Reinforcement Learning (RL) presents a distinct set of optimization challenges compared to the supervised and unsupervised learning paradigms discussed previously. While the goal remains optimization finding parameters that maximize or minimize an objective in RL, this objective is typically the expected cumulative reward obtained by an agent interacting with an environment. The optimization process is complicated by factors like delayed rewards, the need for exploration, and the fact that the data distribution changes as the learning agent's policy improves.

Two primary families of RL algorithms, policy gradient methods and value-based methods (like Q-learning), rely heavily on optimization techniques, often adapting methods covered earlier in this course.

Optimizing Policies Directly: Policy Gradient Methods

Policy gradient methods aim to directly optimize the parameters, denoted by $\theta$ , of the agent's policy $\pi_\theta(a|s)$ . The policy defines the probability of taking action $a$ in state $s$ . The objective is typically to maximize the expected total discounted reward, often written as $J(\theta)$ .

The core challenge lies in estimating the gradient $\nabla_\theta J(\theta)$ because the objective function depends on the interaction trajectories generated by the policy itself within the environment. The Policy Gradient Theorem provides a way forward, offering an expression for this gradient. A common form is:

$\nabla_\theta J(\theta) = \mathbb{E}_{\tau \sim \pi_\theta} \left[ \sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t | s_t) G_t \right]$

Here, $\tau$ represents a trajectory $(s_0, a_0, r_0, s_1, a_1, r_1, ...)$ , $\pi_\theta(a_t | s_t)$ is the probability of taking action $a_t$ in state $s_t$ under policy $\theta$ , and $G_t = \sum_{k=t}^{T} \gamma^{k-t} r_k$ is the total discounted return starting from time step $t$ . The expectation $\mathbb{E}_{\tau \sim \pi_\theta}$ is taken over trajectories generated by following policy $\pi_\theta$ .

This formulation allows us to estimate the gradient using samples from the agent's interactions. The simplest algorithm, REINFORCE, computes Monte Carlo estimates of this gradient and updates the policy parameters using gradient ascent:

$\theta \leftarrow \theta + \alpha \nabla_\theta \log \pi_\theta(a_t | s_t) G_t$

where $\alpha$ is the learning rate.

Optimization Considerations:

High Variance: The gradient estimates obtained via Monte Carlo sampling of $G_t$ often suffer from high variance, leading to slow and unstable learning. This is a major hurdle in policy gradient methods.
Variance Reduction: A standard technique to mitigate high variance is to introduce a baseline $b(s_t)$ that depends only on the state. The gradient estimate becomes: $\nabla_\theta J(\theta) \approx \nabla_\theta \log \pi_\theta(a_t | s_t) (G_t - b(s_t))$ A common choice for the baseline is the value function $V(s_t)$ , which estimates the expected return from state $s_t$ . The term $(G_t - V(s_t))$ is an estimate of the advantage $A(s_t, a_t)$ , leading to Actor-Critic methods where a "critic" learns the value function (or advantage function) to improve the gradient estimate used by the "actor" (the policy).
Choice of Optimizer: Because gradient estimates are noisy, standard Stochastic Gradient Ascent (or Descent if minimizing a negative objective) is often used. Adaptive learning rate methods like Adam or RMSprop (Chapter 3) are particularly popular in RL as they can help manage the noisy gradients and varying parameter sensitivities.
Stability and Step Size: Taking steps that are too large can drastically change the policy, potentially leading to performance collapse. This motivates methods like Trust Region Policy Optimization (TRPO) and Proximal Policy Optimization (PPO). These algorithms constrain the policy update step, ensuring that the new policy doesn't deviate too much from the old one in terms of behavior (often measured by KL divergence). This connects to the ideas of trust regions (Chapter 2) and constrained optimization.

Optimizing Value Functions: Q-Learning

Value-based methods, particularly Q-learning and its deep learning variant Deep Q-Networks (DQN), take a different approach. They aim to learn the optimal action-value function, $Q^*(s, a)$ , which represents the maximum expected return achievable starting from state $s$ , taking action $a$ , and following the optimal policy thereafter.

In DQN, a neural network with parameters $\phi$ is used to approximate $Q^*(s, a)$ , denoted as $Q(s, a; \phi)$ . The optimization problem becomes finding the parameters $\phi$ that minimize the difference between the current Q-value estimate and a target Q-value, typically derived from the Bellman equation. This is often formulated as minimizing the Mean Squared Bellman Error (MSBE):

$L(\phi) = \mathbb{E}_{(s,a,r,s') \sim \mathcal{D}} \left[ (y - Q(s, a; \phi))^2 \right]$

where the target $y = r + \gamma \max_{a'} Q(s', a'; \phi^{-})$ . Here, $\mathcal{D}$ is a replay buffer of past experiences $(s, a, r, s')$ , $\gamma$ is the discount factor, and $\phi^{-}$ represents the parameters of a separate target network, which are periodically updated with the main network's parameters $\phi$ .

Optimization Considerations:

Supervised Learning Analogy: Optimizing the Q-network resembles a supervised regression problem where the network tries to predict the target value $y$ . Standard gradient descent techniques are applicable.
Stability: Training deep Q-networks can be unstable. Two common techniques improve stability:
- Experience Replay: Storing experiences in a buffer $\mathcal{D}$ and sampling mini-batches from it breaks the temporal correlations in sequentially observed data, making samples more independent and identically distributed (i.i.d.), which benefits standard gradient descent methods.
- Target Network: Using a fixed target network $Q(s', a'; \phi^{-})$ provides a stable target $y$ during the gradient update for $Q(s, a; \phi)$ . Without it, the target would change with every update to $\phi$ , potentially leading to oscillations or divergence.
Choice of Optimizer: Similar to policy gradient methods, optimizers like RMSprop and Adam are frequently used for training DQNs due to their effectiveness in deep learning tasks. Learning rate schedules might also be employed.

Connecting RL Optimization to Broader Techniques

The optimization methods used in RL are often specific adaptations or applications of the algorithms discussed throughout this course:

Gradient Descent Variants: SGD, Adam, RMSprop form the backbone for updating both policy parameters and value function approximators.
Constrained Optimization/Trust Regions: Methods like TRPO and PPO explicitly incorporate constraints or trust regions to stabilize policy updates, drawing parallels with techniques discussed earlier in this chapter and in Chapter 2.
Hyperparameter Optimization: Finding good learning rates, discount factors, exploration parameters, network architectures, and optimizer settings is significant. Bayesian optimization (covered earlier in this chapter) is a powerful tool for tuning these hyperparameters efficiently.

In essence, while RL introduces unique objectives and challenges related to agent-environment interaction, the underlying mechanics of updating model parameters rely heavily on the advanced optimization techniques developed for broader machine learning applications. Understanding these optimization algorithms provides a solid foundation for tackling complex RL problems.