Distributed Associative Memory via Online Convex Optimization

Bowen Wang; Matteo Zecchin; Osvaldo Simeone

arxiv: 2509.22321 · v2 · submitted 2025-09-26 · 💻 cs.LG · eess.SP

Distributed Associative Memory via Online Convex Optimization

Bowen Wang , Matteo Zecchin , Osvaldo Simeone This is my paper

Pith reviewed 2026-05-18 13:18 UTC · model grok-4.3

classification 💻 cs.LG eess.SP

keywords associative memorydistributed optimizationonline gradient descentregret boundsrouting treesconvex optimizationmulti-agent learning

0 comments

The pith

Distributed online gradient descent lets agents maintain associative memories with sublinear regret by sharing updates over routing trees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that agents in a distributed network can each keep a local associative memory for recalling their own cues and responses as well as selective data from peers. This is done by having each agent run online gradient descent on convex losses and exchange necessary information along fixed routing trees connecting the network. A sympathetic reader would care because this provides a way to achieve collaborative memory without central coordination or full data sharing. The analysis proves that the total regret grows sublinearly with time, meaning average performance approaches the best possible fixed memory configuration.

Core claim

In this distributed setting, each agent optimizes its local associative memory model by performing online gradient descent updates, communicating the required messages over a spanning routing tree to incorporate information from other agents selectively. The key result is that this protocol guarantees sublinear regret with respect to the best hindsight associative memory, under the assumption of convex loss functions.

What carries the argument

Distributed online gradient descent protocol over routing trees that optimizes local associative memory parameters while propagating information selectively across the network.

If this is right

The protocol achieves sublinear regret in the number of time steps for the online optimization task.
Each local memory can recall both own and selected remote associations through the tree-based communication.
The method outperforms standard online optimization baselines in numerical experiments.
Performance holds for convex losses in networks with static routing trees.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach might be extended to handle dynamic network topologies by periodically rebuilding routing trees, though that would require new analysis.
The sublinear regret property could guide the design of communication-efficient protocols in other distributed learning tasks beyond associative memories.
If the convexity assumption is relaxed to allow non-convex losses common in neural networks, the guarantees may not apply and empirical behavior would need separate study.

Load-bearing premise

Loss functions are convex and the communication network relies on fixed routing trees without considering dynamic changes or failures.

What would settle it

Running experiments with non-convex loss functions or with routing trees that change over time and observing that regret becomes linear or that performance no longer beats baselines would disprove the main guarantees.

read the original abstract

An associative memory (AM) enables cue-response recall, and associative memorization has recently been noted to underlie the operation of modern neural architectures such as Transformers. This work addresses a distributed setting where agents maintain a local AM to recall their own associations as well as selective information from others. Specifically, we introduce a distributed online gradient descent method that optimizes local AMs at different agents through communication over routing trees. Our theoretical analysis establishes sublinear regret guarantees, and experiments demonstrate that the proposed protocol consistently outperforms existing online optimization baselines.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies online gradient descent to distributed associative memory over fixed routing trees and recovers standard sublinear regret under convexity.

read the letter

The main point is that this work takes online convex optimization and uses it for associative memory across agents that share information along routing trees. Each agent runs its own local updates while the tree structure aggregates the subgradients needed for the joint recall task. The regret bound then follows from the usual online gradient descent analysis once the communication protocol is in place. Experiments show the approach beats plain online baselines in the reported settings.

Referee Report

2 major / 2 minor

Summary. The paper proposes a distributed online gradient descent method for optimizing local associative memories maintained by agents in a network. Agents communicate aggregated subgradients over fixed routing trees to enable recall of local and selective remote associations. The central claims are sublinear regret guarantees derived from the online convex optimization framework and consistent outperformance over baseline online optimizers in experiments.

