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Minimal Communication-Cost Statistical Learning

Milad Sefidgaran, Abdellatif Zaidi, Piotr Krasnowski

TL;DR

This work addresses the dual objective of achieving low inference risk and minimal communication cost in distributed learning by jointly designing training and source coding. It introduces a $P_{\hat{W}|W}$-agnostic encoding scheme built on a shared codebook generated from a prior $Q$ and common randomness, with a reconstruction rule $\mathcal{D}(K,W_{\epsilon})=\tilde{W}_{\mathbf{U}}[K]+W_{\epsilon}$ and an index-based transmission framework, plus an optional refinement. The core contribution is a set of in-expectation and one-shot guarantees linking the compressibility of the learned model (via $D_{KL}(P_{\hat{W}|W}||Q)$) to both the empirical risk and the generalization error, as well as to the communication rate, using Ordered Random Coding (ORC) and a one-shot vector-quantizer encoder. Collectively, the results reveal a principled, quantifiable trade-off between model precision, generalization, and communication, enabling practical design of low-cost, high-performance distributed learning systems.

Abstract

A client device which has access to $n$ training data samples needs to obtain a statistical hypothesis or model $W$ and then to send it to a remote server. The client and the server devices share some common randomness sequence as well as a prior on the hypothesis space. In this problem a suitable hypothesis or model $W$ should meet two distinct design criteria simultaneously: (i) small (population) risk during the inference phase and (ii) small 'complexity' for it to be conveyed to the server with minimum communication cost. In this paper, we propose a joint training and source coding scheme with provable in-expectation guarantees, where the expectation is over the encoder's output message. Specifically, we show that by imposing a constraint on a suitable Kullback-Leibler divergence between the conditional distribution induced by a compressed learning model $\widehat{W}$ given $W$ and the prior, one guarantees simultaneously small average empirical risk (aka training loss), small average generalization error and small average communication cost. We also consider a one-shot scenario in which the guarantees on the empirical risk and generalization error are obtained for every encoder's output message.

Minimal Communication-Cost Statistical Learning

TL;DR

This work addresses the dual objective of achieving low inference risk and minimal communication cost in distributed learning by jointly designing training and source coding. It introduces a -agnostic encoding scheme built on a shared codebook generated from a prior and common randomness, with a reconstruction rule and an index-based transmission framework, plus an optional refinement. The core contribution is a set of in-expectation and one-shot guarantees linking the compressibility of the learned model (via ) to both the empirical risk and the generalization error, as well as to the communication rate, using Ordered Random Coding (ORC) and a one-shot vector-quantizer encoder. Collectively, the results reveal a principled, quantifiable trade-off between model precision, generalization, and communication, enabling practical design of low-cost, high-performance distributed learning systems.

Abstract

A client device which has access to training data samples needs to obtain a statistical hypothesis or model and then to send it to a remote server. The client and the server devices share some common randomness sequence as well as a prior on the hypothesis space. In this problem a suitable hypothesis or model should meet two distinct design criteria simultaneously: (i) small (population) risk during the inference phase and (ii) small 'complexity' for it to be conveyed to the server with minimum communication cost. In this paper, we propose a joint training and source coding scheme with provable in-expectation guarantees, where the expectation is over the encoder's output message. Specifically, we show that by imposing a constraint on a suitable Kullback-Leibler divergence between the conditional distribution induced by a compressed learning model given and the prior, one guarantees simultaneously small average empirical risk (aka training loss), small average generalization error and small average communication cost. We also consider a one-shot scenario in which the guarantees on the empirical risk and generalization error are obtained for every encoder's output message.
Paper Structure (7 sections, 2 theorems, 45 equations, 1 figure)

This paper contains 7 sections, 2 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem setup
  3. In-expectation performance
  4. One-shot performance
  5. Proof of Theorem \ref{['th:expectation']}
  6. Proof of Theorem \ref{['th:one-shot']}
  7. Proof of the claim (12)

Key Result

Theorem 1

Suppose that the learning algorithm $\mathcal{A}(S)$ induces $P_{W|S}$. Suppose the loss function is $\mathfrak{L}$-Lipschitz, i.e., $|\ell(z,w)-\ell(z,w')|\leq \mathfrak{L} \|w-w'\|$ for all $w,w'\in \mathcal{W}$ and $z \in \mathcal{Z}$. Consider a quantization set $\mathcal{\hat{W}}\subseteq \math where $\mathbb{E}_K[\cdot]$ denotes the expectation with respect to the stochasticity of the encode

Figures (1)

  • Figure 1: Considered setup for joint local training and remote source coding

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2