pith. sign in

arxiv: 2606.01557 · v1 · pith:4QSUKCYZnew · submitted 2026-06-01 · 💻 cs.LG · eess.SP

Everywhere Learning: Artificial Intelligence with Pointwise Constraints

Ignacio Boero , Ignacio Hounie , Luiz Chamon , Alejandro Ribeiro This is my paper

Pith reviewed 2026-06-28 15:32 UTC · model grok-4.3

classification 💻 cs.LG eess.SP
keywords everywhere learningpointwise constraintsduality theorygeneralization analysisreweightingAI trainingconstraint satisfactionlanguage models
0
0 comments X

The pith

Everywhere learning trains AI to satisfy loss constraints with probability one over the data distribution rather than minimizing average loss.

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

The paper introduces everywhere learning as a paradigm in which AI systems must satisfy loss constraints almost surely. It develops an approximate duality theory to prove that solutions to the empirical everywhere learning problem stay close to solutions of the corresponding statistical problem. Dual variables in this theory reweight the data distribution toward points where constraints are hardest to meet, and the mismatch in concentration of mass between the original distribution and these hard points governs generalization error. A sparse L1 penalty on constraint relaxations provides an additional mechanism to control that error. The framework is demonstrated on an agentic classification task involving language models.

Core claim

We develop an approximate duality theory to substantiate a generalization analysis that establishes the proximity between solutions of empirical and statistical everywhere learning problems. Our results show that dual variables reweigh the data distribution towards points in which loss constraints are more difficult to satisfy and that generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy. We further show that we can control generalization with a sparse L1 penalty on constraint relaxations.

What carries the argument

Approximate duality theory whose dual variables reweight the data distribution toward points where loss constraints are hardest to satisfy.

If this is right

  • Dual variables reweigh the data distribution toward points where loss constraints are more difficult to satisfy.
  • Generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration on difficult points.
  • A sparse L1 penalty on constraint relaxations controls generalization.
  • Empirical and statistical everywhere learning solutions remain proximate under the duality theory.

Where Pith is reading between the lines

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

  • The reweighting view may connect everywhere learning to existing robust optimization methods that emphasize tail behavior.
  • The L1 penalty mechanism could be tested on sequential decision tasks where constraint violations accumulate over time.
  • Extending the duality approximation to non-convex losses would clarify whether the proximity result survives in modern deep models.
  • The concentration-mismatch view suggests that everywhere learning may be easier to apply when data distributions are already somewhat uniform.

Load-bearing premise

The approximate duality theory holds and produces meaningful reweighting that controls generalization via concentration mismatch.

What would settle it

A numerical experiment in which the empirical everywhere learning solution diverges substantially from the statistical solution even after applying the duality-based reweighting.

Figures

Figures reproduced from arXiv: 2606.01557 by Alejandro Ribeiro, Ignacio Boero, Ignacio Hounie, Luiz Chamon.

Figure 1
Figure 1. Figure 1: Fitting polynomials to data generated by adding uniform noise to a sine wave. The relaxed fit corresponds to clipping [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the relationship between the overlap of [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Empirical cumulative distribution function (CDF) of the sample-wise constraint slacks [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Number of samples, dual variables at the end of training, test loss and accuracy in the coding-AF task from FloraBench [36], across both data [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Everywhere learning is a new paradigm whereby Artificial Intelligence (AI) systems are trained to satisfy loss constraints with probability one over the data distribution. This is in contrast to the standard paradigm of training AI systems to minimize average losses. We develop an approximate duality theory to substantiate a generalization analysis that establishes the proximity between solutions of empirical and statistical everywhere learning problems. Our results show that dual variables reweigh the data distribution towards points in which loss constraints are more difficult to satisfy and that generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy. We further show that we can control generalization with a sparse L1 penalty on constraint relaxations. We illustrate the merits of everywhere learning with an experiment in agentic classification for language model tasks.

