Server-Guided CAFe-S in Distributed Learning

• The paper demonstrates that CAFe-S replaces per-client error feedback with a server-generated predictor, enabling aggressive compression while preserving client statelessness and privacy.
• CAFe-S computes a candidate update at the server, which clients use to compress the residual of their local updates, thereby ensuring robust convergence even with biased compressors.
• Empirical results reveal that CAFe-S reduces uplink communication by 30–50% and improves test accuracy by leveraging server data representative of the global distribution.

Server-Guided Compressed Aggregate Feedback (CAFe-S) is a communication-efficient distributed learning framework designed to enable highly compressed, bandwidth-optimized model updates in federated and distributed optimization settings. The key innovation of CAFe-S is replacing traditional per-client error-feedback—associated with privacy risks and stateful operation—with a server-generated, globally shared predictor that is used to compensate client updates prior to compression. This approach enables the use of biased compressors without violation of client statelessness or privacy and provably accelerates convergence, especially when the server's guidance is derived from data representative of the global distribution (Ortega et al., 2024, Ortega et al., 27 Dec 2025).

1. Motivation and Core Principles

CAFe-S addresses a central bottleneck in federated learning (FL) and distributed gradient methods: the communication cost of successive client-server model updates, particularly the uplink from clients to a central server. Many practical compressors are biased (e.g., top-k, quantization), leading to error accumulation unless per-client error-feedback is employed. Traditional error-feedback mechanisms require each client to maintain a control variate—state persisted across rounds—which contravenes the privacy and statelessness assumptions prevalent in cross-device FL. CAFe-S eliminates this requirement by introducing a shared, server-generated predictor to facilitate aggressive compression without per-client state (Ortega et al., 2024, Ortega et al., 27 Dec 2025).

In CAFe-S, the server computes a "candidate update", typically using its own private data or, in its absence, the globally aggregated update from the previous round. Each client compresses the residual between its raw update and the candidate, transmitting the result to the server. The server reconstructs the original update by adding the candidate back post-decompression. This shared, stateless correction process enjoys convergence guarantees even under biased compression schemes.

2. Algorithmic Workflow

The operation of CAFe-S is organized around the following steps:

1. Server-side candidate computation: The server computes either the aggregated client update from the previous round ($\Delta_s^{k-1}$) or, when possessing private data, a fresh candidate update ($\Delta_c^k = -\gamma \nabla f_s(x^k)$).
2. Broadcast: The server transmits $(x^k, \Delta_c^k)$ or $(x^k, \Delta_s^{k-1})$ to all clients.
3. Local update and residual formation: Each client computes its local raw update $\Delta_n^k = -\gamma \nabla f_n(x^k)$, and forms the difference vector $u_n^k = \Delta_n^k - \Delta_c^k$.
4. Encoding and transmission: Each client encodes $u_n^k$ using a possibly biased compressor $E(\cdot)$ and sends $\hat u_n^k$ to the server.
5. Decoding and compensation: The server decodes $\hat u_n^k$ and adds back the candidate, reconstructing $q_n^k = D(\hat u_n^k) + \Delta_c^k$.
6. Aggregation and model update: The server aggregates all $q_n^k$, producing the global update $\Delta_s^k = \frac{1}{N} \sum_{n=1}^N q_n^k$, and updates the global model $x^{k+1} = x^k + \Delta_s^k$.

Clients maintain no persistent state, and all compensation is realized with the shared candidate vector, either from server-side data or as the prior aggregate update (Ortega et al., 27 Dec 2025, Ortega et al., 2024).

3. Mathematical Formulation and Theoretical Guarantees

The global objective is

$f(x) = \frac{1}{N} \sum_{n=1}^N f_n(x)$

where $f_n$ is the local loss at client $n$. The server's private dataset, if available, yields a loss $f_s(x)$ and candidate update $\Delta_c^k = -\gamma \nabla f_s(x^k)$.

Compression is performed with a potentially biased operator $C(\cdot)$ satisfying

$$\mathbb{E}\|C(u) - u\|^2 \leq \omega \|u\|^2, \quad 0 \leq \omega < 1$$

where $\omega$ is the contraction parameter.

