Abstract

The Bellman equation, foundational to reinforcement learning (RL) and dynamic programming, is often celebrated for its elegance and recursive structure, enabling agents to optimize sequential decision-making processes. However, when applied to real-world systems, its assumptions about states, rewards, discounting, and transitions face critical challenges. This post explores the limitations of the Bellman equation in complex environments and proposes alternatives that avoid artificial constraints. We discuss the necessity of optimizing state identification, reward definitions, and action spaces in real-world scenarios, where the Markov property and transition probabilities are often unknown or unidentifiable. Drawing from evolutionary processes, agentic systems, and deep learning, we examine how trial-and-error learning combined with preference optimization can provide a more adaptive approach to solving complex decision-making problems, transcending the limitations of traditional reinforcement learning frameworks.

Introduction

The Bellman equation is widely regarded as a cornerstone of reinforcement learning and dynamic programming, central to algorithms like Q-learning and value iteration. At its core, the Bellman equation captures the recursive nature of decision-making: the value of a state depends on the immediate reward and the value of future states. This framework works well in constrained environments like games or simulations, where the states, actions, rewards, and transitions are explicitly defined. However, the moment we try to apply the Bellman equation to real-world, dynamic systems—such as financial markets, autonomous agents, or biological systems—its assumptions begin to break down.

In the real world, states are rarely well-defined, and identifying the "true state" of a complex system often requires optimization in and of itself. Rewards can be ambiguous or internal, as in the case of human emotions or abstract goals, and are not always provided explicitly by the environment. Moreover, actions in complex systems are far from discrete and can range from subtle micro-interactions to large-scale behaviors, many of which are difficult to enumerate or define in advance. Even the discount factor—critical to the Bellman equation—fails to capture the long-term ramifications of small actions in chaotic or highly sensitive systems.

Furthermore, assumptions like the Markov property—which simplifies decision-making by assuming the future depends only on the current state—can collapse in real-world systems where the past (or memory) plays a significant role in determining future outcomes. Finally, transition probabilities between states are rarely known, and in many cases, they cannot be accurately derived from data.

This post explores the many ways in which the Bellman equation and traditional reinforcement learning frameworks fall short in complex environments, and why evolutionary processes—such as genetic algorithms and trial-and-error learning—offer a more adaptive, flexible approach to decision-making. By drawing from direct preference optimization (DPO) and self-guided learning frameworks, we examine how agentic systems can learn to control their environment and improve autonomously, without relying on the rigid assumptions of standard RL. We also explore how deep learning models, such as transformers, can be augmented with evolutionary mechanisms to evolve their own reasoning chains and self-modify over time, bypassing the limitations imposed by the Bellman equation.

In this post, we will:

Delve into the critical issues with applying the Bellman equation to real-world systems.
Explore alternative approaches inspired by natural selection and evolutionary algorithms, allowing agents to evolve without predefined state spaces or reward functions.
Examine a novel strategy for reasoning chain optimization in large language models (LLMs), integrating trial-and-error learning, DPO, and progressive learning frameworks.

By the end of this post, we aim to challenge conventional thinking around reinforcement learning and offer a new perspective on how to build more adaptive, autonomous systems that can thrive in complex environments without the need for strict assumptions about states, actions, and rewards.

Bellman Equation

The Bellman equation is a key principle in dynamic programming and reinforcement learning, capturing the recursive nature of decision-making in sequential problems. To derive it, we begin with the basic principles underlying optimal control and decision-making over time.

1. The Problem Setup: Sequential Decision-Making

Consider a Markov Decision Process (MDP), where an agent interacts with an environment. The environment is defined by:

States: ( S ) — the set of all possible states in which the agent can find itself.
Actions: ( A ) — the set of all actions the agent can take.
Transition probabilities: ( P(s'|s, a) ) — the probability of transitioning to a new state ( s' ) given the current state ( s ) and action ( a ).
Rewards: ( R(s, a) ) — the reward received when taking action ( a ) in state ( s ).
Discount factor: ( \gamma \in [0, 1] ) — the discount applied to future rewards to model the time preference (with ( \gamma = 1 ) meaning no preference, and ( \gamma = 0 ) meaning only immediate rewards matter).

The objective of the agent is to maximize the expected cumulative reward (or return) over time by choosing the optimal actions in each state.

2. The Value Function

The value of being in a state ( s ) under a policy ( \pi ), denoted $V^\pi(s)$, is the expected total discounted reward starting from ( s ) and following policy ( \pi ). Formally,

$$V^\pi(s) = \mathbb{E}_\pi \left[ \sum_{t=0}^{\infty} \gamma^t r_t \mid s_0 = s \right]$$

Here, ( r_t ) is the reward at time ( t ), and the policy ( \pi ) dictates the action $a_t = \pi(s_t)$.

3. The Principle of Optimality (Bellman's Insight)

Richard Bellman proposed the Principle of Optimality, which states:

An optimal policy has the property that, whatever the initial state and initial decision, the remaining decisions must constitute an optimal policy for the subproblem starting at the state resulting from the first decision.

This leads to the recursive structure of the problem: if you are in state ( s ), the value of following an optimal policy $\pi^*$ (denoted $V^*(s)`$) is equal to the immediate reward plus the value of the next state, given that you act optimally from that point onward.

4. Recursive Decomposition of the Value Function

From the principle of optimality, the value function for an optimal policy $\pi^*$ can be written as:

$$V^*(s) = \max_{a} \mathbb{E} \left[ R(s, a) + \gamma V^*(s') \mid s, a \right]$$

This equation breaks down as follows:

$\max_a$ — The agent selects the action ( a ) that maximizes the expected return.
$R(s, a)$ — The immediate reward for taking action ( a ) in state ( s ).
$\gamma V^*(s')$ — The discounted future value of the next state ( s' ), assuming the agent continues optimally.

