Tomer Koren

EWRL Workshop 2025 Workshop Paper

Convergence and Sample Complexity of First-Order Methods for Agnostic Reinforcement Learning

• Uri Sherman
• Tomer Koren
• Yishay Mansour

We study reinforcement learning (RL) in the agnostic policy learning setting, where the goal is to find a policy whose performance is competitive with the best policy in a given class of interest $\Pi$---crucially, without assuming that $\Pi$ contains the optimal policy. We propose a general policy learning framework that reduces this problem to first-order optimization in a non-Euclidean space, leading to new algorithms as well as shedding light on the convergence properties of existing ones. Specifically, under the assumption that $\Pi$ is convex and satisfies a variational gradient dominance (VGD) condition---an assumption known to be strictly weaker than more standard completeness and coverability conditions---we obtain sample complexity upper bounds for three policy learning algorithms: \emph{(i)} Steepest Descent Policy Optimization, derived from a constrained steepest descent method for non-convex optimization; \emph{(ii)} the classical Conservative Policy Iteration algorithm \citep{kakade2002approximately} reinterpreted through the lens of the Frank-Wolfe method, which leads to improved convergence results; and \emph{(iii)} an on-policy instantiation of the well-studied Policy Mirror Descent algorithm. Finally, we empirically evaluate the VGD condition across several standard environments, demonstrating the practical relevance of our key assumption.

ICML Conference 2025 Conference Paper

Convergence of Policy Mirror Descent Beyond Compatible Function Approximation

• Uri Sherman
• Tomer Koren
• Yishay Mansour

Modern policy optimization methods roughly follow the policy mirror descent (PMD) algorithmic template, for which there are by now numerous theoretical convergence results. However, most of these either target tabular environments, or can be applied effectively only when the class of policies being optimized over satisfies strong closure conditions, which is typically not the case when working with parametric policy classes in large-scale environments. In this work, we develop a theoretical framework for PMD for general policy classes where we replace the closure conditions with a generally weaker variational gradient dominance assumption, and obtain upper bounds on the rate of convergence to the best-in-class policy. Our main result leverages a novel notion of smoothness with respect to a local norm induced by the occupancy measure of the current policy, and casts PMD as a particular instance of smooth non-convex optimization in non-Euclidean space.

ICML Conference 2025 Conference Paper

Dueling Convex Optimization with General Preferences

• Aadirupa Saha
• Tomer Koren
• Yishay Mansour

We address the problem of convex optimization with dueling feedback, where the goal is to minimize a convex function given a weaker form of dueling feedback. Each query consists of two points and the dueling feedback returns a (noisy) single-bit binary comparison of the function values of the two queried points. The translation of the function values to the single comparison bit is through a transfer function. This problem has been addressed previously for some restricted classes of transfer functions, but here we consider a very general transfer function class which includes all functions that admit a series expansion about the origin. Our main contribution is an efficient algorithm with convergence rate of $O(\epsilon^{-4p})$ for smooth convex functions, and an optimal rate of $\widetilde O(\epsilon^{-2p})$ when the objective is both smooth and strongly convex, where $p$ is the minimal degree (with a non-zero coefficient) in the transfer’s series expansion about the origin.

NeurIPS Conference 2025 Conference Paper

Fast Last-Iterate Convergence of SGD in the Smooth Interpolation Regime

• Amit Attia
• Matan Schliserman
• Uri Sherman
• Tomer Koren

We study population convergence guarantees of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, where the noise at optimum is zero or near zero. The behavior of the last iterate of SGD in this setting---particularly with large (constant) stepsizes---has received growing attention in recent years due to implications for the training of over-parameterized models, as well as to analyzing forgetting in continual learning and to understanding the convergence of the randomized Kaczmarz method for solving linear systems. We establish that after $T$ steps of SGD on $\beta$-smooth convex loss functions with stepsize $0 < \eta < 2/\beta$, the last iterate exhibits expected excess risk $\widetilde{O}(\tfrac{1}{\eta (2-\beta \eta) T^{1-\beta\eta/2}} + \tfrac{\eta}{(2-\beta\eta)^2} T^{\beta\eta/2} \sigma_\star^2)$, where $\sigma_\star^2$ denotes the variance of the stochastic gradients at the optimum. In particular, for a well-tuned stepsize we obtain a near optimal $\widetilde{O}(1/T + \sigma_\star/\sqrt T)$ rate for the last iterate, extending the results of Varre et al. (2021) beyond least squares regression; and when $\sigma_\star=0$ we obtain a rate of $O(1/\sqrt T)$ with $\eta=1/\beta$, improving upon the best-known $O(T^{-1/4})$ rate recently established by Evron et al. (2025) in the special case of realizable linear regression.

ICML Conference 2025 Conference Paper

Faster Stochastic Optimization with Arbitrary Delays via Adaptive Asynchronous Mini-Batching

• Amit Attia
• Ofir Gaash
• Tomer Koren

We consider the problem of asynchronous stochastic optimization, where an optimization algorithm makes updates based on stale stochastic gradients of the objective that are subject to an arbitrary (possibly adversarial) sequence of delays. We present a procedure which, for any given $q \in (0, 1]$, transforms any standard stochastic first-order method to an asynchronous method with convergence guarantee depending on the $q$-quantile delay of the sequence. This approach leads to convergence rates of the form $O(\tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems, where $\tau_q$ is the $q$-quantile delay, generalizing and improving on existing results that depend on the average delay. We further show a method that automatically adapts to all quantiles simultaneously, without any prior knowledge of the delays, achieving convergence rates of the form $O(\inf_{q} \tau_q/qT+\sigma/\sqrt{qT})$ for non-convex and $O(\inf_{q} \tau_q^2/(q T)^2+\sigma/\sqrt{qT})$ for convex smooth problems. Our technique is based on asynchronous mini-batching with a careful batch-size selection and filtering of stale gradients.

NeurIPS Conference 2025 Conference Paper

From Contextual Combinatorial Semi-Bandits to Bandit List Classification: Improved Sample Complexity with Sparse Rewards

• 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 of the interaction, the learner observes feedback consisting of 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 $(\varepsilon, \delta)$-PAC variant of the problem for which we design an algorithm that returns an $\varepsilon$-optimal policy with high probability using a sample complexity of $\widetilde{O}\big( (\mathrm{poly}(K/m) + sm / \varepsilon^2) \log (|\Pi|/\delta) \big)$ where $\Pi$ 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 PAC learning algorithm is also computationally efficient given access to an ERM oracle for $\Pi$. 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 $O \big((K^7 + 1/\varepsilon^2)\log (|\mathcal{H}|/\delta)\big)$ sample complexity bound for a finite hypothesis class $\mathcal{H}$, 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 $\widetilde O(|\Pi| + \sqrt{smT \log |\Pi|})$, extending the result of Erez et al. ('24) who consider the simpler single label classification setting.