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.

Referee Report

2 major / 1 minor

Summary. The paper introduces 'everywhere learning,' a paradigm in which AI systems are trained to satisfy loss constraints with probability one over the data distribution, in contrast to minimizing average losses. It develops an approximate duality theory to support a generalization analysis establishing proximity between solutions of empirical and statistical everywhere learning problems. Dual variables are claimed to reweight the data distribution toward points where constraints are harder to satisfy, with generalization controlled by the mismatch in concentration of mass between the data distribution and difficult points; a sparse L1 penalty on constraint relaxations is proposed to control generalization. The approach is illustrated via an experiment on agentic classification for language model tasks.

Significance. If the approximate duality theory holds with valid approximation conditions and produces non-vacuous generalization bounds, the framework could provide a distinct approach to enforcing strict pointwise constraints in machine learning, potentially benefiting robustness in sequential decision-making tasks such as language model agents. The reweighting interpretation and L1 penalty mechanism would offer interpretable levers for generalization that differ from standard empirical risk minimization.

major comments (2)
  1. [Abstract (duality theory claim)] The approximate duality theory is presented as the foundation for the generalization analysis that establishes proximity between empirical and statistical everywhere learning problems, yet no derivation, approximation error bounds, or conditions for validity are supplied; this is load-bearing for all subsequent claims about dual-variable reweighting and concentration mismatch.
  2. [Abstract (generalization analysis)] The statement that 'generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy' depends on the un-derived duality theory; without explicit conditions or a proof sketch, it is impossible to determine whether the bound is meaningful or reduces to a tautology.
minor comments (1)
  1. [Abstract (experiment)] The abstract references an experiment in agentic classification but supplies no details on task definition, baselines, metrics, or quantitative results, which would be needed to evaluate whether the merits are demonstrated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the centrality of the approximate duality theory. The comments correctly identify that the current manuscript does not supply the requested derivation, bounds, or validity conditions. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract (duality theory claim)] The approximate duality theory is presented as the foundation for the generalization analysis that establishes proximity between empirical and statistical everywhere learning problems, yet no derivation, approximation error bounds, or conditions for validity are supplied; this is load-bearing for all subsequent claims about dual-variable reweighting and concentration mismatch.

    Authors: We agree that the abstract states the existence of an approximate duality theory without supplying its derivation, error bounds, or validity conditions. This omission weakens the foundation for the subsequent claims. In the revised version we will (i) add a concise statement of the approximation conditions and error bounds to the abstract and (ii) include an explicit derivation together with the required bounds in the main text. revision: yes

  2. Referee: [Abstract (generalization analysis)] The statement that 'generalization is controlled by the mismatch between the concentration of mass of the data distribution and the concentration of mass on points where constraints are more difficult to satisfy' depends on the un-derived duality theory; without explicit conditions or a proof sketch, it is impossible to determine whether the bound is meaningful or reduces to a tautology.

    Authors: We concur that the generalization claim rests on the un-derived duality result and that, absent conditions or a sketch, its status cannot be assessed. The revision will incorporate the missing derivation and a proof sketch of the generalization bound, together with the explicit conditions under which the bound is non-vacuous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against external benchmarks

full rationale

The abstract outlines an approximate duality theory supporting generalization bounds between empirical and statistical everywhere learning problems, with dual variables reweighting data toward difficult constraints and control via concentration mismatch or L1 penalty. No equations, self-citations, or fitted inputs are presented that reduce the claimed results to definitions or prior author work by construction. The provided text contains no load-bearing steps matching the enumerated circularity patterns, and the central claims rest on the development of the duality theory rather than renaming or self-referential fitting. Absent explicit derivations in the supplied abstract, the analysis is treated as independent.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5668 in / 970 out tokens · 17265 ms · 2026-06-28T15:32:54.641541+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

47 extracted references · 6 canonical work pages

  1. [1]

