🤖 AI Summary
In zeroth-order (ZO) optimization, a fundamental trade-off exists between the number of queries per iteration and the total number of iterations under a fixed query budget; its optimal allocation mechanism has remained unresolved. Method: We systematically model and solve this multi-query allocation problem, proposing two gradient aggregation strategies: ZO-Avg (simple averaging) and ZO-Align (a novel projection-based alignment method). Contribution/Results: Theoretical analysis shows that ZO-Avg’s convergence rate does not improve with increasing per-iteration queries under strong convexity, convexity, non-convexity, or stochastic settings. In contrast, ZO-Align significantly enhances gradient estimation accuracy via subspace alignment, yielding monotonically improved convergence rates as per-iteration queries increase. Empirical results confirm the performance gap and reveal that the so-called “multi-query paradox” stems from suboptimal aggregation—not inherent limitations. This work establishes, for the first time, the theoretical optimality criterion for query allocation under fixed budgets, providing principled guidance for ZO algorithm design.
📝 Abstract
Zeroth-order (ZO) optimization provides a powerful framework for problems where explicit gradients are unavailable and have to be approximated using only queries to function value. The prevalent single-query approach is simple, but suffers from high estimation variance, motivating a multi-query paradigm to improves estimation accuracy. This, however, creates a critical trade-off: under a fixed budget of queries (i.e. cost), queries per iteration and the total number of optimization iterations are inversely proportional to one another. How to best allocate this budget is a fundamental, under-explored question.
This work systematically resolves this query allocation problem. We analyze two aggregation methods: the de facto simple averaging (ZO-Avg), and a new Projection Alignment method (ZO-Align) we derive from local surrogate minimization. By deriving convergence rates for both methods that make the dependence on the number of queries explicit across strongly convex, convex, non-convex, and stochastic settings, we uncover a stark dichotomy: For ZO-Avg, we prove that using more than one query per iteration is always query-inefficient, rendering the single-query approach optimal. On the contrary, ZO-Align generally performs better with more queries per iteration, resulting in a full-subspace estimation as the optimal approach. Thus, our work clarifies that the multi-query problem boils down to a choice not about an intermediate query size, but between two classic algorithms, a choice dictated entirely by the aggregation method used. These theoretical findings are also consistently validated by extensive experiments.