Alice's Adventures in the Markovian World

TL;DR

This paper introduces Alice, an online decision-making algorithm for linear stochastic systems that learns without prior knowledge or training, extending Follow-the-Leader with regularization and constraints, and provides theoretical guarantees.

Contribution

It generalizes Follow-the-Leader to linear Gauss-Markov processes with a novel regularization and online constraint, offering a no-regret proof without estimating system parameters.

Findings

The algorithm achieves no-regret in linear stochastic environments.

Simulations demonstrate Alice's flexibility and superior performance.

Theoretical analysis provides conditions for no-regret guarantees.

Abstract

This paper proposes an algorithm Alice having no access to the physics law of the environment, which is actually linear with stochastic noise, and learns to make decisions directly online without a training phase or a stable policy as initial input. Neither estimating the system parameters nor the value functions online, the proposed algorithm generalizes one of the most fundamental online learning algorithms Follow-the-Leader into a linear Gauss-Markov process setting, with a regularization term similar to the momentum method in the gradient descent algorithm, and a feasible online constraint inspired by Lyapunov's Second Theorem. The proposed algorithm is considered as a mirror optimization to the model predictive control. Only knowing the state-action alignment relationship, with the ability to observe every state exactly, a no-regret proof of the algorithm without state noise is given. The analysis of the general linear system with stochastic noise is shown with a sufficient condition for the no-regret proof. The simulations compare the performance of Alice with another recent work and verify the great flexibility of Alice.

Equations

$${\mathbf{x}}_{t+1}=A{\mathbf{x}}_t+B{\mathbf{u}}_t+{\mathbf{w}}_{t+1},$$

$${\mathbf{u}}_t=K{\mathbf{x}}_t={\Phi }_t\text{vec}(K),$$

$$f_t(K_t)=\frac{\eta }{2}{\mathbf{x}}_t^{\mathrm{T}}{\mathbf{x}}_t+\frac{\beta }{2}{\mathbf{u}}_{t-1}^{\mathrm{T}}{\mathbf{u}}_{t-1},$$

$${\text{Reg}}_T=\sum _{t=1}^T{f}_t(K_t)-\sum _{t=1}^T{f}_t(K^{\ast }),$$

$${\hat{\mathbf{x}}}_i(K_{t+1})\triangleq ({\mathbf{x}}_i-B{\mathbf{u}}_{i-1})+B{\hat{\mathbf{u}}}_{i-1}(K_{t+1}).$$

$${\hat{\mathbf{x}}}_i(K_{t+1})=\mathcal{A}{\mathbf{x}}_{i-1}+B{\hat{\mathbf{u}}}_{i-1}(K_{t+1})+{\mathbf{w}}_i.$$

$${\hat{f}}_i(K_{t+1})$$

$$\begin{array}{rl}{J}_{1:t}(K)& =\sum _{i=1}^t\frac{\eta }{2}{\hat{\mathbf{x}}}_i(K{)}^{\mathrm{T}}{\hat{\mathbf{x}}}_i(K)+\frac{\beta }{2}[B{\hat{\mathbf{u}}}_{i-1}(K_{t+1}){]}^{\mathrm{T}}[B{\hat{\mathbf{u}}}_{i-1}(K_{t+1})].\end{array}$$

$$K_{t+1}$$

$$\text{subject to}\Vert {\hat{\mathbf{x}}}_t(K){\Vert }_2\le \alpha ||{\mathbf{x}}_{t-1}|{|}_2,t\ge t_c$$

$${\mathbf{u}}_t$$

$$\underset{U\triangleq \{{\mathbf{u}}_t,...,{\mathbf{u}}_{t+N-1}\}}{\min }{J}_{t:t+N-1}=\sum _{i=t+1}^{t+N-1}\frac{\eta }{2}{\mathbf{x}}_i^{\mathrm{T}}{\mathbf{x}}_i+\frac{\beta }{2}{\mathbf{u}}_{i-1}^{\mathrm{T}}{\mathbf{u}}_{i-1},$$

