Assignments for Berkeley CS 285: Deep Reinforcement Learning, Decision Making, and Control.

HW1

The analysis part is here: analysis

The code part is here: code_report

The extra credit is still working...

What I learn in HW1

In machine learning, computing the bound of the cost function is important for several reasons:

1. Convergence Analysis : It helps in understanding whether an algorithm will converge to a solution and how quickly. If the bounds of a cost function are known, it's easier to determine if the optimization algorithm (like gradient descent) is making progress towards minimizing the cost.
2. Algorithm Stability : Bounding the cost function ensures the stability of the learning algorithm. It prevents the algorithm from producing extreme values which might indicate a failure to learn or overfitting.
3. Performance Guarantee : Bounds provide a theoretical guarantee on the performance of the learning algorithm. Knowing the upper and lower limits of the cost function helps in setting expectations and understanding the potential effectiveness of the model.
4. Regularization and Generalization : In many cases, bounding the cost function involves regularization terms which help in preventing overfitting and ensuring that the model generalizes well to new, unseen data.
5. Numerical Stability : Bounding the cost function can also be important for numerical stability, preventing issues like underflow or overflow in computations, which are crucial in practical implementations of machine learning algorithms.
6. Comparative Analysis : When different models or algorithms are being compared, knowing the bounds of their cost functions can provide a fair basis for comparison, especially in terms of their potential efficiency and effectiveness.
7. Hyperparameter Tuning : Understanding the bounds can assist in better hyperparameter tuning, as it gives insight into how changes in parameters like learning rate, number of layers, etc., might impact the learning process.

In summary, bounding the cost function in machine learning is crucial for ensuring the effectiveness, stability, and reliability of the learning algorithms. It provides a theoretical foundation that guides practical implementation and application.

In reinforcement learning (RL), the state marginal distribution refers to the distribution over the state space that an agent encounters while following a particular policy. It describes the likelihood of being in each state after many time steps, assuming the agent starts in some initial distribution and follows a specific policy thereafter.

Formally, if �π is the policy that the agent is following and μ(s) is the state marginal distribution under this policy, then �(�)μ(s) provides the probability (or probability density in continuous spaces) of being in state �s while following policy �π over the long run. It is influenced by:

• The initial state distribution , p(s0), which describes the probability of starting in each possible state.
• The transition dynamics , p(st**+1****∣st,at), which describe the probability of transitioning to a new state st+1** given the current state st and action at.
• The policy itself, π(a∣s), which defines the probability of taking action a in state s.

The state marginal distribution is important in RL for several reasons:

1. Policy Evaluation : To evaluate a policy, one often needs to integrate over all possible states weighted by how frequently they are encountered, which is given by the state marginal distribution.
2. Policy Improvement : When improving a policy, it's important to focus on states that are more frequently visited, as changes to the policy in these states will have a larger impact on the overall performance.
3. Algorithm Convergence : Some RL algorithms, such as policy gradient methods, have theoretical guarantees that rely on the assumption that the state marginal distribution will cover the state space adequately.
4. Exploration vs. Exploitation : The state marginal distribution can help determine whether an agent is exploring the state space sufficiently or is overexploiting certain regions.

Mathematically, the state marginal distribution can be difficult to compute, especially in continuous or very large state spaces, and it often requires simulation or sampling methods to estimate. In infinite-horizon problems, this distribution may converge to a stationary distribution, assuming the Markov Decision Process (MDP) is ergodic. This stationary distribution is sometimes referred to as the stationary distribution of the Markov chain induced by the policy.