Since the next state ( s' ) depends on the transition probabilities ( P(s'|s, a) ), the expectation can be written explicitly as:

$$V^*(s) = \max_{a} \sum_{s'} P(s'|s, a) \left[ R(s, a) + \gamma V^*(s') \right]$$

5. The Action-Value Function (Q-function)

An alternative form of the Bellman equation involves the action-value function, $Q^*(s, a)$, which represents the value of taking action ( a ) in state ( s ) and then following the optimal policy from the resulting state:

$$Q^*(s, a) = \mathbb{E} \left[ R(s, a) + \gamma V^*(s') \mid s, a \right]$$

Using this, the Bellman optimality equation for the value function becomes:

$$V^*(s) = \max_a Q^*(s, a)$$

And for the action-value function:

$$Q^*(s, a) = \sum_{s'} P(s'|s, a) \left[ R(s, a) + \gamma \max_{a'} Q^*(s', a') \right]$$

6. Summary of the Bellman Equations

Bellman equation for the value function:

$$V^*(s) = \max_{a} \sum_{s'} P(s'|s, a) \left[ R(s, a) + \gamma V^*(s') \right]$$

Bellman equation for the action-value function:

$$Q^*(s, a) = \sum_{s'} P(s'|s, a) \left[ R(s, a) + \gamma \max_{a'} Q^*(s', a') \right]$$

These equations describe how the value of a state or state-action pair can be decomposed into immediate rewards and the future value of subsequent states, and they form the foundation for many algorithms in dynamic programming and reinforcement learning, such as Value Iteration and Q-Learning.

7. Key Principles Underpinning the Bellman Equation

Markov Property: The next state ( s' ) depends only on the current state ( s ) and action ( a ), not on the history of past states or actions.
Discounting: Future rewards are discounted by a factor ( \gamma ) to reflect the preference for immediate rewards over distant rewards.
Optimality: The value of being in a state and acting optimally is equal to the best immediate action's reward plus the discounted value of future states.

The recursive nature of the Bellman equation is what allows for the systematic decomposition of decision-making problems over time, central to many computational approaches to solving MDPs.

The Limitations of the Bellman Equation in Real-World Systems

The Bellman equation provides a mathematically elegant framework for solving sequential decision-making problems. Its recursive structure allows an agent to determine the optimal action at each state by balancing immediate rewards with the expected future rewards. While this works well for problems like chess, Go, or simulations with well-defined states, rewards, and actions, real-world environments present challenges that the Bellman equation struggles to address.

1. State Identification is Non-Trivial

In classical reinforcement learning, a state (s) is assumed to perfectly encapsulate all relevant information about the environment at a given moment. This simplification is key to the Markov property, which posits that the future is conditionally independent of the past, given the current state. In real-world systems, however, states are not easily identifiable. For instance:

In financial markets, the "state" may depend on a complex interplay of past events, external economic factors, and even psychological behaviors, which cannot be distilled into a singular, static representation.
In autonomous driving, sensor data must be processed to form a meaningful state, but this involves noise, ambiguity, and varying levels of granularity.

The issue is that identifying a state representation that captures all the relevant dynamics of the system is itself an optimization problem. In practical applications, this means a model could lose critical context if the chosen state doesn’t account for key historical data, leading to poor decision-making. Thus, assuming that a perfect state can be predefined often leads to suboptimal policies when applied to more dynamic, complex systems.

2. Rewards Are Often Implicit and Internal

The Bellman equation assumes the existence of an explicit reward function, which the agent seeks to maximize. In games like chess, the reward is clear—win or lose. But in most real-world systems, rewards are often ambiguous or even internal to the agent itself. Consider:

Human decision-making: The reward for a decision could be satisfaction, happiness, or long-term well-being—abstract concepts that don’t come directly from the environment but from internal experiences.
Creative problem solving: The reward might be the discovery of an elegant solution, which is subjective and not immediately quantifiable.

In such cases, the challenge lies in defining a reward function that aligns with the agent’s true goals. Moreover, rewards in the real world are not always immediate or clear. For example, an agent working in long-term planning, such as investment management, may only receive a measurable reward (profit or loss) after months or years. Designing a reward function that accurately reflects these kinds of delayed, uncertain outcomes is non-trivial and often requires optimization over the reward function itself.

3. Discount Factors Fail to Capture Long-Term Sensitivity

In reinforcement learning, the discount factor ( \gamma ) controls how much future rewards are valued compared to immediate ones. A high discount factor (( \gamma \approx 1 )) means that future rewards are almost as valuable as immediate rewards, while a low discount factor prioritizes short-term gains. While this works in structured environments, in chaotic or sensitive systems, even small changes in initial conditions can have significant long-term effects. Consider:

Stock markets: A seemingly insignificant transaction might trigger a cascade of events that dramatically shift the market over time. A low discount factor could cause an agent to miss these effects entirely, while a high discount factor might make it difficult to optimize current actions effectively.
Environmental sustainability: In systems like climate change, small changes (e.g., reducing carbon emissions) might have profound long-term effects, but capturing this in a reward structure using a discount factor is complex and often leads to trade-offs between short-term and long-term outcomes.

The problem is that discounting future rewards equally across all problems introduces artificial constraints that don’t always align with the real-world dynamics, especially when some small actions carry significant long-term consequences. Therefore, in many cases, the discount factor itself needs to be optimized or reconsidered entirely.

4. Actions in Complex Systems Are Not Well-Defined

In real-world environments, actions are not discrete and well-defined like in games. The Bellman equation assumes a fixed set of actions ( A ) that an agent can take in each state. But in the real world, actions may be:

Continuous and subtle: In social interactions, for example, facial expressions, tone of voice, and body language are all subtle actions that can significantly affect outcomes. Identifying these micro-actions is difficult, yet they are often more important than larger, more obvious actions.
Unobservable or latent: In many complex systems, some actions might not be directly observable or measurable, making it hard for an agent to optimize its behavior. For instance, in medical diagnosis, a doctor’s internal reasoning process is an action but is difficult to encode as a discrete set of steps.

Optimizing an agent’s actions in such environments requires far more flexibility than the traditional RL framework provides. Actions must be learned, evolved, or dynamically constructed, rather than predefined.

