Year: 2023

1 paper accepted to ICML

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Our paper on convergence of gradient flow on multi-layer linear networks [1] has been accepted to International Conference on Machine Learning! Congrats Hancheng!

[1] H. Min, R. Vidal, and E. Mallada, “On the Convergence of Gradient Flow on Multi-layer Linear Models,” in International Conference on Machine Learning (ICML), 2023, pp. 1-8.
[Bibtex] [Abstract] [Download PDF]

Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property. Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature

@inproceedings{mvm2023icml,
  abstract = {Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property.
Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature},
  author = {Min, Hancheng and Vidal, Rene and Mallada, Enrique},
  booktitle = {International Conference on Machine Learning (ICML)},
  grants = {TRIPODS-1934979, CAREER-1752362},
  month = {4},
  pages = {1-8},
  pubstate = {accepted, submitted Jan 2023},
  title = {On the Convergence of Gradient Flow on Multi-layer Linear Models},
  url = {https://mallada.ece.jhu.edu/pubs/2023-ICML-MVM.pdf},
  year = {2023}
}

Chengda defended his dissertation

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Chengda Ji, an ME Ph.D. student in our lab, defended his dissertation entitled “Analysis, Control, and Optimization of Networked Dynamical Systems under Uncertainty” on Friday, February 3rd. Congratulations!

1 paper accepted to AISTATS

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Our paper on convergence of gradient descent on overparametrized linear network [1] has been accepted to International Conference on Artificial Intelligence and Statistics!

[1] Z. Xu, H. Min, S. Tarmoun, E. Mallada, and R. Vidal, “Linear Convergence of Gradient Descent For Overparametrized Finite Width Two-Layer Linear Networks with General Initialization,” in International Conference on Artificial Intelligence and Statistics (AISTATS), 2023, pp. 2262-2284.
[Bibtex] [Abstract] [Download PDF]

Recent theoretical analyses of the convergence of gradient descent (GD) to a global minimum for over-parametrized neural networks make strong assumptions on the step size (infinitesimal), the hidden-layer width (infinite), or the initialization (spectral, balanced). In this work, we relax these assumptions and derive a linear convergence rate for two-layer linear networks trained using GD on the squared loss in the case of finite step size, finite width and general initialization. Despite the generality of our analysis, our rate estimates are significantly tighter than those of prior work. Moreover, we provide a time-varying step size rule that monotonically improves the convergence rate as the loss function decreases to zero. Numerical experiments validate our findings.

@inproceedings{xmtmv2023aistats,
  abstract = {Recent theoretical analyses of the convergence of gradient descent (GD) to a global minimum for over-parametrized neural networks make strong assumptions on the step size (infinitesimal), the hidden-layer width (infinite), or the initialization (spectral, balanced). In this work, we relax these assumptions and derive a linear convergence rate for two-layer linear networks trained using GD on the squared loss in the case of finite step size, finite width and general initialization. Despite the generality of our analysis, our rate estimates are significantly tighter than those of prior work. Moreover, we provide a time-varying step size rule that monotonically improves the convergence rate as the loss function decreases to zero. Numerical experiments validate our findings.},
  author = {Xu, Ziqing and Min, Hancheng and Tarmoun, Salma and Mallada, Enrique and Vidal, Rene},
  booktitle = {International Conference on Artificial Intelligence and Statistics (AISTATS)},
  editor = {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
  grants = {No Grant},
  month = {4},
  pages = {2262--2284},
  publisher = {PMLR},
  record = {published, accepted Jan 2023, submitted Oct 2022},
  series = {Proceedings of Machine Learning Research},
  title = {Linear Convergence of Gradient Descent For Overparametrized Finite Width Two-Layer Linear Networks with General Initialization},
  url = {https://mallada.ece.jhu.edu/pubs/2023-AISTATS-XMTMV.pdf},
  volume = {206},
  year = {2023}
}

2 papers accepted to ACC

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Our paper on spectral clustering for weakly-connected network systems [1] and our paper on frequency shaping control [2] have been accepted to American Control Conference 2023!

[1] [doi] H. Min and E. Mallada, “Spectral clustering and model reduction for weakly-connected coherent network systems,” in American Control Conference (ACC), 2023, pp. 2957-2962.
[Bibtex] [Abstract] [Download PDF]

We propose a novel model-reduction methodology for large-scale dynamic networks with tightly-connected components. First, the coherent groups are identified by a spectral clustering algorithm on the graph Laplacian matrix that models the network feedback. Then, a reduced network is built, where each node represents the aggregate dynamics of each coherent group, and the reduced network captures the dynamic coupling between the groups. Our approach is theoretically justified under a random graph setting. Finally, numerical experiments align with and validate our theoretical findings.

