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Emmeran Johnson

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8 papers

2 author rows

EWRL Workshop 2025 Workshop Paper

Does Stochastic Gradient really succeed for bandits?

Dorian Baudry
Emmeran Johnson
Simon Vary
Ciara Pike-Burke
Patrick Rebeschini

Recent works of Mei et al. (2023, 2024) have deepened the theoretical understanding of the Stochastic Gradient Bandit (SGB) policy, showing that using a constant learning rate guarantees asymptotic convergence to the optimal policy, and that sufficiently small learning rates can yield logarithmic regret. However, whether logarithmic regret holds beyond small learning rates remains unclear. In this work, we take a step towards characterizing the regret regimes of SGB as a function of its learning rate. For two--armed bandits, we identify a sharp threshold, scaling with the sub-optimality gap $\Delta$, below which SGB achieves logarithmic regret on all instances, and above which it can incur polynomial regret on some instances. This result highlights the necessity of knowing (or estimating) $\Delta$ to ensure logarithmic regret with a constant learning rate. For general $K$-armed bandits, we further show the learning rate must scale inversely with $K$ to avoid polynomial regret. We introduce novel techniques to derive regret upper bounds for SGB, laying the groundwork for future advances in the theory of gradient-based bandit algorithms.

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NeurIPS Conference 2025 Conference Paper

Does Stochastic Gradient really succeed for bandits?

Dorian Baudry
Emmeran Johnson
Simon Vary
Ciara Pike-Burke
Patrick Rebeschini

Recent works of Mei et al. (2023, 2024) have deepened the theoretical understanding of the *Stochastic Gradient Bandit* (SGB) policy, showing that using a constant learning rate guarantees asymptotic convergence to the optimal policy, and that sufficiently *small* learning rates can yield logarithmic regret. However, whether logarithmic regret holds beyond small learning rates remains unclear. In this work, we take a step towards characterizing the regret *regimes* of SGB as a function of its learning rate. For two--armed bandits, we identify a sharp threshold, scaling with the sub-optimality gap $\Delta$, below which SGB achieves *logarithmic* regret on all instances, and above which it can incur *polynomial* regret on some instances. This result highlights the necessity of knowing (or estimating) $\Delta$ to ensure logarithmic regret with a constant learning rate. For general $K$-armed bandits, we further show the learning rate must scale inversely with $K$ to avoid polynomial regret. We introduce novel techniques to derive regret upper bounds for SGB, laying the groundwork for future advances in the theory of gradient-based bandit algorithms.

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NeurIPS Conference 2025 Conference Paper

On the necessity of adaptive regularisation: Optimal anytime online learning on $\boldsymbol{\ell_p}$-balls

Emmeran Johnson
David Martínez-Rubio
Ciara Pike-Burke
Patrick Rebeschini

We study online convex optimization on $\ell_p$-balls in $\mathbb{R}^d$ for $p > 2$. While always sub-linear, the optimal regret exhibits a shift between the high-dimensional setting ($d > T$), when the dimension $d$ is greater than the time horizon $T$ and the low-dimensional setting ($d \leq T$). We show that Follow-the-Regularised-Leader (FTRL) with time-varying regularisation which is adaptive to the dimension regime is anytime optimal for all dimension regimes. Motivated by this, we ask whether it is possible to obtain anytime optimality of FTRL with fixed non-adaptive regularisation. Our main result establishes that for separable regularisers, adaptivity in the regulariser is necessary, and that any fixed regulariser will be sub-optimal in one of the two dimension regimes. Finally, we provide lower bounds which rule out sub-linear regret bounds for the linear bandit problem in sufficiently high-dimension for all $\ell_p$-balls with $p \geq 1$.

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EWRL Workshop 2025 Workshop Paper

Stochastic Shortest Path with Sparse Adversarial Costs

Emmeran Johnson
Alberto Rumi
Ciara Pike-Burke
Patrick Rebeschini

We study the adversarial Stochastic Shortest Path (SSP) problem with sparse costs under full-information feedback. In the known transition setting, existing bounds based on Online Mirror Descent (OMD) with negative-entropy regularization scale with $\sqrt{\log S A}$, where $SA$ is the size of the state-action space. While we show that this is optimal in the worst-case, this bound fails to capture the benefits of sparsity when only a small number $M \ll SA$ of state-action pairs incur cost. In fact, we also show that the negative-entropy is inherently non-adaptive to sparsity: it provably incurs regret scaling with $\sqrt{\log S}$ on sparse problems. Instead, we propose a novel family of $\ell_r$-norm regularizers ($r \in (1, 2)$) that adapts to the sparsity and achieves regret scaling with $\sqrt{\log M}$ instead of $\sqrt{\log SA}$. We show this is optimal via a matching lower bound, highlighting that $M$ captures the effective dimension of the problem instead of $SA$. Finally, in the unknown transition setting the benefits of sparsity are limited: we prove that even on sparse problems, the minimax regret for any learner scales polynomially with $SA$.

