Liad Erez

Liad Erez, Tal Lancewicki, Uri Sherman et al.

An abundance of recent impossibility results establish that regret minimization in Markov games with adversarial opponents is both statistically and computationally intractable. Nevertheless, none of these results preclude the possibility of regret minimization under the assumption that all parties adopt the same learning procedure. In this work, we present the first (to our knowledge) algorithm for learning in general-sum Markov games that provides sublinear regret guarantees when executed by all agents. The bounds we obtain are for swap regret, and thus, along the way, imply convergence to a correlated equilibrium. Our algorithm is decentralized, computationally efficient, and does not require any communication between agents. Our key observation is that online learning via policy optimization in Markov games essentially reduces to a form of weighted regret minimization, with unknown weights determined by the path length of the agents' policy sequence. Consequently, controlling the path length leads to weighted regret objectives for which sufficiently adaptive algorithms provide sublinear regret guarantees.

Liad Erez, Fan Chen, Alon Cohen et al.

We study contextual bandits in the stochastic i.i.d.\ setting, where a learner observes contexts drawn from an unknown distribution, selects actions from a finite set $A$, and aims to identify an approximately optimal policy from a given class based on bandit feedback. Motivated by bandit multiclass classification with zero-one rewards, we focus on the \emph{$s$-sparse} setting in which, for every context, the reward vector has $L_1$-norm at most $s \ll |A|$. Our main result is the design of algorithms that, with high probability, output an $ε$-optimal policy compared to policy class $Π$ using $\tilde{O} ((s/ε^2 + |A|/ε)\log |Π|/δ)$ samples. We extend this bound to general Natarajan classes and complement it with a matching lower bound (up to logarithmic factors), thereby closing a substantial gap left by prior work (Erez et al., 2024, 2025), which incurred an additional $Θ(|A|^9)$ dependence. We obtain these results via two complementary approaches. First, we analyze contextual bandits through the lens of contextual decision making with structured observations, designing an exploration-by-optimization algorithm whose sample complexity is governed by the \emph{decision-estimation coefficient} (DEC; Foster et al., 2021, 2022). We show that, with $s$-sparse rewards, the induced model class admits a sharp DEC bound that scales with $s$ and directly yields the optimal rate. Since this approach is largely information-theoretic and involves solving complex min-max optimization problems, we also develop a second, more specialized algorithmic method based on a low-variance exploration technique. This approach leads to concrete, tractable algorithms and naturally extends to contextual combinatorial semi-bandits, leading to improved sample complexity guarantees for bandit multiclass list classification.

Liad Erez, Alon Cohen, Tomer Koren et al.

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 $\smash{\widetildeΘ\left(\min \left\{|H| + \sqrt{T}, \sqrt{KT \log |H|} \right\} \right) }$, where $H$ is the underlying (finite) hypothesis class. In particular, we present a new bandit classification algorithm that guarantees regret $\smash{\widetilde{O}(|H|+\sqrt{T})}$, 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.

Alon Cohen, Liad Erez, Steve Hanneke et al.

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.

Orin Levy, Liad Erez, Alon Cohen et al.