NeurIPS Conference 2025 Conference Paper

Multiclass Loss Geometry Matters for Generalization of Gradient Descent in Separable Classification

• Matan Schliserman
• Tomer Koren

We study the generalization performance of unregularized gradient methods for separable linear classification. While previous work mostly deal with the binary case, we focus on the multiclass setting with $k$ classes and establish novel population risk bounds for Gradient Descent for loss functions that decay to zero. In this setting, we show risk bounds that reveal that convergence rates are crucially influenced by the geometry of the loss template, as formalized by Wang and Scott (2024), rather than of the loss function itself. Particularly, we establish risk upper bounds that holds for any decay rate of the loss whose template is smooth with respect to the $p$-norm. In the case of exponentially decaying losses, our results indicates a contrast between the $p=\infty$ case, where the risk exhibits a logarithmic dependence on $k$, and $p=2$ where the risk scales linearly with $k$. To establish this separation formally, we also prove a lower bound in the latter scenario, demonstrating that the polynomial dependence on $k$ is unavoidable. Central to our analysis is a novel bound on the Rademacher complexity of low-noise vector-valued linear predictors with a loss template smooth w. r. t. general $p$-norms.

ICML Conference 2025 Conference Paper

Nearly Optimal Sample Complexity for Learning with Label Proportions

• Róbert Busa-Fekete
• Travis Dick
• Claudio Gentile
• Haim Kaplan
• Tomer Koren
• Uri Stemmer

We investigate Learning from Label Proportions (LLP), a partial information setting where examples in a training set are grouped into bags, and only aggregate label values in each bag are available. Despite the partial observability, the goal is still to achieve small regret at the level of individual examples. We give results on the sample complexity of LLP under square loss, showing that our sample complexity is essentially optimal. From an algorithmic viewpoint, we rely on carefully designed variants of Empirical Risk Minimization, and Stochastic Gradient Descent algorithms, combined with ad hoc variance reduction techniques. On one hand, our theoretical results improve in important ways on the existing literature on LLP, specifically in the way the sample complexity depends on the bag size. On the other hand, we validate our algorithmic solutions on several datasets, demonstrating improved empirical performance (better accuracy for less samples) against recent baselines.

NeurIPS Conference 2025 Conference Paper

Optimal Rates in Continual Linear Regression via Increasing Regularization

• Ran Levinstein
• Amit Attia
• Matan Schliserman
• Uri Sherman
• Daniel Soudry
• Tomer Koren
• Itay Evron

We study realizable continual linear regression under random task orderings, a common setting for developing continual learning theory. In this setup, the worst-case expected loss after $k$ learning iterations admits a lower bound of $\Omega(1/k)$. However, prior work using an unregularized scheme has only established an upper bound of $O(1/k^{1/4})$, leaving a significant gap. Our paper proves that this gap can be narrowed, or even closed, using two frequently used regularization schemes: (1) explicit isotropic $\ell_2$ regularization, and (2) implicit regularization via finite step budgets. We show that these approaches, which are used in practice to mitigate forgetting, reduce to stochastic gradient descent (SGD) on carefully defined surrogate losses. Through this lens, we identify a fixed regularization strength that yields a near-optimal rate of $O(\log k / k)$. Formalizing and analyzing a generalized variant of SGD for time-varying functions, we derive an increasing regularization strength schedule that provably achieves an optimal rate of $O(1/k)$. This suggests that schedules that increase the regularization coefficient or decrease the number of steps per task are beneficial, at least in the worst case.

ICML Conference 2025 Conference Paper

Rapid Overfitting of Multi-Pass SGD in Stochastic Convex Optimization

• Shira Vansover-Hager
• Tomer Koren
• Roi Livni

We study the out-of-sample performance of multi-pass stochastic gradient descent (SGD) in the fundamental stochastic convex optimization (SCO) model. While one-pass SGD is known to achieve an optimal $\Theta(1/\sqrt{n})$ excess population loss given a sample of size $n$, much less is understood about the multi-pass version of the algorithm which is widely used in practice. Somewhat surprisingly, we show that in the general non-smooth case of SCO, just a few epochs of SGD can already hurt its out-of-sample performance significantly and lead to overfitting. In particular, using a step size $\eta = \Theta(1/\sqrt{n})$, which gives the optimal rate after one pass, can lead to population loss as large as $\Omega(1)$ after just one additional pass. More generally, we show that the population loss from the second pass onward is of the order $\Theta(1/(\eta T) + \eta \sqrt{T})$, where $T$ is the total number of steps. These results reveal a certain phase-transition in the out-of-sample behavior of SGD after the first epoch, as well as a sharp separation between the rates of overfitting in the smooth and non-smooth cases of SCO. Additionally, we extend our results to with-replacement SGD, proving that the same asymptotic bounds hold after $O(n \log n)$ steps. Finally, we also prove a lower bound of $\Omega(\eta \sqrt{n})$ on the generalization gap of one-pass SGD in dimension $d = {\widetilde O}(n)$, improving on recent results of Koren et al. (2022) and Schliserman et al. (2024).

NeurIPS Conference 2024 Conference Paper

Fast Rates for Bandit PAC Multiclass Classification

• Liad Erez
• Alon Cohen
• Tomer Koren
• Yishay Mansour
• Shay Moran

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 $O\big( (\operatorname{poly}(K) + 1 / \varepsilon^2) \log (|\mathcal{H}| / \delta) \big)$ for any finite hypothesis class $\mathcal{H}$. 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 $\log |\mathcal{H}|$ 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 $\mathcal{H}$.

ICML Conference 2024 Conference Paper

How Free is Parameter-Free Stochastic Optimization?

• Amit Attia
• Tomer Koren

We study the problem of parameter-free stochastic optimization, inquiring whether, and under what conditions, do fully parameter-free methods exist: these are methods that achieve convergence rates competitive with optimally tuned methods, without requiring significant knowledge of the true problem parameters. Existing parameter-free methods can only be considered “partially” parameter-free, as they require some non-trivial knowledge of the true problem parameters, such as a bound on the stochastic gradient norms, a bound on the distance to a minimizer, etc. In the non-convex setting, we demonstrate that a simple hyperparameter search technique results in a fully parameter-free method that outperforms more sophisticated state-of-the-art algorithms. We also provide a similar result in the convex setting with access to noisy function values under mild noise assumptions. Finally, assuming only access to stochastic gradients, we establish a lower bound that renders fully parameter-free stochastic convex optimization infeasible, and provide a method which is (partially) parameter-free up to the limit indicated by our lower bound.

