source_paper/latex/example_paper.bbl

\begin{thebibliography}{82}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
  \providecommand{\doi}[1]{doi: #1}\else
  \providecommand{\doi}{doi: \begingroup \urlstyle{rm}\Url}\fi

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\newblock Differentiable convex optimization layers.
\newblock \emph{Advances in neural information processing systems}, 32, 2019{\natexlab{a}}.

\bibitem[Agrawal et~al.(2019{\natexlab{b}})Agrawal, Barratt, Boyd, Busseti, and Moursi]{abb+19}
Agrawal, A., Barratt, S., Boyd, S., Busseti, E., and Moursi, W.~M.
\newblock Differentiating through a cone program.
\newblock \emph{Journal of Applied and Numerical Optimization}, 1\penalty0 (2):\penalty0 107--115, 2019{\natexlab{b}}.

\bibitem[Amos \& Kolter(2017)Amos and Kolter]{ak17}
Amos, B. and Kolter, J.~Z.
\newblock {OptNet}: Differentiable optimization as a layer in neural networks.
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Amos, B., Jimenez, I., Sacks, J., Boots, B., and Kolter, J.~Z.
\newblock Differentiable mpc for end-to-end planning and control.
\newblock \emph{Advances in neural information processing systems}, 31, 2018.

\bibitem[ApS(2025)]{mosek}
ApS, M.
\newblock \emph{The MOSEK Python Fusion API manual. Version 11.0.}, 2025.
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\bibitem[Arbel \& Mairal(2022)Arbel and Mairal]{am22}
Arbel, M. and Mairal, J.
\newblock Non-convex bilevel games with critical point selection maps.
\newblock \emph{Advances in Neural Information Processing Systems (NeurIPS)}, 35:\penalty0 8013--8026, 2022.

\bibitem[Arjevani et~al.(2023)Arjevani, Carmon, Duchi, Foster, Srebro, and Woodworth]{acd+23}
Arjevani, Y., Carmon, Y., Duchi, J.~C., Foster, D.~J., Srebro, N., and Woodworth, B.
\newblock Lower bounds for non-convex stochastic optimization.
\newblock \emph{Mathematical Programming}, 199\penalty0 (1):\penalty0 165--214, 2023.

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Bai, T., Luo, J., Zhao, J., Wen, B., and Wang, Q.
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\newblock \doi{10.24963/ijcai.2021/591}.
\newblock URL \url{https://doi.org/10.24963/ijcai.2021/591}.
\newblock Survey Track.

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Bambade, A., Schramm, F., Taylor, A., and Carpentier, J.
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\bibitem[Bertrand et~al.(2022)Bertrand, Klopfenstein, Massias, Blondel, Vaiter, Gramfort, and Salmon]{bkm+22}
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\newblock Implicit differentiation for fast hyperparameter selection in non-smooth convex learning.
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\bibitem[Bracken \& McGill(1973)Bracken and McGill]{bm73}
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\bibitem[Butler(2023)]{b23}
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\newblock On finding small hyper-gradients in bilevel optimization: Hardness results and improved analysis.
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Chu, T., Xu, D., Yao, W., and Zhang, J.
\newblock {SPABA}: A single-loop and probabilistic stochastic bilevel algorithm achieving optimal sample complexity.
\newblock In \emph{Forty-first International Conference on Machine Learning (ICML)}, 2024.
\newblock URL \url{https://openreview.net/forum?id=1YMjzz2g81}.

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\newblock Recent theoretical advances in non-convex optimization.
\newblock In \emph{High-Dimensional Optimization and Probability: With a View Towards Data Science}, pp.\  79--163. Springer, 2022.

\bibitem[Domke(2012)]{d12}
Domke, J.
\newblock Generic methods for optimization-based modeling.
\newblock In \emph{Artificial Intelligence and Statistics}, pp.\  318--326. PMLR, 2012.

\bibitem[Dontchev \& Rockafellar(2009)Dontchev and Rockafellar]{dr09}
Dontchev, A.~L. and Rockafellar, R.~T.
\newblock \emph{Implicit functions and solution mappings}, volume 543.
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Donti, P., Amos, B., and Kolter, J.~Z.
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Elsken, T., Metzen, J.~H., and Hutter, F.
\newblock Neural architecture search: A survey.
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\bibitem[Feurer \& Hutter(2019)Feurer and Hutter]{fh19}
Feurer, M. and Hutter, F.
\newblock Hyperparameter optimization.
\newblock \emph{Automated machine learning: Methods, systems, challenges}, pp.\  3--33, 2019.

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Finn, C., Abbeel, P., and Levine, S.
\newblock Model-agnostic meta-learning for fast adaptation of deep networks.
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Franceschi, L., Frasconi, P., Salzo, S., Grazzi, R., and Pontil, M.
\newblock Bilevel programming for hyperparameter optimization and meta-learning.
\newblock In \emph{International Conference on Machine Learning (ICML)}, pp.\  1568--1577. PMLR, 2018.

