arXiv subjects

Oron Sabag

Publications and source records attributed to Oron Sabag.

At least 19 recordsLinked to original sources

arXiv2026

Regret-Optimal Control for Finite-State Systems

We study the control of finite-state systems driven by exogenous disturbances, and design causal policies that track the performance of a lookahead benchmark controller. This objective is formalized through dynamic regret, so that favorable disturbance sequences are compared against a strong benchmark, while under adverse disturbance sequences the comparison accounts for the benchmark's degraded performance. This benchmark-relative framework provides an alternative to classical MDP formulations, which assume i.i.d. disturbances, and to robust control approaches, which optimize against worst-case disturbances. Our main result is a nested dynamic-programming solution that computes both the optimal worst-case regret and a regret-optimal policy. In particular, we introduce the Regret-Bellman operator, whose fixed-point value function feeds into a finite-horizon dynamic program. Numerical examples show that regret-optimal policies interpolate nicely between MDP-based and robust controllers without requiring knowledge of the disturbance distribution, and can even outperform both under i.i.d. or structured disturbances.

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arXiv2026

Directed Information: Estimation, Optimization and Applications in Communications and Causality

Directed information (DI) is an information measure that attempts to capture directionality in the flow of information from one random process to another. It is closely related to other causal influence measures, such as transfer entropy, Granger causality, and Pearl's causal framework. This monograph provides an overview of DI and its main application in information theory, namely, characterizing the capacity of channels with feedback and memory. We begin by reviewing the definitions of DI, its basic properties, and its relation to Shannon's mutual information. Next, we provide a survey of DI estimation techniques, ranging from classic plug-in estimators to modern neural-network-based estimators. Considering the application of channel capacity estimation, we describe how such estimators numerically optimize DI rate over a class of joint distributions on input and output processes. A significant part of the monograph is devoted to techniques to compute the feedback capacity of finite-state channels (FSCs). The feedback capacity of a strongly connected FSC involves the maximization of the DI rate from the channel input process to the output process. This maximization is performed over the class of causal conditioned probability input distributions. When the FSC is also unifilar, i.e., the next state is given by a time-invariant function of the current state and the new input-output symbol pair, the feedback capacity is the optimal average reward of an appropriately formulated Markov decision process (MDP). This MDP formulation has been exploited to develop several methods to compute exactly, or at least estimate closely, the feedback capacity of a unifilar FSC. This monograph describes these methods, starting from the value iteration algorithm, to Q-graph methods, and reinforcement learning algorithms that can handle large input and output alphabets.

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arXiv2025

Communication over LQG Control Systems: A Convex Optimization Approach to Capacity

We study communication over control systems, where a controller-encoder selects inputs to a dynamical system in order to simultaneously regulate the system and convey a message to an observer that has access to the system's output measurements. This setup reflects implicit communication, as the controller embeds a message in the control signal. The capacity of a control system is the maximal reliable rate of the embedded message subject to a closed-loop control-cost constraint. We focus on linear quadratic Gaussian (LQG) control systems, in which the dynamical system is given by a state-space model with Gaussian noise, and the control cost is a quadratic function of the system inputs and system states. Our main result is a convex optimization upper bound on the capacity of LQG systems. In the case of scalar systems, we prove that the upper bound yields the exact LQG system capacity. The upper bound also recovers all known results, including LQG control, feedback capacity of Gaussian channels with memory, and the LQG system capacity with a state-feedback. For vector LQG control systems, we provide a sufficient condition for tightness of the upper bound, based on the Riccati equation. Numerical simulations indicate the upper bound tightness in all tested examples, suggesting that the upper bound may be equal to the LQG system capacity in the vector case as well.

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arXiv2025

Optimal Online Bookmaking for Any Number of Outcomes

We study the Online Bookmaking problem, where a bookmaker dynamically updates betting odds on the possible outcomes of an event. In each betting round, the bookmaker can adjust the odds based on the cumulative betting behavior of gamblers, aiming to maximize profit while mitigating potential loss. We show that for any event and any number of betting rounds, in a worst-case setting over all possible gamblers and outcome realizations, the bookmaker's optimal loss is the largest root of a simple polynomial. Our solution shows that bookmakers can be as fair as desired while avoiding financial risk, and the explicit characterization reveals an intriguing relation between the bookmaker's regret and Hermite polynomials. We develop an efficient algorithm that computes the optimal bookmaking strategy: when facing an optimal gambler, the algorithm achieves the optimal loss, and in rounds where the gambler is suboptimal, it reduces the achieved loss to the optimal opportunistic loss, a notion that is related to subgame perfect Nash equilibrium. The key technical contribution to achieve these results is an explicit characterization of the Bellman-Pareto frontier, which unifies the dynamic programming updates for Bellman's value function with the multi-criteria optimization framework of the Pareto frontier in the context of vector repeated games.

