Problem-Independent Regret Bounds

• Problem-Independent Regret Bounds are defined as uniform worst-case performance guarantees that do not rely on specific instance parameters, setting a fundamental baseline for algorithm design.
• They guide the development of robust and minimax algorithms by establishing universal rates such as O(√T) in bandit settings, independent of factors like gap or noise levels.
• Methodologies such as packing arguments, information-theoretic reductions, and adversarial constructions are used to derive tight lower bounds that benchmark the limits of online decision-making.

Problem-independent regret bounds are fundamental guarantees in online learning, bandit, and reinforcement learning frameworks. Such bounds characterize the worst-case regret—defined as the difference between the learner’s cumulative loss and the loss of the best fixed action or policy—in a way that is uniform over all environments or problem instances, i.e., they do not depend on specific problem parameters (such as gaps between arm means, noise-levels, etc.). These bounds have played a central role in guiding the design of minimax and robust algorithms and in revealing intrinsic learning-theoretic barriers for various adaptive decision-making problems.

1. Formal Definition and General Principles

A regret bound is said to be problem-independent if, for some class of instances (often the broadest feasible, e.g., all possible reward/loss sequences or all functions in an RKHS ball), it provides an upper (or lower) bound on the regret that holds uniformly over that class, depending only on structural parameters such as number of actions, time horizon, or instance dimension—not on specific unknown parameters of the data-generating process.

Let $R_T(\pi, \mathcal{F})$ denote the regret after $T$ rounds for learner $\pi$ on class $\mathcal{F}$:

$$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$

A problem-independent bound gives $R_T(\pi, \mathcal{F}) \leq C(\mathcal{F}, T)$, where the constant $C$ depends only on generic parameters of $\mathcal{F}$ and $T$, not on unknown environment specifics.

This differs from problem-dependent (or “instance-dependent”) bounds, which incorporate instance-specific quantities such as minimum gap $\Delta$, optimal value, loss variation, etc.

2. Canonical Settings and Minimax Regret Rates

Multi-Armed Bandits (MAB)

For stochastic $T$0-armed bandits with $T$1 rounds, the classical minimax regret bound is

$T$2

achievable up to logarithmic factors by UCB, MOSS, and variants. Problem-independent lower bounds are shown by Information-Theoretic reductions using KL-divergence and Pinsker-type inequalities (Agrawal et al., 2012, Gerchinovitz et al., 2016).

For adversarial $T$3-armed bandits, the minimax regret matches $T$4 with no dependence on the loss sequence specifics, even allowing for adaptive adversaries (Gerchinovitz et al., 2016).

Gaussian Process (GP) Bandits

For sequential optimization over a domain $T$5 of functions $T$6 with RKHS norm constraint $T$7:

• Squared-exponential kernel: Minimizing simple regret $T$8 requires

$T$9

while the average cumulative regret satisfies

$\pi$0

for all $\pi$1 in the RKHS ball (Scarlett et al., 2017, Cai et al., 2020).

• Matérn-$\pi$2 kernel: The lower bound generalizes to scales of $\pi$3, $\pi$4 (Scarlett et al., 2017).

These bounds are fundamentally problem-independent: no algorithm can beat these rates uniformly over all $\pi$5.

Combinatorial Semi-Bandits and Structured Settings

In combinatorial semi-bandits (CMAB), problem-independent distribution-free regret bounds were traditionally $\pi$6; recent refinements exploit variance-modulation and independence to reduce the $\pi$7-dependence to $\pi$8 or remove it, rendering the rates batch-size-independent (Liu et al., 2022).

Online Linear Regression and Self-Normalized Martingales

Problem-independent regret—termed “doubly-uniform” when independent of both covariate scale and comparator norm—is possible with $\pi$9 in $\mathcal{F}$0 dimensions, but not generally for $\mathcal{F}$1 unless smoothness assumptions are imposed (Chen et al., 2 May 2026).

3. Lower Bound Methodology

Problem-independent lower bounds are typically established by:

• Packing/Perturbation Arguments: Constructing a set of hard environments (e.g., “needle-in-haystack” bump functions in RKHS (Scarlett et al., 2017); correlated arm distributions in bandits (Gerchinovitz et al., 2016)), such that distinguishing the true instance requires $\mathcal{F}$2 to scale as dictated by the packing number.
• Information-Theoretic Reductions: Using the chain rule for KL-divergence and tools like Le Cam’s or Fano’s inequality, and coupling arguments to bound regret by the mutual information required to identify key structure.
• Adversarial Constructions: Designing loss or reward sequences that force the learner to explore all arms or states, leading to unavoidable regret proportional to the square-root of $\mathcal{F}$3 (or polynomial in $\mathcal{F}$4 for highly unstructured problems).

