Benjamin Dubois-Taine

Benjamin Dubois-Taine, Francis Bach, Quentin Berthet et al.

We consider the problem of minimizing the sum of two convex functions. One of those functions has Lipschitz-continuous gradients, and can be accessed via stochastic oracles, whereas the other is "simple". We provide a Bregman-type algorithm with accelerated convergence in function values to a ball containing the minimum. The radius of this ball depends on problem-dependent constants, including the variance of the stochastic oracle. We further show that this algorithmic setup naturally leads to a variant of Frank-Wolfe achieving acceleration under parallelization. More precisely, when minimizing a smooth convex function on a bounded domain, we show that one can achieve an $ε$ primal-dual gap (in expectation) in $\tilde{O}(1/ \sqrtε)$ iterations, by only accessing gradients of the original function and a linear maximization oracle with $O(1/\sqrtε)$ computing units in parallel. We illustrate this fast convergence on synthetic numerical experiments.

Sharan Vaswani, Benjamin Dubois-Taine, Reza Babanezhad

We aim to make stochastic gradient descent (SGD) adaptive to (i) the noise $σ^2$ in the stochastic gradients and (ii) problem-dependent constants. When minimizing smooth, strongly-convex functions with condition number $κ$, we prove that $T$ iterations of SGD with exponentially decreasing step-sizes and knowledge of the smoothness can achieve an $\tilde{O} \left(\exp \left( \frac{-T}κ \right) + \frac{σ^2}{T} \right)$ rate, without knowing $σ^2$. In order to be adaptive to the smoothness, we use a stochastic line-search (SLS) and show (via upper and lower-bounds) that SGD with SLS converges at the desired rate, but only to a neighbourhood of the solution. On the other hand, we prove that SGD with an offline estimate of the smoothness converges to the minimizer. However, its rate is slowed down proportional to the estimation error. Next, we prove that SGD with Nesterov acceleration and exponential step-sizes (referred to as ASGD) can achieve the near-optimal $\tilde{O} \left(\exp \left( \frac{-T}{\sqrtκ} \right) + \frac{σ^2}{T} \right)$ rate, without knowledge of $σ^2$. When used with offline estimates of the smoothness and strong-convexity, ASGD still converges to the solution, albeit at a slower rate. We empirically demonstrate the effectiveness of exponential step-sizes coupled with a novel variant of SLS.

Benjamin Dubois-Taine, Sharan Vaswani, Reza Babanezhad et al.

Variance reduction (VR) methods for finite-sum minimization typically require the knowledge of problem-dependent constants that are often unknown and difficult to estimate. To address this, we use ideas from adaptive gradient methods to propose AdaSVRG, which is a more robust variant of SVRG, a common VR method. AdaSVRG uses AdaGrad in the inner loop of SVRG, making it robust to the choice of step-size. When minimizing a sum of n smooth convex functions, we prove that a variant of AdaSVRG requires $\tilde{O}(n + 1/ε)$ gradient evaluations to achieve an $O(ε)$-suboptimality, matching the typical rate, but without needing to know problem-dependent constants. Next, we leverage the properties of AdaGrad to propose a heuristic that adaptively determines the length of each inner-loop in AdaSVRG. Via experiments on synthetic and real-world datasets, we validate the robustness and effectiveness of AdaSVRG, demonstrating its superior performance over standard and other "tune-free" VR methods.