Source-linked AI summary
Adaptive Agent Design
Raj Kiriti Velicheti, Subhonmesh Bose, Tamer Başar
TL;DR
Choosing a useful state representation is difficult when environments are non-Markovian. The paper optimizes an agent’s state-transition kernel and induced policy, showing inner-loop convergence for general environments and outer-design guarantees under POMDP structure.
Problem
In non-Markovian environments, it is unclear how an agent should choose a state representation that captures information needed for decisions.
Source: §1 · p. 1Method
The paper parameterizes the agent’s state-transition kernel, computes a policy for each kernel, and optimizes the resulting objective using zeroth-order or Bayesian optimization.
Source: §2 · pp. 2–3, §7 · p. 17Results
Soft Q-iteration converges almost surely to a stationary-average soft Bellman fixed point in general environments; under POMDP structure, ZOO converges to stationary points and BO has global sub-linear regret.
Source: §5.3 · p. 11, §5.4 · p. 13, §7 · p. 17Takeaways & Limitations
Policy quality is distinct from inner-loop convergence: it depends on state compression and averages induced by the behavioral policy.
Source: §4 · p. 7, §7 · p. 17The inner-loop result does not certify near-optimality in the environment: the state representation limits policies, and its surrogate averages are induced by the behavioral policy.
Source: §4 · p. 7
Abstract
from arXiv · showhide
We consider an agent acting against a general non-Markovian environment. The agent maintains its agent states, but is free to choose a transition kernel across those states and optimize its state-feedback control policies. We study the bi-level agent design problem that optimizes the transition kernel and the policy it induces, given said kernel with offline data of observations and actions obtained via a behavioral policy. For general environments, we show that a soft $Q$-learning algorithm converges almost surely to the fixed point of a soft Bellman equation defined by the stationary averages that the behavioral policy and the chosen kernel induce, and we delineate what separates the resulting policy from an optimal one. In partially observed Markov decision problems, we analyze convergence properties of parametrized transition kernel design via zero-th order and Bayesian optimization techniques.
1 Introduction
Choosing a decision state is difficult when the environment is non-Markovian and the relevant state representation is unclear. This paper formulates adaptive state-kernel design and studies its optimization guarantees.
1 Introduction
- MDPs require a state that captures all information relevant to future decisions, but identifying the right representation is rarely obvious in richer domains.Traditional observability and filtering methods require the practitioner to specify state structure in advance.Source: §1 · p. 1
- The paper builds on work that models agents learning compact internal states in non-Markovian environments and on POMDP research about the effects of limited history.Related work includes the simple agent, complex environment framework, its reinforcement-learning formulation, and POMDP sample-complexity bounds.Source: §1 · p. 2
- The paper optimizes a parameterized state-transition kernel and the policy it induces, addressing how an agent should adapt its state abstraction.The outer step is studied using zeroth-order optimization and Bayesian optimization.Source: §1 · p. 2
2 A Simple Agent in a Non-Markovian Environment
The agent compresses an arbitrarily dependent observation-action history into a finite internal state using a chosen transition kernel. It optimizes that kernel together with a policy that acts on the internal state.
2 A Simple Agent in a Non-Markovian Environment
- The environment may have arbitrary temporal dependence, so the formulation does not require it to be Markovian.The histories contain observations and actions up to time t, with the process assumed to have evolved long enough.Source: §2 · p. 2
- A parameterized stochastic transition kernel fθ updates the finite internal state, compressing the agent’s full interaction history.The quality of this compression depends on θ.Source: §2 · p. 2
- A policy maps internal states to action distributions, and the objective is expected discounted cumulative reward.The reward depends on the internal state, action, and next observation.Source: §2 · pp. 2–3
- The design is bilevel: the policy must be optimal given θ, while the central problem is optimizing the resulting value VD(θ).The objective depends on both the transition parameter and the policy.Source: §2 · p. 3
3 Generic Adaptive Agent Design
The design procedure uses behavioral-policy data to compute and evaluate a policy for each candidate kernel, then updates the kernel using objective evaluations. Its outer optimization can use either zeroth-order optimization or Bayesian optimization.