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arXiv2025

Capacity of the Binary Energy Harvesting Channel

The capacity of a channel with an energy-harvesting (EH) encoder and a finite battery remains an open problem, even in the noiseless case. A key instance of this scenario is the binary EH channel (BEHC), where the encoder has a unit-sized battery and binary inputs. Existing capacity expressions for the BEHC are not computable, motivating this work, which determines the capacity to any desired precision via convex optimization. By modeling the system as a finite-state channel with state information known causally at the encoder, we derive single-letter lower and upper bounds using auxiliary directed graphs, termed $Q$-graphs. These $Q$-graphs exhibit a special structure with a finite number of nodes, $N$, enabling the formulation of the bounds as convex optimization problems. As $N$ increases, the bounds tighten and converge to the capacity with a vanishing gap of $O(N)$. For any EH probability parameter $\eta\in \{0.1,0.2, \dots, 0.9\}$, we compute the capacity with a precision of ${1e-6}$, outperforming the best-known bounds in the literature. Finally, we extend this framework to noisy EH channels with feedback, and present numerical achievable rates for the binary symmetric channel using a Markov decision process.

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arXiv2025

Optimal Online Bookmaking for Binary Games

In online betting, the bookmaker can update the payoffs it offers on a particular event many times before the event takes place, and the updated payoffs may depend on the bets accumulated thus far. We study the problem of bookmaking with the goal of maximizing the return in the worst-case, with respect to the gamblers' behavior and the event's outcome. We formalize this problem as the \emph{Optimal Online Bookmaking game}, and provide the exact solution for the binary case. To this end, we develop the optimal bookmaking strategy, which relies on a new technique called bi-balancing trees, that assures that the house loss is the same for all \emph{decisive} betting sequences, where the gambler bets all its money on a single outcome in each round.

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arXiv2024

Competitive Analysis of Arbitrary Varying Channels

Arbitrary varying channels (AVC) are used to model communication settings in which a channel state may vary arbitrarily over time. Their primary objective is to circumvent statistical assumptions on channel variation. Traditional studies on AVCs optimize rate subject to the worst-case state sequence. While this approach is resilient to channel variations, it may result in low rates for state sequences that are associated with relatively good channels. This paper addresses the analysis of AVCs through the lens of competitive analysis, where solution quality is measured with respect to the optimal solution had the state sequence been known in advance. Our main result demonstrates that codes constructed by a single input distribution do not achieve optimal competitive performance over AVCs. This stands in contrast to the single-letter capacity formulae for AVCs, and it indicates, in our setting, that even though the encoder cannot predict the subsequent channel states, it benefits from varying its input distribution as time proceeds.

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arXiv2024

Coded Kalman Filtering over MIMO Gaussian Channels with Feedback

We consider the problem of remotely stabilizing a dynamical system. A sensor (encoder) co-located with the system communicates with a controller (decoder), whose goal is to stabilize the system, over a noisy communication channel with feedback. To accomplish this, the controller must estimate the system state with finite mean squared error (MSE). The vector-valued dynamical system state follows a Gauss-Markov law with additive control. The channel is a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel with feedback. For such a source, a linear encoder, and a MIMO AWGN channel, the minimal MSE decoder is a Kalman filter. The parameters of the Kalman filter and the linear encoder can be jointly optimized, under a power constraint at the channel input. We term the resulting encoder-decoder pair a coded Kalman filter. We establish sufficient and necessary conditions for the coded Kalman filter to achieve a finite MSE in the real-time estimation of the state. For sufficiency, we introduce a coding scheme where each unstable mode of the state is estimated using the channel outputs of a single sub-channel. We prove a coinciding necessity condition when either the source or channel is scalar and present a matrix-algebraic condition which implies the condition is necessary in general. Finally, we provide a new counter-example demonstrating that linear codes are generally sub-optimal for coding over MIMO channels.

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arXiv2024

The Duality Upper Bound for Finite-State Channels with Feedback

This paper investigates the capacity of finite-state channels (FSCs) with feedback. We derive an upper bound on the feedback capacity of FSCs by extending the duality upper bound method from mutual information to the case of directed information. The upper bound is expressed as a multi-letter expression that depends on a test distribution on the sequence of channel outputs. For any FSC, we show that if the test distribution is structured on a $Q$-graph, the upper bound can be formulated as a Markov decision process (MDP) whose state being a belief on the channel state. In the case of FSCs and states that are either unifilar or have a finite memory, the MDP state simplifies to take values in a finite set. Consequently, the MDP consists of a finite number of states, actions, and disturbances. This finite nature of the MDP is of significant importance, as it ensures that dynamic programming algorithms can solve the associated Bellman equation to establish analytical upper bounds, even for channels with large alphabets. We demonstrate the simplicity of computing bounds by establishing the capacity of a broad family of Noisy Output is the State (NOST) channels as a simple closed-form analytical expression. Furthermore, we introduce novel, nearly optimal analytical upper bounds on the capacity of the Noisy Ising channel.

