Dynamic Improvement Reward (DIR)

• Dynamic Improvement Reward (DIR) is an adaptive framework that continuously refines reward functions based on feedback, data, or optimization signals in sequential decision-making systems.
• It integrates methods like query-based design (e.g., AIRD), dynamic trajectory aggregation, and ensemble techniques to improve uncertainty reduction and transferability.
• DIR minimizes uncertainty, balances multi-objective trade-offs, and reduces issues like catastrophic forgetting to drive robust and efficient policy optimization.

Dynamic Improvement Reward (DIR) formalizes the concept of iteratively and adaptively refining reward functions over time in sequential decision-making systems. Rather than establishing a reward signal in a single static design step, DIR encompasses methods where the reward function evolves based on data, user feedback, environment changes, or optimization signals, with each refinement targeted at reducing uncertainty, increasing transferability, balancing multi-objective trade-offs, or accelerating learning. DIR has emerged as a pivotal concept in reinforcement learning, imitation learning, preference alignment for LLMs, and generative modeling, with instantiations across active query-based inference, dynamic credit assignment, ensemble methods, and co-evolutionary frameworks.

1. Principles and Formal Definition

DIR is characterized by the dynamic update or refinement of the reward function $r_t$ at iteration or time step $t$ in response to new information. The core principle is that the reward specification process is not static but adaptive. The reward update can be formalized as: $$r_{t+1} = \mathrm{Update}(r_t, \text{feedback}_t, \mathcal{D}_t)$$ where $\mathcal{D}_t$ is data (expert demonstrations, user queries, episodic intrinsic rewards, model outputs), and $\mathrm{Update}$ encompasses Bayesian inference, credit assignment schemes, mutual information maximization, dynamic weight adjustment, or meta-learning procedures.

For example, in Active Inverse Reward Design (AIRD) (Mindermann et al., 2018), at each iteration the designer chooses among a set of proxy rewards, and the posterior over the true reward function $r^*$ is updated as: $P(r^*|\mathcal{D}) \propto P(w|r^*, M) P(r^*)$ where $w$ is the chosen proxy, $M$ the environment, and $\mathcal{D}$ reflects accumulated data over queries.

2. Query-Based and Information-Gain-Driven DIR

AIRD (Mindermann et al., 2018) exemplifies query-based DIR, structuring reward design as a sequence of actively selected queries posed to a designer. The designer's responses are maximally informative about the hidden true reward, and are chosen to maximize the expected mutual information (MI) between the query outcome and $r^*$: $$\mathrm{MI}(S_t, D_t) = \mathcal{H}[u|S_t, D_t] - \mathbb{E}_{r^* \sim P(r^*|D_t)}[\mathcal{H}[u|S_t, r^*]]$$ where $u$ is the user's answer and $S_t$ is the query set. This approach leverages both probabilistic inference and optimal experiment design to actively reduce uncertainty over $r^*$. Unlike classical IRD, which passively infers from a single designer-chosen proxy reward, AIRD explores the designer's preferences over suboptimal behaviors, driving dynamic improvement of the reward estimate.

AIRD supports both discrete queries and interpretable feature queries, enabling feedback over linear proxies while allowing the inference of nonlinear true rewards.

3. Dynamic Trajectory Aggregation and Reward Shaping

DIR appears in reward shaping with dynamic trajectory aggregation (Okudo et al., 2021), whereby the state space is dynamically partitioned into abstract states through a subgoal series $(SG, \prec)$. The potential function used for shaping is not hand-designed for the full state space, but learned over the aggregation: $\Phi(s) = V(g(s)) = V(z)$ where $g(s)$ maps a detailed state $s$ to an abstract state $z$, and $V(z)$ is the value over abstract states. The shaping function is: $F(s_t, s_{t+1}) = \gamma \Phi(s_{t+1}) - \Phi(s_t)$ Trajectory segments demarcated by subgoals allow the policy to propagate rewards more efficiently across temporally extended behaviors. This dynamic aggregation minimizes designer effort and extends potential-based shaping to high-dimensional, continuous domains.

4. Transferable and Policy-Independent Reward Learning

DIR frameworks address transferability through mechanisms like dynamics-agnostic discriminator ensembles (Luo et al., 2022). Classical IRL/AIL often entangle the learned reward with environment dynamics or policy histories, hindering application in changed environments. DARL disentangles the reward from dynamics by minimizing mutual information between latent state-action embeddings and next-state information: $I(z; s') \leq I_\text{vCLUB}(z; s')$ and represents rewards as normalized, clipped ensembles of discriminators trained on evolving policy distributions: $$r_E(s,a) = -\log\Bigg(1 - \frac{1}{|\mathcal{C}|} \sum_{i=1}^{|\mathcal{C}|} D_i^{\text{norm\_clip}}(s,a) \Bigg)$$ The ensemble ensures robustness to policy changes and supports learning state-action or state-only rewards, thereby facilitating dynamic reward improvement and transferability.