Significance. If the sublinear regret analysis holds under the stated assumptions, the work offers a theoretically grounded protocol for distributed associative memory that connects online convex optimization with network routing structures. This could inform decentralized learning systems where agents maintain cue-response memories. The experimental results provide supporting evidence of practical gains, and the use of standard OGD regret tools with routing trees is a clear strength.

major comments (2)

[§4 (Regret Analysis, Theorem 1)] §4 (Regret Analysis, Theorem 1): The sublinear regret bound is derived via standard online gradient descent analysis applied to aggregated losses. However, this relies on the assumption that routing trees are static and that each agent receives exact aggregated subgradients at every step. The manuscript provides no extension or robustness analysis for dynamic topology changes, link failures, or packet loss, which are load-bearing for the central claim that the protocol achieves sublinear regret in general networks.
[§3 (Method) and §5 (Experiments)] §3 (Method) and §5 (Experiments): The loss functions for associative memory updates are treated as convex to invoke OGD regret bounds. It is unclear whether the concrete AM loss (e.g., any non-convex embedding or attention-style loss) satisfies this, and the experiments do not include ablations on non-convex variants or dynamic networks. If either assumption fails, the regret guarantee reduces to linear in T, undermining the theoretical contribution.

minor comments (2)

[Abstract] Abstract: The claim of 'sublinear regret guarantees' should specify the rate (e.g., O(√T) or dependence on network diameter) for precision.
[§3] Notation: The definition of the local loss and how the routing tree aggregates subgradients could be clarified with an explicit equation or diagram in §3.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the scope of our assumptions. We respond to each major comment below and indicate planned revisions.

read point-by-point responses

Referee: [§4 (Regret Analysis, Theorem 1)] §4 (Regret Analysis, Theorem 1): The sublinear regret bound is derived via standard online gradient descent analysis applied to aggregated losses. However, this relies on the assumption that routing trees are static and that each agent receives exact aggregated subgradients at every step. The manuscript provides no extension or robustness analysis for dynamic topology changes, link failures, or packet loss, which are load-bearing for the central claim that the protocol achieves sublinear regret in general networks.

Authors: We agree that Theorem 1 relies on static routing trees and exact subgradient aggregation at each step; these are explicit modeling assumptions that enable the direct application of standard OGD regret analysis to the aggregated losses. The manuscript does not claim or analyze robustness to dynamic topologies, link failures, or packet loss. In the revision we will state these assumptions more prominently at the beginning of §4 and add a dedicated limitations paragraph discussing how the protocol could be adapted (e.g., by periodic tree recomputation or error-tolerant aggregation), while noting that a full regret analysis under such conditions lies outside the current scope. revision: partial
Referee: [§3 (Method) and §5 (Experiments)] §3 (Method) and §5 (Experiments): The loss functions for associative memory updates are treated as convex to invoke OGD regret bounds. It is unclear whether the concrete AM loss (e.g., any non-convex embedding or attention-style loss) satisfies this, and the experiments do not include ablations on non-convex variants or dynamic networks. If either assumption fails, the regret guarantee reduces to linear in T, undermining the theoretical contribution.

Authors: The associative-memory loss used in our distributed OGD updates is a convex surrogate (specifically, a squared or hinge loss on cue-response pairs) that satisfies the convexity requirement needed for the OGD regret bound. We will revise §3 to include the explicit loss expression and a short proof of its convexity. The experiments in §5 evaluate the protocol under this convex setting to match the theory; we will add a brief discussion acknowledging that non-convex losses (such as those arising in attention-style embeddings) fall outside the current guarantees and would generally yield only linear regret in the worst case. We will also note the absence of dynamic-network experiments as a limitation. revision: yes

Circularity Check

0 steps flagged

No circularity: regret analysis follows standard OCO results applied to distributed setting

full rationale

The paper proposes a distributed online gradient descent protocol for local associative memories over routing trees and derives sublinear regret bounds. These bounds are obtained by applying established online convex optimization regret analysis to the aggregated subgradients received at each agent. The derivation relies on the convexity of the loss functions and the fixed topology of the routing trees as explicit assumptions, without any reduction of the claimed guarantees to fitted parameters, self-definitions, or load-bearing self-citations. The central result is therefore self-contained against external OCO benchmarks and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard convexity assumptions from online convex optimization; no free parameters, invented entities, or ad-hoc axioms are identifiable from the abstract.

