Quantile-Based Value Function Approximators

• Quantile-based value function approximators are computational frameworks that model entire return distributions using quantile regression to capture uncertainty.
• They extend classic reinforcement learning by learning distributional information, enabling robust and risk-aware decision making in various applications.
• Empirical studies demonstrate their effectiveness in tasks like financial risk forecasting and safe robotic control, underpinned by strong convergence guarantees.

Quantile-based value function approximators are computational frameworks that estimate not just the mean of a value function but the entire value distribution—often parameterized directly through quantile functions. These techniques provide a richer representation of return or cost uncertainty, support robust and risk-sensitive policy design, and underpin advances in both reinforcement learning, control, and functional data analysis. The development and application of quantile-based value function approximators relies on foundational results in quantile regression, novel algorithmic architectures, rigorous convergence guarantees, and empirical evidence from diverse domains.

1. Foundations of Quantile Regression and Value Function Approximation

Quantile regression (QR) forms the statistical basis for quantile-based approximation. In QR, for a random variable $Y$ and covariates $X$, the $\tau$-th conditional quantile function $Q_\tau(Y|X)$ is estimated by solving

$$\min_\theta \sum_i \rho_\tau\left(Y_i - X_i^\top \theta\right), \quad \rho_\tau(u) = u\{\tau - \mathbf{1}_{u<0}\}.$$

This extends regression beyond the mean, allowing value functions to articulate uncertainty in distributional settings and enabling direct inference for risk measures such as Conditional Value-at-Risk (CVaR) (Belloni et al., 2011, Crescenzo et al., 2014).

In the nonparametric QR-series framework, the conditional quantile function is approximated by a linear combination of series terms with quantile-specific coefficients. Uniform convergence rates and strong approximation theory (e.g., coupling with Gaussian processes) underpin inference on function-valued coefficients and their linear functionals. Resampling methods, including pivotal, gradient bootstrap, Gaussian, and weighted bootstrap, are developed for inference over the entire function space (Belloni et al., 2011). Lemmas regarding the uniform deviation of empirical Gram matrices (Guédon and Rudelson) are crucial in establishing convergence and estimator properties.

2. Quantile-Based Distributional Reinforcement Learning

Distributional reinforcement learning (DRL) generalizes classic RL by learning the return distribution for each state–action pair, not just the expected return. QR-based approaches represent this distribution as a collection of quantiles, updated via quantile regression. In QR-DQN (Dabney et al., 2017), a network outputs $N$ quantile estimates $\{\theta_i\}$ and is trained to minimize

$$L(\theta) = \frac{1}{N}\sum_{i=1}^N \mathbb{E}_{z\sim Z_\mathrm{target}}[\rho_{\tau_i}(z-\theta_i)],$$

ensuring convergence to the correct 1-Wasserstein projection due to unbiased gradients. The resulting algorithms enable fine-grained risk modeling, mitigate overestimation bias, and support risk-sensitive policies through direct manipulation of the estimated quantile function (Dabney et al., 2017, Dabney et al., 2018).

Implicit Quantile Networks (IQN) further generalize this by learning the full quantile function as a mapping from quantile level $\tau \in [0,1]$ and state-action features to return, providing a universal value function approximator and supporting flexible risk measures via distortion of the sampling distribution (Dabney et al., 2018).

3. Theoretical Advances: Losses, Projections, and Constraints

A major theoretical advance is the equivalence of the squared Cramér distance (the $L_2$ metric between cumulative distribution functions) and the 1-Wasserstein distance when projecting arbitrary return distributions onto staircase distributions of fixed quantile levels (Lhéritier et al., 2021). For a distribution $F$ projected to a staircase function $F_\theta$, both

$$\min_{\theta} d_C^2(F, F_\theta) \quad \text{and} \quad \min_{\theta} W_1(F, F_\theta)$$

yield the same minimizer. Under non-crossing constraints (enforcing monotonic quantiles), the gradients of the squared Cramér loss and the quantile regression loss are collinear, implying identical optimization trajectories up to a scalar factor for optimizers invariant to rescaling, such as Adam. Efficient computation of the squared Cramér loss via a merged-and-sorted quantile array is possible in $O(N\log N)$ time, enhancing both practical and theoretical properties of value function approximation (Lhéritier et al., 2021).

Non-crossing constraints are essential to prevent invalid quantile orderings, reducing noise in the learning signal and stabilizing training in high-dimensional DRL settings. Networks that enforce monotonicity, such as via appropriate architectural choices, further benefit from improved convergence and robust uncertainty estimates (Lhéritier et al., 2021).