5. Transition Probabilities Are Rarely Known

In the Bellman equation, the transition probabilities ( P(s'|s, a) ) represent the likelihood of moving from one state ( s ) to another ( s' ) after taking action ( a ). These probabilities are often assumed to be known, but in real-world applications, they are typically unknown and highly complex. For instance:

In biological systems, the outcome of an action (e.g., administering a drug) might depend on an intricate web of interactions at the molecular level, many of which are not well understood.
In robotics, transitioning between states is affected by environmental factors like terrain, sensor noise, and mechanical wear, which are difficult to predict or model accurately.

Without knowing these transition probabilities, traditional RL struggles to generate reliable policies. Even with data, estimating these probabilities can be an extremely noisy and uncertain process, which undermines the reliability of solutions based on the Bellman equation.

Beyond the Bellman Equation

While the Bellman equation is foundational in RL, it makes several assumptions that break down in real-world, complex environments. States are difficult to identify, rewards are often implicit, discount factors fail to capture long-term dependencies, actions are not well-defined, and transition probabilities are rarely known. To build truly adaptive systems, we need to move beyond the Bellman equation, a natural next step has been deep RL using neural networks.

Neural Networks as Models for Reinforcement Learning Components

Neural networks have long been utilized in reinforcement learning (RL) as powerful function approximators, but their potential extends beyond merely fitting value functions or policies. Neural networks can also be used to model and learn the underlying components of RL—such as states, actions, rewards, and transitions—especially in complex environments where these components are not readily available or well-defined.

In this section, we explore how neural networks can be employed to learn and optimize the core components of RL that are typically assumed to be predefined, but which may need to be dynamically learned and refined in real-world applications.

1. State Representation Learning

In traditional RL, the agent is assumed to operate in a well-defined state space ( S ), where each state encapsulates all relevant information about the environment at a given time. However, in many real-world scenarios, defining the state space is itself a challenge, as states may be high-dimensional, noisy, or incomplete.

Neural networks can help in learning a compact and meaningful representation of the state. This is particularly important when the raw input (e.g., sensor data, visual inputs) is complex, requiring the network to distill it into a lower-dimensional latent space that captures the essential features for decision-making.

Autoencoders or variational autoencoders (VAEs) can be used to learn low-dimensional latent representations of high-dimensional input spaces, compressing the data into meaningful states.
Recurrent neural networks (RNNs) or transformers can be used to model non-Markovian states, where past states and events have residual effects. These models maintain an internal memory or attention mechanism that helps the agent keep track of important information over time.

By using neural networks to learn a more flexible and optimized state representation, RL agents can operate in environments where states are inherently difficult to define, allowing them to adapt and make decisions in more complex, dynamic settings.

2. Action Space Optimization

In RL, the action space ( A ) is typically predefined, with agents selecting from a fixed set of discrete or continuous actions. However, in many real-world applications, defining the action space is non-trivial, as actions may not be discrete or even well-understood by the agent.

Neural networks can be used to learn and optimize the action space itself, allowing for more flexible and adaptive decision-making.

Action embeddings: Similar to how neural networks learn word embeddings in NLP, they can be used to embed actions in a continuous space, allowing for the exploration of a continuous and contextualized action space. This is particularly useful in scenarios where the actions are not purely discrete (e.g., in robotics, where actions may be small motor movements).
Hierarchical action modeling: Neural networks can be used to learn hierarchical action structures, where low-level actions (e.g., motor commands) are grouped into high-level actions (e.g., walking, grasping). This allows the agent to operate across multiple levels of abstraction, optimizing for high-level actions while still managing low-level details.
Action discovery: Neural networks can help discover new, latent actions that might not be immediately apparent. For instance, in creative problem-solving tasks, neural networks can learn abstract actions (or strategies) that are not predefined by the environment or the agent’s designer.

3. Reward Function Approximation

In standard RL, the agent receives rewards that serve as feedback for the quality of its actions. While rewards are often predefined in toy environments or games, real-world rewards can be more abstract, implicit, or internal to the agent. This is especially true in scenarios where rewards represent intangible outcomes, such as human satisfaction, creativity, or long-term success.

Neural networks can be used to approximate or learn reward functions, particularly when the reward signal is noisy, delayed, or unavailable.

Inverse reinforcement learning (IRL): Neural networks can be used in the context of IRL to infer the underlying reward function based on expert demonstrations. This is useful in tasks where explicit reward signals are not available, but optimal behavior is observable (e.g., imitating expert strategies in autonomous driving).
Learned intrinsic rewards: Instead of relying on predefined extrinsic rewards, neural networks can learn intrinsic reward functions that drive exploration and learning. For instance, an agent might learn to reward itself based on curiosity (e.g., learning new states or exploring new areas) or achieving intermediate goals (e.g., reaching checkpoints).
Continuous reward modeling: Neural networks can handle cases where rewards are delayed or continuous, dynamically learning how to assign credit to actions that contributed to long-term success.

4. Transition Dynamics Modeling

One of the core components of the Bellman equation is the transition probability ( P(s'|s, a) ), which defines how the environment evolves from one state to another after an action is taken. In many real-world applications, transition dynamics are unknown or difficult to model explicitly.

Neural networks can be used to learn the transition dynamics directly from data, allowing RL agents to approximate the environment’s behavior even in the absence of explicit models.

Model-based RL: In model-based reinforcement learning, neural networks can learn the environment’s transition function, enabling agents to simulate the environment internally. This reduces the need for extensive interaction with the environment, making the learning process more sample-efficient.
Uncertainty-aware transition models: Neural networks can be used to model uncertainty in transitions, incorporating probabilistic methods (e.g., Bayesian neural networks or variational inference) to handle noisy or uncertain environments. This allows agents to reason about multiple possible outcomes when planning actions.

By integrating neural networks into these components of RL—state representation, action space, reward modeling, and transition dynamics—agents can better adapt to complex, real-world environments, overcoming some of the rigid assumptions embedded in traditional RL frameworks.

