Directed Rate-Reward Functions

• Directed rate-reward functions are objective formulations that couple scalar rewards with rate-like terms (e.g., mean reward per step, KL penalties) to direct policy behavior.
• They are implemented via methods such as recursive aggregation, KL-regularized control, and rating-weighted divergence penalties across various domains like RL, generative design, and survival analysis.
• This framework provides a flexible balance between reward maximization and regularization to control distribution shifts, risk, and extrapolation in optimization problems.

Searching arXiv for the cited papers and closely related reward-directed formulations. A directed rate-reward function is an objective construction in which optimization is guided not by an undifferentiated cumulative reward alone, but by a reward criterion coupled to a direction-inducing rate term or rate-like statistic. In the cited literature, that role is realized in several formally distinct ways: as a recursive aggregation over reward sequences such as mean reward per step or Sharpe ratio; as a KL-regularized control objective that trades expected reward against deviation from a reference diffusion process; as a log survival-rate reward obtained from multi-step survival probability; and, in a closely related rating-based formulation, as a reward objective augmented by rating-weighted divergence penalties that steer policy distributions away from low-quality behavior classes (Tang et al., 11 Jul 2025, Keramati et al., 2 Aug 2025, Gao et al., 2024, Yoshida, 2016, Wu et al., 13 Jan 2025).

1. Conceptual scope and terminological usage

The sources suggest that the phrase denotes a family of constructions rather than a single canonical formula. In "Recursive Reward Aggregation" (Tang et al., 11 Jul 2025), the term is explicit: directed rate-reward functions are objectives that direct behavior by optimizing a rate of reward relative to some other quantity, such as time, variance, cost, or peak-versus-baseline structure. In "A Reward-Directed Diffusion Framework for Generative Design Optimization" (Keramati et al., 2 Aug 2025), the paper does not explicitly use the phrase, but its formulation is interpretable as reward-directed generation in which a soft value function induces a direction field in sample space while a KL term controls departure from a pre-trained generative prior. In "Reward-Directed Score-Based Diffusion Models via q-Learning" (Gao et al., 2024), the corresponding rate term is a path-space KL penalty between a controlled denoising process and the ideal reverse diffusion. In "On Reward Function for Survival" (Yoshida, 2016), the rate is the one-step survival probability, whose logarithm becomes the per-step reward.

Two technical meanings of "rate" recur. The first is a normalized or ratio-type trajectory functional, such as mean reward per step, variance-regularized return, range, discounted max, or Sharpe ratio. The second is an information-rate or deviation-rate penalty, typically written as a KL term that measures how far a policy or sampler moves away from a reference process. A plausible implication is that the topic sits at the intersection of generalized return design and KL-regularized control: one branch changes how rewards are aggregated over trajectories, while another changes how aggressively a policy or generator is allowed to distort a baseline distribution.

A further distinction is required for rating-based RL. "RbRL2.0: Integrated Reward and Policy Learning for Rating-based Reinforcement Learning" (Wu et al., 13 Jan 2025) develops a directed rating-reward objective rather than a literal rate-reward objective. Its use of rating-weighted KL penalties is nevertheless structurally adjacent, because the directional component again arises from divergence terms that repel the current policy from designated low-quality trajectory distributions.

2. Recursive aggregation over reward sequences

The most explicit formalization appears in the algebraic MDP framework of (Tang et al., 11 Jul 2025). There, an MDP is decomposed into a reward generator and an aggregator. For a fixed policy, the generator recursively produces a reward sequence, and the aggregator folds that sequence into a scalar objective. The standard discounted return is one instance: $G_t=\sum_{k=0}^{\infty}\gamma^k R_{t+k+1},$ but the framework replaces this single aggregation rule by a general recursive statistic aggregation

$\Agg=\post\circ\AggStat,$

with statistic space $T$, initial statistic $\tau_0$, update $\upd:R\times T\to T$, and post-processing $\post:T\to\mathbb R$ (Tang et al., 11 Jul 2025).

