/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

5分で分かる!? 有名論文ナナメ読み：Jacob Devlin et al. : BERT : Pre-training of Deep Bidirectional Transformers for Language Understanding

Harmonic Analysis and the Theory of Probability. By S. Bochner pp. 176. 35s. 1955. (California University Press and Cambridge University Press)

Deep Neural Networks trained as image auto-encoders have recently emerged as a promising direction for advancing the state-of-the-art in image compression. The key challenge in learning such networks is twofold: To deal with quantization, and to control the trade-off between reconstruction error (distortion) and entropy (rate) of the latent image representation. In this paper, we focus on the latter challenge and propose a new technique to navigate the rate-distortion trade-off for an image compression auto-encoder. The main idea is to directly model the entropy of the latent representation by using a context model: A 3D-CNN which learns a conditional probability model of the latent distribution of the auto-encoder. During training, the auto-encoder makes use of the context model to estimate the entropy of its representation, and the context model is concurrently updated to learn the dependencies between the symbols in the latent representation. Our experiments show that this approach, when measured in MS-SSIM, yields a state-of-the-art image compression system based on a simple convolutional auto-encoder.

/pdf/conditional-probability-models-for-deep-image-compression-52lqbky8lh.pdf

Conditional Probability Models for Deep Image Compression

Training large machine learning models requires a distributed computing approach, with communication of the model updates being the bottleneck. For this reason, several methods based on the compression (e.g., sparsification and/or quantization) of updates were recently proposed, including QSGD (Alistarh et al., 2017), TernGrad (Wen et al., 2017), SignSGD (Bernstein et al., 2018), and DQGD (Khirirat et al., 2018). However, none of these methods are able to learn the gradients, which renders them incapable of converging to the true optimum in the batch mode, incompatible with non-smooth regularizers, and slows down their convergence. In this work we propose a new distributed learning method --- DIANA --- which resolves these issues via compression of gradient differences. We perform a theoretical analysis in the strongly convex and nonconvex settings and show that our rates are superior to existing rates. Our analysis of block-quantization and differences between $\ell_2$ and $\ell_\infty$ quantization closes the gaps in theory and practice. Finally, by applying our analysis technique to TernGrad, we establish the first convergence rate for this method.

Distributed Learning with Compressed Gradient Differences.

In this paper we introduce a unified analysis of a large family of variants of proximal stochastic gradient descent ({\tt SGD}) which so far have required different intuitions, convergence analyses, have different applications, and which have been developed separately in various communities. We show that our framework includes methods with and without the following tricks, and their combinations: variance reduction, importance sampling, mini-batch sampling, quantization, and coordinate sub-sampling. As a by-product, we obtain the first unified theory of {\tt SGD} and randomized coordinate descent ({\tt RCD}) methods, the first unified theory of variance reduced and non-variance-reduced {\tt SGD} methods, and the first unified theory of quantized and non-quantized methods. A key to our approach is a parametric assumption on the iterates and stochastic gradients. In a single theorem we establish a linear convergence result under this assumption and strong-quasi convexity of the loss function. Whenever we recover an existing method as a special case, our theorem gives the best known complexity result. Our approach can be used to motivate the development of new useful methods, and offers pre-proved convergence guarantees. To illustrate the strength of our approach, we develop five new variants of {\tt SGD}, and through numerical experiments demonstrate some of their properties.

A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent

We consider an unconstrained problem of minimizing a smooth convex function which is only available through noisy observations of its values, the noise consisting of two parts. Similar to stochastic optimization problems, the first part is of stochastic nature. The second part is additive noise of unknown nature, but bounded in absolute value. In the two-point feedback setting, i.e. when pairs of function values are available, we propose an accelerated derivative-free algorithm together with its complexity analysis. The complexity bound of our derivative-free algorithm is only by a factor of $\sqrt{n}$ larger than the bound for accelerated gradient-based algorithms, where $n$ is the dimension of the decision variable. We also propose a non-accelerated derivative-free algorithm with a complexity bound similar to the stochastic-gradient-based algorithm, that is, our bound does not have any dimension-dependent factor except logarithmic. Notably, if the difference between the starting point and the solution is a sparse vector, for both our algorithms, we obtain a better complexity bound if the algorithm uses an $1$-norm proximal setup, rather than the Euclidean proximal setup, which is a standard choice for unconstrained problems

An Accelerated Method for Derivative-Free Smooth Stochastic Convex Optimization

We consider stochastic convex optimization problems with affine constraints and develop several methods using either primal or dual approach to solve it. In the primal case, we use a special penalization technique to make the initial problem more convenient for using optimization methods. We propose algorithms to solve it based on Similar Triangles Method with Inexact Proximal Step for the convex smooth and strongly convex smooth objective functions and methods based on Gradient Sliding algorithm to solve the same problems in the non-smooth case. We prove the convergence guarantees in the smooth convex case with deterministic first-order oracle. 
We propose and analyze three novel methods to handle stochastic convex optimization problems with affine constraints: SPDSTM, R-RRMA-AC-SA$^2$, and SSTM_sc. All methods use stochastic dual oracle. SPDSTM is the stochastic primal-dual modification of STM and it is applied for the dual problem when the primal functional is strongly convex and Lipschitz continuous on some ball. We extend the result from Dvinskikh & Gasnikov (2019) for this method to the case when only biased stochastic oracle is available. R-RRMA-AC-SA$^2$ is an accelerated stochastic method based on the restarts of RRMA-AC-SA$^2$ from Foster et al. (2019) and SSTM_sc is just stochastic STM for strongly convex problems. Both methods are applied to the dual problem when the primal functional is strongly convex, smooth, and Lipschitz continuous on some ball and use stochastic dual first-order oracle. We develop convergence analysis for these methods for unbiased and biased oracles respectively. 
Finally, we apply all the aforementioned results and approaches to solve the decentralized distributed optimization problem and discuss the optimality of the obtained results in terms of communication rounds and the number of oracle calls per node.

Optimal Decentralized Distributed Algorithms for Stochastic Convex Optimization.

We present a unified framework for analyzing local SGD methods in the convex and strongly convex regimes for distributed/federated training of supervised machine learning models. We recover several known methods as a special case of our general framework, including Local-SGD/FedAvg, SCAFFOLD, and several variants of SGD not originally designed for federated learning. Our framework covers both the identical and heterogeneous data settings, supports both random and deterministic number of local steps, and can work with a wide array of local stochastic gradient estimators, including shifted estimators which are able to adjust the fixed points of local iterations for faster convergence. As an application of our framework, we develop multiple novel FL optimizers which are superior to existing methods. In particular, we develop the first linearly converging local SGD method which does not require any data homogeneity or other strong assumptions.

Eduard Gorbunov

Papers

Distributed Learning with Compressed Gradient Differences.

A Unified Theory of SGD: Variance Reduction, Sampling, Quantization and Coordinate Descent

An Accelerated Method for Derivative-Free Smooth Stochastic Convex Optimization

Optimal Decentralized Distributed Algorithms for Stochastic Convex Optimization.

Local SGD: Unified Theory and New Efficient Methods