NeurIPS Conference 2024 Conference Paper

Private Online Learning via Lazy Algorithms

• Hilal Asi
• Tomer Koren
• Daogao Liu
• Kunal Talwar

We study the problem of private online learning, specifically, online prediction from experts (OPE) and online convex optimization (OCO). We propose a new transformation that transforms lazy online learning algorithms into private algorithms. We apply our transformation for differentially private OPE and OCO using existing lazy algorithms for these problems. Our final algorithms obtain regret which significantly improves the regret in the high privacy regime $\varepsilon \ll 1$, obtaining $\sqrt{T \log d} + T^{1/3} \log(d)/\varepsilon^{2/3}$ for DP-OPE and $\sqrt{T} + T^{1/3} \sqrt{d}/\varepsilon^{2/3}$ for DP-OCO. We also complement our results with a lower bound for DP-OPE, showing that these rates are optimal for a natural family of low-switching private algorithms.

ICML Conference 2024 Conference Paper

Rate-Optimal Policy Optimization for Linear Markov Decision Processes

• Uri Sherman
• Alon Cohen
• Tomer Koren
• Yishay Mansour

We study regret minimization in online episodic linear Markov Decision Processes, and propose a policy optimization algorithm that is computationally efficient, and obtains rate optimal $\widetilde O (\sqrt K)$ regret where $K$ denotes the number of episodes. Our work is the first to establish the optimal rate (in terms of $K$) of convergence in the stochastic setting with bandit feedback using a policy optimization based approach, and the first to establish the optimal rate in the adversarial setup with full information feedback, for which no algorithm with an optimal rate guarantee was previously known.

EWRL Workshop 2024 Workshop Paper

Rate-Optimal Policy Optimization for Linear Markov Decision Processes

• Uri Sherman
• Alon Cohen
• Tomer Koren
• Yishay Mansour

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

ICML Conference 2023 Conference Paper

Improved Regret for Efficient Online Reinforcement Learning with Linear Function Approximation

• Uri Sherman
• Tomer Koren
• Yishay Mansour

We study reinforcement learning with linear function approximation and adversarially changing cost functions, a setup that has mostly been considered under simplifying assumptions such as full information feedback or exploratory conditions. We present a computationally efficient policy optimization algorithm for the challenging general setting of unknown dynamics and bandit feedback, featuring a combination of mirror-descent and least squares policy evaluation in an auxiliary MDP used to compute exploration bonuses. Our algorithm obtains an $\widetilde O(K^{6/7})$ regret bound, improving significantly over previous state-of-the-art of $\widetilde O (K^{14/15})$ in this setting. In addition, we present a version of the same algorithm under the assumption a simulator of the environment is available to the learner (but otherwise no exploratory assumptions are made), and prove it obtains state-of-the-art regret of $\widetilde O (K^{2/3})$.

ICML Conference 2023 Conference Paper

Near-Optimal Algorithms for Private Online Optimization in the Realizable Regime

• Hilal Asi
• Vitaly Feldman
• Tomer Koren
• Kunal Talwar

We consider online learning problems in the realizable setting, where there is a zero-loss solution, and propose new Differentially Private (DP) algorithms that obtain near-optimal regret bounds. For the problem of online prediction from experts, we design new algorithms that obtain near-optimal regret $O \big( \varepsilon^{-1} \mathsf{poly}(\log{d}) \big)$ where $d$ is the number of experts. This significantly improves over the best existing regret bounds for the DP non-realizable setting which are $O \big( \varepsilon^{-1} \min\big\{d, \sqrt{T\log d}\big\} \big)$. We also develop an adaptive algorithm for the small-loss setting with regret $(L^\star+ \varepsilon^{-1}) \cdot O(\mathsf{poly}(\log{d}))$ where $L^\star$ is the total loss of the best expert. Additionally, we consider DP online convex optimization in the realizable setting and propose an algorithm with near-optimal regret $O \big(\varepsilon^{-1} \mathsf{poly}(d) \big)$, as well as an algorithm for the smooth case with regret $O \big( (\sqrt{Td}/\varepsilon)^{2/3} \big)$, both significantly improving over existing bounds in the non-realizable regime.

ICML Conference 2023 Conference Paper

Regret Minimization and Convergence to Equilibria in General-sum Markov Games

• Liad Erez
• Tal Lancewicki
• Uri Sherman
• Tomer Koren
• Yishay Mansour

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 $\textit{swap regret}$, and thus, along the way, imply convergence to a $\textit{correlated}$ 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 $\textit{weighted}$ regret minimization, with $\textit{unknown}$ 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.

ICML Conference 2023 Conference Paper

SGD with AdaGrad Stepsizes: Full Adaptivity with High Probability to Unknown Parameters, Unbounded Gradients and Affine Variance

• Amit Attia
• Tomer Koren

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general “affine variance” noise model and provides sharp rates of convergence in both the low-noise and high-noise regimes.

NeurIPS Conference 2023 Conference Paper

Tight Risk Bounds for Gradient Descent on Separable Data

• Matan Schliserman
• Tomer Koren

We study the generalization properties of unregularized gradient methods applied to separable linear classification---a setting that has received considerable attention since the pioneering work of Soudry et al. (2018). We establish tight upper and lower (population) risk bounds for gradient descent in this setting, for any smooth loss function, expressed in terms of its tail decay rate. Our bounds take the form $\Theta(r_{\ell, T}^2 / \gamma^2 T + r_{\ell, T}^2 / \gamma^2 n)$, where $T$ is the number of gradient steps, $n$ is size of the training set, $\gamma$ is the data margin, and $r_{\ell, T}$ is a complexity term that depends on the tail decay rate of the loss function (and on $T$). Our upper bound greatly improves the existing risk bounds due to Shamir (2021) and Schliserman and Koren (2022), that either applied to specific loss functions or imposed extraneous technical assumptions, and applies to virtually any convex and smooth loss function. Our risk lower bound is the first in this context and establish the tightness of our general upper bound for any given tail decay rate and in all parameter regimes. The proof technique used to show these results is also markedly simpler compared to previous work, and is straightforward to extend to other gradient methods; we illustrate this by providing analogous results for Stochastic Gradient Descent.