This construction admits objectives that are plainly rate- or ratio-like. The paper instantiates mean reward per step by storing count and sum in $T=\mathbb N\times\mathbb R$, variance by storing length, sum, and sum of squares, range by storing max and min, discounted max by a scalar max-recursion, and Sharpe ratio by a post-processing map that divides mean by standard deviation. In this setting, a directed rate-reward function is any objective defined on the full reward sequence and implemented through a bounded-size recursive statistic. The Bellman equation then generalizes from a scalar value to a statistic-valued fixed point: $\tau_\pi(s)= \begin{cases} \tau_0 & s \text{ terminal},\ \upd(r_\pi(s),\tau_\pi(p_\pi(s))) & \text{otherwise}. \end{cases}$

The significance of the framework is that the reward function $r(s,a)$ need not be modified in order to optimize mean reward, risk-adjusted reward, or extremal criteria. The change occurs entirely in the aggregation operator. This is why the paper emphasizes that flexible alignment can be achieved by selecting appropriate reward aggregation functions rather than redesigning the underlying reward signal (Tang et al., 11 Jul 2025).

The same formalism extends to RL algorithms by replacing only the Bellman target. Q-learning updates a statistic-valued $Q$-object; PPO uses a critic that approximates $\Agg=\post\circ\AggStat,$0 instead of a scalar value; TD3 learns statistic functions whose post-processed outputs serve as Q-values. Empirically, the paper reports that different aggregations induce qualitatively different policies: discounted max favors peak-seeking behavior, discounted min yields conservative behavior, mean directly optimizes average reward per step, and Sharpe ratio in the portfolio experiment produces higher test Sharpe ratios than competing methods (Tang et al., 11 Jul 2025).

3. KL-regularized reward direction in diffusion and generative control

In reward-directed diffusion, the rate-reward structure is expressed through KL-regularized control of a denoising process. The discrete-time formulation in (Keramati et al., 2 Aug 2025) treats reverse diffusion as a finite-horizon MDP whose state is $\Agg=\post\circ\AggStat,$1, action is $\Agg=\post\circ\AggStat,$2, and reward is terminal: $\Agg=\post\circ\AggStat,$3 The paper then introduces a KL-regularized objective

$\Agg=\post\circ\AggStat,$4

where $\Agg=\post\circ\AggStat,$5 is the pre-trained reverse kernel and $\Agg=\post\circ\AggStat,$6 is the trade-off temperature (Keramati et al., 2 Aug 2025).

The soft value function $\Agg=\post\circ\AggStat,$7 supplies the directional term. The soft-optimal transition has the Boltzmann-tilted form

$\Agg=\post\circ\AggStat,$8

This is a control-as-inference structure: default denoising transitions are reweighted by exponentiated future reward, while the KL term limits how much information is spent deviating from the base model. The paper implements this both at training time, through reward-weighted maximum likelihood, and at inference time, through soft-value-based importance sampling that evaluates candidate reverse steps by approximate posterior means $\Agg=\post\circ\AggStat,$9 and weights $T$0 (Keramati et al., 2 Aug 2025).

The continuous-time version in (Gao et al., 2024) makes the same structure explicit in path space. There the score itself is treated as the control variable, and the objective is

$T$1

Under Girsanov, the running quadratic penalty is exactly twice the KL between the controlled path measure and the ideal reverse-diffusion path measure. After entropy regularization, the optimal policy is Gaussian,

$T$2

so reward direction appears as a value-gradient correction to the score, while the covariance is fixed by the temperature and diffusion noise (Gao et al., 2024).

