Linear Convergence of Variance-Reduced Stochastic Gradient without Strong Convexity

Stochastic gradient algorithms estimate the gradient based on only one or a few samples and enjoy low computational cost per iteration. They have been widely used in large-scale optimization problems. However, stochastic gradient algorithms are usually slow to converge and achieve sub-linear convergence rates, due to the inherent variance in the gradient computation. To accelerate the convergence, some variance-reduced stochastic gradient algorithms, e.g., proximal stochastic variance-reduced gradient (Prox-SVRG) algorithm, have recently been proposed to solve strongly convex problems. Under the strongly convex condition, these variance-reduced stochastic gradient algorithms achieve a linear convergence rate. However, many machine learning problems are convex but not strongly convex. In this paper, we introduce Prox-SVRG and its projected variant called Variance-Reduced Projected Stochastic Gradient (VRPSG) to solve a class of non-strongly convex optimization problems widely used in machine learning. As the main technical contribution of this paper, we show that both VRPSG and Prox-SVRG achieve a linear convergence rate without strong convexity. A key ingredient in our proof is a Semi-Strongly Convex (SSC) inequality which is the first to be rigorously proved for a class of non-strongly convex problems in both constrained and regularized settings. Moreover, the SSC inequality is independent of algorithms and may be applied to analyze other stochastic gradient algorithms besides VRPSG and Prox-SVRG, which may be of independent interest. To the best of our knowledge, this is the first work that establishes the linear convergence rate for the variance-reduced stochastic gradient algorithms on solving both constrained and regularized problems without strong convexity.

💡 Research Summary

The paper tackles a fundamental limitation of stochastic gradient descent (SGD) – its sub‑linear convergence caused by gradient variance – by extending variance‑reduced methods to settings without strong convexity. The authors focus on two algorithms: Prox‑SVRG, a well‑known variance‑reduced method for regularized problems, and a newly proposed projected variant called VRPSG (Variance‑Reduced Projected Stochastic Gradient) for constrained problems.

The central theoretical contribution is the introduction and rigorous proof of a “Semi‑Strongly Convex” (SSC) inequality. For a feasible point w, the SSC inequality guarantees that the objective gap (f(w)−f⋆ or F(w)−F⋆) lower‑bounds the squared Euclidean distance from w to the optimal solution set, with a constant β>0. Unlike the classic strong convexity condition, SSC only requires that the smooth part h(·) be strongly convex on any compact convex subset of its domain, while the overall problem may be merely convex. The authors show that under mild assumptions (A1‑A3 for constrained problems, and additionally B1 for regularized problems) the SSC inequality holds for a broad class of machine‑learning models, including Lasso, ℓ₁‑constrained logistic regression, and the dual of linear SVM.

With SSC established, the convergence analysis proceeds without invoking full‑gradient based inequalities that are typical for deterministic methods. Both VRPSG and Prox‑SVRG employ a two‑level loop: an outer epoch computes the full gradient ξ̃, while an inner loop of m stochastic updates uses a variance‑reduced gradient estimator
v_{kt}= (∇f_{i_{kt}}(w_{kt-1})−∇f_{i_{kt}}(w̃_{k-1}))/(n p_{i_{kt}})+ξ̃.
VRPSG then projects the tentative step onto the polyhedral feasible set, whereas Prox‑SVRG applies a proximal operator that incorporates the regularizer r(w).

The main theorems state that if the step size η satisfies η<1/(4L_P) (with L_P = max_i L_i/(n p_i)) and the epoch length m is chosen large enough so that a derived contraction factor ρ<1, then the expected objective gap decays linearly:
E