$$\text{subject to}{\mathbf{x}}_{i+1}=A{\mathbf{x}}_i+B{\mathbf{u}}_i,$$

$${\mathbf{x}}_{\min }\le {\mathbf{x}}_i\le {\mathbf{x}}_{\max },i=t+1,...,t+N,$$

$${\mathbf{u}}_{\min }\le {\mathbf{u}}_i\le {\mathbf{u}}_{\max },i=t,...,t+N-1,$$

$$\text{Control update}:{\mathbf{u}}_t={\mathbf{u}}_t^{\ast },$$

$$\begin{array}{rl}& {\mathcal{L}}_t(K,\nu )=\sum _{i=1}^t[\frac{\eta }{2}{\hat{\mathbf{x}}}_i(K{)}^{\mathrm{T}}{\hat{\mathbf{x}}}_i(K)+\frac{\beta }{2}[B{\hat{\mathbf{u}}}_{i-1}(K_{t+1}){]}^{\mathrm{T}}[B{\hat{\mathbf{u}}}_{i-1}(K_{t+1})]]\\ & +\lambda \Vert BK-B{K}_t{\Vert }_2+\nu (\Vert {\hat{\mathbf{x}}}_t(K){\Vert }_2-\alpha \Vert {\mathbf{x}}_{t-1}{\Vert }_2).\end{array}$$

$$\begin{array}{r}\underset{\nu \ge 0}{\sup }\underset{K}{\inf }{\mathcal{L}}_t(K,\nu )=\underset{\nu \ge 0}{\sup }\underset{K}{\inf }{J}_{1:t}(K)+\lambda \Vert BK-B{K}_t{\Vert }_2+\nu (\Vert {\hat{\mathbf{x}}}_t(K){\Vert }_2-\alpha \Vert {\mathbf{x}}_{t-1}{\Vert }_2).\end{array}$$

$${\nabla }_K{\mathcal{L}}_t$$

$$=\eta [B^{\mathrm{T}}\mathcal{A}+(1+\frac{\beta }{\eta })B^{\mathrm{T}}BK]\sum _{i=1}^t{\mathbf{x}}_{i-1}{\mathbf{x}}_{i-1}^{\mathrm{T}},$$

$$K_{t+1}=-\frac{\eta }{\eta +\beta }(B^{\mathrm{T}}B{)}^{†}{B}^{\mathrm{T}}\mathcal{A},$$

$${\mathbf{x}}_{t+1}=\frac{\eta }{\eta +\beta }(\mathcal{A}-B(B^{\mathrm{T}}B{)}^{†}{B}^{\mathrm{T}}\mathcal{A}){\mathbf{x}}_t+\frac{\beta }{\eta +\beta }\mathcal{A}{\mathbf{x}}_t$$

$$\Vert {\mathbf{x}}_{t+1}{\Vert }_2\le (\frac{\eta }{\eta +\beta }\Vert \mathcal{A}-B(B^{\mathrm{T}}B{)}^{†}{B}^{\mathrm{T}}\mathcal{A}{\Vert }_2+\frac{\beta }{\eta +\beta }\Vert \mathcal{A}{\Vert }_2)\Vert {\mathbf{x}}_t{\Vert }_2$$

$$\frac{\eta }{\eta +\beta }\Vert \mathcal{A}+B{K}^{\ast }{\Vert }_2+\frac{\beta }{\eta +\beta }\Vert \mathcal{A}{\Vert }_2=\delta <1,$$

$${\nabla }_{\text{vec}K}{\mathcal{L}}_t(K_t,{\nu }_{t+1}^{♯})=\sum _{i=1}^t[\eta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{i-1}+\beta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{i-1}]\text{vec}(K_t)$$

