Yassir Jedra

Ingvar Ziemann, Anastasios Tsiamis, Bruce Lee et al.

This tutorial serves as an introduction to recently developed non-asymptotic methods in the theory of -- mainly linear -- system identification. We emphasize tools we deem particularly useful for a range of problems in this domain, such as the covering technique, the Hanson-Wright Inequality and the method of self-normalized martingales. We then employ these tools to give streamlined proofs of the performance of various least-squares based estimators for identifying the parameters in autoregressive models. We conclude by sketching out how the ideas presented herein can be extended to certain nonlinear identification problems.

Yassir Jedra, Junghyun Lee, Alexandre Proutière et al.

We investigate the problems of model estimation and reward-free learning in episodic Block MDPs. In these MDPs, the decision maker has access to rich observations or contexts generated from a small number of latent states. We are first interested in estimating the latent state decoding function (the mapping from the observations to latent states) based on data generated under a fixed behavior policy. We derive an information-theoretical lower bound on the error rate for estimating this function and present an algorithm approaching this fundamental limit. In turn, our algorithm also provides estimates of all the components of the MDP. We then study the problem of learning near-optimal policies in the reward-free framework. Based on our efficient model estimation algorithm, we show that we can infer a policy converging (as the number of collected samples grows large) to the optimal policy at the best possible rate. Interestingly, our analysis provides necessary and sufficient conditions under which exploiting the block structure yields improvements in the sample complexity for identifying near-optimal policies. When these conditions are met, the sample complexity in the minimax reward-free setting is improved by a multiplicative factor $n$, where $n$ is the number of possible contexts.

Jerome Taupin, Yassir Jedra, Alexandre Proutiere

We investigate the problem of best policy identification in discounted linear Markov Decision Processes in the fixed confidence setting under a generative model. We first derive an instance-specific lower bound on the expected number of samples required to identify an $\varepsilon$-optimal policy with probability $1-δ$. The lower bound characterizes the optimal sampling rule as the solution of an intricate non-convex optimization program, but can be used as the starting point to devise simple and near-optimal sampling rules and algorithms. We devise such algorithms. One of these exhibits a sample complexity upper bounded by ${\cal O}({\frac{d}{(\varepsilon+Δ)^2}} (\log(\frac{1}δ)+d))$ where $Δ$ denotes the minimum reward gap of sub-optimal actions and $d$ is the dimension of the feature space. This upper bound holds in the moderate-confidence regime (i.e., for all $δ$), and matches existing minimax and gap-dependent lower bounds. We extend our algorithm to episodic linear MDPs.

Yassir Jedra, Sean Mann, Charlotte Park et al.

We consider a variant of matrix completion where entries are revealed in a biased manner. We wish to understand the extent to which such bias can be exploited in improving predictions. Towards that, we propose a natural model where the observation pattern and outcome of interest are driven by the same set of underlying latent (or unobserved) factors. We devise Mask Nearest Neighbor (MNN), a novel two-stage matrix completion algorithm: first, it recovers (distances between) the latent factors by utilizing matrix estimation for the fully observed noisy binary matrix, corresponding to the observation pattern; second, it utilizes the recovered latent factors as features and sparsely observed noisy outcomes as labels to perform non-parametric supervised learning. Our analysis reveals that MNN enjoys entry-wise finite-sample error rates that are competitive with corresponding supervised learning parametric rates. Despite not having access to the latent factors and dealing with biased observations, MNN exhibits such competitive performance via only exploiting the shared information between the bias and outcomes. Finally, through empirical evaluation using a real-world dataset, we find that with MNN, the estimates have 28x smaller mean squared error compared to traditional matrix completion methods, suggesting the utility of the model and method proposed in this work.

Frédéric Zheng, Yassir Jedra, Alexandre Proutiere

We address the problem of estimating a high-dimensional matrix from linear measurements, with a focus on designing optimal rank-adaptive algorithms. These algorithms infer the matrix by estimating its singular values and the corresponding singular vectors up to an effective rank, adaptively determined based on the data. We establish instance-specific lower bounds for the sample complexity of such algorithms, uncovering fundamental trade-offs in selecting the effective rank: balancing the precision of estimating a subset of singular values against the approximation cost incurred for the remaining ones. Our analysis identifies how the optimal effective rank depends on the matrix being estimated, the sample size, and the noise level. We propose an algorithm that combines a Least-Squares estimator with a universal singular value thresholding procedure. We provide finite-sample error bounds for this algorithm and demonstrate that its performance nearly matches the derived fundamental limits. Our results rely on an enhanced analysis of matrix denoising methods based on singular value thresholding. We validate our findings with applications to multivariate regression and linear dynamical system identification.