3 Generic Adaptive Agent Design
- For each candidate kernel, soft Q-learning on internal states generated from behavioral-policy data approximates the policy, which is evaluated in the true environment before the outer update.The data are collected after sufficient mixing; the procedure uses function evaluations rather than assuming access to the gradient of VD.Source: §3 · pp. 3–4
- The analysis assumes a compact convex parameter set, smooth positive internal kernels, and stationary ergodic coverage of every state-action pair under the behavioral policy.The assumptions also require bounded first and second derivatives of the kernel family.Source: §3 · p. 3
3.1 StepOptimize I: Zeroth-Order Optimization
- Zeroth-order optimization estimates the gradient from two evaluations at perturbed parameters and updates the kernel by projected ascent.The two-point estimate uses a sampled unit direction and smoothing radius.Source: §3.1 · p. 4
3.2 StepOptimize II: Bayesian Optimization
- Bayesian optimization fits a Gaussian-process surrogate to past evaluations and selects the next parameter by maximizing an upper-confidence-bound acquisition function.The method uses a Matérn-5/2 covariance over the parameter domain.Source: §3.2 · p. 5
4 Convergence of Inner Soft Q-Learning
The inner soft Q-learning step converges almost surely despite arbitrary environmental memory, tracking an ODE whose stable fixed point is defined by stationary averages. This convergence does not by itself certify that the resulting policy is near-optimal in the environment.
4 Convergence of Inner Soft Q-Learning
- The non-Markovian correction vanishes in expectation under the behavioral policy’s stationary measure, so the iterates track a Markovian ODE.The soft Bellman operator is a contraction, making the ODE globally asymptotically stable at its unique fixed point.Source: §4 · pp. 5, 7
- The fixed point is soft-optimal for the surrogate chain of stationary-average rewards and transitions, not necessarily for the true environment.The stationary averages are induced by the behavioral policy and the chosen transition kernel.Source: §4 · pp. 6–7
- The policy acts on the compressed internal state, so it can be no better than the best policy that the chosen compression admits.The representational gap cannot generally be bounded without additional structure; for POMDP filters with contracting prediction error, filter-stability arguments bound value loss by accumulated expected filtering error.Source: §4 · p. 7
- Because the representational gap depends on the transition parameter and the inner loop does not address it, kernel choice remains a separate optimization problem.The paper analyzes that outer question in POMDPs, where it studies transition parameters obtained by running the procedure.Source: §4 · p. 7
- Soft Q-learning converges almost surely to a well-defined fixed point under Assumption 3 and Robbins–Monro learning-rate conditions.The fixed point is for the soft Bellman equation induced by stationary averages.Source: §4 · pp. 5–7
5 Theoretical Guarantees under a POMDP Environment
The convergence results extend to arbitrary non-Markovian environments, while sharper outer-objective analysis assumes a POMDP. Under that assumption, the paper establishes regularity for the objective and convergence guarantees for both ZOO and Bayesian optimization.
