The Price of Adaptivity in Stochastic Convex Optimization (2402.10898v3)

Abstract: We prove impossibility results for adaptivity in non-smooth stochastic convex optimization. Given a set of problem parameters we wish to adapt to, we define a "price of adaptivity" (PoA) that, roughly speaking, measures the multiplicative increase in suboptimality due to uncertainty in these parameters. When the initial distance to the optimum is unknown but a gradient norm bound is known, we show that the PoA is at least logarithmic for expected suboptimality, and double-logarithmic for median suboptimality. When there is uncertainty in both distance and gradient norm, we show that the PoA must be polynomial in the level of uncertainty. Our lower bounds nearly match existing upper bounds, and establish that there is no parameter-free lunch. En route, we also establish tight upper and lower bounds for (known-parameter) high-probability stochastic convex optimization with heavy-tailed and bounded noise, respectively.

• The paper establishes theoretical lower bounds on the cost of adaptivity, showing suboptimality increases with uncertainty in problem parameters.
• It compares these bounds with state-of-the-art adaptive algorithms, demonstrating near-optimal performance under specific conditions.
• The work reveals intrinsic limits of stochastic first-order methods and motivates future research to improve adaptivity in broader optimization settings.

The Price of Adaptivity in Stochastic Convex Optimization

Introduction

Recent advancements in stochastic optimization methods have underscored the importance of adaptivity in algorithm design, particularly in machine learning applications. Adaptive algorithms, which require minimal or no tuning of parameters, are highly desirable due to their potential to significantly reduce the time, computation, and expertise required to solve optimization problems. This paper investigates the theoretical limits of adaptivity in non-smooth stochastic convex optimization (SCO), with a focus on understanding whether current methods have achieved optimal adaptivity or if there is significant room for improvement.

Theoretical Contributions

The central contribution of this paper is the formal definition and thorough investigation of the "price of adaptivity" (PoA) in the context of non-smooth stochastic convex optimization. PoA is defined as the multiplicative increase in suboptimality—measured in terms of the function-value gap—due to the lack of prior knowledge about certain problem parameters, such as the Lipschitz constant of the objective function or the distance of the initial point from the optimum.

• Lower Bounds: The paper establishes several impossibility results for adaptivity in SCO, showing that adaptivity comes at a significant cost. When uncertainty exists regarding the initial distance to the optimum, the paper proves that the expected suboptimality PoA is at least logarithmic with respect to the level of uncertainty. For the high-probability counterpart, the PoA is shown to exhibit double-logarithmic dependence. Additionally, when both the initial distance and the gradient norm bound are uncertain, the paper presents a polynomial lower bound on the PoA. These lower bounds nearly match existing upper bounds in the literature, underscoring the near-optimality of current adaptive algorithms under certain assumptions.
• Comparison to Upper Bounds: Through a systematic survey of the state-of-the-art adaptive algorithms for SCO, the paper juxtaposes the newly established lower bounds on the PoA with existing upper bounds. The comparison reveals that while current algorithms achieve close to optimal adaptivity under certain conditions, there exists a fundamental cost of adaptivity when problem parameters are unknown in advance.

Implications and Open Questions

The findings underscore several critical insights and open questions in the field of adaptive stochastic optimization:

• Adaptivity vs. Heavy-tailed Noise: The distinction between environments with bounded stochastic gradients and those with bounded second moments of stochastic gradients is stark, with the latter being significantly more challenging for adaptivity. This highlights the elevated cost associated with robustness against heavy-tailed noise in optimization problems.
• Algorithmic Restrictions: The paper's lower bounds expose the intrinsic limitations of stochastic first-order methods under information-theoretic constraints. However, the bounds are less clear for algorithms with unrestricted access to each sample function. Addressing this gap remains an open challenge.
• Beyond Convex Optimization: Extending the concept of PoA to settings beyond non-smooth stochastic convex optimization, including situations with additional structural assumptions (e.g., smoothness or strong convexity) or entirely different problem paradigms (e.g., non-convex optimization), presents a promising direction for future research.

Conclusion

In summary, this work rigorously characterizes the theoretical limitations of adaptivity in non-smooth stochastic convex optimization, revealing that while current algorithms approach optimal adaptivity under specific conditions, fundamental challenges remain. The introduced PoA framework offers a precise metric for quantifying the adaptivity of algorithms and opens numerous avenues for exploring the efficiency of adaptive methods in broader optimization contexts.