Junghyun Lee, Sanghwa Kim, Yassir Jedra et al.

Evaluating large language models increasingly relies on LLM-as-a-judge protocols, but such evaluations remain costly: different judges have different prices and reliabilities, and the difficulty of each prompt-response pair can vary substantially. This raises a basic allocation question: under a fixed budget, how should one distribute evaluation queries across heterogeneous judges and instances to obtain the most accurate score estimates? We formalize this question as *budgeted heteroskedastic multi-judge estimation*. Given $K$ prompt-response pairs, $J$ judges with known costs, and unknown query-judge variances, the goal is to estimate a bounded score vector while minimizing an $\ell_p$-error. Our first contribution is to analyze the inverse-variance weighted estimator (IVWE) and to derive the oracle allocation that minimizes its error rate. Since this allocation depends on the unknown variances, we then address the practical unknown-variance setting by proposing EST-IVWE, an adaptive algorithm that constructs and leverages *optimistically biased* variance estimates to stabilize the empirical allocation. We prove that EST-IVWE matches the oracle IVWE rate up to lower-order terms in the budget. Our second and central theoretical contribution is a matching *local* minimax lower bound, which establishes the instance-optimality of the proposed algorithms. A key technical insight is that Fano-type high-probability arguments are too coarse for this problem: their packing construction loses the local variance structure that governs the optimal allocation. We instead use an Assouad-type in-expectation argument, based on local perturbations, which preserves this structure and yields the sharp allocation-dependent lower bound. Finally, we numerically validate the superiority of our approach over naïve uniform allocation on synthetic and HelpSteer2 datasets.

Stefan Stojanovic, Yassir Jedra, Alexandre Proutiere

We study matrix estimation problems arising in reinforcement learning (RL) with low-rank structure. In low-rank bandits, the matrix to be recovered specifies the expected arm rewards, and for low-rank Markov Decision Processes (MDPs), it may for example characterize the transition kernel of the MDP. In both cases, each entry of the matrix carries important information, and we seek estimation methods with low entry-wise error. Importantly, these methods further need to accommodate for inherent correlations in the available data (e.g. for MDPs, the data consists of system trajectories). We investigate the performance of simple spectral-based matrix estimation approaches: we show that they efficiently recover the singular subspaces of the matrix and exhibit nearly-minimal entry-wise error. These new results on low-rank matrix estimation make it possible to devise reinforcement learning algorithms that fully exploit the underlying low-rank structure. We provide two examples of such algorithms: a regret minimization algorithm for low-rank bandit problems, and a best policy identification algorithm for reward-free RL in low-rank MDPs. Both algorithms yield state-of-the-art performance guarantees.

Yahya Sattar, Yassir Jedra, Sarah Dean

We consider the problem of learning a realization of a partially observed dynamical system with linear state transitions and bilinear observations. Under very mild assumptions on the process and measurement noises, we provide a finite time analysis for learning the unknown dynamics matrices (up to a similarity transform). Our analysis involves a regression problem with heavy-tailed and dependent data. Moreover, each row of our design matrix contains a Kronecker product of current input with a history of inputs, making it difficult to guarantee persistence of excitation. We overcome these challenges, first providing a data-dependent high probability error bound for arbitrary but fixed inputs. Then, we derive a data-independent error bound for inputs chosen according to a simple random design. Our main results provide an upper bound on the statistical error rates and sample complexity of learning the unknown dynamics matrices from a single finite trajectory of bilinear observations.

Yahya Sattar, Yassir Jedra, Maryam Fazel et al.

We consider the problem of learning a realization of a partially observed bilinear dynamical system (BLDS) from noisy input-output data. Given a single trajectory of input-output samples, we provide a finite time analysis for learning the system's Markov-like parameters, from which a balanced realization of the bilinear system can be obtained. Our bilinear system identification algorithm learns the system's Markov-like parameters by regressing the outputs to highly correlated, nonlinear, and heavy-tailed covariates. Moreover, the stability of BLDS depends on the sequence of inputs used to excite the system. These properties, unique to partially observed bilinear dynamical systems, pose significant challenges to the analysis of our algorithm for learning the unknown dynamics. We address these challenges and provide high probability error bounds on our identification algorithm under a uniform stability assumption. Our analysis provides insights into system theoretic quantities that affect learning accuracy and sample complexity. Lastly, we perform numerical experiments with synthetic data to reinforce these insights.

