Chapter VII: Conclusion and Future Work
7.2 Future Work
Continuous-State Dynamics and Inexact Patterns
The impact of Chapter2is that understanding the relationship between system stability and the characteristics of the jump noise allows us to infer the types of model-based controllers we need to design. However, if the system is complex enough, model-based techniques may also be intensive in computation time. For example, the Hamilton-Jacobi-Bellman (HJB) equation is a well-known method for optimal control in deterministic settings with determin-
istic disturbances [66, 13]. Impulse control is a HJB-based method developed for stochastic systems similar to the shot and L´evy noise dynamics described in Chapter 2; extensive dis- cussion on impulse control has been developed in [115] with applications to finance. Yet a well-known limitation of HJB approaches in many scenarios is the large amount of compu- tation time and the inability to analytically solve for the value function from the partial differential equation (PDE). Our implementation and applications of the PLP architecture throughout this thesis have considered discrete values and depend on exact matching be- tween patterns that are observed and patterns that are stored in memory. This does not fully cover cases where the dynamics behave as in Chapter 2, where the possible values the jump could take is a continuous spectrum instead of a discrete set. One way to bypass the scalability to larger pattern spaces is to project multiple patterns into the same equivalence class, similar to the PLMP architecture for the vehicle traffic problem (see Chapter 5). A future point of work would be to design a procedure which constructs a similarity metric in the pattern space that can be used to reduce the dimensionality of the problem prior to control. A relevant branch of literature that could be consulted for this direction is feature extraction in machine learning.
Uncertainty Quantification
The current PLP architecture described in Chapter 3 is just one specific type of imple- mentation among a wide variety of controller frameworks that build upon the concept of learning patterns to reduce time, data consumption, and redundant computation. Typical schemes usingBayesian updates (e.g., Kalman filtering) encode prior knowledge as a proba- bility distribution. While methodically different to the idea of the pattern-occurrence problem from Chapter3, which adopts a more frequentist approach by using martingales to construct closed-form formulas, both ideas are similar in that they avoid redundant computation in handling recurring data. Thus, a natural topic for the future is developing alternative imple- mentations of the pattern-learning component, especially Bayesian approaches to solving the pattern-occurrence problems. One extension is to predict the occurrence of future patterns with more state-of-the-art approaches in uncertainty quantification, especially when some statistics of the dynamics are unknown (e.g., the transition probability matrix of the MJS in Chapter 4). In addition, a combination of the two approaches can be considered: instead of relying entirely on Bayesian means, explicitly learning specific patterns and keeping track of a table (such as the episodic control approach used in Chapter 5) may potentially provide further benefits depending on the application setting.
Other Real-World Applications
The applications considered throughout this thesis have involved extensive software sim- ulations with a number of simplifying assumptions. A broad subject of future work will involve incorporating domain-specific knowledge into the algorithms designed here, as well as a focus on actual deployment to real-world systems. We also aim to design more learning- based stochastic controller and estimation architectures around these systematic model- based/model-free tradeoffs for other applications, including wireless communication, the in- ternet and datacenter resource allocation, biological networks and neuroscience, and in air traffic management, which can be viewed as a 3D extension to the vehicle traffic problem from Chapter 5. A poignant part of future research would be devoted to providing more explicit theoretical guarantees on performance based on these tradeoffs.
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