Issues with Using Reinforcement Learning: Undefined States, Actions, and Rewards

As we explored earlier, the Bellman equation and traditional reinforcement learning (RL) frameworks rely on well-defined states, actions, rewards, and transitions. These assumptions are convenient for simplifying the learning process, but in real-world applications, they often fail. When attempting to apply RL to complex systems, we face the same issues seen with the Bellman equation: none of the core components—states, actions, or rewards—are clearly defined, making it difficult for an agent to learn effectively. In this section, we dive deeper into these issues and why using RL in real-world applications can be problematic without addressing these fundamental ambiguities.

1. Undefined States

In RL, a state ( S ) is assumed to contain all the relevant information about the current situation in an environment. The RL agent bases its decisions on this state, assuming that it perfectly captures the environment's status. However, in most real-world systems, the state is not well-defined and is often hard to model. Some key challenges include:

Complexity and dimensionality: In systems like autonomous driving, financial markets, or healthcare, the “state” might involve complex, high-dimensional sensor inputs, historical data, environmental factors, and even external variables that can’t easily be observed or quantified. For instance, in healthcare, the “state” of a patient might involve subtle physiological signals, medical history, environmental exposure, and socio-economic conditions. How can one encapsulate such a vast amount of data in a singular state representation?
Non-Markovian nature of states: The assumption of the Markov property, which states that the future depends only on the present state, breaks down in real-world applications. Often, the future depends on past states or events that are not captured in the immediate state. For instance, in long-term decision-making problems like investment management, the current state doesn’t fully represent all the relevant information needed to make optimal decisions because it might depend on trends, past behaviors, or latent variables that aren’t observed directly.
State identification as an optimization problem: Instead of assuming a well-defined state, identifying the correct state representation becomes an optimization problem in itself. RL agents often rely on manually engineered features or data preprocessing techniques to create a useful state, but this is challenging, domain-specific, and prone to error.

2. Ill-Defined Actions

In standard RL, the agent is given a set of possible actions ( A ) that it can take in any given state. These actions are usually discrete or continuous variables, and the agent’s goal is to choose the best action in each state to maximize future rewards. However, in real-world applications, the notion of what constitutes an action can be unclear:

Ambiguity in actions: In many environments, the available actions are not easily discernible. Consider human social interactions—actions could include subtle gestures, facial expressions, or changes in tone of voice. Identifying which actions are relevant and important is non-trivial, especially when the effects of certain actions are ambiguous or delayed.
Continuous and hierarchical actions: In tasks like robotics, actions are often continuous rather than discrete (e.g., controlling the torque of a motor or adjusting the angle of a robotic arm). Additionally, actions can be hierarchical—low-level motor actions combine to form higher-level actions like walking or grasping. RL models typically require the action space to be predefined, but this doesn’t capture the complexity of real-world systems where actions could be learned or abstracted at multiple levels of granularity.
Action discovery problem: In many real-world systems, new actions emerge as the agent learns. Actions may not be fully observable at the outset, meaning the RL agent has to discover latent actions or construct abstract strategies over time. Predefining a finite action set from the start can limit the agent’s ability to learn complex behaviors or adapt to dynamic environments.

3. Reward Function Design

In RL, the reward function is the agent’s guiding metric—it receives rewards based on how well it performs a task, and its objective is to maximize cumulative rewards. However, in real-world systems, rewards are often ill-defined or even unavailable. Some challenges include:

Ambiguous or internal rewards: Unlike games or toy simulations, where rewards are explicitly defined (e.g., gaining points or winning), real-world environments often involve internal or implicit rewards. For example, in healthcare, a reward might represent the improvement of a patient’s well-being, which is difficult to quantify. In creative tasks, rewards may relate to satisfaction, aesthetics, or innovation—factors that are highly subjective and cannot be directly measured.
Sparse or delayed rewards: In real-world tasks like scientific discovery or long-term business planning, rewards might only be realized after a long period of time. This delay between action and reward makes it difficult for the agent to learn because it cannot immediately associate its actions with the outcomes.
Reward hacking: In some cases, defining a reward function leads to undesirable behavior where the agent learns to exploit the reward system rather than solving the intended problem. For example, a robot designed to clean a room might learn to push dust under a carpet to receive the cleaning reward without actually improving cleanliness.
Reward definition as an optimization: Designing the reward function is often a manual process, requiring significant domain knowledge. In many applications, defining the reward function becomes an optimization problem, where designers must iteratively refine it based on observed behavior. This trial-and-error approach can lead to unstable or brittle agents, as small changes to the reward function might cause large changes in behavior.

4. Transition Probabilities and Dynamics

In standard RL, the transition probabilities ( P(s'|s, a) ) define how the environment transitions from one state to another after an action is taken. These probabilities are critical for value function estimation and policy optimization. However, in real-world environments, transitions are rarely known or easy to model:

Unknown or complex dynamics: Many real-world systems involve complex, unknown dynamics that are difficult to model or estimate. For instance, in natural ecosystems or biological systems, the interactions between different entities (e.g., species, cells) create highly complex, non-linear dynamics. Estimating transition probabilities in such environments is a massive challenge, often requiring substantial empirical data or sophisticated modeling techniques that are prone to errors.
Stochastic environments: In many real-world scenarios, transitions are stochastic—small changes in actions can lead to dramatically different outcomes. Consider stock market investments, where seemingly minor decisions (e.g., trading at a specific time) can cause significant downstream effects. In these environments, transition probabilities are highly variable, making it difficult for an RL agent to form reliable policies based on prior experiences.
Inability to generalize: Even with extensive data, learned transition probabilities in one environment might not generalize to others. For example, in autonomous driving, an RL agent trained on transition dynamics in a specific city may struggle when deployed in another location due to different road conditions, traffic patterns, or cultural norms.

5. Challenges in Value Function Estimation

In RL, agents rely on value functions (such as Q-values) to estimate the expected cumulative rewards from a given state or state-action pair. However, this requires accurate knowledge of states, actions, rewards, and transitions—none of which are well-defined in real-world applications.

Approximation errors: Even if an agent can estimate value functions, these approximations often break down in real-world environments due to the complexity and uncertainty of transitions and rewards. The value function relies on knowing how the environment will behave over time, which is often not possible in dynamic or unpredictable systems.
Difficulty in long-term reasoning: Value functions typically assume a well-behaved, recursive structure where future rewards can be discounted appropriately. In chaotic or sensitive systems, however, small changes in action or state can have far-reaching consequences, making it hard to predict the long-term effects of an action. As a result, RL agents can struggle to perform long-term reasoning, especially when rewards are sparse or delayed.

