Summary of basic Reinforcement learning algorithms, terms & concepts.
Markov Decision Process (MDP)
Horizon: the number of time steps that we sample or simulate.
Discount factor: discount future rewards.
State-value function (total rewards from state s):
Action-value function (total rewards from taking action a in state s)
Optimal Value Function V*
Linear system with additive Gaussian noise.
Linear Gaussian controller
Linear Gaussian dynamics
Q-learning (SARSA Max)
Monte-Carlo Policy Gradient with a baseline:
Natural Policy Gradient
Backtracking line search:
A3C is the asynchronous version of A2C.
Adding parameter noise for exploration:
Monte Carlo Tree Search (MCTS)
In RL, we may want to estimates the value functions for a policy π but we only collect sample results from π’. Importance sampling allows us to evaluate a policy from another policy.
Importance sampling is a technique of estimating the expected value of f(x) where x has the data distribution p. However, Instead of sampling from p, we calculate the result from sampling q.
We sample data from q and estimate the expected value for p using:
For this to work q(xi) cannot be zero if the corresponding p(xi) is not zero. Let’s demonstrate it with an example and compute 𝔼[f(x)] with data distribution x1 and x2 separately.
Now, we re-estimate the expected value for x2 from sampling results using x1.
In RL, we sample results using the current policy π1 and we can use importance sampling to estimate the result if we change to policy π2.
Advantage function with importance sampling.
Advantage function is the discounted rewards minus a baseline. There are many possible ways to compute value functions (discounted rewards) including Monte Carlo rollout, temporal difference, or a combination of both. For completeness, the following is a k-step look ahead used in A3C to compute the advantage function. This is a k-step rollout. We observe the rewards and baseline it with the function value of the corresponding state which can be approximated by a deep network.
There are many possible baselines. Different baselines can reach the same optimal point as long as it is not depend on the policy model θ. So based on the choice of computing the value functions, you can select the corresponding baseline to reduce the variance of the calculation.
Importance sampling calculates the Advantage function A by sampling Â using the current policy θk. Then the result is recalibrated for the new policy θ using the probability ratio between these two policies.
Trust region method
To sample the kth episode, we use a ε-greedy algorithm below to pick which action to sample next. Here is the data distribution used in selecting the action:
DAgger: Dataset Aggregation
DAgger is an algorithm to argument the training dataset with expert demonstrations for states that may visited by the current learned policy. So even if the current policy is in-accurate or drifted, we can take corrective actions.
Nonlinear least-squares problem with Gauss-Newton method
Newton’s method on optimization:
According to the equation above, the search direction p (a.k.a. Δx), using Newton’s method, can be written as:
Compute the first and second order derivative of f:
In the Gauss-Newton method, we approximate the second derivative above without the Q term. The Q term is smaller than the first term and if the problem has zero residual, r = Q = 0. Gauss-Newton method applies this approximation to the Newton’s method.
The Gauss-Newton method search direction is:
Let’s have an example, we want to determine the growth rate of antelope. We develop a model g and fit data into the the model to compute the residual error r:
We use conjugate gradient method to optimize a quadratic equation or to solve a linear equation.
In a line search, we determine the direction of ascent and then select the step size. For example, in the gradient ascent, we follow the direction with the steepest gradient and take a step with size equal to the gradient times the learning rate. In the example below, the deepest direction, according to the gradient contour, is to move right. The next move may go up and a little bit to the left. Even we are getting closer to the optimal point in every move, we may ask why every move undo part of the previous move (for example, going a little bit left in the second move)
The conjugate gradient method is a line search method with careful choice of coordinated directions. It optimizes a quadratic equation in much fewer step than the gradient ascent. If x is N-dimensional (N parameters), we can find the optimal point in at most n steps. For every next move, we want a direction conjugate to all previous moves. Mathematically, it means any new direction dj must obey the following with any previous direction di:
where A is the matrix in the quadratic equation.
We can visualize this in our example as moving perpendicular to the previous ones.
This is the algorithm. We start with a random (or educated) guess Xo and use the equation below to calculate where should we go next X1. p is the direction to go (equivalent to d above)
Let’s define two terms:
- e is the error between the current guess and the optimal point.
