### Executive Summary

This is a long paper from 2013 that touches on many topics at the intersection of RL and robotics. I’ll list some of my most interesting take-aways here as someone familiar with the RL space and not-so-familiar with the robotics space:

• The physicality of a robotics system complicates learning:
• “Rollouts” are expensive and tedious to obtain
• Sensors can be unreliable or ambiguous
• Everything is continuous so figuring out how to design your representations and approximations can greatly impact the ability of the robot to learn
• You want to protect your hardware so naive exploration is not always possible
• Different robots (even if built to the same spec) will have different physical properties that can affect the policy (think slightly different torque responses in two otherwise identical motors)
• The environment itself is often not constant (think windy days vs calm days)
• You may try to get around these by modelling the system, but this modelling will almost certainly be unable to capture all the relevant dynamics. Thus you need some strategy to deal with model error.

• Crafting a good reward function that can enable learning is often non-obvious and critically important.

• The ability to incorporate prior knowledge is important in making the learning task tractable. This can range from basic models of physics to actual human demonstrations. RL algorithms that allow more localized search “around” expert demonstrations are very valuable in this domain.

• RL (as of 2013) is not easy to apply “out of the box” to robotics task. This is contrasted with supervised learning which can often be applied to a new dataset and quickly produce value.

### 1. Introduction

There is a useful synergy between the fields of Reinforcement Learning (RL) and robotics: RL techniques allow robots to autonomously learn complex-to-engineer behaviors while robotics problems can provide a useful testbed for RL algorithms.

One way to to classify machine learning problems is to look at their complexity of reward structure and their complexity of environmental interaction. RL is relatively high on both scales, and simpler problems (e.g. supervised learning) can often be rephrased to fit into the RL paradigm.

Robotics benchmarks generally have several characteristics that separate them from other interesting benchmarks in the RL domain:

• Problems are often best represented with high-dimensional continuous state and action spaces.
• It is often unrealistic to assume the true state is observable and noise free.
• Experience in physical systems is expensive to obtain.
• As a corollary to the above, reward shaping is important and can be difficult when experience is expensive to obtain.
• Most RL successes in robotics (as of 2013) has been demonstrated with model-based algorithms using policy-search methods (vs value function methods)

### 2. A Concise Introduction to Reinforcement Learning

A crash course in RL:

• An agent tries to maximize the accumulated reward over its lifetime. An environment can either be episodic with well-defined start, stop, and reset points or ongoing.

• An environment has a set of states $$S$$ and a set of actions $$A$$ available to an agent in every state. A state contains all relevant information (i.e. the process is memory-less).

• Transitions between states generate rewards.

• The goal of RL is to find a mapping from states to actions called a policy $$\pi$$ such that the policy maximizes reward. An optimal policy is referred to as $$\pi^ *$$. A policy can be deterministic or probabilistic.

• An RL agent needs to discover the relationship between states, actions, and rewards. Thus RL algorithms will have exploration and exploitation components built into them.

• The above environmental properties taken together are a Markov Decision Process (MDP). Most of classical RL is built against MDPs.

• There are different behaviors that can be optimized in $$\pi^ *$$. One is a finite-time horizon optimization where you are maximizing the expected reward over the next $$h$$ time steps. Another (more common) formulation is that you optimize the expected discounted reward with a manually chosen discount factor $$\gamma$$. Small $$\gamma$$ are greedier and prefer immediate rewards to further out rewards.

• In real-world domains, often you prefer optimizing for average reward (the expected reward as the time horizon approaches infinity) rather than discounted reward.

• Generally, an agent does not know anything about its environment at the start of learning. It must navigate the explore-exploit trade-off as it attempts to learn an optimal policy. While learning algorithms that are polynomial with respect to the action and state space are known, these are generally difficult to apply to robotics (where state is often large and continuous).

• Off-policy methods explore the state space independent of the best-learned policy. On-policy methods explore the state space while following the best-learned policy and thus exploration has to be baked into $$\pi$$.

• An additional complication in the RL setting is that there’s path dependence in that an agent’s earlier actions can affect later rewards. This is referred to as the credit assignment problem.

• Optimizing the primal formulation of the RL problem is known as policy search while optimizing the Lagrangian dual is known as a value function-based approach.