axioms (1)

domain assumption Objective functions are convex.
Required for online convex optimization regret bounds; invoked implicitly in the method description.

pith-pipeline@v0.9.0 · 5605 in / 1073 out tokens · 40124 ms · 2026-05-18T13:18:19.845907+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We adopt an online convex optimization (OCO) framework, making the assumptions that (i) The objective functions f_{m,t}(·) are convex ... (ii) The feasible set X is closed, convex, and bounded ... (iii) The gradients satisfy ∥∇f_{m,t}(X)∥_F ≤ L_m
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

DAM-TOGD applies OGD updates at each agent n based on delayed information received from the other agents ... over trees connecting each agent n to all relevant agents in W_n

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 3 internal anchors

[1]

This principle, fundamental to human cognition, provides a natural abstraction for modeling how information can be efficiently retained, updated, and retrieved

INTRODUCTION Anassociative memory(AM), a classical concept in cogni- tive science, stores cue–response associations, recalling the response when the corresponding cue is presented [1]. This principle, fundamental to human cognition, provides a natural abstraction for modeling how information can be efficiently retained, updated, and retrieved. Recent adva...

work page
[2]

PROBLEM SETTING We study distributed test-time memorization by generaliz- ing the setting in [3, 4, 5] toNagents, where each agent nprocesses a stream ofkeysk n,t ∈R dk (cues) andval- uesv n,t ∈R dv (responses). The goal is to optimize the AM mechanism at eachn-th agent in an online fashion, en- suring that at each timet, agentnnot only recalls its past d...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[3]

PRELIMINARIES In this section, we revisit some conventional methods to solve (2). 3.1. Full-Information DAM In the ideal full-information setting in which all agents have access to the complete set of loss functions{f m,t(·)}m∈N at each timet, the problem can be solved without inter-agent communication via OGD. Specifically, OGD yields the up- date [12, 1...

work page
[4]

PROPOSED METHOD In this section, we study the practical setting in which local agentnhas access only to its local lossf n,t(·)and the logical weight matrixWis not consistent with a consensus solution. 4.1. Protocol We introduce DAM-TOGD, a novel DAM protocol that ad- dresses the problem of minimizing the regret (2) for an arbi- trary matrixWof memorizatio...

work page
[5]

NUMERICAL EXPERIMENTS In this section, we present numerical experiments to illustrate and validate the performance of distributed DAM protocols. 5.1. Setting We consider memorization under the DeltaNet model [9], where memory retrieval is linear and the loss function for agentnat timetis given by the third entry in Table 1. The key vectorsk n,t are genera...

work page 2000
[6]

By leveraging tree-structured communication, DAM-TOGD enables agents to recall both local and cross-agent associa- tions while ensuring sublinear regret

CONCLUSION In this work, we have introduced the DAM problem and proposed DAM-TOGD, a tree-based OGD framework for optimizing associative memorization in distributed settings. By leveraging tree-structured communication, DAM-TOGD enables agents to recall both local and cross-agent associa- tions while ensuring sublinear regret. Numerical experiments confir...

work page
[7]

ACKNOWLEDGMENT This work was partially supported by the European Union’s Horizon Europe project CENTRIC (101096379), by the Open Fellowships of the EPSRC (EP/W024101/1), and by the EP- SRC project (EP/X011852/1)

work page
[8]

Modern methods in associative memory,

Dmitry Krotov, Benjamin Hoover, Parikshit Ram, and Bao Pham, “Modern methods in associative memory,” arXiv preprint arXiv:2507.06211, 2025

work page arXiv 2025
[9]

Dense associative memory for pattern recognition,

Dmitry Krotov and John J Hopfield, “Dense associative memory for pattern recognition,”Advances in neural information processing systems, vol. 29, 2016

work page 2016
[10]

It’s all connected: A journey through test-time memorization, attentional bias, retention, and online optimization.arXiv:2504.13173, 2025

Ali Behrouz, Meisam Razaviyayn, Peilin Zhong, and Vahab Mirrokni, “It’s all connected: A journey through test-time memorization, attentional bias, retention, and online optimization,”arXiv preprint arXiv:2504.13173, 2025

work page arXiv 2025
[11]

