Adaptive aggregation of multiple surrogate gradients

Develop an adaptive method for aggregating a fixed number of surrogate gradients into an optimal directional hint subspace, for instance through an online learning scheme over the surrogates, while avoiding the growth in subspace dimension that results from naively taking their span.

Background

The Control-Variate Zeroth-Order Descent framework processes one low-dimensional hint subspace at each iteration. In practical applications, however, multiple surrogates may be available, and their gradients may accumulate over time. Naively forming the span of these gradients increases the hint dimension, which undermines the computational advantage of low-dimensional directional hints and increases the number of zeroth-order queries required by the method.

The paper identifies adaptive aggregation as an unresolved problem: an effective procedure should combine a fixed number of surrogate gradients in an optimal manner, potentially by treating the choice or weighting of surrogates as an online learning problem. The question is explicitly motivated by preserving a low-dimensional hint representation while exploiting multiple information sources.

References

Several directions remain open. First, the current framework processes a single low-dimensional hint subspace per iteration. In practice, one may have access to multiple surrogates whose gradients accumulate over iterations; naively taking their span as the hint subspace causes the dimension $k$ to grow, eroding low-dimensionality and increasing the required number of queries. A natural question is whether one can adaptively aggregate a fixed number of surrogate gradients in an optimal way---for instance, through an online learning scheme over the surrogates.

— Adaptively Incorporating Directional Hints into Zeroth-Order Optimization  (2609.08277 - Ryabchenko et al., 8 Sep 2026) in Section Discussion and Future Work