NeurIPS Conference 2022 Conference Paper

Benign Underfitting of Stochastic Gradient Descent

• Tomer Koren
• Roi Livni
• Yishay Mansour
• Uri Sherman

We study to what extent may stochastic gradient descent (SGD) be understood as a ``conventional'' learning rule that achieves generalization performance by obtaining a good fit to training data. We consider the fundamental stochastic convex optimization framework, where (one pass, $\textit{without}$-replacement) SGD is classically known to minimize the population risk at rate $O(1/\sqrt n)$, and prove that, surprisingly, there exist problem instances where the SGD solution exhibits both empirical risk and generalization gap of $\Omega(1)$. Consequently, it turns out that SGD is not algorithmically stable in $\textit{any}$ sense, and its generalization ability cannot be explained by uniform convergence or any other currently known generalization bound technique for that matter (other than that of its classical analysis). We then continue to analyze the closely related $\textit{with}$-replacement SGD, for which we show that an analogous phenomenon does not occur and prove that its population risk does in fact converge at the optimal rate. Finally, we interpret our main results in the context of without-replacement SGD for finite-sum convex optimization problems, and derive upper and lower bounds for the multi-epoch regime that significantly improve upon previously known results.

NeurIPS Conference 2022 Conference Paper

Better Best of Both Worlds Bounds for Bandits with Switching Costs

• Idan Amir
• Guy Azov
• Tomer Koren
• Roi Livni

We study best-of-both-worlds algorithms for bandits with switching cost, recently addressed by Rouyer et al. , 2021. We introduce a surprisingly simple and effective algorithm that simultaneously achieves minimax optimal regret bound (up to logarithmic factors) of $\mathcal{O}(T^{2/3})$ in the oblivious adversarial setting and a bound of $\mathcal{O}(\min\{\log (T)/\Delta^2, T^{2/3}\})$ in the stochastically-constrained regime, both with (unit) switching costs, where $\Delta$ is the gap between the arms. In the stochastically constrained case, our bound improves over previous results due to Rouyer et al. , 2021, that achieved regret of $\mathcal{O}(T^{1/3}/\Delta)$. We accompany our results with a lower bound showing that, in general, $\tilde{\mathcal{\Omega}}(\min\{1/\Delta^2, T^{2/3}\})$ switching cost regret is unavoidable in the stochastically-constrained case for algorithms with $\mathcal{O}(T^{2/3})$ worst-case switching cost regret.

EWRL Workshop 2022 Workshop Paper

Rate-Optimal Online Convex Optimization in Adaptive Linear Control

• Asaf B Cassel
• Alon Cohen
• Tomer Koren

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 √ 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.

NeurIPS Conference 2022 Conference Paper

Rate-Optimal Online Convex Optimization in Adaptive Linear Control

• Asaf Benjamin Cassel
• Alon Peled-Cohen
• Tomer Koren

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 $\sqrt{T}$-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.

ICML Conference 2021 Conference Paper

Adversarial Dueling Bandits

• Aadirupa Saha
• Tomer Koren
• Yishay Mansour

We introduce the problem of regret minimization in Adversarial Dueling Bandits. As in classic Dueling Bandits, the learner has to repeatedly choose a pair of items and observe only a relative binary ‘win-loss’ feedback for this pair, but here this feedback is generated from an arbitrary preference matrix, possibly chosen adversarially. Our main result is an algorithm whose $T$-round regret compared to the \emph{Borda-winner} from a set of $K$ items is $\tilde{O}(K^{1/3}T^{2/3})$, as well as a matching $\Omega(K^{1/3}T^{2/3})$ lower bound. We also prove a similar high probability regret bound. We further consider a simpler \emph{fixed-gap} adversarial setup, which bridges between two extreme preference feedback models for dueling bandits: stationary preferences and an arbitrary sequence of preferences. For the fixed-gap adversarial setup we give an $\smash{ \tilde{O}((K/\Delta^2)\log{T}) }$ regret algorithm, where $\Delta$ is the gap in Borda scores between the best item and all other items, and show a lower bound of $\Omega(K/\Delta^2)$ indicating that our dependence on the main problem parameters $K$ and $\Delta$ is tight (up to logarithmic factors). Finally, we corroborate the theoretical results with empirical evaluations.

NeurIPS Conference 2021 Conference Paper

Algorithmic Instabilities of Accelerated Gradient Descent

• Amit Attia
• Tomer Koren

We study the algorithmic stability of Nesterov's accelerated gradient method. For convex quadratic objectives, Chen et al. (2018) proved that the uniform stability of the method grows quadratically with the number of optimization steps, and conjectured that the same is true for the general convex and smooth case. We disprove this conjecture and show, for two notions of algorithmic stability (including uniform stability), that the stability of Nesterov's accelerated method in fact deteriorates exponentially fast with the number of gradient steps. This stands in sharp contrast to the bounds in the quadratic case, but also to known results for non-accelerated gradient methods where stability typically grows linearly with the number of steps.

NeurIPS Conference 2021 Conference Paper

Asynchronous Stochastic Optimization Robust to Arbitrary Delays

• Alon Cohen
• Amit Daniely
• Yoel Drori
• Tomer Koren
• Mariano Schain

We consider the problem of stochastic optimization with delayed gradients in which, at each time step $t$, the algorithm makes an update using a stale stochastic gradient from step $t - d_t$ for some arbitrary delay $d_t$. This setting abstracts asynchronous distributed optimization where a central server receives gradient updates computed by worker machines. These machines can experience computation and communication loads that might vary significantly over time. In the general non-convex smooth optimization setting, we give a simple and efficient algorithm that requires $O( \sigma^2/\epsilon^4 + \tau/\epsilon^2 )$ steps for finding an $\epsilon$-stationary point $x$. Here, $\tau$ is the \emph{average} delay $\frac{1}{T}\sum_{t=1}^T d_t$ and $\sigma^2$ is the variance of the stochastic gradients. This improves over previous work, which showed that stochastic gradient decent achieves the same rate but with respect to the \emph{maximal} delay $\max_{t} d_t$, that can be significantly larger than the average delay especially in heterogeneous distributed systems. Our experiments demonstrate the efficacy and robustness of our algorithm in cases where the delay distribution is skewed or heavy-tailed.