#### Value Function-Based RL

• Value function-based approaches generally attempt to learn a value function $$V^{\pi}(s)$$ or a state-action value function $$Q^{\pi}(s,a)$$ which measure the “goodness” of states and actions. Given correct value functions, then $$\pi^ *$$ becomes selecting the best action in each state as determined by $$V$$ or $$Q$$. Note that:
• $$V$$ and $$Q$$ are parameterized by the policy $$\pi$$. As the policy changes, so too will the value functions.
• $$V$$ can only be used as a basis of a policy if you know the transition probabilities to successor states after the current state.
• Often times $$V$$ and $$Q$$ will be approximated using neural nets or similar.
• Three high-level approaches to value function-based RL:
• Dynamic Programming - usable when you have a model of the transition probabilities and reward function. These methods are model based and include policy iteration and value iteration. They have the flavor of calculating your value function based on a policy and then greedily improving your policy based on this calculation.
• Monte Carlo - These techniques directly sample the environment performing complete rollouts to learn the dynamics and improve the value estimates.
• Temporal Difference Methods - These techniques are similar to Monte Carlo techniques, but learn at each timestep of a rollout rather than waiting for complete traces.

#### Policy Search RL

• Policy search has a number of features that make it amenable to robotics:
• It allows for integration of expert knowledge
• It allows for pre-structured policies
• An optimal policy will often have fewer parameters than an optimal value-function
• Local policy search can often lead to good results
• External constraints can be incorporated naturally
• In general, policy search will optimize a given policy $$\pi$$ parameterized by $$\theta$$ by iteratively calculating the parameter gradient $$\nabla\theta$$ that will increase the expected return: $$\theta_{i+1} = \theta_i + \nabla\theta_i$$

• Calculation of the policy update is the key differentiator between algorithms, and generally comes in two flavors:
• Black box methods - These methods do not leverage the internal structure of the problem, instead relying on sampling to estimate the gradient.
• White box methods - These methods take advantage of specifics of the problem to calculate policy updates. This includes model-based approaches.
• Following the gradient of the expected return $$J$$, that is $$\theta_{i+1} = \theta_i + \alpha\nabla_\theta J_{}$$, is a white box method that has proven particularly useful in a lot of research work. This gradient can be estimated using finite difference methods perturbing $$\theta$$, REINFORCE or likelihood ratio methods, expectation-maximization methods, or dynamic programming methods.

#### Policy Search vs Value Functions

• Value functions are difficult to translate to robotics because the high-dimensional spaces require function approximation out of the box. These approximations are often brittle and expensive.

• Value functions require total coverage of the state space and the largest local error determines the quality of the policy. Additionally, small changes in the value function can result in large changes in the policy that then feedback into the value function. This can result in expensive recalculations as well as instability.

• Policy search often considers only local changes and thus is more amenable to high dimensional spaces. It, however, risks getting caught at local optima.

• Terminology: policy search methods are sometimes called actor methods, while value function search methods are sometimes called critic methods. These can be combined to form actor-critic methods where the policy is used to guide action while the value function is used to decide when to update the policy.

#### Function Approximation

• Typically in RL, function approximation is based on experience collected by interacting with the environment.

• Function approximation is critical in continuous-space problems and often needed even in large discrete-space problems to generalize behavior to similar states.

• There are many flavors of function approximation and these approximators can be used to represent policies, value functions, and/or models.
• Parametric approximators have a fixed set of parameters and are used to fit observed data. Examples include neural nets and linear basis functions.
• Non-parametric approximators expand representational power in proportion to the data collected. Gaussian process regression is an example.
• One general problem of using function approximation techniques developed in the supervised learning world is that they often assume independently and identically distributed data. This is often not feasible in RL settings because the data is often path-dependent and that path can be influenced by the function approximator itself.

### 3. Challenges in Robot Reinforcement Learning

Robotics is a challenging arena for reinforcement learning:

• Continuous state spaces and actions must be represented; do you discretize or approximate? How fine-grained should your controls be? How much dimensionality can your learning algorithm handle?
• The physicality is difficult: samples are expensive, there’s jitter and uncertainty in real world system, there’s maintenance, sometimes a system needs to be manually reset, algorithms must run in real-time.
• Simulation can alleviate some problems with physicality, but must be robust to model errors.
• Goal specification (in the form of specifying a reward function) often requires some thought.

#### The Curse of Dimensionality

• As the number of dimensions grow in a state or action space, the amount of data needed to cover it grows exponentially.
• The high degree of freedom in many robotic assemblies naturally leads to this curse of dimensionality.
• One approach is to impose a hierarchy: e.g. the robot plans at the grid level and then lower-level systems produce motion unaware of the grid.
• Dimensionality reduction can often limit the dynamic capabilities of a robot.

