Accelerating Stochastic Gradient Descent using Predictive Variance Reduction

During the super nice International Conference on Monte Carlo techniques in the beginning of July in Paris at Université Descartes (photo), which featured many outstanding talks, one by Tong Zhang particularly caught my interest. He talked about several variants of Stochastic Gradient Descent (SGD) that basically use variance reduction techniques from Monte Carlo algorithms in order to improve the convergence rate versus vanilla SGD. Even though some of the papers mentioned in the talk do not always point out the connection to Monte Carlo variance reduction techniques.

One of the first works in this line, Accelerating Stochastic Gradient Descent using Predictive Variance Reduction by Johnson and Zhang, suggests using control variates to lower the variance of the loss estimate. Let L_j(\theta_{t-1}) be the loss for the parameter at t-1 and jth data point, then the usual batch gradient descent update is \theta_{t} =\theta_{t-1} - \frac{\eta_t}{N} \sum_{j=1}^N\nabla L_j(\theta_{t-1}) with \eta_t as step size.

In naive SGD instead one picks a data point index uniformly j \sim \mathrm{Unif}(\{1,\dots,N\}) and uses the update \theta_{t} =\theta_{t-1} - \eta_t \nabla L_j(\theta_{t-1}), usually with a decreasing step size \eta_t to guarantee convergence. The expected update resulting from this Monte Carlo estimate of the batch loss is exactly the batch procedure update. However the variance of the estimate is very high, resulting in slow convergence of SGD after the first steps (even in minibatch variants).

The authors choose a well-known solution to this, namely the introduction of a control variate. Keeping a version of the parameter that is close to the optimum, say \tilde\theta, observe that \nabla L_j(\tilde\theta) -  \frac{1}{N} \sum_{i=1}^N  \nabla L_i(\tilde\theta) has an expected value of 0 and is thus a possible control variate. With the possible downside that whenever \tilde\theta is updated, one has to go over the complete dataset.

The contribution, apart from the novel combination of knowledge, is the proof that this improves convergence. This proof assumes smoothness and strong convexity of the overall loss function and convexity of the L_j for the individual data points and then shows that the proposed procedure (termed stochastic variance reduced gradient or SVRG) enjoys geometric convergence. Even though the proof uses a slightly odd version of the algorithm, namely where \tilde\theta \sim\mathrm{Unif}(\{\theta_0,\dots,\theta_{t-1}\}). Rather simply setting  \tilde\theta = \theta_{t-1} should intuitively improve convergence, but the authors could not report a result on that. Overall a very nice idea, and one that has been discussed in more papers quite a bit, among others by Simon Lacoste-Julien and Francis Bach.

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