\bibitem[Ghadimi \& Wang(2018)Ghadimi and Wang]{gw18}
Ghadimi, S. and Wang, M.
\newblock Approximation methods for bilevel programming.
\newblock \emph{arXiv preprint arXiv:1802.02246}, 2018.

\bibitem[Goldstein(1977)]{g77}
Goldstein, A.~A.
\newblock Optimization of lipschitz continuous functions.
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\bibitem[{Gurobi Optimization, LLC}(2025)]{gurobi}
{Gurobi Optimization, LLC}.
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\newblock Differentiating through a quadratic cone program.
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Holmes, C., D{\"u}mbgen, F., and Barfoot, T.~D.
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\newblock \emph{arXiv preprint arXiv:2410.14315}, 2024.

\bibitem[Huang(2024)]{h24}
Huang, F.
\newblock Optimal hessian/jacobian-free nonconvex-pl bilevel optimization.
\newblock In \emph{Forty-first International Conference on Machine Learning (ICML)}, 2024.

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\newblock A differentiable programming system to bridge machine learning and scientific computing.
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Jiang, L., Xiao, Q., Tenorio, V.~M., Real-Rojas, F., Marques, A., and Chen, T.
\newblock A primal-dual-assisted penalty approach to bilevel optimization with coupled constraints.
\newblock In \emph{The Thirty-eighth Annual Conference on Neural Information Processing Systems (NeurIPS)}, 2024.
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\newblock A doubly stochastically perturbed algorithm for linearly constrained bilevel optimization.
\newblock \emph{arXiv preprint arXiv:2504.04545}, 2025.

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Kornowski, G., Padmanabhan, S., Wang, K., Zhang, Z., and Sra, S.
\newblock First-order methods for linearly constrained bilevel optimization.
\newblock In \emph{The Thirty-eighth Annual Conference on Neural Information Processing Systems (NeurIPS)}, 2024.

\bibitem[Krantz \& Parks(2002)Krantz and Parks]{sp02}
Krantz, S.~G. and Parks, H.~R.
\newblock \emph{The implicit function theorem: history, theory, and applications}.
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\bibitem[Kwon et~al.(2023)Kwon, Kwon, Wright, and Nowak]{kkw+23}
Kwon, J., Kwon, D., Wright, S., and Nowak, R.~D.
\newblock A fully first-order method for stochastic bilevel optimization.
\newblock In \emph{International Conference on Machine Learning (ICML)}, pp.\  18083--18113. PMLR, 2023.

\bibitem[Kwon et~al.(2024{\natexlab{a}})Kwon, Kwon, and Lyu]{kkl24}
Kwon, J., Kwon, D., and Lyu, H.
\newblock On the complexity of first-order methods in stochastic bilevel optimization.
\newblock In \emph{International Conference on Machine Learning (ICML)}, 2024{\natexlab{a}}.

\bibitem[Kwon et~al.(2024{\natexlab{b}})Kwon, Kwon, Wright, and Nowak]{kkw+24}
Kwon, J., Kwon, D., Wright, S., and Nowak, R.~D.
\newblock On penalty methods for nonconvex bilevel optimization and first-order stochastic approximation.
\newblock In \emph{International Conference on Learning Representations {(ICLR)}}, 2024{\natexlab{b}}.
\newblock URL \url{https://openreview.net/forum?id=CvYBvgEUK9}.

\bibitem[Lee et~al.(2019)Lee, Maji, Ravichandran, and Soatto]{lmr+19}
Lee, K., Maji, S., Ravichandran, A., and Soatto, S.
\newblock Meta-learning with differentiable convex optimization.
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\bibitem[Liu et~al.(2022)Liu, Ye, Wright, Stone, and Liu]{lyw+22}
Liu, B., Ye, M., Wright, S., Stone, P., and Liu, Q.
\newblock Bome! bilevel optimization made easy: A simple first-order approach.
\newblock \emph{Advances in neural information processing systems}, 35:\penalty0 17248--17262, 2022.

\bibitem[Liu et~al.(2021{\natexlab{a}})Liu, Liu, Yuan, Zeng, and Zhang]{lly+21}
Liu, R., Liu, X., Yuan, X., Zeng, S., and Zhang, J.
\newblock A value-function-based interior-point method for non-convex bi-level optimization.
\newblock In \emph{International Conference on Machine Learning (ICML)}, pp.\  6882--6892. PMLR, 2021{\natexlab{a}}.

\bibitem[Liu et~al.(2021{\natexlab{b}})Liu, Liu, Zeng, and Zhang]{llz+21}
Liu, R., Liu, Y., Zeng, S., and Zhang, J.
\newblock Towards gradient-based bilevel optimization with non-convex followers and beyond.
\newblock \emph{Advances in Neural Information Processing Systems (NeurIPS)}, 34:\penalty0 8662--8675, 2021{\natexlab{b}}.

\bibitem[Liu et~al.(2024)Liu, Liu, Yao, Zeng, and Zhang]{lly+24}
Liu, R., Liu, Z., Yao, W., Zeng, S., and Zhang, J.
\newblock Moreau envelope for nonconvex bi-level optimization: A single-loop and hessian-free solution strategy.
\newblock In \emph{Forty-first International Conference on Machine Learning}, 2024.
\newblock URL \url{https://openreview.net/forum?id=rZD9hV0Bc4}.

