**JacSketch**

We develop a new family of variance reduced stochastic gradient descent methods for minimizing the average of a very large number of smooth functions. Our method –JacSketch– is motivated by novel developments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. In each iteration, JacSketch efficiently updates the Jacobian matrix by first obtaining a random linear measurement of the true Jacobian through (cheap) sketching, and then projecting the previous estimate onto the solution space of a linear matrix equation whose solutions are consistent with the measurement. The Jacobian estimate is then used to compute a variance-reduced unbiased estimator of the gradient. Our strategy is analogous to the way quasi-Newton methods maintain an estimate of the Hessian, and hence our method can be seen as a stochastic quasi-gradient method. We prove that for smooth and strongly convex functions, JacSketch converges linearly with a meaningful rate dictated by a single convergence theorem which applies to general sketches. We also provide a refined convergence theorem which applies to a smaller class of sketches. This enables us to obtain sharper complexity results for variants of JacSketch with importance sampling. By specializing our general approach to specific sketching strategies, JacSketch reduces to the stochastic average gradient (SAGA) method, and several of its existing and many new minibatch, reduced memory, and importance sampling variants. Our rate for SAGA with importance sampling is the current best-known rate for this method, resolving a conjecture by Schmidt et al (2015). The rates we obtain for minibatch SAGA are also superior to existing rates. … **Asynchronous Parallel Sampling Gradient Boosting Decision Tree (asynch-SGBDT)**

With the development of big data technology, Gradient Boosting Decision Tree, i.e. GBDT, becomes one of the most important machine learning algorithms for its accurate output. However, the training process of GBDT needs a lot of computational resources and time. In order to accelerate the training process of GBDT, the asynchronous parallel sampling gradient boosting decision tree, abbr. asynch-SGBDT is proposed in this paper. Via introducing sampling, we adapt the numerical optimization process of traditional GBDT training process into stochastic optimization process and use asynchronous parallel stochastic gradient descent to accelerate the GBDT training process. Meanwhile, the theoretical analysis of asynch-SGBDT is provided by us in this paper. Experimental results show that GBDT training process could be accelerated by asynch-SGBDT. Our asynchronous parallel strategy achieves an almost linear speedup, especially for high-dimensional sparse datasets. … **Quantum Variational Autoencoder (QVAE)**

Variational autoencoders (VAEs) are powerful generative models with the salient ability to perform inference. Here, we introduce a \emph{quantum variational autoencoder} (QVAE): a VAE whose latent generative process is implemented as a quantum Boltzmann machine (QBM). We show that our model can be trained end-to-end by maximizing a well-defined loss-function: a ‘quantum’ lower-bound to a variational approximation of the log-likelihood. We use quantum Monte Carlo (QMC) simulations to train and evaluate the performance of QVAEs. To achieve the best performance, we first create a VAE platform with discrete latent space generated by a restricted Boltzmann machine (RBM). Our model achieves state-of-the-art performance on the MNIST dataset when compared against similar approaches that only involve discrete variables in the generative process. We consider QVAEs with a smaller number of latent units to be able to perform QMC simulations, which are computationally expensive. We show that QVAEs can be trained effectively in regimes where quantum effects are relevant despite training via the quantum bound. Our findings open the way to the use of quantum computers to train QVAEs to achieve competitive performance for generative models. Placing a QBM in the latent space of a VAE leverages the full potential of current and next-generation quantum computers as sampling devices. …

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May 2018

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