Akhil Jalan, Yassir Jedra, Arya Mazumdar et al.

We study transfer learning for matrix completion in a Missing Not-at-Random (MNAR) setting that is motivated by biological problems. The target matrix $Q$ has entire rows and columns missing, making estimation impossible without side information. To address this, we use a noisy and incomplete source matrix $P$, which relates to $Q$ via a feature shift in latent space. We consider both the active and passive sampling of rows and columns. We establish minimax lower bounds for entrywise estimation error in each setting. Our computationally efficient estimation framework achieves this lower bound for the active setting, which leverages the source data to query the most informative rows and columns of $Q$. This avoids the need for incoherence assumptions required for rate optimality in the passive sampling setting. We demonstrate the effectiveness of our approach through comparisons with existing algorithms on real-world biological datasets.

Yassir Jedra, William Réveillard, Stefan Stojanovic et al.

We study contextual bandits with low-rank structure where, in each round, if the (context, arm) pair $(i,j)\in [m]\times [n]$ is selected, the learner observes a noisy sample of the $(i,j)$-th entry of an unknown low-rank reward matrix. Successive contexts are generated randomly in an i.i.d. manner and are revealed to the learner. For such bandits, we present efficient algorithms for policy evaluation, best policy identification and regret minimization. For policy evaluation and best policy identification, we show that our algorithms are nearly minimax optimal. For instance, the number of samples required to return an $\varepsilon$-optimal policy with probability at least $1-δ$ typically scales as ${r(m+n)\over \varepsilon^2}\log(1/δ)$. Our regret minimization algorithm enjoys minimax guarantees typically scaling as $r^{7/4}(m+n)^{3/4}\sqrt{T}$, which improves over existing algorithms. All the proposed algorithms consist of two phases: they first leverage spectral methods to estimate the left and right singular subspaces of the low-rank reward matrix. We show that these estimates enjoy tight error guarantees in the two-to-infinity norm. This in turn allows us to reformulate our problems as a misspecified linear bandit problem with dimension roughly $r(m+n)$ and misspecification controlled by the subspace recovery error, as well as to design the second phase of our algorithms efficiently.

Junghyun Lee, Yassir Jedra, Alexandre Proutière et al.

We study the problem of clustering $T$ trajectories of length $H$, each generated by one of $K$ unknown ergodic Markov chains over a finite state space of size $S$. The goal is to accurately group trajectories according to their underlying generative model. We begin by deriving an instance-dependent, high-probability lower bound on the clustering error rate, governed by the weighted KL divergence between the transition kernels of the chains. We then present a novel two-stage clustering algorithm. In Stage~I, we apply spectral clustering using a new injective Euclidean embedding for ergodic Markov chains -- a contribution of independent interest that enables sharp concentration results. Stage~II refines the initial clusters via a single step of likelihood-based reassignment. Our method achieves a near-optimal clustering error with high probability, under the conditions $H = \tildeΩ(γ_{\mathrm{ps}}^{-1} (S^2 \vee π_{\min}^{-1}))$ and $TH = \tildeΩ(γ_{\mathrm{ps}}^{-1} S^2 )$, where $π_{\min}$ is the minimum stationary probability of a state across the $K$ chains and $γ_{\mathrm{ps}}$ is the minimum pseudo-spectral gap. These requirements provide significant improvements, if not at least comparable, to the state-of-the-art guarantee (Kausik et al., 2023), and moreover, our algorithm offers a key practical advantage: unlike existing approach, it requires no prior knowledge of model-specific quantities (e.g., separation between kernels or visitation probabilities). We conclude by discussing the inherent gap between our upper and lower bounds, providing insights into the unique structure of this clustering problem.

Yahya Sattar, Sunmook Choi, Yassir Jedra et al.

We consider the problem of controlling a linear dynamical system from bilinear observations with minimal quadratic cost. Despite the similarity of this problem to standard linear quadratic Gaussian (LQG) control, we show that when the observation model is bilinear, neither does the Separation Principle hold, nor is the optimal controller affine in the estimated state. Moreover, the cost-to-go is non-convex in the control input. Hence, finding an analytical expression for the optimal feedback controller is difficult in general. Under certain settings, we show that the standard LQG controller locally maximizes the cost instead of minimizing it. Furthermore, the optimal controllers (derived analytically) are not unique and are nonlinear in the estimated state. We also introduce a notion of input-dependent observability and derive conditions under which the Kalman filter covariance remains bounded. We illustrate our theoretical results through numerical experiments in multiple synthetic settings.