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arXiv2023

Regret-Optimal Control under Partial Observability

This paper studies online solutions for regret-optimal control in partially observable systems over an infinite-horizon. Regret-optimal control aims to minimize the difference in LQR cost between causal and non-causal controllers while considering the worst-case regret across all $\ell_2$-norm-bounded disturbance and measurement sequences. Building on ideas from Sabag et al., 2023, on the the full-information setting, our work extends the framework to the scenario of partial observability (measurement-feedback). We derive an explicit state-space solution when the non-causal solution is the one that minimizes the $\mathcal H_2$ criterion, and demonstrate its practical utility on several practical examples. These results underscore the framework's significant relevance and applicability in real-world systems.

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arXiv2023

Coded Kalman Filtering Over Gaussian Channels with Feedback

This paper investigates the problem of zero-delay joint source-channel coding of a vector Gauss-Markov source over a multiple-input multiple-output (MIMO) additive white Gaussian noise (AWGN) channel with feedback. In contrast to the classical problem of causal estimation using noisy observations, we examine a system where the source can be encoded before transmission. An encoder, equipped with feedback of past channel outputs, observes the source state and encodes the information in a causal manner as inputs to the channel while adhering to a power constraint. The objective of the code is to estimate the source state with minimum mean square error at the infinite horizon. This work shows a fundamental theorem for two scenarios: for the transmission of an unstable vector Gauss-Markov source over either a multiple-input single-output (MISO) or a single-input multiple-output (SIMO) AWGN channel, finite estimation error is achievable if and only if the sum of logs of the unstable eigenvalues of the state gain matrix is less than the Shannon channel capacity. We prove these results by showing an optimal linear innovations encoder that can be applied to sources and channels of any dimension and analyzing it together with the corresponding Kalman filter decoder.

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arXiv2023

Competitive Channel-Capacity

We consider communication over channels whose statistics are not known in full, but can be parameterized as a finite family of memoryless channels. A typical approach to address channel uncertainty is to design codes for the worst channel in the family, resulting in the well-known compound channel capacity. Although this approach is robust, it may suffer a significant loss of performance if the capacity-achieving distribution of the worst channel attains low rates over other channels. In this work, we cope with channel uncertainty through the lens of {\em competitive analysis}. The main idea is to optimize a relative metric that compares the performance of the designed code and a clairvoyant code that has access to the true channel. To allow communication rates that adapt to the channel at use, we consider rateless codes with a fixed number of message bits and random decoding times. We propose two competitive metrics: the competitive ratio between the expected rates of the two codes, and a regret defined as the difference between the expected rates. The competitive ratio, for instance, provides a percentage guarantee on the expected rate of the designed code when compared to the rate of the clairvoyant code that knows the channel at hand. Our main results are single-letter expressions for the optimal {\em competitive-ratio} and {\em regret}, expressed as a max-min or min-max optimization. Several examples illustrate the benefits of the competitive analysis approach to code design compared to the compound channel.

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arXiv2023

Capacity of Finite-State Channels with Delayed Feedback

In this paper, we investigate the capacity of finite-state channels (FSCs) in presence of delayed feedback. We show that the capacity of a FSC with delayed feedback can be computed as that of a new FSC with instantaneous feedback and an extended state. Consequently, graph-based methods to obtain computable upper and lower bounds on the delayed feedback capacity of unifilar FSCs are proposed. Based on these methods, we establish that the capacity of the trapdoor channel with delayed feedback of two time instances is given by $\log_2(3/2)$. In addition, we derive an analytical upper bound on the delayed feedback capacity of the binary symmetric channel with a no consecutive ones input constraint. This bound also serves as a novel upper bound on its non-feedback capacity, which outperforms all previously known bounds. Lastly, we demonstrate that feedback does improve the capacity of the dicode erasure channel.

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arXiv2022

Finite-State Channels with Feedback and State Known at the Encoder

We consider finite state channels (FSCs) with feedback and state information known causally at the encoder. This setting is quite general and includes: a memoryless channel with i.i.d. state (the Shannon strategy), Markovian states that include look-ahead (LA) access to the state and energy harvesting. We characterize the feedback capacity of the general setting as the directed information between auxiliary random variables with memory to the channel outputs. We also propose two methods for computing the feedback capacity: (i) formulating an infinite-horizon average-reward dynamic program; and (ii) a single-letter lower bound based on auxiliary directed graphs called $Q$-graphs. We demonstrate our computation methods on several examples. In the first example, we introduce a channel with LA and derive a closed-form, analytic lower bound on its feedback capacity. Furthermore, we show that the mentioned methods achieve the feedback capacity of known unifilar FSCs such as the trapdoor channel, the Ising channel and the input-constrained erasure channel. Finally, we analyze the feedback capacity of a channel whose state is stochastically dependent on the input.