We present regret minimization algorithms for the contextual multi-armed bandit (CMAB) problem over $K$ actions in the presence of delayed feedback, a scenario where loss observations arrive with delays chosen by an adversary. As a preliminary result, assuming direct access to a finite policy class $Π$ we establish an optimal expected regret bound of $ O (\sqrt{KT \log |Π|} + \sqrt{D \log |Π|)} $ where $D$ is the sum of delays. For our main contribution, we study the general function approximation setting over a (possibly infinite) contextual loss function class $ \mathcal{F} $ with access to an online least-square regression oracle $\mathcal{O}$ over $\mathcal{F}$. In this setting, we achieve an expected regret bound of $O(\sqrt{KT\mathcal{R}_T(\mathcal{O})} + \sqrt{ d_{\max} D β})$ assuming FIFO order, where $d_{\max}$ is the maximal delay, $\mathcal{R}_T(\mathcal{O})$ is an upper bound on the oracle's regret and $β$ is a stability parameter associated with the oracle. We complement this general result by presenting a novel stability analysis of a Hedge-based version of Vovk's aggregating forecaster as an oracle implementation for least-square regression over a finite function class $\mathcal{F}$ and show that its stability parameter $β$ is bounded by $\log |\mathcal{F}|$, resulting in an expected regret bound of $O(\sqrt{KT \log |\mathcal{F}|} + \sqrt{d_{\max} D \log |\mathcal{F}|})$ which is a $\sqrt{d_{\max}}$ factor away from the lower bound of $Ω(\sqrt{KT \log |\mathcal{F}|} + \sqrt{D \log |\mathcal{F}|})$ that we also present.

Liad Erez, Tomer Koren

We study the problem of contextual combinatorial semi-bandits, where input contexts are mapped into subsets of size $m$ of a collection of $K$ possible actions. In each round, the learner observes the realized reward of the predicted actions. Motivated by prototypical applications of contextual bandits, we focus on the $s$-sparse regime where we assume that the sum of rewards is bounded by some value $s\ll K$. For example, in recommendation systems the number of products purchased by any customer is significantly smaller than the total number of available products. Our main result is for the $(ε,δ)$-PAC variant of the problem for which we design an algorithm that returns an $ε$-optimal policy with high probability using a sample complexity of $\tilde{O}((poly(K/m)+sm/ε^2) \log(|Π|/δ))$ where $Π$ is the underlying (finite) class and $s$ is the sparsity parameter. This bound improves upon known bounds for combinatorial semi-bandits whenever $s\ll K$, and in the regime where $s=O(1)$, the leading terms in our bound match the corresponding full-information rates, implying that bandit feedback essentially comes at no cost. Our algorithm is also computationally efficient given access to an ERM oracle for $Π$. Our framework generalizes the list multiclass classification problem with bandit feedback, which can be seen as a special case with binary reward vectors. In the special case of single-label classification corresponding to $s=m=1$, we prove an $O((K^7+1/ε^2)\log(|H|/δ))$ sample complexity bound, which improves upon recent results in this scenario. Additionally, we consider the regret minimization setting where data can be generated adversarially, and establish a regret bound of $\tilde O(|Π|+\sqrt{smT\log |Π|})$, extending the result of Erez et al. (2024) who consider the simpler single label classification setting.

Liad Erez, Alon Cohen, Tomer Koren et al.

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 $(\varepsilon,δ)$-PAC version of the problem, with sample complexity of $O\big( (\operatorname{poly}(K) + 1 / \varepsilon^2) \log (|H| / δ) \big)$ 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 $Θ(K)$. 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 $H$.

Liad Erez, Tomer Koren

We study the online learning with feedback graphs framework introduced by Mannor and Shamir (2011), in which the feedback received by the online learner is specified by a graph $G$ over the available actions. We develop an algorithm that simultaneously achieves regret bounds of the form: $\smash{\mathcal{O}(\sqrt{θ(G) T})}$ with adversarial losses; $\mathcal{O}(θ(G)\operatorname{polylog}{T})$ with stochastic losses; and $\mathcal{O}(θ(G)\operatorname{polylog}{T} + \smash{\sqrt{θ(G) C})}$ with stochastic losses subject to $C$ adversarial corruptions. Here, $θ(G)$ is the clique covering number of the graph $G$. Our algorithm is an instantiation of Follow-the-Regularized-Leader with a novel regularization that can be seen as a product of a Tsallis entropy component (inspired by Zimmert and Seldin (2019)) and a Shannon entropy component (analyzed in the corrupted stochastic case by Amir et al. (2020)), thus subtly interpolating between the two forms of entropies. One of our key technical contributions is in establishing the convexity of this regularizer and controlling its inverse Hessian, despite its complex product structure.