Test-time regression: a unifying framework for design- ing sequence models with associative memory,

Ke Alexander Wang, Jiaxin Shi, and Emily B Fox, “Test-time regression: a unifying framework for design- ing sequence models with associative memory,”arXiv preprint arXiv:2501.12352, 2025

work page arXiv 2025
[12]

Understanding transformer from the perspective of as- sociative memory,

Shu Zhong, Mingyu Xu, Tenglong Ao, and Guang Shi, “Understanding transformer from the perspective of as- sociative memory,”arXiv preprint arXiv:2505.19488, 2025

work page arXiv 2025
[13]

Attention is all you need,

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin, “Attention is all you need,”Ad- vances in neural information processing systems, vol. 30, 2017

work page 2017
[14]

Transformers are RNNs: Fast autoregressive transformers with linear atten- tion,

Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pap- pas, and Franc ¸ois Fleuret, “Transformers are RNNs: Fast autoregressive transformers with linear atten- tion,” inInternational conference on machine learning. PMLR, 2020, pp. 5156–5165

work page 2020
[15]

Gated Linear Attention Transformers with Hardware-Efficient Training

Songlin Yang, Bailin Wang, Yikang Shen, Rameswar Panda, and Yoon Kim, “Gated linear attention transformers with hardware-efficient training,”arXiv preprint arXiv:2312.06635, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023
[16]

Linear transformers are secretly fast weight program- mers,

Imanol Schlag, Kazuki Irie, and J ¨urgen Schmidhuber, “Linear transformers are secretly fast weight program- mers,” inInternational conference on machine learning. PMLR, 2021, pp. 9355–9366

work page 2021
[17]

Forgetting transformer: Softmax attention with a forget gate,

Zhixuan Lin, Evgenii Nikishin, Xu Owen He, and Aaron Courville, “Forgetting transformer: Softmax attention with a forget gate,”arXiv preprint arXiv:2503.02130, 2025

work page arXiv 2025
[18]

A Modern Introduction to Online Learning

Francesco Orabona, “A modern introduction to online learning,”arXiv preprint arXiv:1912.13213, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1912
[19]

Online learning and on- line convex optimization,

Shai Shalev-Shwartz et al., “Online learning and on- line convex optimization,”Foundations and Trends® in Machine Learning, vol. 4, no. 2, pp. 107–194, 2012

work page 2012
[20]

A very brief introduction to machine learning with applications to communication systems,

Osvaldo Simeone, “A very brief introduction to machine learning with applications to communication systems,” IEEE Transactions on Cognitive Communications and Networking, vol. 4, no. 4, pp. 648–664, 2018

work page 2018
[21]

Multitask learning over graphs: An approach for distributed, streaming machine learning,

Roula Nassif, Stefan Vlaski, C ´edric Richard, Jie Chen, and Ali H Sayed, “Multitask learning over graphs: An approach for distributed, streaming machine learning,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 14–25, 2020

work page 2020
[22]

Distributed autonomous online learn- ing: Regrets and intrinsic privacy-preserving proper- ties,

Feng Yan, Shreyas Sundaram, S.V .N. Vishwanathan, and Yuan Qi, “Distributed autonomous online learn- ing: Regrets and intrinsic privacy-preserving proper- ties,”IEEE Transactions on Knowledge and Data Engi- neering, vol. 25, no. 11, pp. 2483–2493, 2013

work page 2013
[23]

Steiner tree problems,

Frank K Hwang and Dana S Richards, “Steiner tree problems,”Networks, vol. 22, no. 1, pp. 55–89, 1992

work page 1992
[24]

A fast algorithm for steiner trees,

Lawrence Kou, George Markowsky, and Leonard Berman, “A fast algorithm for steiner trees,”Acta infor- matica, vol. 15, pp. 141–145, 1981

work page 1981
[25]

Distributed associative memory via online convex optimization (online supplementary materials),

Bowen Wang, Matteo Zecchin, and Osvaldo Sime- one, “Distributed associative memory via online convex optimization (online supplementary materials),” https://github.com/BW-Wang/DAM-TOGD, 2025, [Online; accessed 10-Sep-2025]

work page 2025

[1] [1]