    Cam- bridge university press, 2014

    Shai Shalev-Shwartz and Shai Ben-David.Understand- ing machine learning: From theory to algorithms. Cam- bridge university press, 2014

    2014

  2. [2]

    Springer Berlin Hei- delberg, Berlin, Heidelberg, 2008

    Theodor Stewart, Oliver Bandte, Heinrich Braun, Niru- pam Chakraborti, Matthias Ehrgott, Mathias G ¨obelt, Yaochu Jin, Hirotaka Nakayama, Silvia Poles, and Danilo Di Stefano.Real-World Applications of Multiobjective Optimization, pages 285–327. Springer Berlin Hei- delberg, Berlin, Heidelberg, 2008. ISBN 978-3-540- 88908-3. doi: 10.1007/978-3-540-88908-3 ...

    work page doi:10.1007/978-3-540-88908-3 2008

  3. [3]

    Zamir, Alexander Sax, Teresa Yeo, Oguzhan Fatih Kar, Nikhil Cheerla, Rohan Suri, Zhangjie Cao, Jitendra Malik, and Leonidas J

    Amir R. Zamir, Alexander Sax, Teresa Yeo, Oguzhan Fatih Kar, Nikhil Cheerla, Rohan Suri, Zhangjie Cao, Jitendra Malik, and Leonidas J. Guibas. Robust learning through cross-task consistency.CoRR, abs/2006.04096, 2020. URL https://arxiv.org/abs/2006.04096

    arXiv 2006

  4. [4]

    A pareto-efficient algorithm for multiple objective optimization in e-commerce recommendation

    Xiao Lin, Hongjie Chen, Changhua Pei, Fei Sun, Xuanji Xiao, Hanxiao Sun, Yongfeng Zhang, Wenwu Ou, and Peng Jiang. A pareto-efficient algorithm for multiple objective optimization in e-commerce recommendation. InProceedings of the 13th ACM Conference on Recom- mender Systems, RecSys ’19, page 20–28, New York, NY , USA, 2019. Association for Computing Mach...

    work page doi:10.1145/3298689.3346998 2019

  5. [5]

    Multi-task learning as multi-objective optimization.Advances in neural information processing systems, 31, 2018

    Ozan Sener and Vladlen Koltun. Multi-task learning as multi-objective optimization.Advances in neural information processing systems, 31, 2018

    2018

  6. [6]

    A survey on multi-task learning.IEEE transactions on knowledge and data engineering, 34(12):5586–5609, 2021

    Yu Zhang and Qiang Yang. A survey on multi-task learning.IEEE transactions on knowledge and data engineering, 34(12):5586–5609, 2021

    2021

  7. [7]

    Agnostic learning with multiple objectives

    Corinna Cortes, Mehryar Mohri, Javier Gonzalvo, and Dmitry Storcheus. Agnostic learning with multiple objectives. InAdvances in Neural Information Processing Systems, volume 33, pages 20485–20495. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper files/paper/2020/ file/ebea2325dc670423afe9a1f4d9d1aef5-Paper.pdf

    2020

  8. [8]

    Deist, Monika Grewal, Frank J

    Timo M. Deist, Monika Grewal, Frank J. W. M. Dankers, Tanja Alderliesten, and Peter A. N. Bosman. Multi- objective learning to predict pareto fronts using hypervol- ume maximization.CoRR, abs/2102.04523, 2021. URL https://arxiv.org/abs/2102.04523

    arXiv 2021

  9. [9]

    Multi-task learning with deep neural networks: A survey.CoRR, abs/2009.09796, 2020

    Michael Crawshaw. Multi-task learning with deep neural networks: A survey.CoRR, abs/2009.09796, 2020. URL https://arxiv.org/abs/2009.09796

    arXiv 2009

  10. [10]