4. Multivariate and Functional Quantile Approximators

Beyond univariate value distributions, quantile-based architectures have been generalized to:

• Multivariate Quantile Function Forecasters (MQF$^2$), which produce joint quantile forecasts for multi-horizon time series. MQF$^2$ parameterizes the multivariate quantile function as the gradient of an input-convex neural network (PICNN), ensuring monotonicity and avoiding quantile crossing (i.e., for all $u_1,\,u_2$: $(q(u_1)-q(u_2))^\top(u_1-u_2)\geq 0$). Training is performed via either an energy score or maximum likelihood approach, thereby capturing dependencies across time and improving sample path realism (Kan et al., 2022).
• Functional Data Analysis: In function-on-function quantile regression (Beyaztas et al., 2021, Beyaztas et al., 2021), the conditional quantile of a response curve $Y(t)$ given predictors $X_m(s)$ is modeled as

$$Q_\tau[Y_i(t) | X_i] = \sum_{m=1}^M \int_0^1 X_{im}(s)\, \beta_{m\tau}(s, t)\,ds,$$

with parameter estimation transformed to a finite-dimensional problem via basis expansion or functional principal component analysis (FPCA). Iterative procedures (e.g., functional partial quantile regression) and forward variable selection using BIC ensure flexibility, robustness, and predictive accuracy, even under heavy-tailed or contaminated error distributions.

5. Risk-Sensitive, Safety-Aware, and Multi-Agent Extensions

Quantile-based value approximators enable explicit modeling and control of risk. In risk-sensitive RL, CVaR is integrated with quantile regression to regularize policy learning towards safer behavior (Enwerem et al., 8 Jun 2025, Li et al., 2017). The loss function,

$$\mathcal{L}(\theta)=(1-\lambda)\,\mathcal{L}_\mathrm{QR}(\theta)+\lambda\, \mathcal{L}_{\rho}(\hat{\rho}_\beta(C), c_{\mathrm{max}})$$

enforces both approximation of the full cost distribution and penalizes policy options with high tail risk. The risk-sensitive BeLLMan operator used in these algorithms is a contraction in the Wasserstein-1 space, ensuring unique convergence and robust policy updates (Enwerem et al., 8 Jun 2025).

In multi-agent settings, the DFAC framework extends value function factorization to the distributional case. Here, joint return distributions are factorized via quantile mixtures, with individual agents' value functions parameterized as quantiles and combined through a monotonic mixing network. This approach improves robustness and efficiency in highly stochastic environments, such as those simulated in StarCraft Multi-Agent Challenge (SMAC) benchmarks (Sun et al., 2021).

6. Empirical Evidence and Practical Implementation

Empirical studies across diverse environments demonstrate that quantile-based value function approximators:

• Outperform mean-based models on tasks requiring risk awareness or robustness, such as financial value-at-risk (VaR) forecasting (Chen et al., 2017), robot reach-avoid control with safety constraints (Enwerem et al., 8 Jun 2025), and combinatorial generation with risk constraints in GFlowNets (Zhang et al., 2023).
• Exhibit improved sample efficiency and stability in DRL settings, notably in Atari 2600 and DeepMind Control Suite benchmarks (Dabney et al., 2017, Dabney et al., 2018, Luis et al., 2023).
• Support theoretically grounded convergence guarantees due to contraction properties in Wasserstein spaces.
• Are efficiently implementable with code publicly available for leading algorithms (e.g., Epistemic Quantile-Regression with SAC (Luis et al., 2023)).

7. Challenges and Open Problems

Despite their strengths, several challenges persist:

• Choosing the number and placement of quantile fractions or basis functions in finite-dimensional approximations affects both statistical efficiency and computational cost (Yang et al., 2019).
• Enforcing non-crossing constraints may become expensive or nontrivial in high-dimensional, multivariate, or functional settings.
• The exploration of richer distortion risk measures, integration with other regularization techniques, and adaptation to partial observability continue to be active areas of research (Zhang et al., 2023, Kan et al., 2022).
• Practical methods for robust model selection (e.g., via BIC or bootstrapping) and handling computational trade-offs in sequential decision-making contexts warrant further paper.

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In summary, quantile-based value function approximators offer a principled and flexible means to model, estimate, and control uncertainty and risk in sequential decision-making. Their theoretical rigor, algorithmic versatility, and empirical advantages make them central to modern developments in distributional RL, robust control, probabilistic forecasting, and beyond.