$$+[\eta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}({\mathbf{x}}_i-B{\mathbf{u}}_{i-1})]+2{\nu }_{t+1}^{♯}({\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1})\text{vec}(K_t)$$

$$=\sum _{i=1}^{t-1}[\eta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{i-1}+\beta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{i-1}]\text{vec}(K_t)+[\eta {\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}({\mathbf{x}}_i-B{\mathbf{u}}_{i-1})]$$

$$+[\eta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1}+\beta {\Phi }_{t-1}^{\mathrm{T}}{B}^{T}B{\Phi }_{t-1}]\text{vec}(K_t)+[\eta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}({\mathbf{x}}_t-B{\mathbf{u}}_{t-1})]$$

$$+2{\nu }_{t+1}^{♯}({\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1})\text{vec}(K_t)$$

$$=-2{\nu }_t^{♯}({\Phi }_{t-2}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-2})\text{vec}(K_t)+[\eta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1}+\beta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1}]\text{vec}(K_t)$$

$$+[\eta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}({\mathbf{x}}_t-B{\mathbf{u}}_{t-1})]+2{\nu }_{t+1}^{♯}({\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1})\text{vec}(K_t)$$

$$=-2{\nu }_t^{♯}{\Phi }_{t-2}^{\mathrm{T}}{B}^{\mathrm{T}}B{\hat{\mathbf{u}}}_{t-2}+\eta {\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}{\mathbf{x}}_t+(2{\nu }_{t+1}^{♯}+\beta ){\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\mathbf{u}}_{t-1}.$$

$${\nabla }_{\text{vec}K}^2{\mathcal{L}}_t(K_t,{\nu }_{t+1}^{♯})=(\eta +\beta )\sum _{i=1}^t[{\Phi }_{i-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{i-1}]+2{\nu }_{t+1}^{♯}{\Phi }_{t-1}^{\mathrm{T}}{B}^{\mathrm{T}}B{\Phi }_{t-1},$$

$$J_{1:t}(K^{♯})\le J_{1:t}(K_{t+1}),$$

$$J_{1:t}(K_{t+1})+\lambda \Vert B{K}_{t+1}-B{K}_t{\Vert }_2\le J_{1:t}(K_{t+1}^{♯})+\lambda \Vert B{K}_{t+1}^{♯}-B{K}_t^{♯}{\Vert }_2.$$

$$\Vert B{K}_{t+1}-B{K}_t{\Vert }_2$$

$$A=\left[\begin{array}{ccc}1.01& 0.01& 0\\ 0.01& 1.01& 0.01\\ 0& 0.01& 1.01\end{array}\right],B=I_3,\eta =10,\beta =1,\mathbf{\sigma }=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],$$

$$A^{{}^{\prime }}=\left[\begin{array}{ccc}4.01& 0.01& 0\\ 0.01& 1.01& 0.01\\ 0& 0.01& 1.01\end{array}\right],B=I_3,\eta =10,\beta =1,\mathbf{\sigma }=\left[\begin{array}{c}0.1\\ 0.1\\ 0.1\end{array}\right],{\mathbf{x}}_0=\left[\begin{array}{c}5\\ 5\\ 5\end{array}\right].$$

Full text

1 Introduction

*To regret deeply is to live afresh.

• Henry David Thoreau*

Different from the optimal control, which knows the system dynamics, hence has the privilege to stand at the beginning of the time and optimize the cumulative loss up to the terminal time as a function of the control action sequence, different from reinforcement learning, which has the training phase to learn the state-action value function, also different from other online control work which require a stable policy as the initial input to the algorithm (Dean et al., (2018), Abbasi-Yadkori et al., (2019)), the proposed algorithm Alice does not know the physics law of the environment, which is actually linear with stochastic noise, and can only learn to make decisions directly online. She neither estimates the system parameters, nor the value functions.

Only knowing the alignment of her actuators and sensors, with the perfect observer of the state, she would like to achieve a sub-linear regret with respect to a quadratic loss given by the environment at each step after she takes an action.