High-probability lower bounds further employ change-of-measure arguments to capture dependence on the confidence parameter $\mathcal{F}$5, ensuring that no algorithm achieves suboptimal regret with probability exceeding $\mathcal{F}$6 (Cai et al., 2020).

4. Algorithmic Achievability and Tightness

Many problem-independent regret upper bounds are matched, up to logarithmic factors, by explicit algorithms:

• Exp3, UCB1, MOSS: Achieve $\mathcal{F}$7 or $\mathcal{F}$8 for adversarial and stochastic bandits (Agrawal et al., 2012, Gerchinovitz et al., 2016).
• Thompson Sampling: Proven problem-independent $\mathcal{F}$9 regret (Agrawal et al., 2012).
• Online Gradient Descent (Euclidean and Riemannian): $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$0 regret for convex losses, curvature-independent on Hadamard manifolds with h-convexity (Sahinoglu et al., 14 Sep 2025).
• Follow-the-Regularized-Leader (FTRL): With Kullback-Leibler or other $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$1-divergences, yields $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$2 regret against adversarial and possibly unbounded losses, provided moment bounds (Alquier, 2020).

Nevertheless, some minimax lower and upper bounds remain logarithmically separated, notably for Gaussian Process bandits with squared-exponential kernels, where a gap in the exponents of $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$3 persists (Scarlett et al., 2017).

5. Extensions and Variations

Robust and Decentralized Models

Problem-independent bounds extend to robust frameworks (e.g., convex model uncertainty, adversarial corruption), with rates depending solely on model covering numbers or “fuzzy decision-estimation coefficients” (Appel et al., 9 Apr 2025).

In decentralized or networked multi-agent bandit problems, the minimax regret remains $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$4 for well-connected networks, but can degrade to $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$5 in sparse/adversarial communication settings (Xu et al., 2023).

Online Selection and Multi-Secretary Problems

For the multi-secretary problem under i.i.d. finite-support abilities and an adaptive selection policy, there exists a problem-independent $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$6 regret bound—uniform in both horizon $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$7 and budget $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$8—contrasting with the $$R_T(\pi, \mathcal{F}) = \sup_{f \in \mathcal{F}} \mathbb{E}\left[\sum_{t=1}^T \ell_t(a_t) - \min_{a^*} \sum_{t=1}^T \ell_t(a^*)\right].$$9 lower bound for non-adaptive policies (Arlotto et al., 2017).

6. Structural and Geometric Generalizations

On infinite or curved domains (e.g., Riemannian manifolds), problem-independent regret rates matching the Euclidean $R_T(\pi, \mathcal{F}) \leq C(\mathcal{F}, T)$0 or $R_T(\pi, \mathcal{F}) \leq C(\mathcal{F}, T)$1 are achievable when losses are horospherically convex (“h-convex”): the analysis becomes curvature-independent and thus “problem-independent” in the geometric sense (Sahinoglu et al., 14 Sep 2025).

7. Significance and Benchmarks

Problem-independent regret bounds define sharp benchmarks for algorithmic optimum in adversarial and distribution-free settings. They delineate fundamental limits that any algorithm must confront in the absence of exploitable structure, and their tightness (or lack thereof) directly motivates the development of instance-adaptive—problem-dependent—algorithms when better rates are sought.

They further serve as the target in robust, high-confidence, or universal learning problems, and provide a uniform yardstick for evaluating both classical and modern sequential decision-making methods.

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References:

• "Lower Bounds on Regret for Noisy Gaussian Process Bandit Optimization" (Scarlett et al., 2017)
• "Refined Lower Bounds for Adversarial Bandits" (Gerchinovitz et al., 2016)
• "Uniformly bounded regret in the multi-secretary problem" (Arlotto et al., 2017)
• "Batch-Size Independent Regret Bounds for Combinatorial Semi-Bandits with Probabilistically Triggered Arms or Independent Arms" (Liu et al., 2022)
• "Further Optimal Regret Bounds for Thompson Sampling" (Agrawal et al., 2012)
• "Non-exponentially weighted aggregation: regret bounds for unbounded loss functions" (Alquier, 2020)
• "Regret Bounds for Robust Online Decision Making" (Appel et al., 9 Apr 2025)
• "Self-Normalized Martingales and Uniform Regret Bounds for Linear Regression" (Chen et al., 2 May 2026)
• "Regret Lower Bounds in Multi-agent Multi-armed Bandit" (Xu et al., 2023)
• "Online Optimization on Hadamard Manifolds: Curvature Independent Regret Bounds on Horospherically Convex Objectives" (Sahinoglu et al., 14 Sep 2025)
• "On Lower Bounds for Standard and Robust Gaussian Process Bandit Optimization" (Cai et al., 2020)
• "Regret Bounds for Reinforcement Learning via Markov Chain Concentration" (Ortner, 2018)