Sunmook Choi, Yahya Sattar, Yassir Jedra et al.

We study a nonstationary bandit problem where rewards depend on both actions and latent states, the latter governed by unknown linear dynamics. Crucially, the state dynamics also depend on the actions, resulting in tension between short-term and long-term rewards. We propose an explore-then-commit algorithm for a finite horizon $T$. During the exploration phase, random Rademacher actions enable estimation of the Markov parameters of the linear dynamics, which characterize the action-reward relationship. In the commit phase, the algorithm uses the estimated parameters to design an optimized action sequence for long-term reward. Our proposed algorithm achieves $\tilde{\mathcal{O}}(T^{2/3})$ regret. Our analysis handles two key challenges: learning from temporally correlated rewards, and designing action sequences with optimal long-term reward. We address the first challenge by providing near-optimal sample complexity and error bounds for system identification using bilinear rewards. We address the second challenge by proving an equivalence with indefinite quadratic optimization over a hypercube, a known NP-hard problem. We provide a sub-optimality guarantee for this problem, enabling our regret upper bound. Lastly, we propose a semidefinite relaxation with Goemans-Williamson rounding as a practical approach.

Yassir Jedra, Devavrat Shah

The emergence of modern compute infrastructure for iterative optimization has led to great interest in developing optimization-based approaches for a scalable computation of $k$-SVD, i.e., the $k\geq 1$ largest singular values and corresponding vectors of a matrix of rank $d \geq 1$. Despite lots of exciting recent works, all prior works fall short in this pursuit. Specifically, the existing results are either for the exact-parameterized (i.e., $k = d$) and over-parameterized (i.e., $k > d$) settings; or only establish local convergence guarantees; or use a step-size that requires problem-instance-specific oracle-provided information. In this work, we complete this pursuit by providing a gradient-descent method with a simple, universal rule for step-size selection (akin to pre-conditioning), that provably finds $k$-SVD for a matrix of any rank $d \geq 1$. We establish that the gradient method with random initialization enjoys global linear convergence for any $k, d \geq 1$. Our convergence analysis reveals that the gradient method has an attractive region, and within this attractive region, the method behaves like Heron's method (a.k.a. the Babylonian method). Our analytic results about the said attractive region imply that the gradient method can be enhanced by means of Nesterov's momentum-based acceleration technique. The resulting improved convergence rates match those of rather complicated methods typically relying on Lanczos iterations or variants thereof. We provide an empirical study to validate the theoretical results.

Stefan Stojanovic, Yassir Jedra, Alexandre Proutiere

We consider the problem of learning an $\varepsilon$-optimal policy in controlled dynamical systems with low-rank latent structure. For this problem, we present LoRa-PI (Low-Rank Policy Iteration), a model-free learning algorithm alternating between policy improvement and policy evaluation steps. In the latter, the algorithm estimates the low-rank matrix corresponding to the (state, action) value function of the current policy using the following two-phase procedure. The entries of the matrix are first sampled uniformly at random to estimate, via a spectral method, the leverage scores of its rows and columns. These scores are then used to extract a few important rows and columns whose entries are further sampled. The algorithm exploits these new samples to complete the matrix estimation using a CUR-like method. For this leveraged matrix estimation procedure, we establish entry-wise guarantees that remarkably, do not depend on the coherence of the matrix but only on its spikiness. These guarantees imply that LoRa-PI learns an $\varepsilon$-optimal policy using $\widetilde{O}({S+A\over \mathrm{poly}(1-γ)\varepsilon^2})$ samples where $S$ (resp. $A$) denotes the number of states (resp. actions) and $γ$ the discount factor. Our algorithm achieves this order-optimal (in $S$, $A$ and $\varepsilon$) sample complexity under milder conditions than those assumed in previously proposed approaches.

Filippo Vannella, Alexandre Proutiere, Yassir Jedra et al.

Controlling antenna tilts in cellular networks is imperative to reach an efficient trade-off between network coverage and capacity. In this paper, we devise algorithms learning optimal tilt control policies from existing data (in the so-called passive learning setting) or from data actively generated by the algorithms (the active learning setting). We formalize the design of such algorithms as a Best Policy Identification (BPI) problem in Contextual Linear Multi-Arm Bandits (CL-MAB). An arm represents an antenna tilt update; the context captures current network conditions; the reward corresponds to an improvement of performance, mixing coverage and capacity; and the objective is to identify, with a given level of confidence, an approximately optimal policy (a function mapping the context to an arm with maximal reward). For CL-MAB in both active and passive learning settings, we derive information-theoretical lower bounds on the number of samples required by any algorithm returning an approximately optimal policy with a given level of certainty, and devise algorithms achieving these fundamental limits. We apply our algorithms to the Remote Electrical Tilt (RET) optimization problem in cellular networks, and show that they can produce optimal tilt update policy using much fewer data samples than naive or existing rule-based learning algorithms.