Related Work: Abbasi-Yadkori and Szepesvári, (2011) construct a high-probability confidence set around the system parameters based on online least-squares estimation, and derive the regret bound around $\tilde{O}(\sqrt{T})$ for the first time for the linear quadratic control problem. However its implementation requires solving a non-convex optimization problem to precision $\tilde{O}(T^{-1/2})$, which can be computationally intractable. Dean et al., (2018) proposes the first polynomial-time algorithm for the adaptive LQR problem that provides high probability guarantees of sub-linear regret. However, in proving the regret upper bound, a stable initial policy is assumed to be given as input.

Instead of the interplay between regret minimization and parameter estimation online, model-free approaches for reinforcement learning (RL) is applied online in Abbasi-Yadkori et al., (2019) to solve the linear quadratic control problem with regret upper bound$O(T^{\frac{2}{3}})$ proved. Least-squares temporal difference learning Tu and Recht, (2017) is used to approximate the state-action value functions online. However, this algorithm also requires a stable policy as input.

Major Contributions:

1. To the best of our knowledge, the proposed algorithm is the first pure online control algorithm that neither uses any information about the system dynamics, nor owns a stable policy as input, and requires neither parameter tuning nor training phase before testing the algorithm.

2. By deriving the close form expression of the control update, the proposed algorithm is proved to be a no-regret algorithm compared with the optimal control knowing the system dynamics when the linear system has no noise.

3. Theoretical analysis is shown for the general linear system with stochastic noise.

4. The algorithm is compared with Dean et al., (2018) with the exact same experiment and the better flexibility of the proposed algorithm Alice dealing with online adversarial system variations is verified.

2 Preliminaries

The environment is given by

[TABLE]

where the state $\bm{x}_{t}\in\mathbb{R}^{n}$, the agent’s action $\bm{u}_{t}\in\mathbb{R}^{m},$ the state transition matrix $A\in\mathbb{R}^{n\times n}$, the state-action alignment matrix $B\in\mathbb{R}^{n\times m}$. $\bm{w}_{t+1}$ corresponds to state noise drawn i.i.d. from an unknown Gaussian distribution $\mathcal{N}(0,W),$ where the covariance matrix $W\in\mathbb{R}^{n\times n}$ is diagonal and $W[i,i]\triangleq{\sigma}[i]^{2},i=1,..,n,$ and $\bm{\sigma}\in\mathbb{R}^{n}.$ The initial condition $\bm{x}_{0}\sim\mathcal{N}(\bar{\bm{x}}_{0},P_{0})$.

The action $\bm{u}_{t}$ is parameterized as

[TABLE]

where the control gain $K\in\mathbb{R}^{m\times n},\Phi_{t}\triangleq\bm{x}_{t}^{\rm T}\otimes I_{m},$ $\otimes$ denotes the kronecker product, and $\text{vec}(\cdot)$ is the vector operator.

The loss function $f_{t}(K_{t})$ is defined as

[TABLE]

where $t=1,2,...,T,$ $T$ is the terminal time step, $\eta\in\mathbb{R}^{+}$ is the state weighting parameter, and $\beta\in\mathbb{R}^{+}$ is the control weighting parameter. $\frac{\beta}{\eta}=0$ means that Alice would like to achieve her goal at any cost. The larger $\frac{\beta}{\eta}$ is, the less control effort Alice would like to pay to achieve her goal.

The regret up to the terminal step $T$ is defined as

[TABLE]

where $K^{*}=\lim_{T\to\infty}\arg\min_{K}\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}[f_{t}(K)],$ which is the fixed control gain given by the infinite horizon Linear Quadratic Regulator (LQR).

Assumption 1: It is assumed that the state $\bm{x}_{t}$, $\forall t=0,1,2,...,$ can be observed exactly.

