Compression in a Distributed Setting

Motivated by an attempt to understand the formation and development of (human) language, we introduce a "distributed compression" problem. In our problem a sequence of pairs of players from a set of K players are chosen and tasked to communicate messages drawn from an unknown distribution Q. Arguably languages are created and evolve to compress frequently occurring messages, and we focus on this aspect. The only knowledge that players have about the distribution Q is from previously drawn samples, but these samples differ from player to player. The only common knowledge between the players is restricted to a common prior distribution P and some constant number of bits of information (such as a learning algorithm). Letting T_epsilon denote the number of iterations it would take for a typical player to obtain an epsilon-approximation to Q in total variation distance, we ask whether T_epsilon iterations suffice to compress the messages down roughly to their entropy and give a partial positive answer. We show that a natural uniform algorithm can compress the communication down to an average cost per message of O(H(Q) + log (D(P || Q)) in tilde{O}(T_epsilon) iterations while allowing for O(epsilon)-error, where D(. || .) denotes the KL-divergence between distributions. For large divergences this compares favorably with the static algorithm that ignores all samples and compresses down to H(Q) + D(P || Q) bits, while not requiring T_epsilon * K iterations that it would take players to develop optimal but separate compressions for each pair of players. Along the way we introduce a "data-structural" view of the task of communicating with a natural language and show that our natural algorithm can also be implemented by an efficient data structure, whose storage is comparable to the storage requirements of Q and whose query complexity is comparable to the lengths of the message to be compressed. Our results give a plausible mathematical analogy to the mechanisms by which human languages get created and evolve, and in particular highlights the possibility of coordination towards a joint task (agreeing on a language) while engaging in distributed learning

A Bit of Secrecy for Gaussian Source Compression

In this paper, the compression of an independent and identically distributed Gaussian source sequence is studied in an unsecure network. Within a game theoretic setting for a three-party noiseless communication network (sender Alice, legitimate receiver Bob, and eavesdropper Eve), the problem of how to efficiently compress a Gaussian source with limited secret key in order to guarantee that Bob can reconstruct with high fidelity while preventing Eve from estimating an accurate reconstruction is investigated. It is assumed that Alice and Bob share a secret key with limited rate. Three scenarios are studied, in which the eavesdropper ranges from weak to strong in terms of the causal side information she has. It is shown that one bit of secret key per source symbol is enough to achieve perfect secrecy performance in the Gaussian squared error setting, and the information theoretic region is not optimized by joint Gaussian random variables

Centralized and distributed semi-parametric compression of piecewise smooth functions

This thesis introduces novel wavelet-based semi-parametric centralized and distributed compression methods for a class of piecewise smooth functions. Our proposed compression schemes are based on a non-conventional transform coding structure with simple independent encoders and a complex joint decoder. Current centralized state-of-the-art compression schemes are based on the conventional structure where an encoder is relatively complex and nonlinear. In addition, the setting usually allows the encoder to observe the entire source. Recently, there has been an increasing need for compression schemes where the encoder is lower in complexity and, instead, the decoder has to handle more computationally intensive tasks. Furthermore, the setup may involve multiple encoders, where each one can only partially observe the source. Such scenario is often referred to as distributed source coding. In the first part, we focus on the dual situation of the centralized compression where the encoder is linear and the decoder is nonlinear. Our analysis is centered around a class of 1-D piecewise smooth functions. We show that, by incorporating parametric estimation into the decoding procedure, it is possible to achieve the same distortion- rate performance as that of a conventional wavelet-based compression scheme. We also present a new constructive approach to parametric estimation based on the sampling results of signals with finite rate of innovation. The second part of the thesis focuses on the distributed compression scenario, where each independent encoder partially observes the 1-D piecewise smooth function. We propose a new wavelet-based distributed compression scheme that uses parametric estimation to perform joint decoding. Our distortion-rate analysis shows that it is possible for the proposed scheme to achieve that same compression performance as that of a joint encoding scheme. Lastly, we apply the proposed theoretical framework in the context of distributed image and video compression. We start by considering a simplified model of the video signal and show that we can achieve distortion-rate performance close to that of a joint encoding scheme. We then present practical compression schemes for real world signals. Our simulations confirm the improvement in performance over classical schemes, both in terms of the PSNR and the visual quality

Bidirectional compression in heterogeneous settings for distributed or federated learning with partial participation: tight convergence guarantees

We introduce a framework - Artemis - to tackle the problem of learning in a distributed or federated setting with communication constraints and device partial Several workers (randomly sampled) perform the optimization process using a central server to aggregate their computations. To alleviate the communication cost, Artemis allows to compresses the information sent in both directions (from the workers to the server and conversely) combined with a memory It improves on existing algorithms that only consider unidirectional compression (to the server), or use very strong assumptions on the compression operator, and often do not take into account devices partial participation. We provide fast rates of convergence (linear up to a threshold) under weak assumptions on the stochastic gradients (noise's variance bounded only at optimal point) in non-i.i.d. setting, highlight the impact of memory for unidirectional and bidirectional compression, analyze Polyak-Ruppert averaging. We use convergence in distribution to obtain a lower bound of the asymptotic variance that highlights practical limits of compression. And we provide experimental results to demonstrate the validity of our analysis.Comment: 56 pages, 4 theorems, 1 algorithm, code source on GitHu