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Stochastic Depletion Problems: Effective Myopic Policies for a class of Dynamic Optimization Problems
Carri W. Chan, Vivek F. Farias
TL;DR
The paper studies dynamic stochastic optimization problems for which optimal policies can be difficult to compute. It identifies two properties that guarantee a factor-2 myopic policy and verifies them for several problem families, yielding efficient near-optimal policies.
Problem
Method
The paper defines Stochastic Depletion Problems and identifies two properties, VFM and IR, that guarantee a myopic policy’s performance relative to the optimal adaptive policy.
Source: §1 · p. 2Results
The VFM and IR properties hold for submodular-reward and decaying-linear stochastic depletion families, where the myopic policy’s performance analysis is sharp.
Source: §1 · pp. 2–3Takeaways & Limitations
The verified properties yield efficient near-optimal control policies for several dynamic stochastic optimization problems.
Source: §1 · p. 3Identifying additional families that satisfy VFM and IR, and determining the optimality of the approximation schemes, remain open issues.
Source: §7 · p. 23
Abstract
from arXiv · showhide
This paper presents a general class of dynamic stochastic optimization problems we refer to as Stochastic Depletion Problems. A number of challenging dynamic optimization problems of practical interest are stochastic depletion problems. Optimal solutions for such problems are difficult to obtain, both from a pragmatic computational perspective as also from a theoretical perspective. As such, simple heuristics are highly desirable. We isolate two simple properties that, if satisfied by a problem within this class, guarantee that a myopic policy incurs a performance loss of at most 50 % relative to the optimal adaptive control policy for that problem. We are able to verify that these two properties are satisfied for several interesting families of stochastic depletion problems and as a consequence identify efficient near-optimal control policies for a number of interesting dynamic stochastic optimization problems.
1 Introduction
The paper introduces Stochastic Depletion Problems, a class of dynamic stochastic optimization problems for which optimal policies can be difficult to compute. It shows that two properties give myopic policies a factor-2 guarantee and verifies them for several families and applications.
1 Introduction
- The VFM and IR properties guarantee that a myopic policy earns at least half the expected reward of the optimal adaptive policy.The policy maximizes expected reward over the next time step using information available so far.Source: §1 · p. 2
- Both properties hold for submodular-reward and decaying-linear stochastic depletion families, and the myopic policy’s performance analysis is sharp for both.The first family uses a non-decreasing submodular reward; the second gives item rewards that are non-increasing over time.Source: §1 · pp. 2–3
- For call-center queueing models with pre-emption, the framework yields a near-optimal myopic service policy for a broad class of performance metrics.The policy is analogous to c − µ scheduling rules.Source: §1 · p. 3
- For stochastic variants of deterministic online problems, myopic policies have constant-factor guarantees typically no worse, and sometimes better, than guarantees for deterministic counterparts.The paper includes stochastic broadcast scheduling and a stochastic AdWords Assignment scheme with a 3 + ǫ guarantee matching the deterministic version’s best known guarantee.Source: §1 · pp. 3–4
- For dynamic product line design, the analysis concludes that the optimal policy is well approximated by solving a sequence of static product line design problems.Source: §1 · p. 3
- The framework models stochastic selection as attempts whose success is determined by an exogenous stochastic process, extending submodular maximization over simple matroids to stochastic settings.The paper presents the framework as a unified way to analyze such stochastic optimization problems.Source: §1 · pp. 2, 4
2 Model
The model represents a finite-horizon system in which activities deplete item types stochastically and earn rewards. It defines adaptive policies over system states, while noting that computing an optimal policy is generally intractable and that a rational clairvoyant case is NP-hard.
