In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leverage the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the distributed setting, this enables smaller rate-distortion terms which are shown to translate into tighter generalization bounds. The bounds are then applied to the distributed support vector machines (SVM), suggesting that the generalization error of the distributed setting decays faster than that of the centralized one with a factor of $\mathcal{O}(\log(K)/\sqrt{K})$. This finding is validated also experimentally. A similar conclusion is obtained for a multiple-round federated learning setup where each client uses stochastic gradient Langevin dynamics (SGLD).

pdf/rate-distortion-theoretic-bounds-on-generalization-error-for-1th8h101.pdf

Rate-Distortion Theoretic Bounds on Generalization Error for Distributed Learning

In this paper, we provide three applications for $f$-divergences: (i) we introduce Sanov's upper bound on the tail probability of the sum of independent random variables based on super-modular $f$-divergence and show that our generalized Sanov's bound strictly improves over ordinary one, (ii) we consider the lossy compression problem which studies the set of achievable rates for a given distortion and code length. We extend the rate-distortion function using mutual $f$-information and provide new and strictly better bounds on achievable rates in the finite blocklength regime using super-modular $f$-divergences, and (iii) we provide a connection between the generalization error of algorithms with bounded input/output mutual $f$-information and a generalized rate-distortion problem. This connection allows us to bound the generalization error of learning algorithms using lower bounds on the $f$-rate-distortion function. Our bound is based on a new lower bound on the rate-distortion function that (for some examples) strictly improves over previously best-known bounds.

f-divergences and their applications in lossy compression and bounding generalization error

Rate-distortion (R-D) function, a key quantity in information theory, characterizes the fundamental limit of how much a data source can be compressed subject to a fidelity criterion, by any compression algorithm. As researchers push for ever-improving compression performance, establishing the R-D function of a given data source is not only of scientific interest, but also sheds light on the possible room for improving compression algorithms. Previous work on this problem relied on distributional assumptions on the data source (Gibson, 2017) or only applied to discrete data. By contrast, this paper makes the first attempt at an algorithm for sandwiching the R-D function of a general (not necessarily discrete) source requiring only i.i.d. data samples. We estimate R-D sandwich bounds on Gaussian and high-dimension banana-shaped sources, as well as GAN-generated images. Our R-D upper bound on natural images indicates room for improving the performance of state-of-the-art image compression methods by 1 dB in PSNR at various bitrates.

Towards Empirical Sandwich Bounds on the Rate-Distortion Function

Artifical Neural-Network-based (ANN-based) lossy compressors have recently obtained striking results on several sources. Their success may be ascribed to an ability to identify the structure of low-dimensional manifolds in high-dimensional ambient spaces. Indeed, prior work has shown that ANN-based compressors can achieve the optimal entropy-distortion curve for some such sources. In contrast, we determine the optimal entropy-distortion tradeoffs for two low-dimensional manifolds with circular structure and show that state-of-the-art ANN-based compressors fail to optimally compress them. 

/pdf/do-neural-networks-compress-manifolds-optimally-3qnjz8g1.pdf

Do Neural Networks Compress Manifolds Optimally?

In this paper, we use tools from rate-distortion theory to establish new upper bounds on the generalization error of statistical distributed learning algorithms. Specifically, there are $K$ clients whose individually chosen models are aggregated by a central server. The bounds depend on the compressibility of each client's algorithm while keeping other clients' algorithms un-compressed, and leverage the fact that small changes in each local model change the aggregated model by a factor of only $1/K$. Adopting a recently proposed approach by Sefidgaran et al., and extending it suitably to the distributed setting, this enables smaller rate-distortion terms which are shown to translate into tighter generalization bounds. The bounds are then applied to the distributed support vector machines (SVM), suggesting that the generalization error of the distributed setting decays faster than that of the centralized one with a factor of $\mathcal{O}(\log(K)/\sqrt{K})$. This finding is validated also experimentally. A similar conclusion is obtained for a multiple-round federated learning setup where each client uses stochastic gradient Langevin dynamics (SGLD). 

/pdf/rate-distortion-theoretic-bounds-on-generalization-error-for-j4j4ptll.pdf

Rate-Distortion Theoretic Bounds on Generalization Error for Distributed
  Learning

This paper is concerned with a rate-distortion theory for sequences of i.i.d. random variables with general distribution supported on general sets including manifolds and fractal sets. Manifold structures are prevalent in data science, e.g., in compressed sensing, machine learning, image processing, and handwritten digit recognition. Fractal sets find application in image compression and in modeling of Ethernet traffic. We derive a lower bound on the (single-letter) rate-distortion function that applies to random variables $X$ of general distribution $\mu_{X}$ and for continuous $X$ reduces to the classical Shannon lower bound. Moreover, our lower bound is explicit up to a parameter obtained by solving a convex optimization problem in a nonnegative real variable. The only requirement for the bound to apply is the existence of a $\sigma$ -finite reference measure $\mu$ , for $X$ (i.e., a measure $\mu$ with $\mu x\ll\mu$ and such that the generalized entropy $h_{\mu}(X)$ is finite) satisfying a certain subregularity condition. This condition is very general and prevents the reference measure $\mu$ from being highly concentrated on balls of small radii. To illustrate the wide applicability of our result, we evaluate the lower bound for a random variable distributed uniformly on a manifold, namely, the unit circle, and a random variable distributed uniformly on a self-similar set, namely, the middle third Cantor set.

Rate-Distortion Theory for General Sets and Measures

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Rate-Distortion Theory for General Sets and Measures