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Open Problem: Two Riddles in Heavy-Ball Dynamics

Baptiste Goujaud, Adrien Taylor, Aymeric Dieuleveut

TL;DR

The paper investigates two open questions about Polyak's Heavy-Ball method: whether acceleration can occur in one dimension and whether there exist parameter tunings for which HB neither admits a Lyapunov function nor a cycle. It builds on known results that HB cannot achieve acceleration on $\mathcal{F}_{\mu,L}$ in dimensions $d\ge 2$, and then uses a mix of analytical constructions (to exhibit cycles in higher dimensions) and numerical/SDP-based cycle detection to explore the 1D case. It also examines the coexistence of Lyapunov-based convergence and cycling, highlighting regions with no definitive behavior (the “third type” of dynamics) and reporting intriguing structural observations about the borders of these regions. The work leverages Lyapunov searches, PEP/SDP frameworks, and permutation-based cycle construction to map the HB dynamical landscape, and it raises concrete conjectures about a unifying permutation that captures 1D cycles up to large $K$. Overall, the results sharpen our understanding of HB's limits and offer methodological tools that may apply to analyzing other first-order methods in non-quadratic settings.

Abstract

This short paper presents two open problems on the widely used Polyak's Heavy-Ball algorithm. The first problem is the method's ability to exactly \textit{accelerate} in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.

Open Problem: Two Riddles in Heavy-Ball Dynamics

TL;DR

The paper investigates two open questions about Polyak's Heavy-Ball method: whether acceleration can occur in one dimension and whether there exist parameter tunings for which HB neither admits a Lyapunov function nor a cycle. It builds on known results that HB cannot achieve acceleration on in dimensions , and then uses a mix of analytical constructions (to exhibit cycles in higher dimensions) and numerical/SDP-based cycle detection to explore the 1D case. It also examines the coexistence of Lyapunov-based convergence and cycling, highlighting regions with no definitive behavior (the “third type” of dynamics) and reporting intriguing structural observations about the borders of these regions. The work leverages Lyapunov searches, PEP/SDP frameworks, and permutation-based cycle construction to map the HB dynamical landscape, and it raises concrete conjectures about a unifying permutation that captures 1D cycles up to large . Overall, the results sharpen our understanding of HB's limits and offer methodological tools that may apply to analyzing other first-order methods in non-quadratic settings.

Abstract

This short paper presents two open problems on the widely used Polyak's Heavy-Ball algorithm. The first problem is the method's ability to exactly \textit{accelerate} in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.

Paper Structure

This paper contains 14 sections, 7 equations, 8 figures.

Table of Contents

  1. Introduction, background, and open problems
  2. Sketch of the proof.
  3. Open problem 1: $\operatorname{HB}$ behavior in dimension 1
  4. Numerical approach for finding cycles in dimension 1.
  5. Identifying the correct permutation to obtain numerical results for $K> 6$.
  6. Comments on \ref{['fig:cycles_dep_on_sig_K=4', 'fig:cycles_dep_on_sig_K=5']}.
  7. Conclusion on \ref{['fig:cycles_dep_on_sig_K=4', 'fig:cycles_dep_on_sig_K=5']}.
  8. Link to wang2022provable.
  9. Open problem 2: Lyapunov functions, cycles, and third type of behavior
  10. More details on the Lyapunov functions we looked for.
  11. How did we find cycles?
  12. What other type of behavior can we expect?
  13. An interesting, unexplained observation.
  14. Acknowledgments

Figures (8)

  • Figure 1: Comparison of HB behaviors on $\mathcal{Q}_{\mu, L}$ and $\mathcal{F}_{\mu, L}$.
  • Figure 2: $\operatorname{HB}$ behavior as a function of its tuning: Green $\leftrightarrow$ Lyapunov; Purple $\leftrightarrow$ Cycle; Unfilled $\leftrightarrow$ Open Problem 2.
  • Figure 3: Comparison of cycle regions in different dimensions. Border of $\Omega_{\circ\text{-}\mathrm{Cycle}}(\mathcal{F}_{\mu, L})$ in black lines. $\Omega_{\mathrm{Cycle}}(\mathcal{F}_{\mu, L})$ in shades of purples. Both regions are numerically identical.
  • Figure 4: Typical permutation.
  • Figure 5: For $K=4$, $(\mu,L)$ fixed, the set of points $\gamma,\beta$ such that \ref{['eq:lpsigma']} admits a feasible point, depending on $\sigma$.
  • ...and 3 more figures