Assumption 2: There exists $K$, such that $||A+BK||_{2}$ can be placed arbitrarily. (See Abbasi-Yadkori et al., (2014) and Ibrahimi et al., (2012) for the similar assumptions.)

Assumption 3: $\|A\|_{2}<\frac{\eta+\beta}{\beta},$ which intuitively means that there is enough control effort.

3 Alice

The current time in the environment is $t$. Alice now immersed in her fantasy world, is thinking about which fantasy action $\hat{\bm{u}}_{i-1}(K_{t+1})\triangleq K_{t+1}\bm{x}_{i-1}$ she should have taken when she was at state $\bm{x}_{i-1},$ where $i=1,2,...,t.$ The control gain $K_{t+1}$ is a variable to be optimized using (9) at time step $t$ and will be applied as the control gain at time step $t$ in the environment.

After imaging which fantasy action $\hat{\bm{u}}_{i-1}(K_{t+1})$ she should have taken at state $\bm{x}_{i-1}$, Alice imagines that she has landed at a fantasy next state $\hat{\bm{x}}_{i}(K_{t+1})$ by the following operation in her mind

[TABLE]

Since the actual state transition in the environment at time step $i-1$ followed that $\bm{x}_{i}=A\bm{x}_{i-1}+B\bm{u}_{i-1}+{\bm{w}_{i}},$ plugging which into (5) yields that

[TABLE]

where $\mathcal{A}$ denotes the state transition matrix $A$ but inaccessible to Alice and our analysis. Later on $\mathcal{A}$ will be used instead of $A$ to get this paper distinguished from model-based algorithms and analyses. Note (5) reveals that although Alice cannot access $\mathcal{A}$ and ${\bm{w}_{i}}$, she can still make the correct state transition in her mind to the fantasy next state $\hat{\bm{x}}_{i}(K_{t+1})$ since she has access to $\bm{x}_{i}$ and $B\bm{u}_{i-1}$.

The fantasy loss function at step $i$ follows that

[TABLE]

At time step $t$ in the environment, after having travelled again from state $\bm{x}_{0}$ to $\bm{x}_{t}$ in her mind, and imagined having taken a different $\hat{\bm{u}}_{i-1}(K_{t+1})$ at each $\bm{x}_{i-1}$ and received a fantasy next state $\hat{\bm{x}}_{i}(K_{t+1}),$ she sums up all the fantasy losses from $i=1$ to $t$, which yields the following fantasy cost function

[TABLE]

Still at time step $t,$ Alice updates her real action in the environment based on the following optimization, where $\bm{u}_{t}$ are random vectors for $t=0,1,..,t_{w}-1$, and for $t\geq t_{w}$

[TABLE]

where the hyper-parameters $\alpha\in(\frac{1}{\gamma},1),\lambda\geq 0$, $t_{w}\geq 1$ is the initial waiting time, $t_{c}\geq 1$ denotes the waiting time for the hard constraint to function. $\gamma>1$ is a parameter which gives the limit performance that we would like Alice to approach her goal considering the state noise, and $||\mathbb{x}_{T}||_{2}\leq 3\gamma||\bm{\sigma}||_{2}$. $\gamma$ has its natural limit and hence $\gamma>1$ at least.

Remark 1

Disregarding the hard constraint in (9) and setting $\lambda=0$, (9) will be reduced to the online learning algorithm Follow-the-leader (FTL), but is specially formulated in this paper to generalize to a linear Gauss-Markov process setting.

The intuition of Alice’s hard constraint is drawn from Lyapunov’s Second Theorem, which states that the equilibrium is exponentially stable for a linear system if and only if there exists the Lyapunov function (e.g, $\|\bm{x}_{t}\|_{2}$) of the system decreases with time. So to get to the equilibrium point, Alice would like to make sure that the fantasy current Lyapunov function $\|\hat{\bm{x}}_{t}(K)\|_{2}$ in her fantasy decreases compared with the previous $||\bm{x}_{t-1}||_{2}$ in the reality world.

