Improving Multi-Armed Bandits

• Improving Multi-Armed Bandits are sequential allocation models that use monotonic, concave reward curves to capture diminishing returns in effort-based scenarios.
• They leverage parameterized algorithms like PTRR_α and Hybrid_α,B to balance exploration and commitment, achieving instance-adaptive competitive guarantees.
• Instance-adaptive tuning via offline empirical risk minimization allows optimal parameter selection, surpassing worst-case bounds in real-world applications.

The improving multi-armed bandit (MAB) problem addresses sequential allocation under uncertainty with the additional structure that each arm's reward trajectory is monotonically increasing (with diminishing returns), capturing effort-accumulation scenarios such as adaptive R&D, hyperparameter tuning via learning curves, or clinical progression tracking. The algorithmic challenge is to optimize cumulative or final reward when each pull affects the subsequent reward attainable from that arm. The improving MAB model generalizes well-studied concave bandit and best-arm identification frameworks but introduces new complexities in optimization and regret/competitive analysis due to nonstationary, concave reward profiles. Recent developments have sharpened worst-case and data-dependent guarantees, leveraged algorithmic parameterization, and connected the improving bandit setting to offline learning-to-optimize paradigms.

1. Model Formulation and Competitive Ratios

In the improving MAB setting, there are $k$ arms; each arm $i$ is associated with an unknown, nondecreasing, concave reward curve $f_i:\mathbb{N}\rightarrow \mathbb{R}_{\ge0}$ such that $f_i(t+1) - f_i(t) \leq f_i(t) - f_i(t-1)$ for all $t \geq 1$. The learner sequentially allocates at most $T$ total pulls, choosing at each round which arm to play; pulling arm $i$ for the $t$th time generates reward $f_i(t)$. The principal performance metric is the competitive ratio $c$: $$\frac{\mathrm{OPT}_T}{\mathbb{E}[\mathrm{ALG}_T]} \leq c$$ where $\mathrm{OPT}_T = \max_{i}\sum_{t=1}^T f_i(t)$ is the best possible cumulative reward by “fully-committing” to a single arm.

Worst-case lower bounds are $\Omega(k)$ for deterministic and $\Omega(\sqrt{k})$ for randomized algorithms. These rates are tight for linear (i.e., least concave) arm curves. However, the intrinsic structure of real-world $f_i$ often departs significantly from the worst-case, motivating a refined analysis that adapts to instance-specific concavity (Blum et al., 13 Nov 2025).

2. Parameterized Algorithm Families and Adaptive Guarantees

To exploit variable concavity and enable data-driven optimization, recent work has introduced parameterized families of online algorithms that interpolate between worst-case and much sharper, “instance-optimal” guarantees.

2.1 PTRR$_\alpha$ (Power-Thresholded Random Round-Robin)

The PTRR$_\alpha$ family, parameterized by $\alpha \in (0,1]$, executes randomized exploration with an $\alpha$-dependent persistence threshold. Precisely, for each arm $i$:

• The algorithm maintains an estimate $m$ of the maximal final reward and parameter $\tau = T-k$.
• PTRR$_\alpha$ repeats: selects $i$ uniformly among untried arms, and while

$$f_i(t_i) \geq m\left(\frac{t_i}{\tau}\right)^\alpha$$

continues to pull $i$ and increments $t_i$.

Performance: For any instance of “concavity envelope exponent” (CEE) $\beta_I$, i.e., all arms satisfy $f_i(t) \geq f_i(T) (t/T)^{\beta_I}$, PTRR$_\alpha$ achieves

$$\frac{\mathrm{OPT}_T}{\mathbb{E}[\mathrm{ALG}_T]} = O(k^{\alpha/(\alpha+1)})$$

for $\alpha > \beta_I$. As $\alpha \downarrow \beta_I$, the competitive ratio approaches the instance-optimal rate $O(k^{\beta_I/(\beta_I+1)})$; for $\alpha=1$ (least concave), the $O(\sqrt{k})$ barrier is recovered (Blum et al., 13 Nov 2025). This parameterization subsumes prior optimal algorithms and provides a mechanism for leveraging concavity to outperform worst-case rates.

