The Bellman error is a poor proxy for value function accuracy, even with all state-action pairs.
problem The Bellman error is a poor proxy for the accuracy of the value function.
method Study of the Bellman equation as a surrogate objective for value prediction accuracy.
result The magnitude of the Bellman error is only weakly related to the distance to the true value function, even with all state-action pairs.
A new method calibrates value predictions in offline RL to improve reliability.
problem Difficulty in long-horizon value prediction in offline reinforcement learning.
method Bellman calibration, a weak reliability criterion, and Iterated Bellman Calibration.
result Finite-sample guarantees show that Bellman calibration error is controlled at nonparametric rates.
Polynomial-time RL algorithm for constant actions under linear Bellman completeness.
problem Efficient online reinforcement learning with few actions.
method Polynomial-time algorithm based on linear function approximation.
result First computationally efficient algorithm for RL with constant actions under linear Bellman completeness.
Paper studies offline RL with linear approx, focusing on inherent Bellman error.
problem Offline RL with linear approx, focusing on inherent Bellman error.
method Algorithm that succeeds under single-policy coverage condition, leveraging inherent Bellman error.
result Algorithm yields first known guarantee under single-policy coverage, even for linear Bellman completeness.
New method for off-policy evaluation in POMDPs using future-dependent value functions.
problem Curse of horizon in off-policy evaluation for POMDPs.
method Develops future-dependent value functions and minimax learning method.
result PAC result and Bellman completeness for the proposed OPE estimator.
The impact of softmax on the value function itself in reinforcement learning (RL) is often viewed as problematic because it leads to sub-optimal value (or Q) functions and interferes with the contraction properties of the Bellman operator. Surprisingly, despite these concerns, and independent of its effect on explorati…
Paper analyzes distributional reinforcement learning with value function approximation, introducing Bellman unbiasedness and a new algorithm.
problem Improving reinforcement learning by capturing environmental stochasticity and addressing infinite dimensionality.
method Introduces Bellman unbiasedness and proposes SF-LSVI algorithm for provably efficient distributional reinforcement learning.
result Achieves a tight regret bound of O(d_E H^3/2 √K) for distributional reinforcement learning.
In this paper, we consider the stochastic iterative counterpart of the value iteration scheme wherein only noisy and possibly biased approximations of the Bellman operator are available. We call this counterpart as the approximate value iteration (AVI) scheme. Neural networks are often used as function approximators, i…
New method quantifies uncertainty in reinforcement learning models.
problem Quantifying uncertainty over expected cumulative rewards in reinforcement learning.
method Proposes a new uncertainty Bellman equation to more accurately estimate value function variance.
result Our method converges to the true posterior variance over values and improves sample-efficiency.
New method stabilizes FQE by reweighting Bellman targets.
problem Stability guarantees for FQE often rely on Bellman completeness, which can fail with function approximation.
method Proposes stationary-weighted FQE, reweighting Bellman targets by stationary target-to-behavior density ratio.
result Proves finite-sample linear convergence to stationary projected Bellman fixed point without Bellman completeness.
We address the problem of automatic generation of features for value function approximation. Bellman Error Basis Functions (BEBFs) have been shown to improve the error of policy evaluation with function approximation, with a convergence rate similar to that of value iteration. We propose a simple, fast and robust algor…
The paper introduces Bellman-consistent pessimism to improve offline reinforcement learning without overly pessimistic bias.
problem Offline reinforcement learning's challenge of discovering good policies without exhaustive exploration.
method Introduces Bellman-consistent pessimism for function approximation, improving sample complexity and adaptability.
result Improves sample complexity by O(d) in the action space finite case, and automatically adapts to bias-variance tradeoff. We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that case only a terminal condition is needed. Conversely, the case of contracts with …
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
In a discounted reward Markov Decision Process (MDP), the objective is to find the optimal value function, i.e., the value function corresponding to an optimal policy. This problem reduces to solving a functional equation known as the Bellman equation and a fixed point iteration scheme known as the value iteration is u…
We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact…
FORE evaluates occupancy ratios without requiring Bellman completeness.
problem Offline reinforcement learning occupancy ratio estimation.
method Fitted occupancy-ratio evaluation (FORE) using adjoint Bellman recursion.
result FORE achieves convergence in KL without Bellman completeness.