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Lorraine, J., Vicol, P., and Duvenaud, D.
\newblock Optimizing millions of hyperparameters by implicit differentiation.
\newblock In \emph{International conference on artificial intelligence and statistics}, pp.\  1540--1552. PMLR, 2020.

\bibitem[Maclaurin et~al.(2015{\natexlab{a}})Maclaurin, Duvenaud, and Adams]{dda15}
Maclaurin, D., Duvenaud, D., and Adams, R.
\newblock Gradient-based hyperparameter optimization through reversible learning.
\newblock In \emph{International conference on machine learning}, pp.\  2113--2122. PMLR, 2015{\natexlab{a}}.

\bibitem[Maclaurin et~al.(2015{\natexlab{b}})Maclaurin, Duvenaud, and Adams]{mda15}
Maclaurin, D., Duvenaud, D., and Adams, R.
\newblock Gradient-based hyperparameter optimization through reversible learning.
\newblock In \emph{International Conference on Machine Learning (ICML)}, pp.\  2113--2122. PMLR, 2015{\natexlab{b}}.

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Magoon, C.~W., Yang, F., Aigerman, N., and Kovalsky, S.~Z.
\newblock Differentiation through black-box quadratic programming solvers.
\newblock In \emph{The Thirty-ninth Annual Conference on Neural Information Processing Systems}, 2025.
\newblock URL \url{https://openreview.net/forum?id=DvwKWKG1Ul}.

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\newblock Decision-focused learning: Foundations, state of the art, benchmark and future opportunities.
\newblock \emph{Journal of Artificial Intelligence Research}, 80:\penalty0 1623--1701, 2024.

\bibitem[O'Donoghue et~al.(2016)O'Donoghue, Chu, Parikh, and Boyd]{ocpb16}
O'Donoghue, B., Chu, E., Parikh, N., and Boyd, S.
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\newblock \emph{Journal of Optimization Theory and Applications}, 169\penalty0 (3):\penalty0 1042--1068, June 2016.
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Pan, J., Ye, Z., Yang, X., Yang, X., Liu, W., Wang, L., and Bian, J.
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\newblock Automatic differentiation in pytorch.
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Paulus, A., Rol{\'\i}nek, M., Musil, V., Amos, B., and Martius, G.
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\newblock In \emph{International Conference on Machine Learning}, pp.\  8443--8453. PMLR, 2021.

\bibitem[Paulus et~al.(2024)Paulus, Martius, and Musil]{pmm24}
Paulus, A., Martius, G., and Musil, V.
\newblock {LPGD}: A general framework for backpropagation through embedded optimization layers.
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Petrulionyte, I., Mairal, J., and Arbel, M.
\newblock Functional bilevel optimization for machine learning.
\newblock In \emph{The Thirty-eighth Annual Conference on Neural Information Processing Systems (NeurIPS)}, 2024.

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\newblock Theseus: A library for differentiable nonlinear optimization.
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\newblock Meta-learning with implicit gradients.
\newblock \emph{Advances in neural information processing systems}, 32, 2019.

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Razaviyayn, M., Huang, T., Lu, S., Nouiehed, M., Sanjabi, M., and Hong, M.
\newblock Nonconvex min-max optimization: Applications, challenges, and recent theoretical advances.
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Ren, J., Feng, X., Liu, B., Pan, X., Fu, Y., Mai, L., and Yang, Y.
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\newblock A general and streamlined differentiable optimization framework.
\newblock \emph{arXiv preprint arXiv:2510.25986}, 2025.

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Saif, A., Cui, X., Shen, H., Lu, S., Kingsbury, B., and Chen, T.
\newblock Joint unsupervised and supervised training for automatic speech recognition via bilevel optimization.
\newblock In \emph{ICASSP 2024-2024 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)}, pp.\  10931--10935. IEEE, 2024.

\bibitem[Schaller \& Boyd(2025)Schaller and Boyd]{sb25}
Schaller, M. and Boyd, S.
\newblock Code generation for solving and differentiating through convex optimization problems.
\newblock \emph{arXiv preprint arXiv:2504.14099}, 2025.

\bibitem[Shen \& Chen(2023)Shen and Chen]{sc23}
Shen, H. and Chen, T.
\newblock On penalty-based bilevel gradient descent method.
\newblock In \emph{International Conference on Machine Learning (ICML)}, pp.\  30992--31015. PMLR, 2023.

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Shen, H., Yang, Z., and Chen, T.
\newblock Principled penalty-based methods for bilevel reinforcement learning and {RLHF}.
\newblock In \emph{Forty-first International Conference on Machine Learning (ICML)}, 2024.
\newblock URL \url{https://openreview.net/forum?id=Xb3IXEBYuw}.

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\newblock \emph{Mathematical Programming Computation}, 12\penalty0 (4):\penalty0 637--672, 2020.
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\bibitem[Stewart(1977)]{stewart1977perturbation}
Stewart, G.~W.
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