    Alignment of large language models with constrained learning, 2025

    Botong Zhang, Shuo Li, Ignacio Hounie, Osbert Bastani, Dongsheng Ding, and Alejandro Ribeiro. Alignment of large language models with constrained learning, 2025. URL https://arxiv.org/abs/2505.19387

    arXiv 2025

  11. [11]

    Learning with complex loss functions and constraints

    Harikrishna Narasimhan. Learning with complex loss functions and constraints. InProceedings of the Twenty-First International Conference on Artificial In- telligence and Statistics, volume 84 ofProceedings of Machine Learning Research, pages 1646–1654. PMLR, 09–11 Apr 2018. URL https://proceedings.mlr.press/v84/ narasimhan18a.html

    2018

  12. [12]

    Composition and alignment of diffusion models using constrained learning

    Shervin Khalafi, Ignacio Hounie, Dongsheng Ding, and Alejandro Ribeiro. Composition and alignment of diffusion models using constrained learning. InAdvances in Neural Information Processing Systems, volume 38, pages 18629– 18676. Curran Associates, Inc., 2025. URL https: //proceedings.neurips.cc/paper files/paper/2025/file/ 1af991de2d4c4e679bcc5d9e23ac6ba...

    2025

  13. [13]

    Constrained learning with non-convex losses.IEEE Transactions on Infor- mation Theory, 69(3):1739–1760, 2022

    Luiz FO Chamon, Santiago Paternain, Miguel Calvo- Fullana, and Alejandro Ribeiro. Constrained learning with non-convex losses.IEEE Transactions on Infor- mation Theory, 69(3):1739–1760, 2022

    2022

  14. [14]

    Juan Elenter, Luiz F. O. Chamon, and Alejandro Ribeiro. Near-optimal solutions of constrained learning problems,

  15. [15]

    URL https://arxiv.org/abs/2403.11844

    arXiv

  16. [16]

    Springer: New York, 1999

    Vladimir Vapnik.The Nature of Statistical Learning Theory. Springer: New York, 1999

    1999

  17. [17]

    Learning optimal power flow with pointwise constraints, 2025

    Damian Owerko, Anna Scaglione, and Alejandro Ribeiro. Learning optimal power flow with pointwise constraints, 2025. URL https://arxiv.org/abs/2510.20777

    arXiv 2025

  18. [18]

    Feasible learning,

    Juan Ramirez, Ignacio Hounie, Juan Elenter, Jose Gallego-Posada, Meraj Hashemizadeh, Alejandro Ribeiro, and Simon Lacoste-Julien. Feasible learning,

  19. [19]

    URL https://arxiv.org/abs/2501.14912

    arXiv

  20. [20]

    Luiz F. O. Chamon and Alejandro Ribeiro. Probably approximately correct constrained learning, 2021. URL https://arxiv.org/abs/2006.05487

    arXiv 2021

  21. [21]

    Ben-Tal, L.E

    A. Ben-Tal, L.E. Ghaoui, and A. Nemirovski.Robust Optimization. Princeton Series in Applied Mathematics. Princeton University Press, 2009. ISBN 9781400831050. URL https://books.google.com/books?id=DttjR7IpjUEC

    2009

  22. [22]

    Brown, and Constantine Caramanis

    Dimitris Bertsimas, David B. Brown, and Constantine Caramanis. Theory and applications of robust optimiza- tion, 2010. URL https://arxiv.org/abs/1010.5445

    Pith/arXiv arXiv 2010

  23. [23]

    Robust opti- mization using machine learning for uncertainty sets,

    Theja Tulabandhula and Cynthia Rudin. Robust opti- mization using machine learning for uncertainty sets,

  24. [24]

    URL https://arxiv.org/abs/1407.1097

    Pith/arXiv arXiv

  25. [25]

    Chance- constrained programming.Management science, 6(1): 73–79, 1959

    Abraham Charnes and William W Cooper. Chance- constrained programming.Management science, 6(1): 73–79, 1959

    1959

  26. [26]