ICML Conference 2021 Conference Paper

Dueling Convex Optimization

• Aadirupa Saha
• Tomer Koren
• Yishay Mansour

We address the problem of convex optimization with preference (dueling) feedback. Like the traditional optimization objective, the goal is to find the optimal point with the least possible query complexity, however, without the luxury of even a zeroth order feedback. Instead, the learner can only observe a single noisy bit which is win-loss feedback for a pair of queried points based on their function values. % The problem is certainly of great practical relevance as in many real-world scenarios, such as recommender systems or learning from customer preferences, where the system feedback is often restricted to just one binary-bit preference information. % We consider the problem of online convex optimization (OCO) solely by actively querying $\{0, 1\}$ noisy-comparison feedback of decision point pairs, with the objective of finding a near-optimal point (function minimizer) with the least possible number of queries. %a very general class of monotonic, non-decreasing transfer functions, and analyze the problem for any $d$-dimensional smooth convex function. % For the non-stationary OCO setup, where the underlying convex function may change over time, we prove an impossibility result towards achieving the above objective. We next focus only on the stationary OCO problem, and our main contribution lies in designing a normalized gradient descent based algorithm towards finding a $\epsilon$-best optimal point. Towards this, our algorithm is shown to yield a convergence rate of $\tilde O(\nicefrac{d\beta}{\epsilon \nu^2})$ ($\nu$ being the noise parameter) when the underlying function is $\beta$-smooth. Further we show an improved convergence rate of just $\tilde O(\nicefrac{d\beta}{\alpha \nu^2} \log \frac{1}{\epsilon})$ when the function is additionally also $\alpha$-strongly convex.

NeurIPS Conference 2021 Conference Paper

Never Go Full Batch (in Stochastic Convex Optimization)

• Idan Amir
• Yair Carmon
• Tomer Koren
• Roi Livni

We study the generalization performance of $\text{\emph{full-batch}}$ optimization algorithms for stochastic convex optimization: these are first-order methods that only access the exact gradient of the empirical risk (rather than gradients with respect to individual data points), that include a wide range of algorithms such as gradient descent, mirror descent, and their regularized and/or accelerated variants. We provide a new separation result showing that, while algorithms such as stochastic gradient descent can generalize and optimize the population risk to within $\epsilon$ after $O(1/\epsilon^2)$ iterations, full-batch methods either need at least $\Omega(1/\epsilon^4)$ iterations or exhibit a dimension-dependent sample complexity.

ICML Conference 2021 Conference Paper

Online Policy Gradient for Model Free Learning of Linear Quadratic Regulators with √T Regret

• Asaf B. Cassel
• Tomer Koren

We consider the task of learning to control a linear dynamical system under fixed quadratic costs, known as the Linear Quadratic Regulator (LQR) problem. While model-free approaches are often favorable in practice, thus far only model-based methods, which rely on costly system identification, have been shown to achieve regret that scales with the optimal dependence on the time horizon T. We present the first model-free algorithm that achieves similar regret guarantees. Our method relies on an efficient policy gradient scheme, and a novel and tighter analysis of the cost of exploration in policy space in this setting.

NeurIPS Conference 2021 Conference Paper

Optimal Rates for Random Order Online Optimization

• Uri Sherman
• Tomer Koren
• Yishay Mansour

We study online convex optimization in the random order model, recently proposed by Garber et al. (2020), where the loss functions may be chosen by an adversary, but are then presented to the online algorithm in a uniformly random order. Focusing on the scenario where the cumulative loss function is (strongly) convex, yet individual loss functions are smooth but might be non-convex, we give algorithms that achieve the optimal bounds and significantly outperform the results of Garber et al. (2020), completely removing the dimension dependence and improve their scaling with respect to the strong convexity parameter. Our analysis relies on novel connections between algorithmic stability and generalization for sampling without-replacement analogous to those studied in the with-replacement i. i. d. setting, as well as on a refined average stability analysis of stochastic gradient descent.

ICML Conference 2021 Conference Paper

Private Stochastic Convex Optimization: Optimal Rates in L1 Geometry

• Hilal Asi
• Vitaly Feldman
• Tomer Koren
• Kunal Talwar

Stochastic convex optimization over an $\ell_1$-bounded domain is ubiquitous in machine learning applications such as LASSO but remains poorly understood when learning with differential privacy. We show that, up to logarithmic factors the optimal excess population loss of any $(\epsilon, \delta)$-differentially private optimizer is $\sqrt{\log(d)/n} + \sqrt{d}/\epsilon n. $ The upper bound is based on a new algorithm that combines the iterative localization approach of Feldman et al. (2020) with a new analysis of private regularized mirror descent. It applies to $\ell_p$ bounded domains for $p\in [1, 2]$ and queries at most $n^{3/2}$ gradients improving over the best previously known algorithm for the $\ell_2$ case which needs $n^2$ gradients. Further, we show that when the loss functions satisfy additional smoothness assumptions, the excess loss is upper bounded (up to logarithmic factors) by $\sqrt{\log(d)/n} + (\log(d)/\epsilon n)^{2/3}. $ This bound is achieved by a new variance-reduced version of the Frank-Wolfe algorithm that requires just a single pass over the data. We also show that the lower bound in this case is the minimum of the two rates mentioned above.

ICML Conference 2021 Conference Paper

Stochastic Multi-Armed Bandits with Unrestricted Delay Distributions

• Tal Lancewicki
• Shahar Segal
• Tomer Koren
• Yishay Mansour

We study the stochastic Multi-Armed Bandit (MAB) problem with random delays in the feedback received by the algorithm. We consider two settings: the {\it reward dependent} delay setting, where realized delays may depend on the stochastic rewards, and the {\it reward-independent} delay setting. Our main contribution is algorithms that achieve near-optimal regret in each of the settings, with an additional additive dependence on the quantiles of the delay distribution. Our results do not make any assumptions on the delay distributions: in particular, we do not assume they come from any parametric family of distributions and allow for unbounded support and expectation; we further allow for the case of infinite delays where the algorithm might occasionally not observe any feedback.

NeurIPS Conference 2021 Conference Paper

Towards Best-of-All-Worlds Online Learning with Feedback Graphs

• 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: $O(\sqrt{\theta(G) T})$ with adversarial losses; $O(\theta(G)\mathrm{polylog}{T})$ with stochastic losses; and $O(\theta(G)\mathrm{polylog}{T} + \sqrt{\theta(G) C})$ with stochastic losses subject to $C$ adversarial corruptions. Here, $\theta(G)$ 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.

