Regret Guarantees for Linear Contextual Stochastic Shortest Path

We define the problem of linear Contextual Stochastic Shortest Path (CSSP), where at the beginning of each episode, the learner observes an adversarially chosen context that determines the MDP through a fixed but unknown linear function. The learner's objective is to reach a designated goal state with minimal expected cumulative loss, despite having no prior knowledge of the transition dynamics, loss functions, or the mapping from context to MDP. In this work, we propose LR-CSSP, an algorithm that achieves a regret bound of $\widetilde{O}(K^{2/3} d^{2/3} |S| |A|^{1/3} B_\star^2 T_\star \log (1/ δ))$, where $K$ is the number of episodes, $d$ is the context dimension, $S$ and $A$ are the sets of states and actions respectively, $B_\star$ bounds the optimal cumulative loss and $T_\star$, unknown to the learner, bounds the expected time for the optimal policy to reach the goal. In the case where all costs exceed $\ell_{\min}$, LR-CSSP attains a regret of $\widetilde O(\sqrt{K \cdot d^2 |S|^3 |A| B_\star^3 \log(1/δ)/\ell_{\min}})$. Unlike in contextual finite-horizon MDPs, where limited knowledge primarily leads to higher losses and regret, in the CSSP setting, insufficient knowledge can also prolong episodes and may even lead to non-terminating episodes. Our analysis reveals that LR-CSSP effectively handles continuous context spaces, while ensuring all episodes terminate within a reasonable number of time steps.

Sample Complexity of Agnostic Multiclass Classification: Natarajan Dimension Strikes Back

The fundamental theorem of statistical learning states that binary PAC learning is governed by a single parameter -- the Vapnik-Chervonenkis (VC) dimension -- which determines both learnability and sample complexity. Extending this to multiclass classification has long been challenging, since Natarajan's work in the late 80s proposing the Natarajan dimension (Nat) as a natural analogue of VC. Daniely and Shalev-Shwartz (2014) introduced the DS dimension, later shown by Brukhim et al. (2022) to characterize multiclass learnability. Brukhim et al. also showed that Nat and DS can diverge arbitrarily, suggesting that multiclass learning is governed by DS rather than Nat. We show that agnostic multiclass PAC sample complexity is in fact governed by two distinct dimensions. Specifically, we prove nearly tight agnostic sample complexity bounds that, up to log factors, take the form $\frac{DS^{1.5}}ε + \frac{Nat}{ε^2}$ where $ε$ is the excess risk. This bound is tight up to a $\sqrt{DS}$ factor in the first term, nearly matching known $Nat/ε^2$ and $DS/ε$ lower bounds. The first term reflects the DS-controlled regime, while the second shows that the Natarajan dimension still dictates asymptotic behavior for small $ε$. Thus, unlike binary or online classification -- where a single dimension (VC or Littlestone) controls both phenomena -- multiclass learning inherently involves two structural parameters. Our technical approach departs from traditional agnostic learning methods based on uniform convergence or reductions to realizable cases. A key ingredient is a novel online procedure based on a self-adaptive multiplicative-weights algorithm performing a label-space reduction, which may be of independent interest.

Fast Rates for Bandit PAC Multiclass Classification

We study multiclass PAC learning with bandit feedback, where inputs are classified into one of $K$ possible labels and feedback is limited to whether or not the predicted labels are correct. Our main contribution is in designing a novel learning algorithm for the agnostic $(\varepsilon,\delta)$-PAC version of the problem, with sample complexity of $O\big( (\operatorname{poly}(K) + 1 / \varepsilon^2) \log (|H| / \delta) \big)$ for any finite hypothesis class $H$. In terms of the leading dependence on $\varepsilon$, this improves upon existing bounds for the problem, that are of the form $O(K/\varepsilon^2)$. We also provide an extension of this result to general classes and establish similar sample complexity bounds in which $\log |H|$ is replaced by the Natarajan dimension. This matches the optimal rate in the full-information version of the problem and resolves an open question studied by Daniely, Sabato, Ben-David, and Shalev-Shwartz (2011) who demonstrated that the multiplicative price of bandit feedback in realizable PAC learning is $\Theta(K)$. We complement this by revealing a stark contrast with the agnostic case, where the price of bandit feedback is only $O(1)$ as $\varepsilon \to 0$. Our algorithm utilizes a stochastic optimization technique to minimize a log-barrier potential based on Frank-Wolfe updates for computing a low-variance exploration distribution over the hypotheses, and is made computationally efficient provided access to an ERM oracle over $H$.

The Real Price of Bandit Information in Multiclass Classification

We revisit the classical problem of multiclass classification with bandit feedback (Kakade, Shalev-Shwartz and Tewari, 2008), where each input classifies to one of $K$ possible labels and feedback is restricted to whether the predicted label is correct or not. Our primary inquiry is with regard to the dependency on the number of labels $K$, and whether $T$-step regret bounds in this setting can be improved beyond the $\smash{\sqrt{KT}}$ dependence exhibited by existing algorithms. Our main contribution is in showing that the minimax regret of bandit multiclass is in fact more nuanced, and is of the form $\smash{\widetilde{\Theta}\left(\min \left\{|\mathcal{H}| + \sqrt{T}, \sqrt{KT \log |{\mathcal{H}|}} \right\} \right) }$, where $\mathcal{H}$ is the underlying (finite) hypothesis class. In particular, we present a new bandit classification algorithm that guarantees regret $\smash{\widetilde{O}(|\mathcal{H}|+\sqrt{T})}$, improving over classical algorithms for moderately-sized hypothesis classes, and give a matching lower bound establishing tightness of the upper bounds (up to log-factors) in all parameter regimes.

