Adaptive Sampling line search for local stochastic optimization with integer variables

Abstract

We consider optimization problems with an objective function that is estimable using a Monte Carlo oracle, constraint functions that are known deterministically through a constraint-satisfaction oracle, and integer decision variables. Seeking an appropriately defined local minimum, we propose an iterative adaptive sampling algorithm that, during each iteration, performs a statistical local optimality test followed by a line search when the test detects a stochastic descent direction. We prove a number of results. First, the true function values at the iterates generated by the algorithm form an almost-supermartingale process, and the iterates are absorbed with probability one into the set of local minima in finite time. Second, such absorption happens exponentially fast in iteration number and in oracle calls. This result is analogous to non-standard rate guarantees in stochastic continuous optimization contexts that involve sharp minima. Third, the oracle complexity of the proposed algorithm increases linearly in the dimensionality of the local neighborhood. As a solver, primarily due to combining line searches that use common random numbers with statistical tests for local optimality, the proposed algorithm is effective on a variety of problems. We illustrate such performance using three problem suites, on problems ranging from 25 to 200 dimensions.

Appendices

Procedure LI: line search

Procedure LI, listed in Algorithm 3, is straightforward. Starting with the candidate next iterate $\widetilde{X}_k$, LI successively observes objective estimates obtained with sample size $M_k$ at points that are “closest” to the line $\widetilde{X}_k + t \hat{d}_k, t \in \mathbb {R}$, as long as the observed objective estimates are monotone decreasing. More precisely, given the starting point $W_{0}{:}{=}\widetilde{X}_k$ and direction $\hat{d}_k$, LI obtains objective estimates at points $W_{\ell } {:}{=}{{\,\mathrm{\arg \!\min }\,}}\bigl \{ \Vert x - (\widetilde{X}_k + 2^{\ell -1}\, s_0 \, \hat{d}_k) \Vert :x \in \mathcal {X}\setminus \{\widetilde{X}_k\} \bigr \},$ $\ell = 1,2,3,\ldots$ where $s_0$ is a fixed constant that defaults to $s_0 = 1$. The ${{\,\mathrm{\arg \!\min }\,}}$ operation is computationally trivial since the neighbors of the point $\widetilde{X}_k + t \hat{d}_k$ can be obtained by rounding. The line search proceeds as long as the sequence ${\bar{F}}(W_{\ell },M_k), \ell =0,1,2,\ldots$ is non-increasing, or a pre-specified limit on the maximum number of steps in the line search is reached. Finally, Procedure LI performs a simple bisection search to find a better point between the penultimate point and the last point, as the last step size may be large.

Procedure DA: estimating a descent direction

Procedure DA estimates a descent direction $\hat{d}_k$ at the candidate iterate or finds that is an estimated local minimizer in its $\mathcal {N}_1$-neighborhood. DA performs the following steps:

• 1. DA enumerates the neighbors of looking for (a) at least $d+1$ feasible neighbors that form a simplex with volume in d dimensions, and (b) at least one better neighbor. (See Algorithm 4 steps 1–14.)

• 2. If there are no better neighbors, DA returns with $\mathcal {B}$ is true. (See Algorithm 4 step 15.)

• 3. Otherwise, DA constructs an estimated descent cone and estimated descent direction. (See Algorithm 4 steps 17–21.)

Finally, if DA identifies an estimated better neighbor, it updates the candidate next iterate. As our analysis holds with or without this “hop,” for simplicity, we omit it from Sect. 4.