NeurIPS Conference 2020 Conference Paper

Bandit Linear Control

• Asaf Cassel
• Tomer Koren

We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only the cost incurred by the learner is observed. We present a new and efficient algorithm that, for strongly convex and smooth costs, obtains regret that grows with the square root of the time horizon T. We also give extensions of this result to general convex, possibly non-smooth costs, and to non-stochastic system noise. A key component of our algorithm is a new technique for addressing bandit optimization of loss functions with memory.

NeurIPS Conference 2020 Conference Paper

Can Implicit Bias Explain Generalization? Stochastic Convex Optimization as a Case Study

• Assaf Dauber
• Meir Feder
• Tomer Koren
• Roi Livni

The notion of implicit bias, or implicit regularization, has been suggested as a means to explain the surprising generalization ability of modern-days overparameterized learning algorithms. This notion refers to the tendency of the optimization algorithm towards a certain structured solution that often generalizes well. Recently, several papers have studied implicit regularization and were able to identify this phenomenon in various scenarios. We revisit this paradigm in arguably the simplest non-trivial setup, and study the implicit bias of Stochastic Gradient Descent (SGD) in the context of Stochastic Convex Optimization. As a first step, we provide a simple construction that rules out the existence of a \emph{distribution-independent} implicit regularizer that governs the generalization ability of SGD. We then demonstrate a learning problem that rules out a very general class of \emph{distribution-dependent} implicit regularizers from explaining generalization, which includes strongly convex regularizers as well as non-degenerate norm-based regularizations. Certain aspects of our constructions point out to significant difficulties in providing a comprehensive explanation of an algorithm's generalization performance by solely arguing about its implicit regularization properties.

ICML Conference 2020 Conference Paper

Logarithmic Regret for Learning Linear Quadratic Regulators Efficiently

• Asaf B. Cassel
• Alon Cohen
• Tomer Koren

We consider the problem of learning in Linear Quadratic Control systems whose transition parameters are initially unknown. Recent results in this setting have demonstrated efficient learning algorithms with regret growing with the square root of the number of decision steps. We present new efficient algorithms that achieve, perhaps surprisingly, regret that scales only (poly-)logarithmically with the number of steps, in two scenarios: when only the state transition matrix A is unknown, and when only the state-action transition matrix B is unknown and the optimal policy satisfies a certain non-degeneracy condition. On the other hand, we give a lower bound which shows that when the latter condition is violated, square root regret is unavoidable.

NeurIPS Conference 2020 Conference Paper

Prediction with Corrupted Expert Advice

• Idan Amir
• Idan Attias
• Tomer Koren
• Yishay Mansour
• Roi Livni

We revisit the fundamental problem of prediction with expert advice, in a setting where the environment is benign and generates losses stochastically, but the feedback observed by the learner is subject to a moderate adversarial corruption. We prove that a variant of the classical Multiplicative Weights algorithm with decreasing step sizes achieves constant regret in this setting and performs optimally in a wide range of environments, regardless of the magnitude of the injected corruption. Our results reveal a surprising disparity between the often comparable Follow the Regularized Leader (FTRL) and Online Mirror Descent (OMD) frameworks: we show that for experts in the corrupted stochastic regime, the regret performance of OMD is in fact strictly inferior to that of FTRL.

STOC Conference 2020 Conference Paper

Private stochastic convex optimization: optimal rates in linear time

• Vitaly Feldman
• Tomer Koren
• Kunal Talwar

NeurIPS Conference 2020 Conference Paper

Stochastic Optimization with Laggard Data Pipelines

• Naman Agarwal
• Rohan Anil
• Tomer Koren
• Kunal Talwar
• Cyril Zhang

State-of-the-art optimization is steadily shifting towards massively parallel pipelines with extremely large batch sizes. As a consequence, CPU-bound preprocessing and disk/memory/network operations have emerged as new performance bottlenecks, as opposed to hardware-accelerated gradient computations. In this regime, a recently proposed approach is data echoing (Choi et al. , 2019), which takes repeated gradient steps on the same batch while waiting for fresh data to arrive from upstream. We provide the first convergence analyses of "data-echoed" extensions of common optimization methods, showing that they exhibit provable improvements over their synchronous counterparts. Specifically, we show that in convex optimization with stochastic minibatches, data echoing affords speedups on the curvature-dominated part of the convergence rate, while maintaining the optimal statistical rate.

ICML Conference 2019 Conference Paper

Learning Linear-Quadratic Regulators Efficiently with only √T Regret

• Alon Cohen
• Tomer Koren
• Yishay Mansour

We present the first computationally-efficient algorithm with $\widetilde{O}(\sqrt{T})$ regret for learning in Linear Quadratic Control systems with unknown dynamics. By that, we resolve an open question of Abbasi-Yadkori and Szepesvari (2011) and Dean, Mania, Matni, Recht, and Tu (2018).

NeurIPS Conference 2019 Conference Paper

Memory Efficient Adaptive Optimization

• Rohan Anil
• Vineet Gupta
• Tomer Koren
• Yoram Singer

Adaptive gradient-based optimizers such as Adagrad and Adam are crucial for achieving state-of-the-art performance in machine translation and language modeling. However, these methods maintain second-order statistics for each parameter, thus introducing significant memory overheads that restrict the size of the model being used as well as the number of examples in a mini-batch. We describe an effective and flexible adaptive optimization method with greatly reduced memory overhead. Our method retains the benefits of per-parameter adaptivity while allowing significantly larger models and batch sizes. We give convergence guarantees for our method, and demonstrate its effectiveness in training very large translation and language models with up to 2-fold speedups compared to the state-of-the-art.

NeurIPS Conference 2019 Conference Paper

Robust Bi-Tempered Logistic Loss Based on Bregman Divergences

• Ehsan Amid
• Manfred K. Warmuth
• Rohan Anil
• Tomer Koren

We introduce a temperature into the exponential function and replace the softmax output layer of the neural networks by a high-temperature generalization. Similarly, the logarithm in the loss we use for training is replaced by a low-temperature logarithm. By tuning the two temperatures, we create loss functions that are non-convex already in the single layer case. When replacing the last layer of the neural networks by our bi-temperature generalization of the logistic loss, the training becomes more robust to noise. We visualize the effect of tuning the two temperatures in a simple setting and show the efficacy of our method on large datasets. Our methodology is based on Bregman divergences and is superior to a related two-temperature method that uses the Tsallis divergence.