6. RL and the Bellman Equation Assumptions

All of these issues—undefined states, ill-defined actions, ambiguous rewards, and uncertain transitions—point to the fact that standard RL and the Bellman equation make unrealistic assumptions about how an agent interacts with the environment. In reality:

States are not perfect representations.
Actions are not always predefined and can evolve over time.
Rewards are not simple, immediate, or well-defined in many domains.
Transition probabilities are often complex or unknown.

When the core assumptions of the Bellman equation break down, it becomes clear that RL as traditionally formulated cannot provide a robust solution in real-world scenarios without major modifications or adaptive mechanisms.

Beyond RL

While neural networks offer the potential to model components of RL more flexibly, the fundamental assumptions behind RL still limit its applicability in real-world, dynamic environments. To address these limitations, alternative approaches, such as Neural Architecture Search (NAS) and evolutionary algorithms, can help push the boundaries beyond traditional RL.

1. Neural Architecture Search (NAS)

Neural Architecture Search (NAS) automates the process of finding the best neural network architecture for a given task. Instead of assuming a predefined architecture, NAS dynamically explores a search space of possible architectures, optimizing for performance on a given task.

NAS can be applied to RL in several ways:

Architecture discovery for RL: RL agents typically rely on predefined architectures (e.g., DQNs, actor-critic networks), but NAS can be used to discover better architectures for policy networks or value functions. This could result in architectures that are more adaptive to the complexities of the environment, learning more efficiently than manually designed networks.
Task-specific optimization: NAS can be used to tailor architectures for specific tasks, optimizing both the neural network structure and the action-selection strategy simultaneously. This is especially useful in environments where the complexity of the task varies over time, requiring more dynamic, flexible architectures.
Progressive NAS: NAS can be integrated into a progressive learning framework, where simpler architectures are used initially, and more complex architectures are explored as the agent encounters more difficult tasks. This allows for more efficient learning over time, avoiding overfitting to early-stage environments.

2. Evolutionary Algorithms and Genetic Programming

Another approach beyond RL is inspired by natural selection—using evolutionary algorithms or genetic programming to optimize neural network architectures, policies, or even the RL components themselves.

Evolving architectures and strategies: Evolutionary algorithms can be used to evolve neural network architectures, starting with simple networks and allowing the fittest architectures (those that perform best on tasks) to propagate over generations. This process is inherently trial-and-error, similar to RL but without the assumption of predefined states or actions.
Co-evolution of policies and environments: Evolutionary algorithms allow for the co-evolution of policies (how the agent behaves) and environments (the tasks the agent faces). This creates a dynamic learning environment where agents evolve to solve progressively harder problems, and the environment evolves to challenge the agent.
Self-modifying agents: Inspired by evolutionary biology, self-modifying agents can be designed to evolve not just their behavior but their entire architecture. These agents would learn how to modify their own neural network structures based on feedback from the environment, allowing for continuous adaptation and self-improvement.

3. Meta-Learning

Meta-learning, or "learning to learn," offers a powerful framework for training agents that can adapt to new tasks more quickly. Instead of training an agent from scratch for each task, meta-learning focuses on optimizing the agent’s learning algorithm itself.

Few-shot adaptation: Meta-learning allows agents to adapt to new environments using few-shot learning, where they learn new tasks with very few examples. This is crucial in real-world environments where data is often sparse or tasks change frequently.
Model-agnostic meta-learning (MAML): Approaches like MAML train agents to learn efficiently across a distribution of tasks, equipping them with the ability to quickly adapt to new environments without extensive retraining.

4. Self-Supervised Learning for Policy Discovery

Self-supervised learning allows agents to train without explicit rewards or task-specific labels, learning by predicting future states or self-generating tasks.

Policy pre-training: Self-supervised learning can be used to **

pre-train policies** for RL agents by learning the general structure of the environment and action space. Agents can explore and learn about the environment independently before formal task-specific training begins, reducing the reliance on external rewards.

Task-agnostic exploration: Agents can use self-supervised methods to explore the environment and generate intrinsic rewards, allowing them to discover latent patterns and actions without predefined goals.

Beyond RL conclusions

While traditional RL frameworks, centered around the Bellman equation, rely on rigid assumptions about states, actions, rewards, and transitions, neural networks provide a more flexible means of modeling these components in complex, real-world environments. However, to truly transcend the limitations of RL, we can look toward Neural Architecture Search (NAS), evolutionary algorithms, and meta-learning as ways to automatically discover, adapt, and optimize policies, architectures, and strategies. These approaches unlock the potential for agents to evolve autonomously, tackle more dynamic and unpredictable tasks, and operate beyond the limitations of predefined states and rewards.

Synthetic Data Generation for Reasoning Tasks

In the context of enhancing reasoning abilities in large language models (LLMs) and overcoming the limitations of reinforcement learning frameworks like those centered around the Bellman equation, synthetic data generation plays a critical role. While real-world data is often scarce or incomplete for tasks requiring structured, step-by-step reasoning, synthetic data allows us to generate trial-and-error-based reasoning chains that can help fine-tune models and improve their reasoning capabilities.

Building on concepts from direct preference optimization (DPO), trial-and-error learning, and self-guided progression, synthetic data generation provides the framework for iterating over reasoning tasks. In this section, we will explore how synthetic data generation can be structured to create meaningful reasoning chains that gradually improve the model’s reasoning process without relying on the artificial constructs of traditional RL.

1. Purpose of Synthetic Data Generation

In complex reasoning tasks, like solving math problems or understanding the causal dynamics of physical systems, the absence of explicit reasoning traces in the training data presents a major bottleneck. While ground truth answers may be known (e.g., the final solution to a math problem), the intermediate reasoning steps are often unavailable. This is where synthetic data generation becomes invaluable.