New approach uses PDE learning for faster RL fine-tuning.
problem Learning optimal control policy for diffusion process.
method Solves variational inequality based on HJB equations.
result Shows fine-tuning can be done via supervised regression.
A new estimator combines bootstrapping and rollout methods in RL.
problem Combining strengths of bootstrapping and rollout methods in RL.
method Subgraph Bellman operators and fixed point solving.
result Upper bound on error approaches optimal TD variance with additional term.
Selective state-adaptive regularization improves offline RL performance.
problem Extrapolation errors and value overestimation in static dataset RL.
method State-adaptive regularization coefficients trust Bellman-driven results selectively.
result Significant improvement in performance on D4RL benchmark.
Trading strategy mimics optimal control with simple heuristic.
problem Maximizing profit in markets with limited optimal control.
method Using the gradient of the Bellman value function as effective microstructure alphas.
result Trading strategy approximates optimal behavior in long-term trades.
Deep learning method proves convergence for high-dimensional PDEs.
problem Solving high-dimensional nonlinear PDEs for mean field control problems.
method Deep Galerkin method (DGM) for Hamilton-Jacobi-Bellman (HJB) equations.
result DGM converges to the true value function of mean field control problems.
New method improves stability of soft FQI for offline RL.
problem Stability issues in soft FQI under function approximation.
method Stationary reweighting to align operator norms.
result Local linear convergence proved under certain conditions.
Sequential decision making in the presence of uncertainty and stochastic dynamics gives rise to distributions over state/action trajectories in reinforcement learning (RL) and optimal control problems. This observation has led to a variety of connections between RL and inference in probabilistic graphical models (PGMs)…
In this paper, we consider the problem of online learning of Markov decision processes (MDPs) with very large state spaces. Under the assumptions of realizable function approximation and low Bellman ranks, we develop an online learning algorithm that learns the optimal value function while at the same time achieving ve…
In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear paraboli…
This paper aims at theoretically and empirically comparing two standard optimization criteria for Reinforcement Learning: i) maximization of the mean value and ii) minimization of the Bellman residual. For that purpose, we place ourselves in the framework of policy search algorithms, that are usually designed to maximi…
Paper introduces v-CMC linking causality and utility.
problem Linking causality and utility for value theory.
method Developed a new causal independence principle (v-CMC) and proved its equivalence.
result Equivalence of local, global, and decomposition versions of v-CMC.
Direct and indirect RL methods classified and compared.
problem Classifying RL methods for sequential decision making.
method Direct RL solves optimal policy directly, indirect RL solves Bellman equation.
result Direct and indirect RL methods are equivalent and can be unified.
A new method improves policy evaluation in RL by tracking value uncertainties.
problem Limitations in existing policy evaluation methods for deep RL tasks.
method KOVA (Kalman Optimization for Value Approximation) based on extended Kalman filter.
result KOVA minimizes a regularized objective function that considers parameter and noisy return uncertainties.
Optimizes portfolio in volatile markets with jumps, providing accurate formulas.
problem Optimizing wealth in a volatile financial market with jumps.
method Analyzes an incomplete stochastic volatility model, derives closed-form portfolio formulas using HJB equation and super-solution/sub-solution.
result Proves accuracy of derived portfolio formulas for both small and finite time horizons.
Study on LOB dynamics using mean-field game theory.
problem Modeling liquidity dynamics in limit order books.
method Mean-field stochastic differential equation and control problem formulation.
result Equilibrium density function of LOB can be derived.