ICML Conference 2019 Conference Paper

Semi-Cyclic Stochastic Gradient Descent

• Hubert Eichner
• Tomer Koren
• H. Brendan McMahan
• Nathan Srebro
• Kunal Talwar

We consider convex SGD updates with a block-cyclic structure, i. e. , where each cycle consists of a small number of blocks, each with many samples from a possibly different, block-specific, distribution. This situation arises, e. g. , in Federated Learning where the mobile devices available for updates at different times during the day have different characteristics. We show that such block-cyclic structure can significantly deteriorate the performance of SGD, but propose a simple approach that allows prediction with the same guarantees as for i. i. d. , non-cyclic, sampling.

ICML Conference 2018 Conference Paper

Online Linear Quadratic Control

• Alon Cohen
• Avinatan Hassidim
• Tomer Koren
• Nevena Lazic
• Yishay Mansour
• Kunal Talwar

We study the problem of controlling linear time-invariant systems with known noisy dynamics and adversarially chosen quadratic losses. We present the first efficient online learning algorithms in this setting that guarantee $O(\sqrt{T})$ regret under mild assumptions, where $T$ is the time horizon. Our algorithms rely on a novel SDP relaxation for the steady-state distribution of the system. Crucially, and in contrast to previously proposed relaxations, the feasible solutions of our SDP all correspond to “strongly stable” policies that mix exponentially fast to a steady state.

ICML Conference 2018 Conference Paper

Shampoo: Preconditioned Stochastic Tensor Optimization

• Vineet Gupta 0001
• Tomer Koren
• Yoram Singer

Preconditioned gradient methods are among the most general and powerful tools in optimization. However, preconditioning requires storing and manipulating prohibitively large matrices. We describe and analyze a new structure-aware preconditioning algorithm, called Shampoo, for stochastic optimization over tensor spaces. Shampoo maintains a set of preconditioning matrices, each of which operates on a single dimension, contracting over the remaining dimensions. We establish convergence guarantees in the stochastic convex setting, the proof of which builds upon matrix trace inequalities. Our experiments with state-of-the-art deep learning models show that Shampoo is capable of converging considerably faster than commonly used optimizers. Surprisingly, although it involves a more complex update rule, Shampoo’s runtime per step is comparable in practice to that of simple gradient methods such as SGD, AdaGrad, and Adam.

NeurIPS Conference 2017 Conference Paper

Affine-Invariant Online Optimization and the Low-rank Experts Problem

• Tomer Koren
• Roi Livni

We present a new affine-invariant optimization algorithm called Online Lazy Newton. The regret of Online Lazy Newton is independent of conditioning: the algorithm's performance depends on the best possible preconditioning of the problem in retrospect and on its \emph{intrinsic} dimensionality. As an application, we show how Online Lazy Newton can be used to achieve an optimal regret of order $\sqrt{rT}$ for the low-rank experts problem, improving by a $\sqrt{r}$ factor over the previously best known bound and resolving an open problem posed by Hazan et al (2016).

NeurIPS Conference 2017 Conference Paper

Multi-Armed Bandits with Metric Movement Costs

• Tomer Koren
• Roi Livni
• Yishay Mansour

We consider the non-stochastic Multi-Armed Bandit problem in a setting where there is a fixed and known metric on the action space that determines a cost for switching between any pair of actions. The loss of the online learner has two components: the first is the usual loss of the selected actions, and the second is an additional loss due to switching between actions. Our main contribution gives a tight characterization of the expected minimax regret in this setting, in terms of a complexity measure $\mathcal{C}$ of the underlying metric which depends on its covering numbers. In finite metric spaces with $k$ actions, we give an efficient algorithm that achieves regret of the form $\widetilde(\max\set{\mathcal{C}^{1/3}T^{2/3}, \sqrt{kT}})$, and show that this is the best possible. Our regret bound generalizes previous known regret bounds for some special cases: (i) the unit-switching cost regret $\widetilde{\Theta}(\max\set{k^{1/3}T^{2/3}, \sqrt{kT}})$ where $\mathcal{C}=\Theta(k)$, and (ii) the interval metric with regret $\widetilde{\Theta}(\max\set{T^{2/3}, \sqrt{kT}})$ where $\mathcal{C}=\Theta(1)$. For infinite metrics spaces with Lipschitz loss functions, we derive a tight regret bound of $\widetilde{\Theta}(T^{\frac{d+1}{d+2}})$ where $d \ge 1$ is the Minkowski dimension of the space, which is known to be tight even when there are no switching costs.

ICML Conference 2016 Conference Paper

Online Learning with Feedback Graphs Without the Graphs

• Alon Cohen
• Tamir Hazan
• Tomer Koren

We study an online learning framework introduced by Mannor and Shamir (2011) in which the feedback is specified by a graph, in a setting where the graph may vary from round to round and is \emphnever fully revealed to the learner. We show a large gap between the adversarial and the stochastic cases. In the adversarial case, we prove that even for dense feedback graphs, the learner cannot improve upon a trivial regret bound obtained by ignoring any additional feedback besides her own loss. In contrast, in the stochastic case we give an algorithm that achieves \widetildeΘ(\sqrtαT) regret over T rounds, provided that the independence numbers of the hidden feedback graphs are at most α. We also extend our results to a more general feedback model, in which the learner does not necessarily observe her own loss, and show that, even in simple cases, concealing the feedback graphs might render the problem unlearnable.

NeurIPS Conference 2016 Conference Paper

Online Pricing with Strategic and Patient Buyers

• Michal Feldman
• Tomer Koren
• Roi Livni
• Yishay Mansour
• Aviv Zohar

We consider a seller with an unlimited supply of a single good, who is faced with a stream of $T$ buyers. Each buyer has a window of time in which she would like to purchase, and would buy at the lowest price in that window, provided that this price is lower than her private value (and otherwise, would not buy at all). In this setting, we give an algorithm that attains $O(T^{2/3})$ regret over any sequence of $T$ buyers with respect to the best fixed price in hindsight, and prove that no algorithm can perform better in the worst case.

STOC Conference 2016 Conference Paper

The computational power of optimization in online learning

• Elad Hazan
• Tomer Koren

NeurIPS Conference 2016 Conference Paper

The Limits of Learning with Missing Data

• Brian Bullins
• Elad Hazan
• Tomer Koren

We study regression and classification in a setting where the learning algorithm is allowed to access only a limited number of attributes per example, known as the limited attribute observation model. In this well-studied model, we provide the first lower bounds giving a limit on the precision attainable by any algorithm for several variants of regression, notably linear regression with the absolute loss and the squared loss, as well as for classification with the hinge loss. We complement these lower bounds with a general purpose algorithm that gives an upper bound on the achievable precision limit in the setting of learning with missing data.