    Society for Indus- trial and Applied Mathematics, Philadelphia, PA, 2021

    Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczynski.Lectures on Stochastic Programming: Modeling and Theory, Third Edition. Society for Indus- trial and Applied Mathematics, Philadelphia, PA, 2021. doi: 10.1137/1.9781611976595. URL https://epubs.siam. org/doi/abs/10.1137/1.9781611976595

    work page doi:10.1137/1.9781611976595 2021

  27. [27]

    Strong duality in risk-constrained nonconvex functional pro- gramming.arXiv preprint arXiv:2206.11948, 2022

    Dionysis Kalogerias and Spyridon Pougkakiotis. Strong duality in risk-constrained nonconvex functional pro- gramming.arXiv preprint arXiv:2206.11948, 2022

    arXiv 2022

  28. [28]

    Golub and Charles F

    Marco C. Campi and Simone Garatti.Introduction to the Scenario Approach. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2018. doi: 10.1137/1. 9781611975444. URL https://epubs.siam.org/doi/abs/10. 1137/1.9781611975444

    work page doi:10.1137/1 2018

  29. [29]

    Springer London, London, 2006

    Arkadi Nemirovski and Alexander Shapiro.Scenario Approximations of Chance Constraints, pages 3–47. Springer London, London, 2006. ISBN 978-1-84628- 095-5. doi: 10.1007/1-84628-095-8 1. URL https: //doi.org/10.1007/1-84628-095-8 1

    work page doi:10.1007/1-84628-095-8 2006

  30. [30]

    Henderson

    Sujin Kim, Raghu Pasupathy, and Shane G. Henderson. A guide to sample average approximation. 2015. URL https://api.semanticscholar.org/CorpusID:17021796

    2015

  31. [31]

    A sample ap- proximation approach for optimization with probabilistic constraints.SIAM Journal on Optimization, 19(2):674– 699, 2008

    James Luedtke and Shabbir Ahmed. A sample ap- proximation approach for optimization with probabilistic constraints.SIAM Journal on Optimization, 19(2):674– 699, 2008. doi: 10.1137/070702928. URL https://doi. org/10.1137/070702928

    work page doi:10.1137/070702928 2008

  32. [32]

    Generalization bounds for stochastic saddle point problems, 2020

    Junyu Zhang, Mingyi Hong, Mengdi Wang, and Shuzhong Zhang. Generalization bounds for stochastic saddle point problems, 2020. URL https://arxiv.org/abs/ 2006.02067

    arXiv 2020

  33. [33]

    What is a good metric to study gen- eralization of minimax learners?, 2022

    Asuman Ozdaglar, Sarath Pattathil, Jiawei Zhang, and Kaiqing Zhang. What is a good metric to study gen- eralization of minimax learners?, 2022. URL https: //arxiv.org/abs/2206.04502

    arXiv 2022

  34. [34]

    Train simulta- neously, generalize better: Stability of gradient-based minimax learners, 2020

    Farzan Farnia and Asuman Ozdaglar. Train simulta- neously, generalize better: Stability of gradient-based minimax learners, 2020. URL https://arxiv.org/abs/2010. 12561

    2020

  35. [35]

    Why l1 is a good approximation to l0: A geometric JOURNAL OF LATEX CLASS FILES, VOL

    Carlos Ramirez, Vladik Kreinovich, and Miguel Argaez. Why l1 is a good approximation to l0: A geometric JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 13 explanation. 2013

    2020

  36. [36]

    Functional nonlinear sparse models.IEEE Transactions on Signal Processing, 68:2449–2463, 2020

    Luiz FO Chamon, Yonina C Eldar, and Alejandro Ribeiro. Functional nonlinear sparse models.IEEE Transactions on Signal Processing, 68:2449–2463, 2020

    2020

  37. [37]