2.2 The Hybrid$_{\alpha,B}$ Family (Best-of-Both-Worlds BAI)

The Hybrid$_{\alpha,B}$ family, with parameters $\alpha\in(\beta_I,1]$ and budget $B\leq T/2$, combines an envelope-based best-arm identification (BAI) routine with a backup approximate BAI procedure. In the initial $B$ steps, it maintains confidence updates $L_i(t_i)$ and $U_i(t_i)$ for each arm, using a UCB-style check for provable BAI. If no certificate is found, it switches to PTRR$_\alpha$ on the remaining budget, thus guaranteeing either exact best-arm output on “easy” instances (envelope separation in $B$ rounds), or reverting to the competitive PTRR$_\alpha$ guarantee otherwise.

Best-of-Both-Worlds: When the instance admits early BAI (i.e., a “gap-clearance” condition holding within $B$ rounds), exact best-arm is identified; otherwise, Hybrid$_{\alpha,B}$ returns an arm $\hat{i}$ with

$$\mathbb{E}[f_{\hat{i}}(T)] \geq \Omega\left(k^{-\alpha/(\alpha+1)}\right) f^*(T)$$

reverting to the optimal-in-$k$ approximation (Blum et al., 13 Nov 2025).

3. Instance-Adaptive Tuning and Data-Driven Learning

Both PTRR$_\alpha$ and Hybrid$_{\alpha,B}$ enable adaptive selection of algorithm parameters via offline empirical risk minimization. The loss landscape (e.g., competitive ratio as a function of $\alpha$ or $(\alpha,B)$) is piecewise-constant with low combinatorial complexity—$O(kT)$ regions for PTRR$_\alpha$, $O(kT^2)$ for Hybrid$_{\alpha,B}$. Formal VC-type uniform convergence results establish that

$$N = O\left(\frac{H^2}{\epsilon^2}\bigl(\log(kT) + \log(1/\delta)\bigr)\right)$$

IID samples suffice to tune these parameters within $\epsilon$ of optimal with probability $1-\delta$ for best-in-class performance under an instance distribution $\mathcal{D}$ (Blum et al., 13 Nov 2025). This framework allows practitioners to select algorithm parameters a priori for a given application environment, sidestepping the need for online testing of structural assumptions (e.g., concavity envelope exponent).

4. Relation to Classical and Bayesian Approaches

In the classical stationary bandit model, regret minimization is typically achieved via index policies (UCB, Thompson Sampling, etc.) and their distributional optimality or minimax guarantees are characterized under i.i.d. reward assumptions. In the improving bandit setting:

• Standard regret-minimizing policies are suboptimal: deterministic algorithms are $\Omega(k)$-competitive, and the optimal randomized is $O(\sqrt{k})$-competitive only when the worst-case “linear arms” regime holds.
• PTRR$_\alpha$ and related algorithms eliminate the need for forced exploration/exploitation balancing, instead adapting the persistence threshold to the observed progress curve structure.
• The data-driven paradigm is reminiscent of hyperparameter selection via offline validation in machine learning, but here the sample complexity is sharply characterized and independent of any stationarity or gap condition.

No Bayesian prior or parametric reward distribution tuning is required: all performance guarantees are instance-adaptive and supported by tight, explicit competitive ratio or sample complexity bounds (Blum et al., 13 Nov 2025).

5. Extension to Related Structured Bandit Models

The improving bandit problem complements a spectrum of structured bandit models including:

• Concave bandits (where the arms are concave functions of pulls, but not necessarily nondecreasing)—the improving model imposes monotonicity, further restricting the feasible reward developments.
• Non-concave and generalized parametric bandits (where the regularity may help in identification but require sophisticated subspace or tensor iteration methods for minimax rates) (Huang et al., 2021).
• Best-arm identification under nonstationary and nonparametric settings, often using elaborate envelope or elimination testing for early stopping.
• Regional or grouped bandits, where information can be shared within arms, but the improving dynamic uniquely combines exploration and commitment (Wang et al., 2018).

6. Outlook and Open Challenges

The most recent parameterized algorithms for improving MABs already reach the jointly optimal competitive ratios as a function of both the number of arms and the instance's effective concavity. Fundamental open questions persist:

• Characterization of strong minimax lower bounds under more general effort-gain functions or partial observability.
• Efficient, parameter-free, online selection of thresholds $\alpha$ and $B$ without prior offline data or knowledge of concavity parameters.
• Extensions to settings with stochastic or adversarial disturbances in $f_i$, or inter-arm dependencies.
• Theoretical understanding of the statistical rates for learning parametric best-arm selectors in high dimensions.

The field is rapidly developing, with the unification of competitive-ratio analysis, online adaptivity, and offline learning-to-optimize machinery likely to yield further advances in effort-allocation under uncertainty and time-varying reward settings (Blum et al., 13 Nov 2025).