A note on sample complexity of multistage stochastic programs

Abstract

We derive a lower bound for the sample complexity of the Sample Average Approximation method for a certain class of multistage stochastic optimization problems. In previous works, upper bounds for such problems were derived. We show that the dependence of the lower bound with respect to the complexity parameters and the problem’s data are comparable to the upper bound’s estimates. Like previous results, our lower bound presents an additional multiplicative factor showing that it is unavoidable for certain stochastic problems.

Introduction

Consider the following $T$-stage stochastic programming problem represented in the nested form

$$\underset{x_1\in X_1}{min}\{f\left(x_1\right)≔F_1\left(x_1\right)+{\mathbb{E}}_{|{\xi }_1}[\underset{x_2\in X_2(x_1,{\xi }_2)}{inf}{F}_2(x_2,{\xi }_2)+{\mathbb{E}}_{|{\xi }_{\left[2\right]}}[\cdots +{\mathbb{E}}_{|{\xi }_{[T-1]}}\left[\underset{x_T\in X_T(x_{T-1},{\xi }_T)}{inf}{F}_T(x_T,{\xi }_T)\right]]]\}\text{,}$$

driven by the random data process ${\xi }_1,\dots ,{\xi }_T$. Here, $x_t\in {\mathbb{R}}^{n_t},t=1,\dots ,T$, are decisions variables, $F_t:{\mathbb{R}}^{n_t}\times {\mathbb{R}}^{d_t}\to \mathbb{R}$ are continuous functions and $X_t:{\mathbb{R}}^{n_{t-1}}\times {\mathbb{R}}^{d_t}\rightrightarrows {\mathbb{R}}^{n_t}$, $t=2,\dots ,T$, are measurable multifunctions. The (continuous) function $F_1:{\mathbb{R}}^{n_1}\to \mathbb{R}$, the (nonempty) closed set $X_1$ and the vector ${\xi }_1$ are deterministic. Moreover, ${\xi }_{\left[t\right]}≔({\xi }_1,\dots ,{\xi }_t)$ denotes the history (information) available until stage $t$ by the decision maker.

If the (conditional) distribution of ${\xi }_t$ (given ${\xi }_{[t-1]}$) is continuous, problem (1) cannot be addressed directly, except for some trivial cases. In fact, the (conditional) expected value operators are multidimensional integrals on ${\mathbb{R}}^{d_t}$, that are typically impossible to evaluate with high accuracy even for moderate values of the dimension.

Hence, one usually makes a discretization of the random data of problem (1) building a scenario tree. A classical idea is to construct the tree via Monte Carlo conditional sampling techniques. Given the scenario tree, one solves the SAA problem, that is, problem (1) with the discrete random data. This is the basic idea of the SAA method.

In general, even if we solve the SAA problem exactly, its first-stage optimal decision will not be optimal for the true problem. So, there exists an error that comes from the fact that we are approximating the true stochastic process. Suppose that the true stochastic problem has an optimal solution. One can investigate sufficient conditions on the stage sample sizes $N_2,\dots ,N_T$ in order to guarantee that the following conditions happen (jointly) with probability at least $1-\alpha$: (i) any first-stage $\delta$-optimal solution of the SAA problem is a first-stage $ϵ$-optimal solution of the true problem, and (ii) the set of first-stage $\delta$-optimal solutions of the SAA problem is nonempty; where $ϵ>0$, $\delta \in [0,ϵ)$, and $\alpha \in (0,1)$ are specified parameters that we refer to as complexity parameters. Let us point out that this notion of complexity (with condition (ii) being implicitly assumed) was proposed and studied in  [3], [6], [8].

In  [4], it was given an explicit definition of the sample complexity of SAA method for instances and classes of $T$-stage stochastic optimization problems. In the same reference, it was argued that estimates of the sample sizes derived in  [8], [6] are upper bounds estimates for the sample complexity of static and multistage problems, respectively, satisfying some reasonable regularity conditions. In  [4] it was obtained an explicit upper bound’s estimate for the complexity of $T$-stage problems under relaxed regularity conditions. We will see later that it was important to relax these conditions in order to make a fair comparison between the upper and lower bounds estimates of the sample complexity.

In Section  2, we state the definition of sample complexity for $T$-stage stochastic problems and some extensions on the complexity’s upper bounds obtained in  [4]. In Section  3, we present a family of $T$-stage convex stochastic optimization problems where it is possible to derive a lower bound for the sample complexity of each one of these problems. We apply this result to derive our lower bound for the sample complexity of a family of convex $T$-stage problems. In Section  4, we compare our lower bound with the one derived for multistage financial optimization problems through no-arbitrage reasoning arguments. We also indicate one possible way to extend our results to the class of linear multistage optimization problems. This section is followed by a technical appendix.