The purpose of generating synthetic data is twofold:

Generate reasoning chains that simulate how an expert might approach the problem in a step-by-step manner.
Provide training examples for the model to fine-tune its ability to reason, by iterating over multiple possible chains and progressively refining the model’s reasoning logic based on both the correct final output and the efficiency of reasoning steps.

2. Structure of Generated Reasoning Chains

To effectively generate synthetic data, we define a structure where reasoning chains are demarcated by tokens such as <cot> (chain-of-thought), allowing the model to sequentially generate and optimize its thought process. Here's how the reasoning chain structure can be organized:

Task Prompt: The model is presented with a reasoning task, such as solving a math equation or answering a physics problem.
Reasoning Chains: The model generates a series of reasoning steps, each separated by a <cot> token. These steps simulate how a human might decompose a complex problem into smaller parts.

Example:
```
<cot> Identify key variables <cot> Apply Newton's second law <cot> Rearrange equation to solve for mass <cot> Substitute known values <cot> Compute the result
```
Constrained Output: After generating reasoning chains, the model produces a final constrained output between <begin_constrained_output> and <end_constrained_output> tokens. This is the final answer that should match the known ground truth.

Example:
```
<begin_constrained_output> m = 5 kg <end_constrained_output>
```
General Response: Additionally, the model generates a general explanatory response between <begin_response> and <end_response> tokens, which provides a high-level explanation of the solution.

Example:
```
<begin_response> The mass of the object was determined by applying Newton's second law, solving for mass, and substituting the known values. <end_response>
```

By structuring the synthetic data in this manner, the model learns to generate both detailed reasoning steps and final outputs that match ground truth solutions.

3. Trial-and-Error Generation of Reasoning Chains

The key aspect of synthetic data generation is the trial-and-error process, which is akin to deep reinforcement learning (RL) in some respects but differs in key ways. Instead of generating action sequences within an environment, the model generates reasoning sequences within a structured framework. Here’s how this process works:

Multiple Reasoning Paths: For each task, the model generates multiple possible reasoning chains, exploring different ways to approach the problem. Even when the model is not able to solve the problem initially (e.g., in zero-shot settings), it still produces reasoning chains that can later be evaluated and refined.
Error Correction via Feedback: The generated reasoning chains are then compared to known ground truth or evaluated for internal logical consistency. Chains that lead to the correct final output, while being efficient (fewer steps), are positively reinforced. Chains that are incorrect or overly complex are penalized. This is akin to the credit assignment problem in RL but is handled more explicitly here through direct preference optimization (DPO).
Progressive Refinement: As the model generates more reasoning chains for similar tasks, it learns to prune inefficient steps and optimize its reasoning process, gradually converging to the most logical and concise chains.

4. Using Ground Truth to Guide Learning

Unlike in RL, where rewards are often delayed or sparse, synthetic data generation directly benefits from the availability of ground truth solutions. Even though ground truth may only represent the final answer, it can still guide the trial-and-error generation of intermediate reasoning chains.

Constrained Outputs with Ground Truth Matching: By using <begin_constrained_output> and <end_constrained_output> tokens to contain the final output, the model is explicitly encouraged to generate reasoning chains that eventually match the ground truth. This creates a more goal-oriented learning loop where reasoning chains are optimized not just for logical consistency but also for reaching the correct final output.
Intermediate Feedback: The model is also guided by the structure of the reasoning chains. During training, if any of the intermediate steps in the chain are identified as incorrect, the model can be penalized early, reducing the chance of carrying forward incorrect steps into subsequent reasoning chains.

5. Differences from Deep Reinforcement Learning

While synthetic data generation and deep RL share some similarities—both involve trial-and-error learning and iterative improvement—there are crucial differences:

Explicit Feedback vs. Rewards: In synthetic data generation, the feedback is direct (matching against ground truth), whereas in deep RL, feedback comes in the form of rewards that might be delayed or sparse. The RL agent must infer which actions contributed to success, while synthetic data generation provides step-by-step guidance through reasoning chains.
Action vs. Reasoning: In deep RL, the agent explores an action space to navigate an environment. In synthetic data generation, the model explores a reasoning space, generating chains of thought that simulate logical progression rather than physical actions.
Efficiency vs. Exploration: While RL must balance exploration and exploitation, synthetic data generation places a greater emphasis on efficiency, encouraging the model to minimize unnecessary reasoning steps while still producing correct answers. This is integrated into the loss function through mechanisms like DPO, which biases the model toward shorter, correct reasoning chains.

6. Automating the Selection of Reasoning Tasks

An important component of synthetic data generation is automating the process of selecting progressively harder problems. As the model improves, it should be presented with more challenging tasks that push its reasoning abilities further. This can be implemented as an optimization problem:

Dynamic Task Difficulty: The system can dynamically adjust the difficulty of tasks by analyzing the model’s performance. For instance, if the model consistently generates correct and efficient reasoning chains, the system can automatically increase the complexity of the next set of tasks.
Self-guided Progression: As we discussed, this process can be self-guided by using metrics like reasoning chain length, logical consistency, and correctness to determine when the model is ready for more difficult problems. This is similar to curriculum learning but integrated within the trial-and-error process.

7. Scaling Synthetic Data Generation

To scale this process across large datasets and diverse domains (e.g., math, physics, ciphers), we need to implement a robust task sampling mechanism that ensures the model is exposed to a variety of reasoning challenges. This can be done by:

Generating diverse task types: Ensuring that the model encounters reasoning tasks from different domains, each requiring different types of logic and problem-solving strategies.
Balancing difficulty levels: Dynamically adjusting the sampling strategy to expose the model to a balance of easy, intermediate, and hard tasks, ensuring continuous learning without overloading the model with overly difficult problems too early.

8. Benefits of Synthetic Data Generation for Reasoning

Synthetic data generation provides a powerful tool for training LLMs to develop better reasoning capabilities through structured, trial-and-error learning. By generating reasoning chains with predefined token structures like <cot>, and directly integrating ground truth into the training process with constrained outputs, we can create an efficient learning pipeline that refines the model’s reasoning abilities. Unlike deep RL, which relies on rewards and exploratory action sequences, synthetic data generation offers more structured, goal-oriented guidance, making it ideal for complex reasoning tasks that require both accuracy and efficiency.