- r measures how far we are from the correct value b. We can view that as the error e transformed by A to the space of b.
We can derive:
Say our next point is computed as:
In order for the future direction not to undo progress from the previous directions, let’s try d and e to be orthogonal:
α depends on e which we don’t know its value. So instead of orthogonal, let’s try all searching direction d to be A-orthogonal:
To fulling these conditions, we will move xi in the direction d to the optimal point in the search direction d.
Using the A-orthogonal requirement, α is computed as:
So conjugate gradient is about finding an A-orthogonal search direction every time and move to the optimal point in that search direction. These vectors are independent of each other and span a N-dimensional space. So we can solve in at most n steps.
In vector form:
where g is the Jacobian matrix and H is the Hessian matrix.
In deep learning, we want a model predicting data distribution Q resembling the distribution P from the data. Such difference between two probability distributions can be measured by KL Divergence which is defined as:
Matrix A is positive-definite if
for all real non-zero vector z.
Supervised learning v.s. unsupervised learning v.s. reinforcement learning
Supervised learning is learning from a training dataset provided with labels from knowledgable sources (e.g. the human). For example, we want to learn the object class (label) of an object inside a picture (a data sample). When the system is presented with a new sample, it should generalize what it learns to answer that question “what is the label?”.
Unsupervised learning explores the hidden structure of the data. For example, what is the latent features of the data sample or what category (cluster) the data belong to.
Reinforcement learning observers the environment and takes actions to maximize the rewards (goals). It mainly deals with exploration, exploitation, trial-and-error search, delayed rewards, system dynamics (how the environment reacts to the actions taken) and defining goals.
The system (like robots) that interacts and acts on the environment.
A policy defines how the learning agent may act from a specific state. Mathematically, it is the probability of taking an action a given the state s. π = p(a | s)
A reward signal (r(s, a, s’)) defines the goal in a reinforcement learning problem. For example, that is the score in a game or turning a door knob for a robotic arm. The total rewards is the return.
Value function (state-value function)
The value of a state is the total amount of rewards that an agent can collect from now to the end V(s).
The value of taking an action from a state is the total amount of rewards that an agent can collect from now to the end by taken a specific action Q(s, a).
Model of the environment
A model describe how an environment may change upon an action from a state p(s, a, s’).
An episode plays through the whole sequence (e.g. a game) until reaching the terminal state.
Discount rate γ
Discount rate values the future rewards at the present value.
Model v.s. model free
In model free RL, the system dynamic is unknown or not needed.
Monte Carlo Method
Monte Carlo methods play through a completed episode. It computes the average of the sample returns from multiple episodes to estimate value functions. The following creates a running average using the Monte Carlo method.
Monte Carlo control
Use Monte Carlo methods to evaluate the current policy. Using the estimated value functions, we improve the current policy.
On-policy learning v.s. off-policy learning
We use a policy to determine which actions to take. In on-policy learning, we optimize the current policy and use it to determine what actions to explore next. Since the current policy is not optimized or sub-optimized, it still allows some form of exploration. Off-policy learning allows a second policy, like the ε-greedy policy, to explicitly define the exploration it wants. Off-policy learning optimizes the current policy but use another policy to explore actions. Off-policy provides greater choice but slower to converge and have higher complexity. Off-policy learning mostly utilizes importance sampling in evaluating policy.
Temporal-Difference Learning (TD)
Instead of completing the whole episode like the Monte Carlo:
We rollout k-steps and collect the rewards. We estimate the value function based on the collected rewards and the value function after k-steps. Below is an 1-step TD learning. We find our the reward after taking one action and find out the value function for the new state.
We use the system dynamics (model) to generate simulated experience and use them to refit the value functions or the policy.
The difference between learning and planning is one from real experience generated by the environment and one from simulated experience by a model.
The following is the Dyna-Q algorithm which it uses the learned model to refit the Q-function values.
Optimize trajectory based on an open-loop system. Observe the initial (first) state & optimize the corresponding actions.
For a stochastic system, this is suboptimal because we do not readjust the actions based on the observed states where we transit to.
Optimize trajectory based on a closed-loop system which we take actions based on the observed states.