This principle, fundamental to human cognition, provides a natural abstraction for modeling how information can be efficiently retained, updated, and retrieved

INTRODUCTION Anassociative memory(AM), a classical concept in cogni- tive science, stores cue–response associations, recalling the response when the corresponding cue is presented [1]. This principle, fundamental to human cognition, provides a natural abstraction for modeling how information can be efficiently retained, updated, and retrieved. Recent adva...

work page

[2] [2]

PROBLEM SETTING We study distributed test-time memorization by generaliz- ing the setting in [3, 4, 5] toNagents, where each agent nprocesses a stream ofkeysk n,t ∈R dk (cues) andval- uesv n,t ∈R dv (responses). The goal is to optimize the AM mechanism at eachn-th agent in an online fashion, en- suring that at each timet, agentnnot only recalls its past d...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[3] [3]

PRELIMINARIES In this section, we revisit some conventional methods to solve (2). 3.1. Full-Information DAM In the ideal full-information setting in which all agents have access to the complete set of loss functions{f m,t(·)}m∈N at each timet, the problem can be solved without inter-agent communication via OGD. Specifically, OGD yields the up- date [12, 1...

work page

[4] [4]

PROPOSED METHOD In this section, we study the practical setting in which local agentnhas access only to its local lossf n,t(·)and the logical weight matrixWis not consistent with a consensus solution. 4.1. Protocol We introduce DAM-TOGD, a novel DAM protocol that ad- dresses the problem of minimizing the regret (2) for an arbi- trary matrixWof memorizatio...

work page

[5] [5]

NUMERICAL EXPERIMENTS In this section, we present numerical experiments to illustrate and validate the performance of distributed DAM protocols. 5.1. Setting We consider memorization under the DeltaNet model [9], where memory retrieval is linear and the loss function for agentnat timetis given by the third entry in Table 1. The key vectorsk n,t are genera...

work page 2000

[6] [6]

By leveraging tree-structured communication, DAM-TOGD enables agents to recall both local and cross-agent associa- tions while ensuring sublinear regret

CONCLUSION In this work, we have introduced the DAM problem and proposed DAM-TOGD, a tree-based OGD framework for optimizing associative memorization in distributed settings. By leveraging tree-structured communication, DAM-TOGD enables agents to recall both local and cross-agent associa- tions while ensuring sublinear regret. Numerical experiments confir...

work page

[7] [7]

ACKNOWLEDGMENT This work was partially supported by the European Union’s Horizon Europe project CENTRIC (101096379), by the Open Fellowships of the EPSRC (EP/W024101/1), and by the EP- SRC project (EP/X011852/1)

work page

[8] [8]

Modern methods in associative memory,

Dmitry Krotov, Benjamin Hoover, Parikshit Ram, and Bao Pham, “Modern methods in associative memory,” arXiv preprint arXiv:2507.06211, 2025

work page arXiv 2025

[9] [9]

Dense associative memory for pattern recognition,

Dmitry Krotov and John J Hopfield, “Dense associative memory for pattern recognition,”Advances in neural information processing systems, vol. 29, 2016

work page 2016

[10] [10]

It’s all connected: A journey through test-time memorization, attentional bias, retention, and online optimization.arXiv:2504.13173, 2025

Ali Behrouz, Meisam Razaviyayn, Peilin Zhong, and Vahab Mirrokni, “It’s all connected: A journey through test-time memorization, attentional bias, retention, and online optimization,”arXiv preprint arXiv:2504.13173, 2025

work page arXiv 2025

[11] [11]

Test-time regression: a unifying framework for design- ing sequence models with associative memory,

Ke Alexander Wang, Jiaxin Shi, and Emily B Fox, “Test-time regression: a unifying framework for design- ing sequence models with associative memory,”arXiv preprint arXiv:2501.12352, 2025

work page arXiv 2025

[12] [12]

