Regularity-Aware Stochastic MGDA with Adaptive Conflict-Avoidant Update Direction Control
The paper identifies that vanilla Stochastic Multi-Gradient Descent (SMG) suffers from suboptimal convergence due to gradient noise affecting the continuity of the conflict-avoidant (CA) direction. It proves that while the CA direction is generally 1/2-Hölder continuous, it achieves Lipschitz continuity under specific regularity conditions. A new method, MoRe (Stochastic Multi-Objective Regularity-Aware), is proposed to exploit this Lipschitz continuity when the problem is regular, switching to
Analysis
TL;DR
- The paper identifies that vanilla Stochastic Multi-Gradient Descent (SMG) suffers from suboptimal convergence due to gradient noise affecting the continuity of the conflict-avoidant (CA) direction.
- It proves that while the CA direction is generally 1/2-Hölder continuous, it achieves Lipschitz continuity under specific regularity conditions.
- A new method, MoRe (Stochastic Multi-Objective Regularity-Aware), is proposed to exploit this Lipschitz continuity when the problem is regular, switching to fixed scalarization otherwise.
- Theoretical analysis shows MoRe improves the nonconvex convergence rate from $\widetilde{\mathcal O}(T^{-1/4})$ to $\widetilde{\mathcal O}(T^{-1/2})$.
- Empirical results confirm the theoretical convergence rates and demonstrate improved multi-task performance.
Why It Matters
This research addresses a fundamental bottleneck in multi-objective optimization, specifically the slow convergence of stochastic methods due to gradient noise. By providing a rigorous theoretical bound and a practical algorithmic solution, it enables faster and more stable training for models requiring simultaneous optimization of multiple conflicting goals, such as in multi-task learning.
Technical Details
- Problem Analysis: The study analyzes the bias in update directions caused by mini-batch sampling noise in SMG, linking convergence speed to the continuity properties of the CA direction with respect to the Jacobian matrix.
- Continuity Proof: It establishes that the CA direction is $1/2$-Hölder continuous in the worst case but can be upgraded to Lipschitz continuity under additional regularity conditions.
- Algorithm Design: The MoRe algorithm dynamically switches between CA direction updates (when gradient conflict is high and regularity holds) and linear scalarization updates (otherwise), leveraging the improved continuity for faster convergence.
- Convergence Rate: The method theoretically guarantees a convergence rate of $\widetilde{\mathcal O}(T^{-1/2})$ in nonconvex settings, doubling the exponent compared to previous stochastic MGDA methods.
- Validation: Experiments verify the per-iterate conflict-avoidance guarantees and empirical convergence behavior aligns with the derived theoretical rates.
Industry Insight
- Practitioners working on multi-task learning or multi-objective reinforcement learning should consider adopting regularity-aware strategies to accelerate convergence without sacrificing stability.
- Understanding the continuity properties of optimization directions can guide the design of more robust stochastic optimizers for complex, noisy environments.
- The switch mechanism between conflict-avoidant and scalarized updates offers a practical heuristic for balancing exploration of trade-offs versus efficient descent in multi-objective scenarios.
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