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NeurIPS Conference 2025 Conference Paper

Stochastic Shortest Path with Sparse Adversarial Costs

Emmeran Johnson
Alberto Rumi
Ciara Pike-Burke
Patrick Rebeschini

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ICLR Conference 2024 Conference Paper

Sample-Efficiency in Multi-Batch Reinforcement Learning: The Need for Dimension-Dependent Adaptivity

Emmeran Johnson
Ciara Pike-Burke
Patrick Rebeschini

We theoretically explore the relationship between sample-efficiency and adaptivity in reinforcement learning. An algorithm is sample-efficient if it uses a number of queries $n$ to the environment that is polynomial in the dimension $d$ of the problem. Adaptivity refers to the frequency at which queries are sent and feedback is processed to update the querying strategy. To investigate this interplay, we employ a learning framework that allows sending queries in $K$ batches, with feedback being processed and queries updated after each batch. This model encompasses the whole adaptivity spectrum, ranging from non-adaptive `offline' ($K=1$) to fully adaptive ($K=n$) scenarios, and regimes in between. For the problems of policy evaluation and best-policy identification under $d$-dimensional linear function approximation, we establish $\Omega(\log \log d)$ lower bounds on the number of batches $K$ required for sample-efficient algorithms with $n = O(poly(d))$ queries. Our results show that just having adaptivity ($K>1$) does not necessarily guarantee sample-efficiency. Notably, the adaptivity-boundary for sample-efficiency is not between offline reinforcement learning ($K=1$), where sample-efficiency was known to not be possible, and adaptive settings. Instead, the boundary lies between different regimes of adaptivity and depends on the problem dimension.

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NeurIPS Conference 2023 Conference Paper

Optimal Convergence Rate for Exact Policy Mirror Descent in Discounted Markov Decision Processes

Emmeran Johnson
Ciara Pike-Burke
Patrick Rebeschini

Policy Mirror Descent (PMD) is a general family of algorithms that covers a wide range of novel and fundamental methods in reinforcement learning. Motivated by the instability of policy iteration (PI) with inexact policy evaluation, unregularised PMD algorithmically regularises the policy improvement step of PI without regularising the objective function. With exact policy evaluation, PI is known to converge linearly with a rate given by the discount factor $\gamma$ of a Markov Decision Process. In this work, we bridge the gap between PI and PMD with exact policy evaluation and show that the dimension-free $\gamma$-rate of PI can be achieved by the general family of unregularised PMD algorithms under an adaptive step-size. We show that both the rate and step-size are unimprovable for PMD: we provide matching lower bounds that demonstrate that the $\gamma$-rate is optimal for PMD methods as well as PI and that the adaptive step-size is necessary to achieve it. Our work is the first to relate PMD to rate-optimality and step-size necessity. Our study of the convergence of PMD avoids the use of the performance difference lemma, which leads to a direct analysis of independent interest. We also extend the analysis to the inexact setting and establish the first dimension-optimal sample complexity for unregularised PMD under a generative model, improving upon the best-known result.

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EWRL Workshop 2023 Workshop Paper

Optimal Convergence Rate for Exact Policy Mirror Descent in Discounted Markov Decision Processes

Emmeran Johnson
Ciara Pike-Burke
Patrick Rebeschini

Policy Mirror Descent (PMD) is a general family of algorithms that covers a wide range of novel and fundamental methods in reinforcement learning. Motivated by the instability of policy iteration (PI) with inexact policy evaluation, PMD algorithmically regularises the policy improvement step of PI. With exact policy evaluation, PI is known to converge linearly with a rate given by the discount factor $\gamma$ of a Markov Decision Process. In this work, we bridge the gap between PI and PMD with exact policy evaluation and show that the dimension-free $\gamma$-rate of PI can be achieved by the general family of unregularised PMD algorithms under an adaptive step-size. We show that both the rate and step-size are unimprovable for PMD: we provide matching lower bounds that demonstrate that the $\gamma$-rate is optimal for PMD methods as well as PI and that the adaptive step-size is necessary to achieve it. Our work is the first to relate PMD to rate-optimality and step-size necessity. Our study of the convergence of PMD avoids the use of the performance difference lemma, which leads to a direct analysis of independent interest. We also extend the analysis to the inexact setting and establish the first dimension-optimal sample complexity for unregularised PMD under a generative model, improving upon the best-known result.

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