[Paper Review] An adaptive stochastic optimization algorithm for resource allocation

Xavier Fontaine, Shie Mannor+1 more|arXiv (Cornell University)|Feb 12, 2019

Advanced Bandit Algorithms Research9 references4 citations

TL;DR

This paper proposes an adaptive stochastic optimization algorithm for sequential resource allocation under diminishing returns, leveraging noisy gradient feedback and the Łojasiewicz inequality to achieve problem-adaptive convergence rates. The algorithm dynamically adjusts to the regularity of reward functions—measured by the Łojasiewicz exponent—achieving optimal regret rates for strongly concave (fast) and linear (classical bandit) cases, with improved performance in intermediate regimes without prior knowledge of the regularity parameter.

ABSTRACT

We consider the classical problem of sequential resource allocation where a decision maker must repeatedly divide a budget between several resources, each with diminishing returns. This can be recast as a specific stochastic optimization problem where the objective is to maximize the cumulative reward, or equivalently to minimize the regret. We construct an algorithm that is {\em adaptive} to the complexity of the problem, expressed in term of the regularity of the returns of the resources, measured by the exponent in the Łojasiewicz inequality (or by their universal concavity parameter). Our parameter-independent algorithm recovers the optimal rates for strongly-concave functions and the classical fast rates of multi-armed bandit (for linear reward functions). Moreover, the algorithm improves existing results on stochastic optimization in this regret minimization setting for intermediate cases.

Motivation & Objective

• To address the challenge of sequential resource allocation with unknown, concave reward functions under noisy gradient feedback.
• To design an algorithm that adapts to the intrinsic regularity of the reward functions, measured by the Łojasiewicz exponent or universal concavity parameter.
• To achieve optimal regret rates across all levels of regularity—from strongly concave to linear—without requiring prior knowledge of the problem's complexity.
• To improve upon existing stochastic optimization methods in regret minimization by leveraging gradient feedback and adaptive sampling strategies.

Proposed method

• The algorithm uses a binary search (dichotomy) framework to recursively partition the budget space and refine allocations based on noisy gradient observations.
• It employs a recursive decomposition of the optimization problem into subproblems at each level of a binary tree structure, with adaptive sampling based on gradient magnitude estimates.
• Sampling counts for each candidate allocation are determined by the inverse of the estimated gradient norm, ensuring higher exploration in regions of lower gradient magnitude.
• The algorithm uses confidence intervals and logarithmic scaling to control error propagation across recursive levels, ensuring high-probability guarantees.
• It dynamically adjusts the number of samples per point based on the depth in the binary tree and the estimated gradient strength, achieving adaptive convergence.
• The method is parameter-free, relying only on the Łojasiewicz inequality to guide sampling and convergence, without requiring knowledge of the smoothness or strong convexity parameters.

Experimental results

Research questions

• RQ1 Can an adaptive stochastic optimization algorithm achieve optimal regret rates across all levels of regularity in resource allocation problems without prior knowledge of the Łojasiewicz exponent?
• RQ2 How can noisy gradient feedback be leveraged to accelerate convergence in stochastic optimization for concave reward functions?
• RQ3 Does a parameter-independent algorithm exist that recovers both fast rates for strongly concave functions and classical bandit rates for linear functions?
• RQ4 Can the algorithm achieve regret bounds that scale as $T^{-\beta/2}$ for functions satisfying the Łojasiewicz inequality with exponent $\beta \in [1,2]$?
• RQ5 What is the performance gap between theoretical upper bounds and empirical regret in intermediate regularity regimes?

Key findings

• The algorithm achieves regret bounds that scale as $T^{-\beta/2}$ for $\beta \in [1,2]$, matching the theoretical lower bound up to logarithmic factors.
• For $\beta \geq 2$, the regret scales as $\log(T)/T$, which is faster than the classical $\widetilde{\mathcal{O}}(\sqrt{T})$ bandit rate.
• The algorithm recovers the optimal $\widetilde{\mathcal{O}}(T^{-1})$ rate for strongly concave functions (corresponding to $\beta=2$) without prior knowledge of the strong convexity parameter.
• Empirical results confirm that the regret is tightly squeezed between $T^{-\beta/2}$ and $(T/\log^2 T)^{-\beta/2}$ for $\beta \leq 2$, validating the theoretical scaling.
• In the $\log$-log scale, the slope of the regret curve matches $-\beta/2$, confirming the correctness of the convergence rate across different values of $\beta$.
• The algorithm outperforms existing methods in intermediate regularity regimes, demonstrating improved regret compared to non-adaptive or non-gradient-based approaches.