Automated Generation of Reasoning Chains and Progressive Learning

One of the key challenges in training large language models (LLMs) to improve their reasoning abilities lies in generating reasoning chains that the model can iteratively refine through trial-and-error, and doing so in a way that maintains the difficulty of tasks just at the edge of the model's current capabilities. By automating the generation of reasoning chains, measuring task difficulty, and progressively presenting harder tasks, we can create a self-guided learning framework that efficiently trains the model to improve reasoning without relying on manually curated data.

In this section, we’ll cover how to integrate reasoning chain generation, task difficulty measurement, progressive learning, and the required loss function into a unified process.

1. Generating Reasoning Chains

The first step is to have the LLM generate reasoning chains. Each reasoning chain represents a sequence of intermediate steps that lead from the problem statement to the final solution. These chains are annotated with special tokens to separate individual steps and mark the final, constrained output.

Reasoning Chain Structure: For each task, the model generates multiple reasoning steps demarcated by <cot> (chain-of-thought) tokens. Each <cot> token indicates the beginning of a new reasoning step.

Example:
```
<cot> Break the equation into smaller parts <cot> Solve for x <cot> Substitute known values <cot> Simplify the expression <cot> Compute the final value
```
Constrained Output: The final solution is marked with <begin_constrained_output> and <end_constrained_output> tokens, which help define the exact answer that should match the ground truth.

Example:
```
<begin_constrained_output> x = 5 <end_constrained_output>
```
General Response: Additionally, the model generates a general description of the solution between <begin_response> and <end_response> tokens. This is a conversational explanation aimed at clarity and user understanding, fine-tuned through normal supervised learning and Direct Preference Optimization (DPO).

Example:
```
<begin_response> The value of x was calculated by solving the equation step by step and substituting the known values. <end_response>
```

The reasoning chain provides a clear, step-by-step thought process that the model can refine and optimize, while the constrained output ensures that the final solution is correct and matches the ground truth.

2. Measuring Task Difficulty

To effectively guide the model’s learning process, we need to estimate the difficulty of each task and dynamically adjust the difficulty based on the model's current performance. The difficulty score reflects how well the model can handle reasoning tasks at various levels of complexity.

Performance-based Difficulty: One way to measure difficulty is based on the model's ability to generate correct reasoning chains. For each task, difficulty can be scored by analyzing:
- The number of steps in the reasoning chain.
- The proportion of intermediate steps that match ground truth reasoning.
- The correctness of the final constrained output.
Dynamic Difficulty Adjustment: Tasks that the model consistently solves correctly with minimal reasoning steps are considered easier and receive a lower difficulty score. Conversely, tasks where the model struggles—by either producing incorrect chains or requiring more steps—are considered harder and receive a higher difficulty score.

This difficulty score can be dynamically adjusted based on the model’s performance, allowing the system to continually estimate the edge of the model’s capabilities.

3. Progressive Learning from Edge-Case Tasks

To maximize learning efficiency, the model should focus on tasks that are just beyond its current abilities—tasks that are difficult but still solvable with effort. These are referred to as edge-case tasks.

Task Selection Mechanism: Based on the measured difficulty score, the system selects tasks that are at the boundary of what the model can solve. These tasks should be hard enough to push the model's reasoning abilities but not so difficult that they lead to constant failure. This aligns with the principle of progressive learning or curriculum learning, where the model is exposed to increasingly complex tasks as it improves.
Feedback Loop for Task Progression: As the model improves, the difficulty score of tasks it can handle is recalculated, and the task selection mechanism progressively introduces more complex problems. This feedback loop ensures that the model is continuously challenged without being overwhelmed by problems far beyond its current reasoning abilities.

4. Unified Loss Function for Reasoning Chains and Progression

To optimize this process, we need a unified loss function that takes into account:

The correctness and efficiency of reasoning chains.
The accuracy of the final constrained output.
The progressive learning from tasks at the edge of the model’s abilities.

(i) Reasoning Chain Loss ($\mathcal{L}_{\text{reason}}$)

This component of the loss encourages the model to generate reasoning chains that match the ground truth and take fewer steps.

$$\mathcal{L}_{\text{reason}} = - \sum_{i=1}^{n} \mathbb{I}(r_i = \text{ground truth step}) \cdot f_{\text{eff}}(r_i)$$

$r_i$ represents each reasoning step in the chain.
$f_{\text{eff}}(r_i)$ penalizes unnecessary steps, encouraging the model to find more efficient paths to the solution.
The indicator function $\mathbb{I}(r_i = \text{ground truth step})$ is 1 if the reasoning step matches the ground truth and 0 otherwise.

(ii) Constrained Output Loss ($\mathcal{L}_{\text{constrained}}$)

This term ensures that the final output generated between <begin_constrained_output> and <end_constrained_output> matches the correct ground truth answer.

$$\mathcal{L}_{\text{constrained}} = - \mathbb{I}(o = \text{ground truth output})$$

$o$ represents the model’s final constrained output.

This term forces the model to produce an output that aligns with the known ground truth for the task.

(iii) General Response Loss ($\mathcal{L}_{\text{response}}$)

The general response loss encourages the model to generate a coherent and user-friendly explanation of the solution, guided by traditional supervised fine-tuning and DPO principles.

$$\mathcal{L}_{\text{response}} = \text{Supervised loss (cross-entropy)} + \text{DPO bias}$$

This term helps the model provide clear explanations in a natural language format, which is important for human-AI interaction.

(iv) Difficulty Matching Loss ($\mathcal{L}_{\text{diff}}$)

This term ensures that the tasks presented to the model are appropriately challenging, pushing the model just beyond its current capabilities while avoiding overwhelming difficulty.

$$\mathcal{L}_{\text{diff}} = \left| D_{\text{model}} - D_{\text{task}} \right|$$

$D_{\text{model}}$: The model’s current difficulty level, derived from recent task performance.
$D_{\text{task}}$: The difficulty score of the task.