Locally Optimal Descent for Dynamic Stepsize Scheduling

We introduce a novel dynamic learning-rate scheduling scheme grounded in theory with the goal of simplifying the manual and time-consuming tuning of schedules in practice. Our approach is based on estimating the locally-optimal stepsize, guaranteeing maximal descent in the direction of the stochastic gradient of the current step. We first establish theoretical convergence bounds for our method within the context of smooth non-convex stochastic optimization, matching state-of-the-art bounds while only assuming knowledge of the smoothness parameter. We then present a practical implementation of our algorithm and conduct systematic experiments across diverse datasets and optimization algorithms, comparing our scheme with existing state-of-the-art learning-rate schedulers. Our findings indicate that our method needs minimal tuning when compared to existing approaches, removing the need for auxiliary manual schedules and warm-up phases and achieving comparable performance with drastically reduced parameter tuning.

Rate-Optimal Policy Optimization for Linear Markov Decision Processes

We study regret minimization in online episodic linear Markov Decision Processes, and obtain rate-optimal $\widetilde O (\sqrt K)$ regret where $K$ denotes the number of episodes. Our work is the first to establish the optimal (w.r.t.~$K$) rate of convergence in the stochastic setting with bandit feedback using a policy optimization based approach, and the first to establish the optimal (w.r.t.~$K$) rate in the adversarial setup with full information feedback, for which no algorithm with an optimal rate guarantee is currently known.

APART: Diverse Skill Discovery using All Pairs with Ascending Reward and DropouT

We study diverse skill discovery in reward-free environments, aiming to discover all possible skills in simple grid-world environments where prior methods have struggled to succeed. This problem is formulated as mutual training of skills using an intrinsic reward and a discriminator trained to predict a skill given its trajectory. Our initial solution replaces the standard one-vs-all (softmax) discriminator with a one-vs-one (all pairs) discriminator and combines it with a novel intrinsic reward function and a dropout regularization technique. The combined approach is named APART: Diverse Skill Discovery using All Pairs with Ascending Reward and Dropout. We demonstrate that APART discovers all the possible skills in grid worlds with remarkably fewer samples than previous works. Motivated by the empirical success of APART, we further investigate an even simpler algorithm that achieves maximum skills by altering VIC, rescaling its intrinsic reward, and tuning the temperature of its softmax discriminator. We believe our findings shed light on the crucial factors underlying success of skill discovery algorithms in reinforcement learning.

Efficient Rate Optimal Regret for Adversarial Contextual MDPs Using Online Function Approximation

We present the OMG-CMDP! algorithm for regret minimization in adversarial Contextual MDPs. The algorithm operates under the minimal assumptions of realizable function class and access to online least squares and log loss regression oracles. Our algorithm is efficient (assuming efficient online regression oracles), simple and robust to approximation errors. It enjoys an $\widetilde{O}(H^{2.5} \sqrt{ T|S||A| ( \mathcal{R}(\mathcal{O}) + H \log(\delta^{-1}) )})$ regret guarantee, with $T$ being the number of episodes, $S$ the state space, $A$ the action space, $H$ the horizon and $\mathcal{R}(\mathcal{O}) = \mathcal{R}(\mathcal{O}_{\mathrm{sq}}^\mathcal{F}) + \mathcal{R}(\mathcal{O}_{\mathrm{log}}^\mathcal{P})$ is the sum of the regression oracles' regret, used to approximate the context-dependent rewards and dynamics, respectively. To the best of our knowledge, our algorithm is the first efficient rate optimal regret minimization algorithm for adversarial CMDPs that operates under the minimal standard assumption of online function approximation.

Counterfactual Optimism: Rate Optimal Regret for Stochastic Contextual MDPs

We present the UC$^3$RL algorithm for regret minimization in Stochastic Contextual MDPs (CMDPs). The algorithm operates under the minimal assumptions of realizable function class, and access to offline least squares and log loss regression oracles. Our algorithm is efficient (assuming efficient offline regression oracles) and enjoys an $\widetilde{O}(H^3 \sqrt{T |S| |A|(\log (|\mathcal{F}|/\delta) + \log (|\mathcal{P}|/ \delta) )})$ regret guarantee, with $T$ being the number of episodes, $S$ the state space, $A$ the action space, $H$ the horizon, and $\mathcal{P}$ and $\mathcal{F}$ are finite function classes, used to approximate the context-dependent dynamics and rewards, respectively. To the best of our knowledge, our algorithm is the first efficient and rate-optimal regret minimization algorithm for CMDPs, which operates under the general offline function approximation setting.

Rate-Optimal Online Convex Optimization in Adaptive Linear Control

We consider the problem of controlling an unknown linear dynamical system under adversarially changing convex costs and full feedback of both the state and cost function. We present the first computationally-efficient algorithm that attains an optimal $\smash{\sqrt{T}}$-regret rate compared to the best stabilizing linear controller in hindsight, while avoiding stringent assumptions on the costs such as strong convexity. Our approach is based on a careful design of non-convex lower confidence bounds for the online costs, and uses a novel technique for computationally-efficient regret minimization of these bounds that leverages their particular non-convex structure.