The soft constraint $\lambda\|BK-BK_{t}\|_{2}$ encourages Alice’s control gain $K_{t+1}$ to stay close to $K_{t}$ as time increasing. This is similar to the momentum method in the gradient decent algorithm.

$\textit{Alice}(\bm{x}_{0},B,T,\eta,\beta,\alpha,\lambda,\gamma)$

 

For $t=0$ to $t_{w}-1:$

   Alice takes a random action $\bm{u}_{t}$

   Alice observes $\bm{x}_{t+1},{f}_{t+1}$

End For

For $t=t_{w}$ to $T:$

   If $||\mathbb{x}_{t}||_{2}\leq 3\gamma||\bm{\sigma}||_{2}$:

      Break;

     Alice takes action $\bm{u}_{t}=0.$

   Else:

     Alice chooses $K_{t+1}$ according to (9)

     Alice takes action $\bm{u}_{t}$ according to (10)

   End If

  Alice observes $\bm{x}_{t+1}$ and receives ${f}_{t+1}(K_{t+1}),\hat{f}_{t+1}(K_{t+1})$

End For

3.1 Alice, Model Predictive Control and Follow-the-leader

Another optimization considered as the mirror optimization to Alice is the model predictive control (MPC). Written in accordance with the terms defined in this paper yields that, at each time step $t$ with $\bm{x}_{t}$ observed,

[TABLE]

where the hard constraints (13) and (14) are optional, $N$ is the horizon of the optimization.

Unlike Alice, who thinks in retrospect minimizing the cumulative fantasy loss in her mind (in other words, regrets for her past), MPC minimizes the cumulative loss of the future $N$ steps (in other words, predicts and plans for the future). But since the nature of the future is its uncertainty, MPC, making a hidden but strong assumption on the model for the future (12), is very sensitive to model uncertainties. To the contrary, due to the nature of the past, which is its certainty, Alice can always get the most "genuine" fantasy states, see (6) and (5).

Another nature of the past is its irreversibility, so Alice can only regret in her mind. She hopes to enforce Lyapunov’s Second Theorem to get the sequence $\|\bm{x_{t}}\|_{2}$exponentially decayed, which also implies a no-regret algorithm, but because she does not have the ability to predict the future, she can only enforce it in her fantasy: she always thinks in retrospect that she should have taken a different action to make the fantasy current state $\|\hat{\bm{x}}_{t}(K)\|_{2}$ smaller than the previous $||\bm{x}_{t-1}||_{2}$.

Given the ability to predict the future, Alice’s mirror optimization MPC has the advantage in terms of planning ahead. For example, MPC can plan to make the state and control actions being bounded as wanted in (13) and (14), but Alice cannot even surely bound the state asymptotically in the following analysis in section 4.3.

Although keeps regretting, Alice is not a "loser" because she can use her experience from (9) to improve her future decision making with (10). Such spirit is also described by the online learning algorithm FTL Shalev-Shwartz et al., (2012), and perhaps, Henry David Thoreau. The regret bound of FTL is $O(\log T)$ in that the loss function $f_{t}(K_{t})$ is assumed to be $L_{t}$ Lipschitz upper bounded by a constant $L,$ which is equivalent to saying that the state $\bm{x}_{t}$ is assumed to be bounded in our paper according to the definition (3). However, it is not a valid assumption for a Markov-decision process framework, which can be seen from (1) that $\bm{x}_{t}$ can blow up exponentially as time increasing. To reconcile such conflict, other recent work on online control thus assume a stable policy as input to enable the estimation of system parameters or state-action value functions online. However, our work has fundamental difference from theirs since we do not make such strong an assumption and the goal of Alice is to find a stable policy directly online, which is considered as the input to the algorithms in Dean et al., (2018) and Abbasi-Yadkori et al., (2019).