5 Theoretical Guarantees under a POMDP Environment
- Under Assumption 4, the latent pair (x_t, s_t) is Markovian, although observations and internal states alone are not, enabling V_D(θ) to be expressed as a joint-chain value and differentiated.Under Assumptions 1–4, ∇θV_D(θ) is Lipschitz continuous on Θ.Source: §5 · p. 8, §5.1 · p. 10
5.2 Bounded Variance of the Monte Carlo Estimator
- The Monte Carlo estimator is unbiased for the H-truncated return, has sub-Gaussian noise, and its variance shrinks at rate 1/M while growing as γ approaches 1.These noise properties support the convergence analyses, including the GP-UCB confidence bounds used for BO.Source: §5.2 · p. 11
5.3 Convergence of Algorithm 1 with ZOO
- The ZOO analysis controls bias from finite smoothing and variance from Monte Carlo evaluations to obtain its convergence guarantee.The two-point estimator’s errors are balanced by the choices of smoothing radius and step size.Source: §5.3 · pp. 11–12
- Under Assumptions 1–4, ZOO converges to a first-order stationary point of V_D at an O(1/K) rate.The theorem uses constant step size η ≤ 1/(2L) and smoothing radius δ > 0; choosing η = O(K^-1/2) and δ = O(K^-1/4) gives the stated rate.Source: §5.3 · pp. 11–12
5.4 Convergence of Algorithm 1 with Bayesian Optimization
- With convex-combination parametrization and a Matérn-5/2 kernel, BO’s sublinear information gain yields vanishing average regret and asymptotic convergence to the global optimum.This guarantee comes with stronger parametrization and sub-Gaussian-noise requirements than ZOO.Source: §5.4 · pp. 12–13
- The BO guarantee’s RKHS norm bound B can scale aggressively with the number of base kernels N.The paper attributes this growth to repeated chain-rule application through the Bellman matrix inversion.Source: §5.4 · p. 13
6 Numerical Experiments
The experiments examine how optimizer choice, kernel parametrization, behavioral-data exploration, and internal-state capacity affect adaptive-agent reward. In the routing task, reward rises with capacity until the agent can represent the three hidden targets.
6 Numerical Experiments
- The routing task delays target observations with null outputs between junctions, so the agent must retain the hidden target across steps.Correct actions receive asymmetric rewards, while incorrect actions are penalized.Source: §6 · p. 14
- Three basis kernels map target observations to internal states, while null observations preserve state; the adaptive kernel combines these bases on a probability simplex.Source: §6 · p. 14
6.1 Landscape Geometry and Optimizer Performance
- BO bypasses local-gradient traps in suboptimal kernel mixtures and reaches the finite exhaustive optimum, while ZOO can remain trapped.The comparison uses |S| = 2 and K = 3.Source: Table 1 · p. 14, §6.1 · p. 14
6.2 Impact of Kernel Parametrization
- Redundant copies of basis kernels eliminate optimization potential, while noisy independent bases drive rewards into severe penalties.The experiments indicate that seeding the design with informative basis kernels matters.Source: §6.2 · p. 15, Table 2 · p. 16
6.3 Role of Exploratory Behavioral Policy
- Biased or deterministic behavioral policies narrow state-action data and cripple soft Q estimates regardless of candidate kernel; uniform random exploration secures global bounds.The reported bounds apply across limited dimensionalities.Source: §6.3 · pp. 15–16, Table 3 · p. 16
6.4 State Capacity Bounds
- Reward grows linearly with internal-state capacity until |S| exceeds 3, then plateaus because three states suffice for the environment's targets.Additional dimensionality provides no further tracking progression in this task.Source: §6.4 · p. 16, Figure 2 · p. 17
7 Conclusions
The paper studies adaptive transition-kernel choice in unknown, possibly non-Markovian environments and reports soft Q-iteration convergence under behavioral-policy stationary averages. It also gives POMDP convergence results for zeroth-order and Bayesian optimization, while identifying fixed-data, fixed-capacity, and stationary-environment boundaries.
7 Conclusions
- For each transition parameter, soft Q-iteration converges almost surely to a soft Bellman fixed point built from stationary averages induced by the behavioral policy.The paper also delineates how the resulting policy differs from an optimal one.Source: §7 · p. 17
- For POMDPs, zeroth-order optimization converges to stationary points, while Bayesian optimization achieves global sub-linear regret.Source: §7 · p. 17
- The study fixes the behavioral-policy dataset and agent-state cardinality, and considers a simple agent in a stationary non-Markovian environment.The authors identify adapting data collection, changing state-space size, and dynamic multi-agent environments as future directions.Source: §7 · p. 17