⚡ electrical engineering

Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Con-vergence Rate and Optimality Gap

This paper investigates the trade-off between convergence speed and optimality gap in distributed machine learning, demonstrating through simulations that while non-Lipschitz signum-based functions accelerate convergence in distributed regression, they inevitably introduce significant steady-state optimality gaps compared to linear methods.

Original authors: Mohammadreza Doostmohammadian, Amir Ahmad Ghods, Alireza Aghasi, Zulfiya R. Gabidullina, Hamid R. Rabiee

Published 2026-08-04

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CC BY 4.0

Original authors: Mohammadreza Doostmohammadian, Amir Ahmad Ghods, Alireza Aghasi, Zulfiya R. Gabidullina, Hamid R. Rabiee

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where thousands of tiny computers, scattered across a city like fireflies in a jar, need to solve a giant math puzzle together. They can't all talk to one central boss; instead, they only whisper to their immediate neighbors. This is the heart of distributed optimization, a field where scientists teach these networks to learn and make decisions without a single leader. The goal is often machine learning, where the network tries to find the perfect "rule" (like a line that fits a scatter of dots) that explains all the data everyone has collected.

To make this happen, the computers usually follow a gentle, steady rhythm, taking small steps toward the answer. This is like a group of hikers slowly adjusting their path to meet at a campfire. But what if they could move faster? What if they could sprint toward the meeting point? This is where non-Lipschitz functions come in. Think of these as a special kind of "super-speed" rule. Instead of walking gently, the computers use a sharp, aggressive push—like a magnet snapping two pieces together instantly—to reach agreement in record time. For years, researchers hoped this "snap" would make learning both fast and perfect. But here's the twist: in the real, messy world of digital computers, that same snap might cause the hikers to overshoot the campfire, shaking back and forth right next to the goal without ever quite settling down.

This paper, titled "Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning: Trade-off Between Convergence Rate and Optimality Gap," dives into that exact dilemma. The authors, a team of researchers from Iran, the US, and Russia, set out to test whether using these "super-speed" signum-based functions is a magic bullet or a double-edged sword. They didn't just guess; they built a digital playground to watch these algorithms in action.

The researchers simulated a distributed linear regression problem, which is essentially a game where many computers try to agree on the best straight line to fit a cloud of data points. They compared the old, steady "walking" method against the new, aggressive "snapping" method. Their simulations, run on datasets ranging from 100 to 12,000 data points across networks of 10 to 100 agents, revealed a clear and somewhat disappointing truth: speed comes at a cost.

While the signum-based functions did indeed make the computers reach the general area of the solution much faster—sometimes achieving what looks like "finite-time" convergence—they found that the system never truly stopped moving. Instead of settling perfectly on the best possible line, the computers began to vibrate or "chatter" around the answer. This creates what the authors call an optimality gap: a small but persistent error where the final result is close, but not quite perfect. The paper suggests that the more aggressive the "snap" (controlled by specific mathematical parameters), the faster the initial speed, but the larger the final error.

Crucially, the authors found that this isn't a bug you can simply ignore; it's a fundamental trade-off. In their simulations, using a fixed step size with these fast functions guaranteed a permanent gap between the result and the true best answer. However, they did discover a way to shrink this gap: using a diminishing step size. Imagine the hikers sprinting at first but then slowing down to a tiny, careful shuffle as they get closer to the campfire. This method allowed the system to eventually settle closer to the perfect answer, but it sacrificed that initial burst of speed.

The paper concludes that while these non-Lipschitz, signum-based functions are powerful tools for scenarios where getting close quickly is more important than being perfect (like in noisy environments or when dealing with outliers), they are not a universal upgrade. If you need the mathematically perfect solution, the "snap" might actually keep you from ever quite getting there. The authors suggest that future work should focus on balancing these speeds, perhaps using hybrid approaches that combine the best of both worlds, but for now, the lesson is clear: in the digital dance of distributed learning, you can't always have it both fast and flawless.

Technical Summary: Using Non-Lipschitz Signum-based Functions for Distributed Optimization and Machine Learning

Problem Statement
The paper addresses the challenge of balancing convergence speed and solution accuracy in distributed optimization and machine learning (ML), specifically within distributed linear regression (DLR) frameworks. While existing linear consensus-based algorithms offer asymptotic convergence, they often exhibit slow convergence rates. Recently, non-Lipschitz continuous optimization algorithms utilizing signum-based functions have been proposed to achieve faster, finite-time, or fixed-time convergence. However, the authors identify a critical issue: in discrete-time implementations, these non-Lipschitz dynamics often lead to "chattering" phenomena (steady-state oscillations) around the optimal point. This results in a persistent optimality gap (steady-state residual) in the objective function, preventing the system from reaching the exact optimal solution. The core problem investigated is the trade-off between the accelerated convergence rate offered by signum-based functions and the degradation of optimality in discrete-time setups.

