$ , $\tau _ { 2 }$ of the NFR layer belong to $\{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . 7 , 0 . 9 \}$ . We use the Adam optimizer (Kingma & Ba, 2014) for all models. Without specification, we report the average results based on the best of each parameter.
+
+# F.2. More Descriptions about the GNN Model
+
+• GCN (Kipf & Welling, 2017) applies spectral graph convolution by propagating node features using the Laplacian matrix. The core update rule is: $H ^ { ( l + 1 ) } = \sigma ( \tilde { D } ^ { ^ { \bullet } - 1 / 2 } \tilde { A } \breve { \tilde { D } } ^ { ^ { \bullet } - ^ { \bullet } 1 / 2 } H ^ { ( l ) } W ^ { ( l ) } )$ , where $\stackrel {  } { A } = A + I$ is the adjacency matrix with self-loops, $\tilde { D }$ is its degree matrix, $W ^ { ( l ) }$ is the trainable weight matrix, and $\sigma$ is a activation function.
+
+• GraphSAGE (Hamilton et al., 2017) samples a fixed number of neighbors and aggregates their features, making it scalable for large graphs. The general update rule is: $h _ { v } ^ { ( l + 1 ) } = \sigma ( W ^ { ( l ) } \cdot \mathrm { A G G R E G A T E } ( \{ h _ { u } ^ { ( l ) } , \forall u \in \mathcal { N } ( v ) \} ) )$ , where AGGREGATE $( \cdot )$ represents the aggregation operation.
+
+• GAT (Hamilton et al., 2017) employs self-attention to dynamically assign importance to neighbors. The update rule is: $\begin{array} { r } { h _ { v } ^ { ( l + 1 ) } = \sigma \left( \sum _ { u \in \mathcal { N } ( v ) } \alpha _ { v u } W ^ { ( l ) } \bar { h } _ { u } ^ { ( l ) } \right) } \end{array}$ , where the attention coefficient $\alpha _ { v u }$ is computed as:
+
+$$
+\alpha _ { v u } = \frac { \exp ( \mathrm { L e a k y R e L U } ( a ^ { T } [ W ^ { ( l ) } h _ { v } ^ { ( l ) } \| W ^ { ( l ) } h _ { u } ^ { ( l ) } ] ) ) } { \sum _ { k \in \mathcal { N } ( v ) } \exp ( \mathrm { L e a k y R e L U } ( a ^ { T } [ W ^ { ( l ) } h _ { v } ^ { ( l ) } \| W ^ { ( l ) } h _ { k } ^ { ( l ) } ] ) ) } .
+$$
+
+Here, $\alpha$ is a learnable vector, and $\parallel$ denotes concatenation.
+
+# F.3. Impact of Average Node Degree of Dataset on HOA
+
+As shown in Table 1, the social network datasets, Facebook (Rozemberczki et al., 2021) and LastFM (Rozemberczki & Sarkar, 2020), exhibit a higher average node degree $( D e g . )$ compared to citation networks (Cora (Yang et al., 2016) and CiteSeer (Yang et al., 2016)). For Facebook and LastFM, as illustrated in Fig. 6, the accuracy leveled off after $K = 1$ , despite further increases in $K$ . This suggests that for social networks with higher average node degrees, the initial aggregation step provides a sufficient representation of node information. In contrast, for lower-degree citation networks Cora and CiteSeer, the accuracy continues to improve until $K = 6 4$ after $K = 1$ . This indicates that in low-degree datasets, HOA requires more steps to aggregate enough neighboring nodes for better accuracy.
+
+# F.4. Scalability on Heterophilic Graphs
+
+When applied to heterophilic graphs (e.g., Flickr (Huang et al., 2017) and Reddit (Hamilton et al., 2017); see Table 3 for detailed statistics), UPGNET continues to demonstrate superior utility compared to other baselines, as shown in Fig. 7. The HOA layer, by preserving Dirichlet energy and enabling personalized aggregation, effectively calibrates noise in heterophilic graphs, as evidenced by the experiments in Fig. 8.
+
+# F.5. Ablation Study on HOA layer
+
+Table 3. Statistics of heterophilic graph datasets.   
+
+<table><tr><td>DATASET</td><td>#CLASSES</td><td>#NODES</td><td>#EDGES</td><td>#FEATURES</td></tr><tr><td>FLICKR</td><td>7</td><td>89,250</td><td>899,756</td><td>500</td></tr><tr><td>REDDIT</td><td>41</td><td>232,965</td><td>114,615,892</td><td>602</td></tr></table>
+
+![](images/73a5dd2695b7f8ee0769d7845ae5c05828a5ce9e4ae0da72bb2a4fdbdfcd65fa.jpg)  
+Figure 7. Performance comparison of UPGNET and other baselines on Flickr and Reddit. X-axis represents $\epsilon$ and yaxis represents test accuracy $( \% )$ . UPGNET exhibits superior performance compared to other baselines.
+
+![](images/5039a8289c37571e54a0c96ebc56e5d9bb82b32104017d7c010749ae11303eb1.jpg)  
+Figure 8. Effect of HOA vs. SKA on graph learning performance across various steps $K \in \{ 2 , 4 , 8 , 1 6 , 3 2 , 6 4 \}$ . HOA demonstrates its superior denoising capability on heterophilic datasets (Flickr and Reddit).
+
+Fig. 9 presents an additional experiment demonstrating its superiority in private graph learning compared to residual connections (RC) (Liu et al., 2021). RC primarily addresses vanishing gradients but remains ineffective in suppressing LDP noise. In contrast, HOA mitigates oversmoothing by preserving Dirichlet energy, thereby effectively calibrating noise.
+
+# F.6. NFR vs. other Regularization Approaches
+
+As stated in Thm. 3, a key aspect of noise calibration in LDPGNN lies in reducing the effective feature dimensions. NFR employs L1 regularization, which directly facilitates feature selection and enhances fine-grained noise calibration. In contrast, Dropout (Baldi & Sadowski, 2013) and Group Lasso (Meier et al., 2008) fail to achieve precise feature selection tailored for noise calibration. Dropout introduces randomness by stochastically deactivating neurons, leading to instability in feature selection, while Group Lasso enforces sparsity at a predefined group level, requiring carefully designed group definitions that may not align with optimal noise calibration. These limitations reduce their effectiveness in mitigating noise impact. Empirical results (Table 4) further validate that NFR outperforms Dropout and Group Lasso in preserving learning utility.
+
+![](images/322b35f86590a50c8255a907c233c438007a9286fc5727d9450e186455ed1afb.jpg)  
+Figure 9. Comparison of HOA and Residual Connection (RC) (Dataset: Cora, $\epsilon = 0 . 0 1 $ ). HOA demonstrates significantly better classification accuracy in private graph learning compared to RC.
+
+Table 4. Comparison of our NFR (without HOA) with dropout and group Lasso $\epsilon = 0 . 0 1$ , GCN). The values in the table represent accuracy $( \% )$ . NFR demonstrates significantly better learning utility compared to other sparsity-inducing techniques.   
+
+<table><tr><td>BASELINE</td><td>CORA</td><td>CITESEER</td><td>LASTFM</td><td>FACEBOOK</td></tr><tr><td>DROPOUT</td><td>63.4</td><td>52.7</td><td>61.3</td><td>77.6</td></tr><tr><td>GROUP LASSO</td><td>57.6</td><td>50.5</td><td>57.1</td><td>78.9</td></tr><tr><td>OURS</td><td>71.3</td><td>57.2</td><td>67.1</td><td>84.9</td></tr></table>