We study the minimax estimation error for distributed covariance matrix estimation in the vertical-split (feature-split) setting, where two agents each observe different coordinates of $m$ i.i.d. sub-Gaussian samples and communicate a limited number of bits to a central server. While Rahmani et al. [2025] established nearly tight bounds for dense (unstructured) cross-covariance matrices, we investigate whether imposing elementwise $s$-sparsity on the cross-covariance $C_{21}$ can reduce the required communication and sample complexity. In contrast to the horizontal-split setting, where Braverman et al. [2016] showed that sparsity does not reduce communication cost for mean estimation, we prove that sparsity does help for cross-covariance estimation in the vertical split. Specifically, we establish minimax lower bounds showing that the communication budget per agent scales as $B_k = Ω(σ^4 d_k\, s' \log(d_1 d_2/s')/\varepsilon^2)$ and the sample complexity for cross-covariance estimation as $m = Ω(σ^4\, s' \log(d_1 d_2/s')/\varepsilon^2)$, where $s' = s \wedge d_{\min}$. For the $1$-sparse case, this yields an exponential improvement from $d_1 d_2$ to $\log(d_1 d_2)$ compared to the dense rate. Our lower bounds are established via Fano's method with an explicit sparse packing using a Varshamov--Gilbert-type argument for signed partial permutation matrices combined with the Conditional Strong Data Processing Inequality of Rahmani et al. [2025]. We show the bounds are tight with a matching achievable scheme, based on covering-net quantization and entry-wise hard thresholding, that attains the $s$-sparse lower bound up to polylogarithmic factors.

Neural Information Processing Systems http://nips.cc/

Gradient compression is a widely-established remedy to tackle the communication bottleneck in distributed training of large deep neural networks (DNNs). Under the error-feedback framework, Top-k sparsification, sometimes with k as little as 0.1% of the gradient size, enables training to the same model quality as the uncompressed case for a similar iteration count. From the optimization perspective, we find that Top-k is the communication-optimal sparsifier given a per-iteration k element budget. We argue that to further the benefits of gradient sparsification, especially for DNNs, a different perspective is necessary -- one that moves from per-iteration optimality to consider optimality for the entire training. We identify that the total error -- the sum of the compression errors for all iterations -- encapsulates sparsification throughout training.

Federated Learning (FL) is constrained by the communication and energy limitations of decentralized edge devices. While gradient sparsification via Top-K magnitude pruning effectively reduces the communication payload, it remains inherently energy-agnostic. It assumes all parameter updates incur identical downstream transmission and memory-update costs, ignoring hardware realities. We formalize the pruning process as an energy-constrained projection problem that accounts for the hardware-level disparities between memory-intensive and compute-efficient operations during the post-backpropagation phase. We propose Cost-Weighted Magnitude Pruning (CWMP), a selection rule that prioritizes parameter updates based on their magnitude relative to their physical cost. We demonstrate that CWMP is the optimal greedy solution to this constrained projection and provide a probabilistic analysis of its global energy efficiency. Numerical results on a non-IID CIFAR-10 benchmark show that CWMP consistently establishes a superior performance-energy Pareto frontier compared to the Top-K baseline.

Neural Information Processing Systems http://nips.cc/

We identify that the total error -- the sum of the compression errors for all iterations -- encapsulates sparsification throughout training.

Biased gradient compression with error feedback (EF) reduces communication in federated learning (FL), but under non-IID data, the residual error can decay slowly, causing gradient mismatch and stalled progress in the early rounds. We propose step-ahead partial error feedback (SA-PEF), which integrates step-ahead (SA) correction with partial error feedback (PEF). SA-PEF recovers EF when the step-ahead coefficient α = 0 and step-ahead EF (SAEF) when α = 1. For non-convex objectives and δ-contractive compressors, we establish a second-moment bound and a residual recursion that guarantee convergence to stationar-ity under heterogeneous data and partial client participation. To balance SAEF's rapid warm-up with EF's long-term stability, we select α near its theory-predicted optimum. Experiments across diverse architectures and datasets show that SA-PEF consistently reaches target accuracy faster than EF. Modern large-scale machine learning increasingly relies on distributed computation, where both data and compute are spread across many devices. Federated learning (FL) enables model training in this setting without centralizing raw data, enhancing privacy and scalability under heterogeneous client distributions (McMahan et al., 2017; Kairouz et al., 2021). In each synchronous FL round, the server broadcasts the current global model to a subset of clients. These clients perform several steps of stochastic gradient descent (SGD) on their local data and return updates to the server, which aggregates them to form the next global iterate (Huang et al., 2022; Wang & Ji, 2022; Li et al., 2024). Although FL leverages rich distributed data, it faces two key challenges.