Proximal point algorithm revisited, episode 1. The proximally guided subgradient method

This is episode 1 of the three part series that revisits the classical proximal point algorithm. See the previous post for introduction and notation.

The proximally guided subgradient method

As the first example of contemporary applications of the proximal point method, consider the problem of minimizing the expectation:1

\[\min_{x\in {\mathbb R}^d}~ F(x)=\mathbb{E}_{\zeta} f(x,\zeta).\]

Here, \(\zeta\) is a random variable, and the only access to \(F\) is by sampling \(\zeta\). It is difficult to overstate the importance of this problem class (often called stochastic approximation) in large-scale optimization; see e.g. Bottou and Bousquet (2008); Bartlett, Jordan, and McAuliffe (2006).

When the problem is convex, the stochastic subgradient method (Polyak and Juditsky 1992; Robbins and Monro 1951; Nemirovski et al. 2008) has strong theoretical guarantees and is often the method of choice. In contrast, when applied to nonsmooth and nonconvex problems, the behavior of the method is poorly understood. The recent paper (Davis and Grimmer 2017) shows how to use the proximal point method to guide the subgradient iterates in this broader setting, with rigorous guarantees.

Henceforth, assume that the function \(x\mapsto f(x,\zeta)\) is \(\rho\)-weakly convex and \(L\)-Lipschitz for each \(\zeta\). Davis and Grimmer (2017) proposed the scheme outlined below.

Proximally guided stochastic subgradient method

  • Data: \(x_0\in {\mathbb R}^d\), \(\{j_t\}\subset\mathbb{N}\), \(\{\alpha_j\}\subset{\mathbb R}_{++}\)
  • For \(t=0,\ldots,T\) do
    • Set \(y_0=x_t\)
    • For \(j=0,\ldots,j_t-2\) do
      • Sample \(\zeta\) and choose \(v_j\in\partial (f(\cdot,\zeta)+\rho\|\cdot-x_t\|^2)(y_j)\)
      • Set \(y_{j+1}= y_j-\alpha_jv_j\)
    • Set \(x_{t+1}= \frac{1}{j_t}\sum_{j=0}^{j_t-1}y_j\)

The method proceeds by applying a proximal point method with each subproblem approximately solved by a stochastic subgradient method. The intuition is that each proximal subproblem is \(\rho/2\)-strongly convex and therefore according to well-known results (e.g. Lacoste-Julien, Schmidt, and Bach (2012); Rakhlin, Shamir, and Sridharan (2012); Hazan and Kale (2011); Juditsky and Nesterov (2014)), the stochastic subgradient method should converge at the rate \(O(\frac{1}{T})\) on the subproblem, in expectation. This intuition is not quite correct because the objective function of the subproblem is not globally Lipschitz – a key assumption for the \(O(\frac{1}{T})\) rate. Nonetheless, the authors show that warm-starting the subgradient method for each proximal subproblem with the current proximal iterate corrects this issue, yielding a favorable guarantees (Davis and Grimmer 2017 Theorem 1).

To describe the rate of convergence, set \(j_t=t+\lceil 648\log(648)\rceil\) and \(\alpha_j=\tfrac{2}{\rho(j+49)}\) in the Proximally guided stochastic subgradient method. Then the scheme will generate an iterate \(x\) satisfying

\[\mathbb{E}_{\zeta}[\|\nabla F_{2\rho}(x)\|^2]\leq \varepsilon\]

after at most

\[O\left(\frac{\rho^2(F(x_0)-\inf F)^2}{\varepsilon^2}+\frac{L^4 \log^{4}(\varepsilon^{-1})}{\varepsilon^2}\right)\]

subgradient evaluations. This rate agrees with analogous guarantees for stochastic gradient methods for smooth nonconvex functions (Ghadimi and Lan 2013). It is also worth noting that convex constraints on \(x\) can be easily incorporated into the Proximally guided stochastic subgradient method by introducing a nearest-point projection in the definition of \(y_{j+1}\).


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  1. For simplicity of the exposition, the minimization problem is unconstrained. Simple constraints can be accommodated using a projection operation.