Study optimal consumption and investment strategies with leverage constraints using Epstein-Zin utility.
problem Optimal portfolio choice under leverage constraints and Epstein-Zin utility.
method Established viscosity solution to HJB equation, demonstrated smoothness, characterized optimal strategies, derived explicit solutions.
result Explicit solutions for optimal consumption and investment strategies under leverage constraints.
Optimizes dividend and reinsurance strategies for correlated insurance lines.
problem Stochastic control of optimal reinsurance and dividend policies for multiple insurance lines.
method Maximizes cumulative discounted dividends using a Hamilton-Jacobi-Bellman equation and finite difference method.
result Provides optimal strategies for transferring risk among reinsurers.
New methods for off-policy evaluation in reinforcement learning.
problem Improving off-policy evaluation in reinforcement learning.
method Developed MWL and MQL estimators for importance weights and value functions.
result MWL and MQL provide a unified view of algorithms in reinforcement learning.
One-step Bellman alignment improves online RL by reducing task mismatch.
problem Online RL struggles with task similarity defined by rewards or transitions.
method One-step Bellman alignment and re-weighted targeting (RWT) to correct task mismatch.
result Regret bounds show task shift complexity, not target MDP, affects performance.
The paper develops RL methods for optimal switching between multiple states.
problem Optimal switching between multiple states in continuous time.
method Entropy-regularized exploration, HJB equations, policy improvement, value function convergence.
result The RL algorithm converges to optimal policies as temperature parameter vanishes.
In this paper, we extend the jump-diffusion model proposed by Davis and Lleo to include jumps in asset prices as well as valuation factors. The criterion, following earlier work by Bielecki, Pliska, Nagai and others, is risk-sensitive optimization (equivalent to maximizing the expected growth rate subject to a constrai…
We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value functi…
Optimal dividend strategy in insurance with limited rate decrease.
problem Maximizing expected discounted dividends up to ruin in a risk model.
method Two-dimensional optimal control problem solved using Hamilton-Jacobi-Bellman equation.
result Value function is unique viscosity solution and can be approximated by ratcheting strategies.
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
KBB algorithm reduces sample complexity for policy evaluation in general state spaces.
problem Policy evaluation in large state spaces with high sample complexity.
method Alternates between fitting Bellman residual and estimating value function via adaptive feature set growth.
result Super-linear convergence rates demonstrated, with reductions in sample complexity.
CEFOL uses deep learning for dynamic programming with recursive utility.
problem Challenges in solving dynamic programming problems with recursive utility.
method Introduces a separate neural network for certainty equivalent, uses first-order optimality conditions to learn value and policy functions.
result CEFOL achieves high accuracy in learning value and policy functions, matching VFI benchmarks.
This paper introduces a set of algorithms for Monte-Carlo Bayesian reinforcement learning. Firstly, Monte-Carlo estimation of upper bounds on the Bayes-optimal value function is employed to construct an optimistic policy. Secondly, gradient-based algorithms for approximate upper and lower bounds are introduced. Finally…
We obtain the classical Hanner inequalities by the Bellman function method. These inequalities give sharp estimates for the moduli of convexity of Lebesgue spaces. Easy ideas from differential geometry help us to find the Bellman function using neither "magic guesses" nor calculations.
Value function learning plays a central role in many state-of-the-art reinforcement-learning algorithms. Many popular algorithms like Q-learning do not optimize any objective function, but are fixed-point iterations of some variant of Bellman operator that is not necessarily a contraction. As a result, they may easily …
Study tight offline learning bounds for linear MDPs using variance information.
problem Understanding statistical limits with linear function representations in offline reinforcement learning.
method Variance-aware pessimistic value iteration (VAPVI) that reweights Bellman residuals based on estimated variances.
result Improved offline learning bounds expressed in terms of system quantities.
This paper is a continuation of Ishitani and Kato (2015), in which we derived a continuous-time value function corresponding to an optimal execution problem with uncertain market impact as the limit of a discrete-time value function. Here, we investigate some properties of the derived value function. In particular, we …