#### The Curse of Real-World Samples

• Robots are expensive and repairs take time and money. Safe exploration is an understudied field.
• Often the environment is not fully captured by the state space, and thus learning may not converge as conditions change (e.g. a strong wind affecting the mobility of a robot).
• Often human involvement is needed to reset a robot so it can re-sample the environment. This is slow and expensive. Sample efficiency is important to ameliorate this.
• Real time requirements can complicate the sampling and learning procedures.
• Time discretization and variable signal processing delays can affect the ability to learn and control a robot.

#### The Curse of Under-Modeling and Uncertainty

• Modeling can be used to ameliorate some of the difficulties of the physical world, but brings its own issues.
• Model errors can compound and thus make behavior transfer difficult
• Tasks that are self-stabilizing (e.g. the robot doesn’t need to actively control itself to avoid crash-and-burn) allow for better transfer of modelled learning. This is less true for tasks that require constant control (e.g. pole-balancing).

#### The Curse of Goal Specification

• Goals are implicitly specified by the reward function. Crafting a good reward function is often non-trivial.
• Variance in reward must be able to be leveraged to improve the policy, otherwise no learning will occur.
• A sparse reward can be overly difficult for a robot to use as a learning cue so sometimes intermediate rewards will be added.
• The difficulty of programming the control algorithm is partially transferred to the shaping of the reward.
• Inverse optimal control - trying to learn a reward function from a series of expert demonstrations.

### 4. Tractability Through Representation

• Success in robotics RL has been achieved by leveraging:
• Effective representations
• Approximate models
• Prior knowledge

#### Smart State-Action Discretization

• Hand crafted discretization is the standard. Care is taken to balance expressiveness against state space reduction.

• One can also attempt to automatically learn a discretization. This seemingly complicates the task because you are optimizing the state representation AND the learning performance simultaneously.

• Meta-actions/options - automatically constructing high-level actions has “fascinated” RL researchers. A number of success have been demonstrated in the hierarchical RL space, although most seem to occur in fairly constrained environments and/or on toy tasks.

#### Value Function Approximation

• Function approximation can be used to model the value function OR to model the system.

• Unfortunately, unstable behavior and divergence is often observed in practice when using function approximators (some linear models are immune to this, though).

• If good features are known, value function approximation can use a linear approximator. However, this relies on the quality of the features and is not a general solution (in fact, it can make some problems impossible).

• Neural nets have (unsurprisingly) shown to be useful for some tasks, but obviously divergence must be contended with.

• Other tactics:

• Generalization heuristics for neighboring states
• Using local models for particular parts of the state space
• Gaussian methods (Gaussian Process Regression)

#### Pre-structured Policies

• Picking a good approximator to represent the policy is important. For example, trade-offs between speed of learning and representational power might be considered.

• Depending on the task, a number of different approximators might be appropriate. Researchers have seen success using:

• Linear models
• Motor primitives
• Gaussian mixture models
• Neural nets
• Non parametric approximators

### 5. Tractibility through Prior Knowledge

• Incorporating prior knowledge can constrain the search space and dramatically increase the ability to learn effective policies. This prior knowledge can look like:
• Initial policies
• Demonstrations
• Initial models
• Physical constraints (e.g. torque limits)
• Sometimes hard constraints (such as ones designed to protect robot hardware) pose difficulties for standard RL algos (i.e. hitting a discontinuous wall)

#### Prior Knowledge Through Demonstration

• People and animals often learn from imitation as well as trial and error. In the RL world, this is termed imitation and/or apprenticeship learning.

• Demonstrations can remove the need for global exploration (i.e. by telling the learner the critical states to focus on) and allow for much less expensive local optimization. Learning a good global solution, however, requires a good demonstration (see “Fosbury Flop” in Olympic High Jump).

• Both value function and policy search methods seem to work best in practice when they’re constrained to making small changes to the distribution over states while learning.

• When teaching a robot, the teacher can either demonstrate directly or control the robot. Direct demonstration requires translation to “robot world” while controlling the robot requires the teacher learning how to control it.

• When teaching is not straightforward, a hard-coded policy can be used as demonstration. This has been shown to work in robotic walking tasks.

#### Prior Knowledge Through Task Structuring

• Decomposing tasks into simpler tasks (as in Hierarchical RL) and composing simpler tasks into more complicated actions can both be used to learn better policies.

#### Directing Exploration with Prior Knowledge

• Prior knowledge can be used to fine tune the trade-off between exploration and exploitation and increase overall reward.

### 6. Tractability Through Models

• Many robotics RL problems can be made tractable by learning approximate models of the transition system (vs attempting to directly learn value function approximations or policies from live interaction).

