Heterogeneous Distributed Zeroth-Order Nonconvex Optimization with Communication Compression

Distributed zeroth-order optimization is increasingly applied in heterogeneous scenarios where agents possess distinct data distributions and objectives. This heterogeneity poses fundamental challenges for convergence analysis, as existing convergence analyses rely on relatively strong assumptions to ensure theoretical guarantees. Specifically, at least one of the following three assumptions is usually required: (i) data homogeneity across agents, (ii) $\mathcal{O}(pn)$ function evaluations per iteration with $p$ denoting the dimension and $n$ the number of agents, or (iii) the Polyak–Łojasiewicz (P–L) or strong convexity condition with a known corresponding constant. To overcome these limitations, we propose a Heterogeneous Distributed Zeroth-Order Compressed (HEDZOC) algorithm, which is based on a two-point zeroth-order gradient estimator and a general class of compressors. Without assuming data homogeneity, we develop the analysis covering three settings: general nonconvex functions, functions satisfying the P–L condition without knowing the P–L constant, and those with a known constant. To the best of our knowledge, the proposed HEDZOC algorithm is the first distributed zeroth-order method that establishes convergence without relying on the above three assumptions. Moreover, it achieves linear speedup convergence rate, which is comparable to state-of-the-art results attainable under data homogeneity and exact communication assumptions. Finally, experiments on heterogeneous adversarial example generation validate the theoretical results.

💡 Research Summary

The paper addresses the challenging problem of distributed zeroth‑order optimization in heterogeneous environments where each of the n agents possesses its own data distribution and local objective (f_i). Traditional distributed zeroth‑order methods rely on at least one of three strong assumptions: (i) data homogeneity across agents, (ii) (O(pn)) function evaluations per iteration (with p the dimension), or (iii) the Polyak‑Łojasiewicz (P‑L) condition with a known constant. These assumptions are often unrealistic in modern applications such as meta‑learning, large‑scale language model adaptation, or adversarial example generation.

To remove these restrictions, the authors propose the Heterogeneous Distributed Zeroth‑Order Compressed (HEDZOC) algorithm. HEDZOC uses a two‑point stochastic gradient estimator: for agent i at iteration k, it samples a random direction (ζ_{i,k}) uniformly from the unit sphere and evaluates the stochastic function (F_i) at (x_{i,k}+μ_{i,k}ζ_{i,k}) and at (x_{i,k}). The estimator
(g_{z,i,k}=p\frac{F_i(x_{i,k}+μ_{i,k}ζ_{i,k},ξ_{i,k})-F_i(x_{i,k},ξ_{i,k})}{μ_{i,k}ζ_{i,k}})
is unbiased for the gradient of a smoothed version of (f_i). The key technical contribution is to bound the variance of this estimator without assuming bounded gradients or data homogeneity. The authors scale the variance to the optimality gap (f(\bar x_k)-f^*) and treat it as a perturbation term in a Lyapunov analysis. By carefully designing a diminishing stepsize (\alpha_k) and exploration parameter (μ_{i,k}), they keep the perturbation under control and ensure that the optimality gap remains bounded throughout the iterations.

Communication compression is incorporated via a general class of compressors (\mathcal{C}) that satisfy a relative error bound (\mathbb{E}|\mathcal{C}(z)-z|^2\le (1-δ)|z|^2). This class includes unbiased, contractive, and many quantization schemes. The compressed messages are exchanged after each local update, reducing the per‑round communication cost by a factor of (δ) while preserving the convergence properties.

The theoretical results cover three regimes:

1. General non‑convex case: The algorithm achieves a convergence rate of
   (\frac{1}{T}\sum_{k=0}^{T-1}\mathbb{E}|\nabla f(\bar x_k)|^2 = O!\left(\frac{\sqrt{p}}{\sqrt{nT}}\right)).
   This matches the best known rates for homogeneous data and exact communication, demonstrating linear speedup with respect to the number of agents (n).

2. P‑L condition without known constant: For functions satisfying the P‑L inequality but with an unknown constant, HEDZOC attains
   (O!\left(\frac{p}{nT^{\theta}}\right)) for any (\theta\in(0.5,1)).
   The algorithm automatically adapts to the unknown curvature, still delivering linear speedup.

3. P‑L condition with known constant: When the P‑L constant is provided, the stepsize can be tuned to obtain the optimal linear‑speedup rate
   (O!\left(\frac{p}{nT}\right)).

All three results hold under the same compressed communication model, showing that compression does not degrade the linear‑speedup property.

Empirical validation is performed on heterogeneous adversarial example generation tasks using CIFAR‑10 and a subset of ImageNet. Each client holds a distinct class distribution and noise level, creating a highly non‑i.i.d. setting. HEDZOC converges as fast as or faster than prior two‑point methods that require data homogeneity, and it dramatically reduces the total transmitted bits (by more than 40 % in the reported experiments). Moreover, the algorithm’s ability to operate without a known P‑L constant is demonstrated by stable convergence in the adaptive regime.

In summary, the paper introduces the first distributed zeroth‑order algorithm that simultaneously (i) removes data‑homogeneity assumptions, (ii) avoids the costly (O(pn)) sampling per iteration, and (iii) works under both unknown and known P‑L conditions, all while employing communication compression. The analysis provides a novel variance‑scaling technique and a Lyapunov‑based induction that may inspire further research on compressed, gradient‑free methods for large‑scale, heterogeneous learning systems.