Under the assumptions:

• L-smoothness: $f$ is differentiable, $\nabla f$ is $L$-Lipschitz, and $f(x) \geq f^\star$.
• Bounded local dissimilarity: There exists $B^2 \geq 1$ such that

$$\frac{1}{N} \sum_{n=1}^N \|\nabla f_n(x)\|^2 \leq B^2 \|\nabla f(x)\|^2$$

• Bounded server-client dissimilarity: There exists $G^2 \geq 0$ such that

$$\frac{1}{N} \sum_{n=1}^N \|\nabla f_n(x) - \nabla f_s(x)\|^2 \leq G^2 \frac{1}{N} \sum_{n=1}^N \|\nabla f_n(x)\|^2$$

the CAFe-S update iterates satisfy

$$\frac{1}{K} \sum_{k=0}^{K-1} \mathbb{E} \|\nabla f(x^k)\|^2 \leq \frac{2(f(x^0)-f^\star)}{\gamma K (1 - \omega G^2 B^2)}$$

for step-size $\gamma \leq 1/L$ and $\omega G^2 B^2 < 1$. The convergence rate to an $\varepsilon$-stationary point is thus

$$\mathcal{O}\left(\frac{1}{\varepsilon (1 - \omega G^2 B^2)}\right)$$

This result demonstrates an advantage for CAFe-S: if the server's data is highly representative (small $G^2$), the compression penalty is minimal, and the convergence rate approaches that of uncompressed distributed gradient descent (Ortega et al., 27 Dec 2025, Ortega et al., 2024).

4. Comparison with Related Frameworks

A comparison of leading distributed learning compression frameworks elucidates CAFe-S's distinct features:

Method	Predictor Used	Step-size	Error-term factor	Statefulness
DCGD	None	$\gamma \leq 1/L$	$\omega B^2$	Per-client memory
CAFe	Previous aggregate ($\Delta_s^{k-1}$)	$\gamma \leq \frac{1-\omega}{L(1+\omega)}$	$\omega B^2 \cdot (1-\omega)$	Stateless
CAFe-S	Server candidate ($\Delta_c^k$)	$\gamma \leq 1/L$	$\omega G^2 B^2$	Stateless

In contrast to DCGD, which requires per-client control variates for error feedback, CAFe and CAFe-S use a single predictor—either the global aggregate or a server-computed candidate—shared across all clients. CAFe-S's use of up-to-date, data-driven candidates ensures smaller residuals and reduced compression error, especially as the representativeness of the server's data improves. In scenarios with significant downlink bandwidth and asymmetric uplink constraints, the additional cost of broadcasting the candidate update from the server is justified by a significant reduction in uplink communication and improved convergence (Ortega et al., 27 Dec 2025, Ortega et al., 2024).

5. Empirical Evaluation and Practical Performance

Experiments on standard benchmarks (MNIST, EMNIST, CIFAR-100) with both IID and non-IID data distributions demonstrate that CAFe-S achieves superior test accuracy and faster convergence compared to both direct compression and classical error-feedback under highly aggressive compression regimes. In typical FL and distributed learning settings:

• CAFe-S achieves communication-round savings of 30–50% at fixed accuracy compared to DCGD.
• The "compression gain ratio" $\|\Delta_n^k - \Delta_c^k\| / \|\Delta_n^k\|$ is well below 1 for much of training, confirming the predictor efficacy.
• CAFe-S performance improves nearly monotonically with the representativeness of the server's data, as varied by a parameter $\beta \in [0,1]$ controlling overlap with the global distribution.
• When the server dataset is too small or unrepresentative, CAFe (which uses the aggregate, albeit stale, as predictor) may outperform CAFe-S due to lower variance in the predictor update (Ortega et al., 27 Dec 2025, Ortega et al., 2024).

6. Limitations, Extensions, and Future Directions

CAFe-S requires the server to broadcast either the previous aggregate or a fresh candidate update per round, adding limited downlink overhead (one full-precision vector per round). Clients can mitigate this by re-computing aggregates locally if minor additional memory is acceptable.

The theoretical guarantees assume bounded gradient dissimilarity—if this parameter is large (extreme heterogeneity), the compression penalty can dominate. However, empirical results indicate CAFe-S retains robustness in practice. CAFe-S generalizes to more advanced optimization protocols, including local multi-step training (as in FedAvg), momentum, decentralized settings, and can be integrated with adaptive (side-information-based) compressors or differential privacy mechanisms (Ortega et al., 2024, Ortega et al., 27 Dec 2025).

The principal future direction is the systematic exploitation of server-guided predictors beyond small centralized datasets, potentially leveraging self-supervised pre-training or synthetic data to further improve representativeness and overall communication efficiency.