• It is desirable to build a model esp. because simulation or rehearsal is much faster than gathering actual experience in the physical world.

#### Core Issues and General Techniques

• Model-based methods that learn a model from data can be much more sample efficient. However, there can be problems with:
• Expansive compute resource requirements
• Simulation biases
• Real world stochasticity
• Difficulties sampling from the simulator

Simulation Biases

• Any simulation will not capture all the dynamics of the real world. If these dynamics are important to the task, then this can break the learning algorithm and generate policies that work in the simulator but not in the real world.

• An interesting distinction is the type of simulator errors that compound vs non-compounding errors.

• Adding noise to the model can help reduce tendencies to overfit policies to quirks of the simulator.

• Uncertainties about environmental dynamics maintained in the model itself can be leveraged to create policies that generalize better.

• Tricks around model randomization have also proven to be useful.

#### Successful Approaches for Learning Forward Models

How can we obtain a candidate policy from a forward model?

• Rollouts using the model can be used to calculate rough gradients to update policies or learn approximate value functions.

• The model can be directly interrogated to generate plausible policies using techniques from control systems research.

### 7. A Case Study: Ball-in-a-Cup

• This section describes a case study of teaching a robot to catch a ball in a cup. The cup is held in one hand, the ball is attached to the bottom of the cup with a string, and initially the ball is at a dead hang under the cup. The cup is quickly moved to fling the ball above the cup and then caught with the cup.

• The naive rendering of the scenario would have an intractable-for-RL ~20 state and ~7 action dimensions.

• Reward shaping is important: the “catch the ball”-only reward was too sparse to learn effectively. The “closeness to cup” reward got stuck in local minimums (hitting the ball with the cup edge).

• Creating a faithful simulator is difficult.

• The policy is represented as a dynamical system of motor primitives. Over the course of successive research projects, the policy representation was distilled. The particular policy representation choice has the nice property that it is easy to include knowledge learned from demonstration.

• An initial demonstration was done by having a human directly manipulate the robot arm. RL was then used to search the local space “around” the demonstrated movement to optimize.

• Policy search methods are better suited for episodic tasks that need local optimization (thanks to the demo) with a large, high-dimensional state/action space. An EM method was employed instead of a gradient based method to be more sample efficient and require less hyperparameter tuning. This algorithm performs a local search around the demonstrated policy.

• They used a simulator for rehearsal training, but often a good simulator policy would just miss getting the ball into the cup. Thus the ability to switch between the simulator and the physical system and incorporate data from both was important.

• Getting data from the physical system was slow and tedious.

• Ultimately, the policy converged and the robot was regularly able to get the ball into the cup after ~100 episodes.

• A different value-function-based approach was also tested (perhaps by another team?) where the task was decomposed into two separate phases (swing-up and catch). The catch phase was hard coded, but a swing-up policy was learned using SARSA and state-action discretization.

### 8. Discussion

• RL in the robotics domain is not yet (as of 2013) straightforward to apply, it’s part art and part science at the moment.
• Users must decide when enough prior knowledge has been baked in and when to allow learning to take over.
• Reward function shaping and domain representation is very important.

#### Open Questions (as of 2013)

• How can we choose representations automatically?
• How do you approximate states, value functions, and domains (or any combination of those three) for a given problem?
• How can you generate good reward functions automatically?
• How much prior knowledge is useful?
• How can we encode prior knowledge automatically? Esp. when human demonstrations are difficult.
• How can RL better cope with the noisy, ambiguous, and incomplete sensor data that is often found in robotics domains?
• How can we reduce the need for hyperparameter tuning?
• How can we better deal with under-modeling and model errors?
• Modelling is attractive given the cost of obtaining real world data, but bad models (or poorly coping algorithms) can lead to bad policies.

#### Practical Challenges

• How can we better transfer learning (e.g. simple tasks to more complex tasks) and better exploit the data we have.
• Reproducibility and benchmarks for the research community.

#### Lessons from Robotics to RL researchers

• The challenges inherent to the robotics domain may not be fully appreciated in much of the RL field (multi-dimensional continuous action spaces, continuously drifting noise, frequent changes in the hardware and the environment, and the inevitability of undermodeling)

• Incorporating domain knowledge (e.g. basic models of the physical world) has been successful in robotics. Better techniques for doing this would be valuable.

• Local optimality may be more important than global optimality (esp. when you’re starting from a reasonable baseline like a human demonstration).

• More research into reward shaping, esp. physically motivated reward shaping.