2 Model
- At each discrete time step, an activity causes random depletion of available items, with depletion statistics governed by an exogenous process.The state tracks remaining items, time, and histories of the processes; the activity chosen affects the depletion distribution.Source: §2 · p. 5
- The objective is to choose activities adaptively to maximize expected reward over the first T time steps.Rewards are non-negative and depend on items depleted and time; the model sets terminal-time reward to zero.Source: §2 · p. 5
- The formulation permits item arrivals and depletion deadlines by restricting when a type can be depleted.It assumes a known bound on total arrivals in any period.Source: §2 · p. 6
- The optimal value function satisfies a Bellman equation, but dynamic programming is computationally intractable because the state space is exponentially large.This holds even when the exogenous processes are known in advance.Source: §2 · p. 6
- The rational clairvoyant Stochastic Depletion problem is NP-hard.The paper establishes this result by reduction from Set-Cover.Source: §2 · pp. 6–7
3 A Myopic Heuristic for Stochastic Depletion
The myopic policy chooses the activity maximizing expected reward in the next time step, while adapting to available information but ignoring effects on future states. Under Value Function Monotonicity and Immediate Rewards, it is a 2-approximation; an approximate myopic oracle also yields a stated approximation guarantee.
3 A Myopic Heuristic for Stochastic Depletion
- The myopic policy maximizes expected one-step reward using current information, but ignores how its choice affects rewards in future states.The action set may be exponentially large, though some problems admit efficient implicit representations for the maximization.Source: §3 · p. 7
3.1 The Myopic Heuristic is an online 2-Approximation Algorithm: Sufficient Conditions
- Value Function Monotonicity requires that, at the same time, having more items available does not reduce the optimal value.Immediate Rewards requires that depleting items without using a time step is at least as valuable as taking a time step to deplete them.Source: §3.1 · p. 8
- When the optimal value function satisfies both properties, the myopic policy is guaranteed to achieve at least half the optimal value.The proof compares against a clairvoyant policy that knows the sample paths of the exogenous P_t processes but not the realization of X_t until time t + 1, yielding a 2-competitive online guarantee.Source: §3.1 · pp. 8–9
3.1.1 Performance with an approximate Myopic Oracle:
- An α-approximation oracle for the myopic subproblem yields a 1+α approximation for the original problem, provided the two value-function properties hold.The paper notes that such an oracle is useful when the myopic maximization itself is difficult.Source: §3.1.1 · p. 10
4 Families Satisfying the VFM and IR Properties
The paper identifies two families of stochastic depletion problems satisfying the properties needed for a factor-2 myopic guarantee. It also gives an instance showing that this bound is sharp.
4 Families Satisfying the VFM and IR Properties
- The sufficient properties are difficult to check generally, so the paper establishes these two families as tractable settings for the myopic guarantee.The authors note that other families may also satisfy the properties.Source: §4 · p. 10
4.1 Submodular Stochastic Depletion Problems
- For submodular stochastic depletion problems, the myopic heuristic is a 2-approximation.The reward function is monotone and submodular in depleted items; the analysis verifies both Immediate Rewards and Value Function Monotonicity for this family.Source: §4.1 · pp. 10–13
4.2 Linear Decaying Stochastic Depletion Problems
- For linear decaying stochastic depletion problems, the myopic heuristic is also a 2-approximation.These problems have nonnegative linear rewards that are non-increasing over time, and satisfy both required properties.Source: §4.2 · pp. 13–14
4.3 A Worst Case Example
- The two identified families guarantee a factor-2 bound, and the analysis is sharp: an instance in both families approaches that loss.The optimal policy can earn a factor of 2 − ε more than the myopic policy, with ε arbitrarily small.Source: §4.3 · pp. 14–15
5 Applications: Stochastic Control
The applications establish uniform performance guarantees for myopic policies across several stochastic-control problems. They include queueing, broadcast scheduling, and dynamic product-line design, with a factor-2 guarantee in the cited settings.
