Information-theoretic lower bounds on the oracle complexity of stochastic convex optimization (1009.0571v3)

Abstract: Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for various function classes.

• The paper establishes fundamental lower bounds on oracle query complexity for various convex function classes using advanced information-theoretic techniques.
• It reveals that complexity scales with problem dimension, Lipschitz continuity, and strong convexity parameters, underscoring nuanced optimization challenges.
• It demonstrates that sparsity in optima can reduce oracle complexity, offering practical benchmarks for designing efficient algorithms in large-scale machine learning.

Overview of Information-Theoretic Lower Bounds on Stochastic Convex Optimization

The paper investigates the theoretical limitations inherent in stochastic convex optimization by exploring lower bounds on oracle complexity in an oracle model of computation. The authors, Alekh Agarwal, Peter L. Bartlett, Pradeep Ravikumar, and Martin J. Wainwright, focus on deriving the fundamental difficulty of optimization problems that are central to machine learning and statistics. This contrasts the vast body of existing literature that primarily centers on upper bounds, thereby providing a critical understanding of the computational complexity involved in such optimization tasks.

The research delineates tight minimax complexity estimates for various function classes, effectively establishing lower bounds on the queries required in stochastic optimization processes. By applying advanced information-theoretic techniques, the paper enhances our comprehension of how problem dimensionality and function class characteristics contribute to optimization challenges.

Key Results

The paper presents several significant findings:

1. Convex Lipschitz Functions:
   • For convex Lipschitz functions, the paper establishes lower bounds that exhibit a dependency on the problem dimension and Lipschitz constant. Specifically, for function classes characterized by parameters $p \in [1, 2]$ and $p > 2$, the complexity scales differently, thus emphasizing the nuances brought by the varying Lipschitz conditions and oracle responses.
2. Strongly Convex Functions:
   • In dealing with strongly convex functions, the paper finds that the oracle complexity is influenced by both the Lipschitz continuity and the strong convexity parameter. The presence of strong convexity allows for potentially smaller complexity bounds, indicative of how the structural properties of loss functions influence optimization processes.
3. Functions with Sparse Optima:
   • The work also considers functions whose minimizers are sparse, leading to the insight that optimization complexity diminishes with increased sparsity. This finding aligns with the broader understanding in statistical estimation that sparsity can simplify computational tasks.

The theoretical tools used include packing arguments and information measures such as Fano's inequality, which relate the task of optimization to multi-hypothesis testing — underscoring the statistical dimension of understanding convex optimization. Such methods bolster the paper's comprehensive approach, illustrating that complexity bounds for stochastic oracles must consider multivariate information dependencies inherent within the function's structure.

Theoretical and Practical Implications

The lower bounds on oracle complexity have significant implications:

• Theoretical Impact:
  • By establishing these bounds, the research provides a benchmark for evaluating the efficiency of stochastic optimization algorithms. This can foster development in algorithm design, guiding researchers towards achieving or approaching these theoretical lower limits.
• Practical Impact:
  • Understanding these bounds helps in realizing the trade-offs between computational cost and optimization accuracy, thereby influencing decisions in areas like large-scale machine learning where resources are constrained.
• Future Directions:
  • The paper invites exploration into complexity constraints when considered in conjunction with other computational models. An example could be investigating memory-limited or distributed settings to uncover additional bounds or properties that affect computational sustainability.

The research underscores the complexity-theoretic aspects of convex optimization, contributing critical insights into computational resource allocation in stochastic environments. As machine learning models grow in scope and complexity, such foundational research proves indispensable for paving the way toward more efficient resource-aware algorithms.