    Boyd and L

    S.P. Boyd and L. Vandenberghe.Convex Opti- mization. Number pt. 1 in Berichte ¨uber verteilte messysteme. Cambridge University Press, 2004. ISBN 9780521833783. URL https://books.google.com/books? id=mYm0bLd3fcoC

    2004

  38. [38]

    Multilayer feedforward networks are universal approxi- mators.Neural networks, 2(5):359–366, 1989

    Kurt Hornik, Maxwell Stinchcombe, and Halbert White. Multilayer feedforward networks are universal approxi- mators.Neural networks, 2(5):359–366, 1989

    1989

  39. [39]

    Gnns as predictors of agentic workflow performances.arXiv preprint arXiv:2503.11301, 2025

    Yuanshuo Zhang, Yuchen Hou, Bohan Tang, Shuo Chen, Muhan Zhang, Xiaowen Dong, and Siheng Chen. Gnns as predictors of agentic workflow performances.arXiv preprint arXiv:2503.11301, 2025

    arXiv 2025

  40. [40]

    Evaluating large language models trained on code

    Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde De Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas Joseph, Greg Brockman, et al. Evaluating large language models trained on code. arXiv preprint arXiv:2107.03374, 2021

    Pith/arXiv arXiv 2021

  41. [41]

    Program synthesis with large language models.arXiv preprint arXiv:2108.07732, 2021

    Jacob Austin, Augustus Odena, Maxwell Nye, Maarten Bosma, Henryk Michalewski, David Dohan, Ellen Jiang, Carrie Cai, Michael Terry, Quoc Le, et al. Program synthesis with large language models.arXiv preprint arXiv:2108.07732, 2021

    Pith/arXiv arXiv 2021

  42. [42]

    Aflow: Au- tomating agentic workflow generation.arXiv preprint arXiv:2410.10762, 2024

    Jiayi Zhang, Jinyu Xiang, Zhaoyang Yu, Fengwei Teng, Xionghui Chen, Jiaqi Chen, Mingchen Zhuge, Xin Cheng, Sirui Hong, Jinlin Wang, et al. Aflow: Au- tomating agentic workflow generation.arXiv preprint arXiv:2410.10762, 2024

    Pith/arXiv arXiv 2024

  43. [43]

    Springer, 1991

    Michel Ledoux and Michel Talagrand.Probability in Banach Spaces: isoperimetry and processes, volume 23. Springer, 1991

    1991

  44. [44]

    Springer Science & Business Media, 3rd edition, 2006

    Charalambos D Aliprantis and Kim C Border.Infinite Dimensional Analysis: A Hitchhiker’s Guide. Springer Science & Business Media, 3rd edition, 2006

    2006

  45. [45]

    Bertsekas.Convex Optimization Theory

    D. Bertsekas.Convex Optimization Theory. Athena Scientific optimization and computation series. Athena Scientific, 2009. ISBN 9781886529311. URL https: //books.google.com/books?id=lC1EEAAAQBAJ. JOURNAL OF LATEX CLASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 14 APPENDIXA RADAMACHERCOMPLEXITY ANDUCRATE DERIVATION In Assumption 1 we assume the existence of an u...

    2009

  46. [46]

    Probably approximately supra-optimal: E(x,y)∼D ℓ0(fθ(x), y) ≤P ⋆ + 2ζ0(N, δ),(55)

  47. [47]

    1 N sup fθ∈H NX i=1 σig(zi)ℓ(fθ, zi) # (96) =E σ

    Probably approximately feasible: P r ℓ(fθ(x),y)≤c i ≥1−ζ I(N, δ).(56) Proof.Feasibility. We first prove that ˆfθ is feasible for ˆP. Suppose it is not; then ˆP=∞. ButP≤ ∞imply that ∃fθ feasible for P. As the constraint holdsD−a.e.,f θ must be feasible for ˆP. Combined with the boundedness ofℓ 0 it implies thatf θ is a feasible point with bounded objective...

    2020