  Federated optimization, an emerging paradigm which finds wide real-world
applications such as federated learning, enables multiple clients (e.g., edge
devices) to collaboratively optimize a global function. The clients do not
share their local datasets and typically only share their local gradients.
However, the gradient information is not available in many applications of
federated optimization, which hence gives rise to the paradigm of federated
zeroth-order optimization (ZOO). Existing federated ZOO algorithms suffer from
the limitations of query and communication inefficiency, which can be
attributed to (a) their reliance on a substantial number of function queries
for gradient estimation and (b) the significant disparity between their
realized local updates and the intended global updates. To this end, we (a)
introduce trajectory-informed gradient surrogates which is able to use the
history of function queries during optimization for accurate and
query-efficient gradient estimation, and (b) develop the technique of adaptive
gradient correction using these gradient surrogates to mitigate the
aforementioned disparity. Based on these, we propose the federated zeroth-order
optimization using trajectory-informed surrogate gradients (FZooS) algorithm
for query- and communication-efficient federated ZOO. Our FZooS achieves
theoretical improvements over the existing approaches, which is supported by
our real-world experiments such as federated black-box adversarial attack and
federated non-differentiable metric optimization.