@inproceedings{mm2023acc,
  abstract = {We propose a novel model-reduction methodology
for large-scale dynamic networks with tightly-connected components.
First, the coherent groups are identified by a spectral
clustering algorithm on the graph Laplacian matrix that models
the network feedback. Then, a reduced network is built, where
each node represents the aggregate dynamics of each coherent
group, and the reduced network captures the dynamic coupling
between the groups. Our approach is theoretically justified
under a random graph setting. Finally, numerical experiments
align with and validate our theoretical findings.},
  author = {Min, Hancheng and Mallada, Enrique},
  booktitle = {American Control Conference (ACC)},
  doi = {10.23919/ACC55779.2023.10156212},
  grants = {CAREER-1752362, CPS-2136324},
  month = {6},
  pages = {2957-2962},
  record = {presented Jun 2023, accepted Jan 2023, submitted Sep 2022},
  title = {Spectral clustering and model reduction for weakly-connected coherent network systems},
  url = {https://mallada.ece.jhu.edu/pubs/2023-ACC-MM.pdf},
  year = {2023}
}
[2] B. K. Poolla, Y. Lin, A. Bernstein, E. Mallada, and D. Groß, “Frequency shaping control for weakly-coupled grid-forming IBRs,” in American Control Conference (ACC), 2023, pp. 1-6.
[Bibtex] [Abstract] [Download PDF]

We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models –based on frequency weighted balanced truncation– that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.

@inproceedings{plbmg2023acc,
  abstract = {We introduce a novel framework to approximate the aggregate frequency dynamics of coherent synchronous generators. By leveraging recent results on dynamics concentration of tightly connected networks, we develop a hierarchy of reduced order models --based on frequency weighted balanced truncation-- that accurately approximate the aggregate system response. Our results outperform existing aggregation techniques and can be shown to monotonically improve the approximation as the hierarchy order increases.},
  author = {Poolla, Bala Kameshwar and Lin, Yashen and Bernstein, Andrey and Mallada, Enrique and Groß, Dominic},
  booktitle = {American Control Conference (ACC)},
  grants = {CAREER-1752362, CPS-2136324},
  month = {1},
  pages = {1-6},
  record = {presented Jun 2023, accepted Jan 2023, submitted Sep 2022},
  title = {Frequency shaping control for weakly-coupled grid-forming IBRs},
  url = {https://mallada.ece.jhu.edu/pubs/2023-ACC-PLBMG.pdf},
  year = {2023}
}

1 paper accepted to TAC

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Our paper on safe reinforcement learning with almost sure constraints [1] has been accepted to Transactions on Automatic Control Special Issue on Learning for Control!

[1] [doi] A. Castellano, H. Min, J. Bazerque, and E. Mallada, “Learning to Act Safely with Limited Exposure and Almost Sure Certainty,” IEEE Transactions on Automatic Control, vol. 68, iss. 5, pp. 2979-2994, 2023.
[Bibtex] [Abstract] [Download PDF]

This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty. Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines. We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.

@article{cmbm2023tac,
  abstract = {This paper aims to put forward the concept that learning to take safe actions in unknown environments, even with probability one guarantees, can be achieved without the need for an unbounded number of exploratory trials, provided that one is willing to navigate trade-offs between optimality, level of exposure to unsafe events, and the maximum detection time of unsafe actions. We illustrate this concept in two complementary settings. We first focus on the canonical multi-armed bandit problem and seek to study the intrinsic trade-offs of learning safety in the presence of uncertainty.  Under mild assumptions on sufficient exploration, we provide an algorithm that provably detects all unsafe machines in an (expected) finite number of rounds. The analysis also unveils a trade-off between the number of rounds needed to secure the environment and the probability of discarding safe machines.  We then consider the problem of finding optimal policies for a Markov Decision Process (MDP) with almost sure constraints. 
We show that the (action) value function satisfies a barrier-based decomposition which allows for the identification of feasible policies independently of the reward process. Using this decomposition, we develop a Barrier-learning algorithm, that identifies such unsafe state-action pairs in a finite expected number of steps. Our analysis further highlights a trade-off between the time lag for the underlying MDP necessary to detect unsafe actions, and the level of exposure to unsafe events. Simulations corroborate our theoretical findings, further illustrating the aforementioned trade-offs, and suggesting that safety constraints can further speed up the learning process.},
  author = {Castellano, Agustin and Min, Hancheng and Bazerque, Juan and Mallada, Enrique},
  doi = {10.1109/TAC.2023.3240925},
  grants = {CAREER-1752362, TRIPODS-1934979, CPS-2136324},
  journal = {IEEE Transactions on Automatic Control},
  month = {5},
  number = {5},
  pages = {2979-2994},
  record = {published, online May 2023, accepted Jan 2023, revised Oct 2022, submitted May 2021},
  title = {Learning to Act Safely with Limited Exposure and Almost Sure Certainty},
  url = {https://mallada.ece.jhu.edu/pubs/2023-TAC-CMBM.pdf},
  volume = {68},
  year = {2023}
}