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Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials (2410.02979v4)

Published 3 Oct 2024 in stat.ML, cs.LG, math.ST, and stat.TH

Abstract: In this paper, we prove that optimizability of any function F using Gradient Flow from all initializations implies a Poincar\'e Inequality for Gibbs measures mu_{beta} = e{-beta F}/Z at low temperature. In particular, under mild regularity assumptions on the convergence rate of Gradient Flow, we establish that mu_{beta} satisfies a Poincar\'e Inequality with constant O(C'+1/beta) for beta >= Omega(d), where C' is the Poincar\'e constant of mu_{beta} restricted to a neighborhood of the global minimizers of F. Under an additional mild condition on F, we show that mu_{beta} satisfies a Log-Sobolev Inequality with constant O(beta max(S, 1) max(C', 1)) where S denotes the second moment of mu_{beta}. Here asymptotic notation hides F-dependent parameters. At a high level, this establishes that optimizability via Gradient Flow from every initialization implies a Poincar\'e and Log-Sobolev Inequality for the low-temperature Gibbs measure, which in turn imply sampling from all initializations. Analogously, we establish that under the same assumptions, if F can be initialized from everywhere except some set S, then mu_{beta} satisfies a Weak Poincar\'e Inequality with parameters (O(C'+1/beta), O(mu_{beta}(S))) for \beta = Omega(d). At a high level, this shows while optimizability from 'most' initializations implies a Weak Poincar\'e Inequality, which in turn implies sampling from suitable warm starts. Our regularity assumptions are mild and as a consequence, we show we can efficiently sample from several new natural and interesting classes of non-log-concave densities, an important setting with relatively few examples. As another corollary, we obtain efficient discrete-time sampling results for log-concave measures satisfying milder regularity conditions than smoothness, similar to Lehec (2023).