5.1 Service Policies for Simple Queueing Models
- The queueing model’s myopic policy earns expected rewards within a factor of 2 of the optimum, even against an optimal policy with full job-arrival knowledge.Jobs are matched to servers, and completion rewards are non-negative and non-increasing with delay.Source: §5.1 · pp. 15–16
5.2 Stochastic Broadcast Scheduling
- The broadcast-scheduling myopic policy is within a factor of 2 of optimal; with known requests, common deadlines, and static channels, it is optimal.For general broadcast scheduling, the myopic rule treats channels as static and assumes no further arrivals; repeated channel re-estimation supports this design approach.Source: §5.2 · pp. 17–18
5.3 Dynamic Product Line Design
- For dynamic product-line design, a myopic policy achieves expected revenues within a factor of 2 of optimal by solving a static design problem with revised demand and market-size estimates.The model allows general demand seasonality and explicitly represents the effect of current sales on future demand.Source: §5.3 · pp. 18–19
6 Applications: Stochastic variants of submodular maximization problems over Matroids
The section applies stochastic depletion to submodular maximization and stochastic AdWords assignment. It gives factor-2 myopic guarantees for several settings and a 3 + ǫ guarantee for a rectangle-packing variant of AdWords.
6 Applications: Stochastic variants of submodular maximization problems over Matroids
- For non-decreasing submodular maximization, the local greedy heuristic is a 2-approximation over partition matroids and the myopic heuristic is a 2-approximation over cardinality matroids.The cardinality-matroid guarantee is weaker than the known e−1 approximation ratio cited in the paper.Source: §6 · pp. 20–21
- In the stochastic selection problem, which allows k attempts with probabilistic success, the myopic heuristic is a 2-approximation.When success probabilities are constant across attempts and elements, the paper also states an e−1 approximation for the myopic heuristic.Source: §6 · p. 21
6.1 Cost-per-Click AdWords Assignment
- The myopic heuristic is a 2-approximation for Cost-per-Click AdWords Assignment, where advertisers pay only when assigned ads are clicked.The stochastic formulation also allows exogenous advertiser arrivals and departures.Source: §6.1 · p. 22
- 3 + ǫ is the myopic guarantee for rectangle-constrained AdWords assignment, matching the best known guarantee for the deterministic problem.The guarantee uses a 2 + ǫ approximation algorithm for the myopic rectangle-packing subproblem.Source: §6.1 · pp. 22–23
7 Concluding Remarks
The paper introduces a broad stochastic-depletion framework in which simple myopic policies can be near-optimal under stated conditions. It also identifies open questions about extending the framework and establishing the limits of its guarantees.
7 Concluding Remarks
- The framework provides conditions under which a myopic policy is near-optimal and yields approximation algorithms for dynamic optimization applications.The authors also verify those conditions for broad families of stochastic depletion problems.Source: §7 · p. 23
- The paper proposes studying k-step lookahead policies and asks whether, or under what conditions, they may dominate one-step myopic policies.Source: §7 · p. 23
- The stochastic generalization of generalized assignment with random successful item placement does not reduce to a stochastic depletion problem, despite its similarity.Source: §7 · p. 23
- Identifying additional families that satisfy VFM and IR, and conditions under which either property may fail, remains open.Source: §7 · p. 23
- The authors ask whether a factor-2 approximation is optimal for either problem family covered by their guarantee, noting that e−1 is the best guarantee one may expect unless P = NP.A simple myopic scheme comes close to that guarantee, which may be established in the generality of the stochastic depletion framework.Source: §7 · p. 23
A Miscellaneous Technical Proofs
The technical appendix supports the Immediate Rewards property for linear decaying stochastic depletion problems and includes arguments about optimal first actions and reward bounds. It also records an equality case when the probabilities are constant across activities.
A Miscellaneous Technical Proofs
- For linear decaying stochastic depletion problems, Lemma 3 states the Immediate Rewards property for all states and eligible α values.Source: A Miscellaneous Technical Proofs · p. 25
- The reward-bound proof partitions jobs into fictitious and real jobs, bounds fictitious-job rewards using non-increasing weights, and bounds real-job rewards by the optimal value from the reduced state.Source: A Miscellaneous Technical Proofs · pp. 25–26
- When the relevant probability is constant across activities, the stated myopic policy is optimal for every state.Source: A Miscellaneous Technical Proofs · p. 26
- An optimal policy can be chosen to transmit a packet in P* at the first time step, by constructing a policy that contradicts the assumption that no optimal policy does so.The proof compares policies up to the first such transmission and concludes that the constructed policy is also optimal.Source: A Miscellaneous Technical Proofs · p. 26