4 Analysis of Alice

4.1 The Existence of the Optimal Solution and the Feasibility of the Constraints in (9)

The Lagrangian $\mathcal{L}_{t}$ associated with the optimization (9) at time step $t$ follows that

[TABLE]

The dual problem to (9) follows that

[TABLE]

Theorem 1

Under Assumption 1 and 2, Slater’s condition holds true for the optimization (9) at any time step $t$ with probability at least $\delta=0.997,$ which also shows that the hard constraints in (9) are feasible w.p. at least $\delta$. Hence at any time step with probability at least $\delta$, (9) and its dual (17) have strong duality, and such optimal $\nu^{*}$ and $K^{*}$ exist that obey the KKT conditions.

Proof: It follows from Assumption 2 that there exists $K$ such that $||\mathcal{A}+BK||_{2}$ can be placed arbitrarily. Hence there exists $K$ such that $||\mathcal{A}+BK||_{2}={\delta}<{\alpha-\frac{1}{\gamma}},$ where $\alpha>\frac{1}{\gamma}$ from the definition around (9). If follows from (5) that $||\hat{\bm{x}}_{t}(K)||_{2}=||(\mathcal{A}+BK)\bm{x}_{t-1}+\bm{w}_{t}||_{2}\leq||\mathcal{A}+BK||_{2}||\bm{x}_{t-1}||_{2}+||\bm{w}_{t}||_{2}<\alpha||\bm{x}_{t-1}||_{2}-\frac{||\bm{x}_{t-1}||_{2}}{\gamma}+||\bm{w}_{t}||_{2}.$ Since $||\bm{w}_{t}||_{2}\leq||3\bm{\sigma}||_{2}$ w.p at least $\delta,$ and $||\bm{x}_{t-1}||_{2}>3\gamma||\bm{\sigma}||_{2},$ we could know that the hard constraint $\|\hat{\bm{x}}_{t}(K)\|_{2}\leq\alpha||\bm{x}_{t-1}||_{2},t\geq 1$ can be inactive w.p at least $\delta$ at any time step $t$, which means that Slater’s condition holds true. Hence the strong duality and KKT conditions are satisfied w.p at least $\delta$ (Boyd and Vandenberghe, (2004)).

4.2 Special Case: Linear System with No Noise

Theorem 2

Set $t_{w}=n,t_{c}=T,\lambda=0,$ and under Assumption 1, 2 and 3, Alice is a no-regret algorithm given $\|\bm{\sigma}\|_{2}=0$.

Proof: For this case, $\nabla_{K}\mathcal{L}_{t}$ at time step $t\geq n$ yields that

[TABLE]

since $\mathcal{A}\bm{x}_{i-1}=\bm{x}_{i}-B\bm{u}_{i-1}.$ As stated around (9), $\bm{u}_{i}$ are random vectors for $i=0,1,..,n-1$, thus with probability 1, $\sum_{i=1}^{t}\bm{x}_{i-1}\bm{x}_{i-1}^{\rm T}$ is invertible, $t\geq n$. It follows from $\nabla_{K}\mathcal{L}_{t}=0$ that $B^{\rm T}\mathcal{A}+(1+\frac{\beta}{\eta})B^{\rm T}BK=0.$ The optimal $K_{t+1}$ with the minimal norm follows that

[TABLE]

where $(\cdot)^{\dagger}$ denotes the pseudo inverse. Substituting (18) into $\bm{x}_{t+1}=(\mathcal{A}+BK_{t+1})\bm{x}_{t}$ yields that

[TABLE]

Hence

[TABLE]

It follows from the regression Montgomery et al., (2012) that $\|\mathcal{A}+BX\|_{2}$ is minimized by $BX=-B(B^{\rm T}B)^{\dagger}B^{\rm T}A$. Thus $\|\mathcal{A}-B(B^{\rm T}B)^{\dagger}B^{\rm T}\mathcal{A}\|_{2}\leq\|\mathcal{A}+BK^{*}\|_{2}$, $\forall K^{*}.$ It follows from Assumption 2 and 3 that there exists such $K^{*}$ that