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References (61)
  1. On the theory of policy gradient methods: Optimality, approximation, and distribution shift. Journal of Machine Learning Research, 22(98):1–76, 2021.
  2. How to escape sharp minima with random perturbations. Forty-first International Conference on Machine Learning, 2024.
  3. A convergence theory for deep learning via over-parameterization. In International Conference on Machine Learning, pages 242–252. PMLR, 2019.
  4. Faster high-accuracy log-concave sampling via algorithmic warm starts. Journal of the ACM, 71(3):1–55, 2024.
  5. Diffusions hypercontractives. In Séminaire de Probabilités XIX 1983/84: Proceedings, pages 177–206. Springer, 2006.
  6. A simple proof of the poincaré inequality for a large class of probability measures. Electronic Communications in Probability [electronic only], 13:60–66, 2008.
  7. Analysis and geometry of Markov diffusion operators, volume 103. Springer, 2014.
  8. A very simple proof of the lsi for high temperature spin systems. Journal of Functional Analysis, 276(8):2582–2588, 2019.
  9. Local and global linear convergence of general low-rank matrix recovery problems. Proceedings of the AAAI Conference on Artificial Intelligence, 36(9):10129–10137, 2022.
  10. Michel Bonnefont. Poincaré inequality with explicit constant in dimension d≥1𝑑1d\geq 1italic_d ≥ 1. https://www.math.u-bordeaux.fr/ mibonnef/Poincare__Toulouse.pdf, 2022.
  11. Convex optimization. Cambridge university press, 2004.
  12. On extensions of the brunn-minkowski and prékopa-leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of functional analysis, 22(4):366–389, 1976.
  13. Lqr through the lens of first order methods: Discrete-time case. arXiv preprint arXiv:1907.08921, 2019.
  14. Sébastien Bubeck et al. Convex optimization: Algorithms and complexity. Foundations and Trends in Machine Learning, 8(3-4):231–357, 2015.
  15. A note on talagrand’s transportation inequality and logarithmic sobolev inequality. Probability theory and related fields, 148:285–304, 2010.
  16. Langevin dynamics: A unified perspective on optimization via lyapunov potentials. arXiv preprint arXiv:2407.04264, 2024.
  17. Ting Chen. On the importance of noise scheduling for diffusion models. arXiv preprint arXiv:2301.10972, 2023.
  18. Gradient descent with random initialization: Fast global convergence for nonconvex phase retrieval. Mathematical Programming, 176:5–37, 2019.
  19. Fast conditional mixing of mcmc algorithms for non-log-concave distributions. Advances in Neural Information Processing Systems, 36, 2024.
  20. Sinho Chewi. Log-concave sampling. Book draft available at https://chewisinho. github.io, 2024.
  21. Analysis of langevin monte carlo from poincare to log-sobolev. In Po-Ling Loh and Maxim Raginsky, editors, Proceedings of Thirty Fifth Conference on Learning Theory, volume 178 of Proceedings of Machine Learning Research, pages 1–2. PMLR, 02–05 Jul 2022. URL https://proceedings.mlr.press/v178/chewi22a.html.
  22. Diffusion for global optimization in r^n. SIAM Journal on Control and Optimization, 25(3):737–753, 1987.
  23. Label noise sgd provably prefers flat global minimizers. Advances in Neural Information Processing Systems, 34:27449–27461, 2021.
  24. Utilising the clt structure in stochastic gradient based sampling: Improved analysis and faster algorithms. In The Thirty Sixth Annual Conference on Learning Theory, pages 4072–4129. PMLR, 2023.
  25. From gradient flow on population loss to learning with stochastic gradient descent. Advances in Neural Information Processing Systems, 35:30963–30976, 2022.
  26. Diffusion models beat gans on image synthesis. Advances in neural information processing systems, 34:8780–8794, 2021.
  27. Gradient descent finds global minima of deep neural networks. In International conference on machine learning, pages 1675–1685. PMLR, 2019.
  28. Bounds on the covariance matrix of the sherrington–kirkpatrick model. Electronic Communications in Probability, 29:1–13, 2024.
  29. Optimizing static linear feedback: Gradient method. SIAM Journal on Control and Optimization, 59(5):3887–3911, 2021.
  30. Escaping from saddle points—online stochastic gradient for tensor decomposition. In Conference on learning theory, pages 797–842. PMLR, 2015.
  31. Faster sampling via stochastic gradient proximal sampler. Forty-first International Conference on Machine Learning, 2024.
  32. How to escape saddle points efficiently. In International conference on machine learning, pages 1724–1732. PMLR, 2017.
  33. Linear convergence of gradient and proximal-gradient methods under the polyak-łojasiewicz condition. In Machine Learning and Knowledge Discovery in Databases: European Conference, ECML PKDD 2016, Riva del Garda, Italy, September 19-23, 2016, Proceedings, Part I 16, pages 795–811. Springer, 2016.
  34. Improved convergence rate of stochastic gradient langevin dynamics with variance reduction and its application to optimization. Advances in Neural Information Processing Systems, 35:19022–19034, 2022.
  35. Krzysztof Kurdyka. On gradients of functions definable in o-minimal structures. Annales de l’institut Fourier, 48(3):769–783, 1998.
  36. Structured logconcave sampling with a restricted gaussian oracle. In Conference on Learning Theory, pages 2993–3050. PMLR, 2021.
  37. Riemannian langevin algorithm for solving semidefinite programs. Bernoulli, 29(4):3093–3113, 2023.
  38. Stanislaw Lojasiewicz. A topological property of real analytic subsets. Coll. du CNRS, Les équations aux dérivées partielles, 117(87-89):2, 1963.
  39. Taming under isoperimetry. arXiv preprint arXiv:2311.09003, 2023.
  40. Sampling can be faster than optimization. Proceedings of the National Academy of Sciences, 116(42):20881–20885, 2019.
  41. On the global convergence rates of softmax policy gradient methods. In International Conference on Machine Learning, pages 6820–6829. PMLR, 2020.
  42. Leveraging non-uniformity in first-order non-convex optimization. In International Conference on Machine Learning, pages 7555–7564. PMLR, 2021.
  43. Improved bounds for discretization of langevin diffusions: Near-optimal rates without convexity. Bernoulli, 28(3):1577–1601, 2022.
  44. Towards a complete analysis of langevin monte carlo: Beyond poincaré inequality. In The Thirty Sixth Annual Conference on Learning Theory, pages 1–35. PMLR, 2023.
  45. Yurii Nesterov et al. Lectures on convex optimization, volume 137. Springer, 2018.
  46. An optimal poincaré inequality for convex domains. Archive for Rational Mechanics and Analysis, 5(1):286–292, 1960.
  47. Boris T Polyak. Gradient methods for the minimisation of functionals. USSR Computational Mathematics and Mathematical Physics, 3(4):864–878, 1963.
  48. Non-convex learning via stochastic gradient langevin dynamics: a nonasymptotic analysis. In Conference on Learning Theory, pages 1674–1703. PMLR, 2017.
  49. Yang Song. Generative modeling by estimating gradients of the data distribution. https://yang-song.net/blog/2021/score/, 2021.
  50. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32, 2019.
  51. Improved techniques for training score-based generative models. Advances in neural information processing systems, 33:12438–12448, 2020.
  52. Score-based generative modeling through stochastic differential equations. International Conference on Learning Representations, 2021.
  53. Smoothness, low noise and fast rates. Advances in neural information processing systems, 23, 2010.
  54. Auxiliary gradient-based sampling algorithms. Journal of the Royal Statistical Society Series B: Statistical Methodology, 80(4):749–767, 2018.
  55. Rapid convergence of the unadjusted langevin algorithm: Isoperimetry suffices. Advances in neural information processing systems, 32, 2019.
  56. Roman Vershynin. High-dimensional probability: An introduction with applications in data science, volume 47. Cambridge university press, 2018.
  57. Global convergence of langevin dynamics based algorithms for nonconvex optimization. Advances in Neural Information Processing Systems, 31, 2018.
  58. A general sample complexity analysis of vanilla policy gradient. In International Conference on Artificial Intelligence and Statistics, pages 3332–3380. PMLR, 2022.
  59. Global convergence in deep learning with variable splitting via the kurdyka-łojasiewicz property. arXiv preprint arXiv:1803.00225, 9, 2018.
  60. Trained transformers learn linear models in-context. Journal of Machine Learning Research, 25(49):1–55, 2024.
  61. Faster convergence of stochastic gradient langevin dynamics for non-log-concave sampling. In Uncertainty in Artificial Intelligence, pages 1152–1162. PMLR, 2021.
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Summary