Theoretical analysis in "Reward-Directed Conditional Diffusion: Provable Distribution Estimation and Reward Improvement" (Yuan et al., 2023) sharpens the same trade-off. For target reward $T$3, the reward optimality gap is bounded by a decomposition

$T$4

where $T$5 is reward-learning error, $T$6 is on-support diffusion error, and $T$7 is off-support diffusion error. The paper relates $T$8 to off-policy bandit regret in the latent feature subspace, shows that $T$9 worsens with distribution shift as target reward $\tau_0$0 increases, and controls $\tau_0$1 by off-support deviation from the learned subspace (Yuan et al., 2023). This suggests that, in diffusion settings, a directed rate-reward function is not merely a heuristic steering term: it is a quantitative balance between reward improvement, distribution shift, and extrapolation cost.

4. Engineering design optimization as a reward-directed generative process

The design-optimization instantiation in (Keramati et al., 2 Aug 2025) shows how the abstract rate-reward construction becomes operational in a non-differentiable engineering setting. Designs are encoded as parametric vectors: a 384-dimensional airfoil representation built from 192 $\tau_0$2 points, and a 44-parameter ship-hull representation. Rewards are supplied by non-differentiable XGBoost surrogates: normalized $\tau_0$3 for airfoils with $\tau_0$4, and negative total resistance across multiple speeds and drafts for ship hulls with $\tau_0$5. The ship reward includes feasibility penalties,

$\tau_0$6

where $\tau_0$7 penalizes self-intersection, coordinates outside $\tau_0$8, and violations of hull-parameter bounds (Keramati et al., 2 Aug 2025).

The framework is specifically designed for cases in which rewards come from costly simulations or surrogates that are non-differentiable or prohibitively expensive to differentiate. Training is gradient-free with respect to reward: reward values enter only as scalar weights in a weighted DDPM fine-tuning loss. At inference, reward-directed importance sampling uses a small duplication factor $\tau_0$9 and no backward passes, so the reward signal acts through sample reweighting rather than through $\upd:R\times T\to T$0 (Keramati et al., 2 Aug 2025).

Empirically, the framework generates out-of-distribution high-performance designs. For 2D airfoils, reward-weighted MLE plus value-based sampling shifts the $\upd:R\times T\to T$1 distribution to the right, produces designs more than 10 percent above training maxima, and includes a design with $\upd:R\times T\to T$2, with XFOIL validation confirming aerodynamic plausibility. For ship hulls, reward-directed sampling achieves more than 25 percent reduction in total calm-water resistance relative to training data and vanilla DDPM outputs, while MAXSURF hydrostatics and large-angle stability satisfy IMO intact stability criteria and Holtrop-Mennen resistance curves remain smooth and physically plausible (Keramati et al., 2 Aug 2025).

In this application, the rate term is not a throughput or time-normalized quantity. It is the KL cost of moving the reverse chain away from the pre-trained design prior. The directed rate-reward function therefore regulates extrapolation in design space: reward pushes toward unseen high-performance regions, and the KL term constrains that push to remain near physically meaningful generative structure.

5. Rating-conditioned directional penalties and neighboring formulations

RbRL2.0 introduces a closely related but distinct mechanism in policy space (Wu et al., 13 Jan 2025). The environment is modeled as a reward-free MDP augmented with human-provided ratings on trajectory segments. Ratings define discrete performance classes, and the method learns a reward model $\upd:R\times T\to T$3 from those labels while simultaneously using the rating structure to shape policy updates. The integrated objective is

$\upd:R\times T\to T$4

where $\upd:R\times T\to T$5 is the Gaussian approximation to the trajectory-feature distribution of rating class $\upd:R\times T\to T$6, $\upd:R\times T\to T$7 is the current policy distribution, and the weights satisfy

$\upd:R\times T\to T$8

The highest rating class is excluded from the KL penalty, so the policy is repelled from lower-rated behaviors most strongly and less strongly from less-bad behaviors (Wu et al., 13 Jan 2025).

The paper describes this as a directed rating-reward notion. A plausible implication is that it functions as a neighboring directed rate-reward construction in the broader sense: reward learning supplies the scalar performance term, while weighted KL penalties provide the directional field in distribution space. The rate-like component here is not a temporal rate or risk ratio, but a divergence-weighted geometric displacement away from undesirable trajectory classes.