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arXiv2022

Feedback capacity of Gaussian channels with memory

We consider the feedback capacity of a MIMO channel whose channel output is given by a linear state-space model driven by the channel inputs and a Gaussian process. The generality of our state-space model subsumes all previous studied models such as additive channels with colored Gaussian noise, and channels with an arbitrary dependence on previous channel inputs or outputs. The main result is a computable feedback capacity expression that is given as a convex optimization problem subject to a detectability condition. We demonstrate the capacity result on the auto-regressive Gaussian noise channel, where we show that even a single time-instance delay in the feedback reduces the feedback capacity significantly in the stationary regime. On the other hand, for large regression parameters (in the non-stationary regime), the feedback capacity can be approached with delayed feedback. Finally, we show that the detectability condition is satisfied for scalar models and conjecture that it is true for MIMO models.

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arXiv2022

Optimal Competitive-Ratio Control

Inspired by competitive policy designs approaches in online learning, new control paradigms such as competitive-ratio and regret-optimal control have been recently proposed as alternatives to the classical $\mathcal{H}_2$ and $\mathcal{H}_\infty$ approaches. These competitive metrics compare the control cost of the designed controller against the cost of a clairvoyant controller, which has access to past, present, and future disturbances in terms of ratio and difference, respectively. While prior work provided the optimal solution for the regret-optimal control problem, in competitive-ratio control, the solution is only provided for the sub-optimal problem. In this work, we derive the optimal solution to the competitive-ratio control problem. We show that the optimal competitive ratio formula can be computed as the maximal eigenvalue of a simple matrix, and provide a state-space controller that achieves the optimal competitive ratio. We conduct an extensive numerical study to verify this analytical solution, and demonstrate that the optimal competitive-ratio controller outperforms other controllers on several large scale practical systems. The key techniques that underpin our explicit solution is a reduction of the control problem to a Nehari problem, along with a novel factorization of the clairvoyant controller's cost. We reveal an interesting relation between the explicit solutions that now exist for both competitive control paradigms by formulating a regret-optimal control framework with weight functions that can also be utilized for practical purposes.

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arXiv2022

The Feedback Capacity of Noisy Output is the STate (NOST) Channels

We consider finite-state channels (FSCs) where the channel state is stochastically dependent on the previous channel output. We refer to these as Noisy Output is the STate (NOST) channels. We derive the feedback capacity of NOST channels in two scenarios: with and without causal state information (CSI) available at the encoder. If CSI is unavailable, the feedback capacity is $C_{\text{FB}}= \max_{P(x|y')} I(X;Y|Y')$, while if it is available at the encoder, the feedback capacity is $C_{\text{FB-CSI}}= \max_{P(u|y'),x(u,s')} I(U;Y|Y')$, where $U$ is an auxiliary RV with finite cardinality. In both formulas, the output process is a Markov process with stationary distribution. The derived formulas generalize special known instances from the literature, such as where the state is i.i.d. and where it is a deterministic function of the output. $C_{\text{FB}}$ and $C_{\text{FB-CSI}}$ are also shown to be computable via convex optimization problem formulations. Finally, we present an example of an interesting NOST channel for which CSI available at the encoder does not increase the feedback capacity.

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arXiv2021

Reducing the LQG Cost with Minimal Communication

We study the linear quadratic Gaussian (LQG) control problem, in which the controller's observation of the system state is such that a desired cost is unattainable. To achieve the desired LQG cost, we introduce a communication link from the observer (encoder) to the controller. We investigate the optimal trade-off between the improved LQG cost and the consumed communication (information) resources, measured with the conditional directed information, across all encoding-decoding policies. The main result is a semidefinite programming formulation for that optimization problem in the finite-horizon scenario, which applies to time-varying linear dynamical systems. This result extends a seminal work by Tanaka et al., where the only information the controller knows about the system state arrives via a communication channel, to the scenario where the controller has also access to a noisy observation of the system state. As part of our derivation to show the optimiality of an encoder that transmits a memoryless Gaussian measurement of the state, we show that the presence of the controller's observations at the encoder can not reduce the minimal directed information. For time-invariant systems, where the optimal policy may be time-varying, we show in the infinite-horizon scenario that the optimal policy is time-invariant and can be computed explicitly from a solution of a finite-dimensional semidefinite programming. The results are demonstrated via examples that show that even low-quality measurements can have a significant impact on the required communication resources.

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