Yassir Jedra, Alexandre Proutiere

We consider the problem of online learning in Linear Quadratic Control systems whose state transition and state-action transition matrices $A$ and $B$ may be initially unknown. We devise an online learning algorithm and provide guarantees on its expected regret. This regret at time $T$ is upper bounded (i) by $\widetilde{O}((d_u+d_x)\sqrt{d_xT})$ when $A$ and $B$ are unknown, (ii) by $\widetilde{O}(d_x^2\log(T))$ if only $A$ is unknown, and (iii) by $\widetilde{O}(d_x(d_u+d_x)\log(T))$ if only $B$ is unknown and under some mild non-degeneracy condition ($d_x$ and $d_u$ denote the dimensions of the state and of the control input, respectively). These regret scalings are minimal in $T$, $d_x$ and $d_u$ as they match existing lower bounds in scenario (i) when $d_x\le d_u$ [SF20], and in scenario (ii) [lai1986]. We conjecture that our upper bounds are also optimal in scenario (iii) (there is no known lower bound in this setting). Existing online algorithms proceed in epochs of (typically exponentially) growing durations. The control policy is fixed within each epoch, which considerably simplifies the analysis of the estimation error on $A$ and $B$ and hence of the regret. Our algorithm departs from this design choice: it is a simple variant of certainty-equivalence regulators, where the estimates of $A$ and $B$ and the resulting control policy can be updated as frequently as we wish, possibly at every step. Quantifying the impact of such a constantly-varying control policy on the performance of these estimates and on the regret constitutes one of the technical challenges tackled in this paper.

Yassir Jedra, Alexandre Proutiere

We study the problem of best-arm identification with fixed confidence in stochastic linear bandits. The objective is to identify the best arm with a given level of certainty while minimizing the sampling budget. We devise a simple algorithm whose sampling complexity matches known instance-specific lower bounds, asymptotically almost surely and in expectation. The algorithm relies on an arm sampling rule that tracks an optimal proportion of arm draws, and that remarkably can be updated as rarely as we wish, without compromising its theoretical guarantees. Moreover, unlike existing best-arm identification strategies, our algorithm uses a stopping rule that does not depend on the number of arms. Experimental results suggest that our algorithm significantly outperforms existing algorithms. The paper further provides a first analysis of the best-arm identification problem in linear bandits with a continuous set of arms.

Yassir Jedra, Alexandre Proutiere

We present a new finite-time analysis of the estimation error of the Ordinary Least Squares (OLS) estimator for stable linear time-invariant systems. We characterize the number of observed samples (the length of the observed trajectory) sufficient for the OLS estimator to be $(\varepsilon,δ)$-PAC, i.e., to yield an estimation error less than $\varepsilon$ with probability at least $1-δ$. We show that this number matches existing sample complexity lower bounds [1,2] up to universal multiplicative factors (independent of ($\varepsilon,δ)$ and of the system). This paper hence establishes the optimality of the OLS estimator for stable systems, a result conjectured in [1]. Our analysis of the performance of the OLS estimator is simpler, sharper, and easier to interpret than existing analyses. It relies on new concentration results for the covariates matrix.

Yassir Jedra, Alexandre Proutiere

This paper establishes problem-specific sample complexity lower bounds for linear system identification problems. The sample complexity is defined in the PAC framework: it corresponds to the time it takes to identify the system parameters with prescribed accuracy and confidence levels. By problem-specific, we mean that the lower bound explicitly depends on the system to be identified (which contrasts with minimax lower bounds), and hence really captures the identification hardness specific to the system. We consider both uncontrolled and controlled systems. For uncontrolled systems, the lower bounds are valid for any linear system, stable or not, and only depend of the system finite-time controllability gramian. A simplified lower bound depending on the spectrum of the system only is also derived. In view of recent finitetime analysis of classical estimation methods (e.g. ordinary least squares), our sample complexity lower bounds are tight for many systems. For controlled systems, our lower bounds are not as explicit as in the case of uncontrolled systems, but could well provide interesting insights into the design of control policy with minimal sample complexity.