Algorithmically, the method estimates class-wise Gaussian means and covariances from buffers $\upd:R\times T\to T$9, computes KL divergences to the current policy distribution, and combines their gradients with the policy-gradient term based on learned cumulative reward. The paper reports experiments on DeepMind Control Suite tasks—HalfCheetah, Walker, and Quadruped—with $\post:T\to\mathbb R$0 rating classes. RbRL2.0 is similar to RbRL on HalfCheetah, consistently outperforms it on Walker, and significantly outperforms it on Quadruped, especially for $\post:T\to\mathbb R$1, with improved convergence and overall performance over the existing rating-based method that uses reward learning alone (Wu et al., 13 Jan 2025).

6. Survival probability, multiplicative objectives, and limits of the concept

The survival formulation of (Yoshida, 2016) shows an older and more literal rate interpretation. The objective is to maximize the multi-step survival probability

$\post:T\to\mathbb R$2

where $\post:T\to\mathbb R$3. Because the survival event factorizes along a trajectory,

$\post:T\to\mathbb R$4

taking logs converts a multiplicative criterion into an additive RL return. The resulting reward is

$\post:T\to\mathbb R$5

The paper further shows that maximizing the associated average-reward RL objective is equivalent, up to a factor of $\post:T\to\mathbb R$6, to maximizing a variational lower bound on $\post:T\to\mathbb R$7 (Yoshida, 2016).

This construction is instructive because it makes directionality explicit. States with high one-step survival probability receive rewards near zero, whereas states with low survival probability incur large negative rewards. For small death probability $\post:T\to\mathbb R$8, the reward satisfies $\post:T\to\mathbb R$9, so the reward is approximately negative instantaneous hazard. The rate-reward function is therefore the log of a temporal survival rate, and long-horizon survival emerges from accumulation of these local log-rates (Yoshida, 2016).

The grid-world experiment illustrates the mechanism. A 3×3 environment with food, poison, and a battery level $T=\mathbb N\times\mathbb R$0 uses

$T=\mathbb N\times\mathbb R$1

with $T=\mathbb N\times\mathbb R$2 after eating poison and $T=\mathbb N\times\mathbb R$3 otherwise. Using tabular Sarsa$T=\mathbb N\times\mathbb R$4, the agent learns survival behavior: median survival time increases, battery at death concentrates around $T=\mathbb N\times\mathbb R$5, food consumption increases, and poison consumption remains essentially zero (Yoshida, 2016).

Taken together, the literature also delineates the limits of the concept. The recursive-aggregation framework requires recursive computability with a fixed-size statistic $T=\mathbb N\times\mathbb R$6; non-recursive functionals such as true medians or semivariance without approximation are not directly supported, and nonlinear ratios such as Sharpe raise contraction and distributional-RL questions in stochastic environments (Tang et al., 11 Jul 2025). Reward-directed diffusion faces the cost of extrapolation: larger target reward or stronger guidance increases distribution shift and off-support error, so actual reward can plateau or decline when the generator moves beyond well-covered regions (Yuan et al., 2023). RbRL2.0 provides no formal convergence theorem and depends on Gaussian approximations of rating-conditioned trajectory distributions and on the choice of $T=\mathbb N\times\mathbb R$7 (Wu et al., 13 Jan 2025). The survival formulation assumes the temporal survival probability is known and may not be optimal for learning speed, even if it is principled for objective specification (Yoshida, 2016).

These limitations clarify the status of directed rate-reward functions in current research. They are best understood not as a single standardized object, but as a class of objective designs that combine a directional reward criterion with a rate-like normalization or regularization mechanism. In RL, this can mean recursive aggregation of returns. In diffusion and generative design, it can mean KL-budgeted deviation from a learned prior or from the true reverse process. In survival, it can mean a log-rate reward derived from multiplicative viability. The common thread is that the function does not merely score outcomes; it specifies how aggressively optimization should move through trajectory space or sample space in order to obtain those outcomes.