5. Multi-Objective and Step-Level DIR

DIR is also instantiated in frameworks optimizing multiple rewards and balancing trade-offs dynamically. Bandit-based methods such as DynaOpt and C-DynaOpt (Min et al., 20 Mar 2024) employ multi-armed bandits to adapt reward weights: $$p_t(i) = (1-\gamma)\frac{a_{t,i}}{\sum_j a_{t,j}} + \frac{\gamma}{N+1}$$ with weight updates via: $$w_{t+1,i} = w_{t,i} \exp(\frac{\gamma \hat{r}_{t,i}^{BW}}{K})$$ The reward combination is rebalanced as training proceeds, supporting dynamic improvement in multi-criteria objectives such as fluency, coherence, and reflection quality.

Concurrently, step-level credit assignment in RL-driven T2I fine-tuning (Liao et al., 25 May 2025) tracks cosine similarity increments between intermediate and final images, reshaping a trajectory-level reward into per-step contributions: $\hat{R}(s_t, a_t) = w_t \cdot r(x_0, c)$ with $w_t$ determined by the normalized impact of each denoising window. The shaped reward conforms to a potential-based shaping framework, thus maintaining optimal policy invariance while enabling dynamic improvement.

6. DIR in Co-Evolutionary and Incremental Reward Frameworks

In skill acquisition and policy evolution, DIR is realized by co-evolutionary reward-policy frameworks (Huang et al., 18 Dec 2024). ROSKA leverages LLMs to dynamically generate candidate reward functions informed by task returns and prior best functions. Policy populations are hybridized between the best-known parameters and random initializations, with the fusion parameter $\alpha$ optimized via Bayesian methods: $$\theta_f^m(\alpha) = \alpha \theta_\text{best}^{(m-1)} + (1-\alpha) \theta_0$$ The iterative loop guarantees continuous improvement, capturing the symbiotic evolution of rewards and policies under data-efficient constraints.

RID (Wang et al., 26 Nov 2024) targets incremental learning of downstream reward objectives in generative models, preventing catastrophic forgetting by freezing task-specific adapters and employing last-step EMA-momentum distillation: $$\max_{A_t, B_t} \sum_{c \in C_\text{train}} [R_t(f(z_1 | c)) - \lambda \|f(z_1 | c) - f^T(z_1 | c)\|^2]$$ This structure ensures stable and consistent reward optimization across sequential tasks.

7. Theoretical Underpinnings and Empirical Efficacy

Papers provide theoretical guarantees for DIR methods, including bounds on suboptimality gaps that decompose into off-policy bandit regret and diffusion-based distribution shift (Yuan et al., 2023): $$\text{SubOpt}(\hat{P}_a; y^* = a) \leq \mathcal{A}_1 + \mathcal{A}_2 + \mathcal{A}_3$$ where terms correspond to regression error, diffusion process deviation, and off-support penalty.

Empirical studies across domains—robotic skill acquisition, safe policy improvement in LLMs, RL-driven text or image generation—demonstrate consistent improvements. For instance, Active Inverse Reward Design (Mindermann et al., 2018) substantially reduces test regret in unseen environments compared to vanilla IRD, while ensemble methods (Luo et al., 2022) yield higher reward consistency and transferability.

Summary Table: DIR Mechanism Types

Mechanism	Adjustment Signal	Key Objective
Query-based (AIRD)	Mutual information gain	Uncertainty reduction
Bandit-based (DynaOpt)	Reward-weight updates	Multi-objective alignment
Ensemble-based (DARL)	Policy/discriminator history	Transferability
Trajectory Aggregation	Subgoal changes	Efficient reward shaping
Incremental Distillation (RID)	Adapter + EMA distillation	Catastrophic forgetting mitigation

Each of these mechanisms realizes the DIR principle by adaptively refining the reward specification or allocation based on ongoing data, feedback, or optimization signals.

• Active Inverse Reward Design (Mindermann et al., 2018)
• Reward Shaping with Dynamic Trajectory Aggregation (Okudo et al., 2021)
• Transferable Reward Learning by Dynamics-Agnostic Discriminator Ensemble (Luo et al., 2022)
• Dynamic Reward Adjustment in Multi-Reward Reinforcement Learning (Min et al., 20 Mar 2024)
• Reward Incremental Learning in Text-to-Image Generation (Wang et al., 26 Nov 2024)
• Efficient Language-instructed Skill Acquisition via Reward-Policy Co-Evolution (Huang et al., 18 Dec 2024)
• Step-level Reward for Free in RL-based T2I Diffusion Model Fine-tuning (Liao et al., 25 May 2025)
• Learning to Optimize Multi-Objective Alignment Through Dynamic Reward Weighting (Lu et al., 14 Sep 2025)

Conclusion

Dynamic Improvement Reward methods constitute a foundational shift from static to adaptive reward schemes in learning systems. By integrating mechanisms for iterative update, transferability, multi-objective balancing, credit assignment, and co-evolution, DIR supports more robust, efficient, and generalizable policy optimization in reinforcement learning, generative modeling, and real-world interactive agents. The mathematical formalism and empirical results emphasize the necessity of dynamic reward adjustment for state-of-the-art performance in complex, uncertain, and evolving environments.