NeurIPS Conference 2015 Conference Paper

Bandit Smooth Convex Optimization: Improving the Bias-Variance Tradeoff

• Ofer Dekel
• Ronen Eldan
• Tomer Koren

Bandit convex optimization is one of the fundamental problems in the field of online learning. The best algorithm for the general bandit convex optimization problem guarantees a regret of $\widetilde{O}(T^{5/6})$, while the best known lower bound is $\Omega(T^{1/2})$. Many attemptshave been made to bridge the huge gap between these bounds. A particularly interesting special case of this problem assumes that the loss functions are smooth. In this case, the best known algorithm guarantees a regret of $\widetilde{O}(T^{2/3})$. We present an efficient algorithm for the banditsmooth convex optimization problem that guarantees a regret of $\widetilde{O}(T^{5/8})$. Our result rules out an $\Omega(T^{2/3})$ lower bound and takes a significant step towards the resolution of this open problem.

NeurIPS Conference 2015 Conference Paper

Fast Rates for Exp-concave Empirical Risk Minimization

• Tomer Koren
• Kfir Levy

We consider Empirical Risk Minimization (ERM) in the context of stochastic optimization with exp-concave and smooth losses---a general optimization framework that captures several important learning problems including linear and logistic regression, learning SVMs with the squared hinge-loss, portfolio selection and more. In this setting, we establish the first evidence that ERM is able to attain fast generalization rates, and show that the expected loss of the ERM solution in $d$ dimensions converges to the optimal expected loss in a rate of $d/n$. This rate matches existing lower bounds up to constants and improves by a $\log{n}$ factor upon the state-of-the-art, which is only known to be attained by an online-to-batch conversion of computationally expensive online algorithms.

STOC Conference 2014 Conference Paper

Bandits with switching costs: T 2/3 regret

• Ofer Dekel
• Jian Ding
• Tomer Koren
• Yuval Peres

FOCS Conference 2014 Conference Paper

Chasing Ghosts: Competing with Stateful Policies

• Uriel Feige
• Tomer Koren
• Moshe Tennenholtz

We consider sequential decision making in a setting where regret is measured with respect to a set of stateful reference policies, and feedback is limited to observing the rewards of the actions performed (the so called “bandit” setting). If either the reference policies are stateless rather than stateful, or the feedback includes the rewards of all actions (the so called “expert” setting), previous work shows that the √ optimal regret grows like Θ(√T) in terms of the number of decision rounds T. The difficulty in our setting is that the decision maker unavoidably loses track of the internal states of the reference policies, and thus cannot reliably attribute rewards observed in a certain round to any of the reference policies. In fact, in this setting it is impossible for the algorithm to estimate which policy gives the highest (or even approximately highest) total reward. Nevertheless, we design an algorithm that achieves expected regret that is sublinear in T, of the form O(T/ log 1/4 T). Our algorithm is based on a certain local repetition lemma that may be of independent interest. We also show that no algorithm can guarantee expected regret better than O(T/ log 3/2 T).

NeurIPS Conference 2014 Conference Paper

The Blinded Bandit: Learning with Adaptive Feedback

• Ofer Dekel
• Elad Hazan
• Tomer Koren

We study an online learning setting where the player is temporarily deprived of feedback each time it switches to a different action. Such model of \emph{adaptive feedback} naturally occurs in scenarios where the environment reacts to the player's actions and requires some time to recover and stabilize after the algorithm switches actions. This motivates a variant of the multi-armed bandit problem, which we call the \emph{blinded multi-armed bandit}, in which no feedback is given to the algorithm whenever it switches arms. We develop efficient online learning algorithms for this problem and prove that they guarantee the same asymptotic regret as the optimal algorithms for the standard multi-armed bandit problem. This result stands in stark contrast to another recent result, which states that adding a switching cost to the standard multi-armed bandit makes it substantially harder to learn, and provides a direct comparison of how feedback and loss contribute to the difficulty of an online learning problem. We also extend our results to the general prediction framework of bandit linear optimization, again attaining near-optimal regret bounds.

ICML Conference 2013 Conference Paper

Almost Optimal Exploration in Multi-Armed Bandits

• Zohar Shay Karnin
• Tomer Koren
• Oren Somekh

We study the problem of exploration in stochastic Multi-Armed Bandits. Even in the simplest setting of identifying the best arm, there remains a logarithmic multiplicative gap between the known lower and upper bounds for the number of arm pulls required for the task. This extra logarithmic factor is quite meaningful in nowadays large-scale applications. We present two novel, parameter-free algorithms for identifying the best arm, in two different settings: given a target confidence and given a target budget of arm pulls, for which we prove upper bounds whose gap from the lower bound is only doubly-logarithmic in the problem parameters. We corroborate our theoretical results with experiments demonstrating that our algorithm outperforms the state-of-the-art and scales better as the size of the problem increases.

NeurIPS Conference 2013 Conference Paper

Distributed Exploration in Multi-Armed Bandits

• Eshcar Hillel
• Zohar Karnin
• Tomer Koren
• Ronny Lempel
• Oren Somekh

We study exploration in Multi-Armed Bandits (MAB) in a setting where~$k$ players collaborate in order to identify an $\epsilon$-optimal arm. Our motivation comes from recent employment of MAB algorithms in computationally intensive, large-scale applications. Our results demonstrate a non-trivial tradeoff between the number of arm pulls required by each of the players, and the amount of communication between them. In particular, our main result shows that by allowing the $k$ players to communicate \emph{only once}, they are able to learn $\sqrt{k}$ times faster than a single player. That is, distributing learning to $k$ players gives rise to a factor~$\sqrt{k}$ parallel speed-up. We complement this result with a lower bound showing this is in general the best possible. On the other extreme, we present an algorithm that achieves the ideal factor $k$ speed-up in learning performance, with communication only logarithmic in~$1/\epsilon$.

ICML Conference 2012 Conference Paper

Linear Regression with Limited Observation

• Elad Hazan
• Tomer Koren

NeurIPS Conference 2011 Conference Paper

Beating SGD: Learning SVMs in Sublinear Time

• Elad Hazan
• Tomer Koren
• Nati Srebro

We present an optimization approach for linear SVMs based on a stochastic primal-dual approach, where the primal step is akin to an importance-weighted SGD, and the dual step is a stochastic update on the importance weights. This yields an optimization method with a sublinear dependence on the training set size, and the first method for learning linear SVMs with runtime less then the size of the training set required for learning!