This term minimizes the gap between the model’s capabilities and the complexity of the tasks it is given.

(v) Final Loss Function ($\mathcal{L}_{\text{total}}$)

By combining all components, the unified loss function ensures that the model generates correct and efficient reasoning chains, matches ground truth outputs, generates clear responses, and continuously learns from tasks at the appropriate difficulty level:

$$\mathcal{L}_{\text{total}} = \alpha \cdot \mathcal{L}_{\text{reason}} + \beta \cdot \mathcal{L}_{\text{constrained}} + \gamma \cdot \mathcal{L}_{\text{response}} + \delta \cdot \mathcal{L}_{\text{diff}}$$

$\alpha$, $\beta$, $\gamma$, and $\delta$ are hyperparameters controlling the weight of each loss term.

5. Learning Flow

Task Presentation: The model is presented with a task and generates reasoning chains using <cot> tokens.
Reasoning Chain Generation: The model attempts to solve the problem step-by-step, generating multiple reasoning chains for evaluation.
Evaluation and Feedback: The generated reasoning chains and the final output are evaluated against ground truth. The loss function is computed, and the model’s parameters are updated.
Task Difficulty Adjustment: Based on performance, the task selection mechanism adjusts the difficulty of the next task to match the model’s improving capabilities.
Iterative Learning: This process repeats, with the model progressively learning from harder tasks and improving its reasoning abilities.

By integrating the generation of reasoning chains, dynamic task difficulty measurement, and progressive learning into a single process, we can train LLMs to improve their reasoning abilities in a structured and scalable way. The unified loss function ensures that the model continuously refines its reasoning, output correctness, and general response quality, all while being challenged by tasks at the edge of its capabilities. This approach pushes beyond the limitations of traditional RL and creates a more adaptive and efficient learning pipeline for complex reasoning tasks.

Summary and Conclusion

In this post, we explored the limitations of the traditional Bellman equation and reinforcement learning (RL) frameworks when applied to real-world systems, where assumptions about states, actions, rewards, and transitions often fail. We delved into the inherent challenges of RL, such as the difficulty of defining states, the ambiguity of rewards, the failure of the discount factor to capture long-term dynamics, and the impracticality of predefined action spaces. These limitations necessitate a shift in how we design and optimize intelligent systems, especially in environments that are dynamic, complex, or require more sophisticated forms of reasoning.

To address these issues, we outlined several key strategies for moving beyond the limitations of traditional RL:

Neural Networks as Models for RL Components:
- State Representation: Neural networks can dynamically learn state representations through autoencoders or recurrent models, ensuring that the agent captures the relevant information in complex environments.
- Action Space Optimization: Instead of relying on predefined actions, neural networks can learn hierarchical or continuous actions, helping agents explore and optimize within more flexible action spaces.
- Reward Function Learning: Neural networks can model intrinsic rewards or infer reward functions through inverse reinforcement learning, allowing agents to navigate environments where rewards are implicit or delayed.
- Transition Modeling: Neural networks can be used to learn transition dynamics, especially when environment models are unknown or uncertain, allowing for model-based RL approaches.
Beyond RL: Alternative Approaches:
- Neural Architecture Search (NAS): NAS allows us to automate the process of discovering optimal neural network architectures for RL agents, optimizing their structure and behavior based on task-specific needs. This provides greater flexibility and performance than predefined architectures.
- Evolutionary Algorithms: Inspired by natural selection, evolutionary algorithms allow for the co-evolution of policies and architectures, enabling agents to evolve autonomously. These algorithms can explore diverse strategies and environments, producing more adaptive and resilient agents.
- Meta-Learning: Meta-learning equips agents with the ability to quickly adapt to new environments and tasks by optimizing how they learn. This allows for few-shot adaptation and rapid policy learning, reducing the need for extensive retraining in changing environments.
- Self-Supervised Learning: Agents can use self-supervised learning to pre-train on tasks without explicit rewards, allowing them to explore and learn patterns in their environment before formal training begins.
Automated Reasoning Chain Generation and Progressive Learning:
- Reasoning Chain Generation: We proposed a structured system where the model generates reasoning chains demarcated by <cot> tokens, simulating a step-by-step logical thought process. Each chain culminates in a final output wrapped in <begin_constrained_output> and <end_constrained_output> tokens, ensuring that the solution aligns with ground truth.
- Task Difficulty Measurement: To drive progressive learning, we implemented a system that dynamically estimates task difficulty based on the model's performance. Tasks are selected at the edge of the model's capabilities, ensuring continuous improvement without overwhelming the model with overly complex problems.
- Progressive Learning and Task Selection: The model progressively learns by solving tasks at the boundary of its current abilities. As the model improves, the difficulty of tasks increases automatically, creating a self-guided curriculum learning system.
- Unified Loss Function: We introduced a unified loss function that integrates reasoning chain correctness, final output accuracy, general response generation, and task difficulty matching. This loss ensures the model refines its reasoning while being continuously challenged.

Conclusion

Traditional reinforcement learning frameworks, rooted in the Bellman equation, face significant limitations when applied to real-world environments where states, rewards, and actions are not clearly defined, transitions are uncertain, and complexity abounds. To overcome these limitations, we’ve explored alternative approaches that focus on dynamic learning, evolution, and self-guided progression.

By leveraging neural networks to model RL components, Neural Architecture Search to optimize architectures, evolutionary algorithms for adaptive policy evolution, and meta-learning for rapid task adaptation, we can create systems that go beyond the constraints of RL and build truly autonomous, self-improving agents.

Additionally, the process of automated reasoning chain generation offers a powerful approach for LLMs to iteratively refine their reasoning skills. By measuring task difficulty, progressively presenting harder problems, and optimizing reasoning chains through structured loss functions, we can develop models that are capable of improving their logical thinking and problem-solving abilities over time.

Ultimately, this integrated framework moves us closer to the goal of creating general-purpose agents that can operate in complex, uncertain environments without the need for rigid, predefined assumptions about the world. These agents will not only reason effectively but also evolve and adapt, paving the way for more intelligent, adaptive systems across a variety of domains.

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