Methodology
The study focuses on a distributed linear regression problem formulated as a consensus-constrained optimization task over a multi-agent network. The authors propose and analyze a Gradient-Tracking (GT) based distributed algorithm enhanced with non-Lipschitz signum-based functions.

Algorithm Design: The authors modify standard linear GT dynamics (which use linear differences between agent states) by replacing the linear terms with signum-based functions of the form $sgn_v(x) = x|x|^{v-1}$ $s g n_{v} (x) = x ∣ x ∣^{v - 1}$ . The proposed continuous-time dynamics for the state $x_i$ $x_{i}$ and auxiliary gradient-tracking variable $y_i$ $y_{i}$ incorporate two types of signum terms:
- $v_1 < 1$ : Provides fast convergence when states are close to the equilibrium (finite-time behavior).
- $v_2 > 1$ : Provides fast convergence when states are far from the equilibrium.
- Similar parameters ( $u_1, u_2$ ) are applied to the gradient tracking update.
Discretization: The continuous-time dynamics are discretized for implementation. The authors analyze how the discretization step-size and the non-Lipschitz nature of the functions interact, noting that fixed step-sizes in this context can lead to persistent steady-state errors.
Mitigation Strategy: To address the optimality gap, the paper proposes using a diminishing step-size sequence (e.g., $\eta_k = \eta_0 / \sqrt{k+1}$ ) rather than a fixed step-size. This approach aims to attenuate the bias and oscillations caused by non-Lipschitz terms in later iterations while maintaining faster initial convergence.
Simulation Setup: The proposed method is evaluated through extensive simulations on:
- A randomly generated dataset for distributed linear regression.
- A large-scale synthetic dataset ( $N=1000$ points, $n=100$ agents).
- A real-world dataset (MNIST) using logistic regression over a network of 16 nodes.
  The performance is compared across different parameter configurations ( $v_1, v_2, u_1, u_2$ ) and against existing algorithms (GP, SGP, S-ADDOPT).

Key Contributions

Identification of the Trade-off: The paper explicitly demonstrates that while non-Lipschitz signum-based functions can accelerate convergence (potentially achieving finite-time or fixed-time convergence in continuous time), they introduce a steady-state optimality gap in discrete-time implementations due to chattering.
Gradient-Tracking with Nonlinearity: The authors extend gradient-tracking distributed optimization to include signum-based nonlinearities. They prove that the optimal point remains an invariant state under these dynamics for continuous-time systems, provided the functions are odd and sign-preserving.
Parameter Analysis: The study systematically analyzes how parameters $v_1, v_2, u_1, u_2$ affect both the convergence rate and the magnitude of the optimality gap. It establishes that aggressive nonlinearity (parameters far from 1) increases convergence speed but also increases the steady-state residual.
Step-Size Mitigation: The paper proposes and validates that using a diminishing step-size sequence can reduce the optimality gap to vanishing levels as $k \to \infty$ , albeit at the cost of slowing down the asymptotic convergence rate compared to fixed step-sizes.

Results

Convergence vs. Residual: Simulations confirm that signum-based dynamics converge significantly faster than linear dynamics in the transient phase. However, with fixed step-sizes, the system settles into a steady-state oscillation around the optimal solution rather than converging to it exactly. The size of this gap depends on the specific values of the signum parameters and the step-size.
Gradient Tracking Impact: The introduction of signum functions into the gradient-tracking component (the $y_i$ update) was found to be particularly sensitive, often resulting in larger steady-state errors compared to applying it only to the consensus term.
Diminishing Step-Size: Using a diminishing step-size successfully reduced the optimality gap, allowing the algorithm to approach the true optimum more closely, though the convergence trajectory became slower in the later stages.
Real-World Application: In the MNIST logistic regression experiment, the signum-based algorithm (with parameters moderately close to 1) achieved fast convergence with a sufficiently low optimality gap, outperforming or matching existing algorithms like GP, SGP, and S-ADDOPT in terms of the speed-accuracy balance.

Significance and Claims
The paper claims that its findings contribute to the ongoing discourse on distributed algorithms by highlighting a critical, often overlooked trade-off: the gain in convergence speed via non-Lipschitz functions comes at the cost of exact optimality in discrete-time settings. The authors argue that these solutions are practical for scenarios where rapid convergence is prioritized over exact precision, or where the system can tolerate a small steady-state error. Conversely, for applications requiring high precision, the paper suggests that the use of diminishing step-sizes or careful parameter tuning is necessary to mitigate the optimality gap. The work provides a framework for decision-making in distributed estimation, detection, and resource allocation, suggesting that the choice between linear and non-Lipschitz dynamics should be driven by the specific requirements of the application regarding speed versus accuracy.

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