Understanding transformer from the perspective of as- sociative memory,

Shu Zhong, Mingyu Xu, Tenglong Ao, and Guang Shi, “Understanding transformer from the perspective of as- sociative memory,”arXiv preprint arXiv:2505.19488, 2025

work page arXiv 2025

[13] [13]

Attention is all you need,

Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin, “Attention is all you need,”Ad- vances in neural information processing systems, vol. 30, 2017

work page 2017

[14] [14]

Transformers are RNNs: Fast autoregressive transformers with linear atten- tion,

Angelos Katharopoulos, Apoorv Vyas, Nikolaos Pap- pas, and Franc ¸ois Fleuret, “Transformers are RNNs: Fast autoregressive transformers with linear atten- tion,” inInternational conference on machine learning. PMLR, 2020, pp. 5156–5165

work page 2020

[15] [15]

Gated Linear Attention Transformers with Hardware-Efficient Training

Songlin Yang, Bailin Wang, Yikang Shen, Rameswar Panda, and Yoon Kim, “Gated linear attention transformers with hardware-efficient training,”arXiv preprint arXiv:2312.06635, 2023

work page internal anchor Pith review Pith/arXiv arXiv 2023

[16] [16]

Linear transformers are secretly fast weight program- mers,

Imanol Schlag, Kazuki Irie, and J ¨urgen Schmidhuber, “Linear transformers are secretly fast weight program- mers,” inInternational conference on machine learning. PMLR, 2021, pp. 9355–9366

work page 2021

[17] [17]

Forgetting transformer: Softmax attention with a forget gate,

Zhixuan Lin, Evgenii Nikishin, Xu Owen He, and Aaron Courville, “Forgetting transformer: Softmax attention with a forget gate,”arXiv preprint arXiv:2503.02130, 2025

work page arXiv 2025

[18] [18]

A Modern Introduction to Online Learning

Francesco Orabona, “A modern introduction to online learning,”arXiv preprint arXiv:1912.13213, 2019

work page internal anchor Pith review Pith/arXiv arXiv 1912

[19] [19]

Online learning and on- line convex optimization,

Shai Shalev-Shwartz et al., “Online learning and on- line convex optimization,”Foundations and Trends® in Machine Learning, vol. 4, no. 2, pp. 107–194, 2012

work page 2012

[20] [20]

A very brief introduction to machine learning with applications to communication systems,

Osvaldo Simeone, “A very brief introduction to machine learning with applications to communication systems,” IEEE Transactions on Cognitive Communications and Networking, vol. 4, no. 4, pp. 648–664, 2018

work page 2018

[21] [21]

Multitask learning over graphs: An approach for distributed, streaming machine learning,

Roula Nassif, Stefan Vlaski, C ´edric Richard, Jie Chen, and Ali H Sayed, “Multitask learning over graphs: An approach for distributed, streaming machine learning,” IEEE Signal Processing Magazine, vol. 37, no. 3, pp. 14–25, 2020

work page 2020

[22] [22]

Distributed autonomous online learn- ing: Regrets and intrinsic privacy-preserving proper- ties,

Feng Yan, Shreyas Sundaram, S.V .N. Vishwanathan, and Yuan Qi, “Distributed autonomous online learn- ing: Regrets and intrinsic privacy-preserving proper- ties,”IEEE Transactions on Knowledge and Data Engi- neering, vol. 25, no. 11, pp. 2483–2493, 2013

work page 2013

[23] [23]

Steiner tree problems,

Frank K Hwang and Dana S Richards, “Steiner tree problems,”Networks, vol. 22, no. 1, pp. 55–89, 1992

work page 1992

[24] [24]

A fast algorithm for steiner trees,

Lawrence Kou, George Markowsky, and Leonard Berman, “A fast algorithm for steiner trees,”Acta infor- matica, vol. 15, pp. 141–145, 1981

work page 1981

[25] [25]

Distributed associative memory via online convex optimization (online supplementary materials),

Bowen Wang, Matteo Zecchin, and Osvaldo Sime- one, “Distributed associative memory via online convex optimization (online supplementary materials),” https://github.com/BW-Wang/DAM-TOGD, 2025, [Online; accessed 10-Sep-2025]

work page 2025