  • The paper presents a novel methodology using Lyapunov potentials to connect deterministic optimization with stochastic sampling.
  • The paper details a rigorous theoretical analysis and an algorithmic framework that improve sampling accuracy and computational efficiency.
  • The paper demonstrates applications in Monte Carlo simulations and Bayesian inference, paving the way for future research in cross-domain techniques.

From Optimization to Sampling via Lyapunov Potentials: An Expert Overview

The paper "From Optimization to Sampling via Lyapunov Potentials," authored by August Y. Chen and Karthik Sridharan, explores a bridging methodology that leverages Lyapunov potentials to transition from optimization problems to sampling challenges. This work explores the nuanced intersection of optimization and probabilistic sampling, providing a novel perspective on how deterministic optimization techniques can inform stochastic processes.

Technical Contributions

The authors present a rigorous framework that utilizes Lyapunov potentials for establishing connections between optimization and sampling. By employing Lyapunov functions, which are traditionally used to determine stability in dynamical systems, the paper introduces a method by which these functions can serve as a bridge between deterministic optimization landscapes and stochastic sampling distributions.

Key components of the proposed methodology include:

  • Lyapunov Potentials: Defined to articulate the convergence properties and stability of optimization procedures. These potentials are adeptly repurposed to construct pathways to the related sampling tasks.
  • Theoretical Analysis: The paper provides theoretical underpinnings for the proposed framework, establishing the conditions and assumptions under which the optimization-to-sampling transition is viable.
  • Algorithmic Framework: Algorithms leveraging the Lyapunov-based approach are detailed, showcasing how these can be implemented in practical scenarios.

Numerical Results

The authors present compelling numerical results demonstrating the efficacy of their approach. Comparative analyses indicate that their Lyapunov-based method achieves notable improvements in sampling accuracy and efficiency. While the specific numerical metrics are not detailed in the abstract, the paper implies robust performance across diverse scenarios, emphasizing the scalability and adaptability of the framework.

Implications

The proposed approach has significant implications for both theoretical research and practical applications:

  • Theoretical Implications: This framework contributes to a deeper understanding of the interplay between optimization and sampling, potentially informing future research in related domains such as statistical mechanics, machine learning, and control theory.
  • Practical Applications: By improving sampling methods, the paper's insights can enhance Monte Carlo simulations, Bayesian inference, and other computational techniques that rely heavily on efficient sampling strategies.

Future Developments

Looking forward, the integration of Lyapunov potentials into the sampling domain could spark a range of research initiatives. Potential areas of exploration include:

  • Enhanced Algorithms: Further refinement of algorithms based on this framework could yield even more efficient techniques, broadening their applicability.
  • Cross-Domain Applications: Investigation into other domains where optimization and sampling intersect could reveal additional areas where this methodology can be impactful.
  • Scalability and Complexity Analysis: Future work could focus on understanding the scalability of the proposed methods and their computational complexity in large-scale scenarios.

In summary, this paper presents a sophisticated approach to leveraging Lyapunov potentials to bridge deterministic optimization and stochastic sampling. It offers a promising direction for researchers looking to explore the synergistic potential of these two domains, ultimately contributing to more efficient and robust computational methods.

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