[TABLE]

Thus $\|\bm{x}_{t+1}\|_{2}\leq\delta\|\bm{x}_{t}\|_{2}$. Hence $\|\bm{x}_{t+1}\|_{2}$ will decay exponentially, $\forall t\geq n.$ It follows from (3) that $f_{t}(K_{t})$ will decay exponentially, hence $\sum_{t=1}^{T}f_{t}(K_{t})$ is bounded by a constant. It follows from the property of LQR Kwakernaak and Sivan, (1972) that $\sum_{t=1}^{T}f_{t}(K^{*})$ is also bounded by a constant, thus it follows from (4) that $\text{Reg}_{T}$ is bounded by a constant. Thus Alice is a no-regret algorithm.

4.3 General Linear System With Stochastic Noise

Set $t_{w}=t_{c}=1.$

Lemma 4.1

For a strongly convex function $\mathcal{L}_{t}(K),||K^{*}-K||\leq\frac{2||\nabla\mathcal{L}_{t}(K)||_{2}}{\min{||\nabla^{2}\mathcal{L}_{t}(K)||_{2}}},$ where $\nabla\mathcal{L}_{t}(K^{*})=0.$ Boyd and Vandenberghe, (2004)

Theorem 3

At any time step $t\geq 2,$ with probability at least $\delta^{2}$, $||BK^{\sharp}_{t+1}-BK^{\sharp}_{t}||_{2}\leq\frac{2b||-2\nu^{\sharp}_{t}\Phi_{t-2}^{\rm T}B^{\rm T}B\hat{\bm{u}}_{t-2}+\eta\Phi_{t-1}^{\rm T}B^{\rm T}\bm{x}_{t}+(2\nu^{\sharp}_{t+1}+\beta)\Phi_{t-1}^{\rm T}B^{\rm T}B\bm{u}_{t-1}||_{2}}{\|(\eta+\beta)\sum_{i=1}^{\rm T}[\Phi_{i-1}^{\rm T}B^{\rm T}B\Phi_{i-1}]+2\nu^{\sharp}_{t+1}\Phi_{t-1}^{\rm T}B^{\rm T}B\Phi_{t-1}\|_{2}}\triangleq\zeta_{t},$ where $\nabla_{\text{vec}K}\mathcal{L}_{t-1}(K^{\sharp}_{t},\nu^{\sharp}_{t};\lambda\equiv 0)=0,$ $\nabla_{\text{vec}K}\mathcal{L}_{t}(K^{\sharp}_{t+1},\nu^{\sharp}_{t+1};\lambda\equiv 0)=0,$ and $\|B\|_{2}\triangleq b.$

Proof:

with probability at least $\delta^{2}$, $\nabla_{\text{vec}K}\mathcal{L}_{t-1}(K^{\sharp}_{t},\nu^{\sharp}_{t};\lambda\equiv 0)=0,$ $\nabla_{\text{vec}K}\mathcal{L}_{t}(K^{\sharp}_{t+1},\nu^{\sharp}_{t+1};\lambda\equiv 0)=0$ are true. Using $\nabla_{\text{vec}K}\mathcal{L}_{t-1}(K^{\sharp}_{t},\nu^{\sharp}_{t};\lambda\equiv 0)=0,\nabla_{\text{vec}K}\mathcal{L}_{t}(K^{\sharp}_{t},\nu^{\sharp}_{t+1};\lambda\equiv 0)$ follows that

[TABLE]

The hessian $\nabla^{2}_{\text{vec}K}\mathcal{L}_{t}(K_{t},\nu^{\sharp}_{t+1})$ follows that

[TABLE]

which is a constant. It follows from Lemma 4.1 that $\|B(K_{t+1}-K_{t})\|_{2}\leq\zeta_{t}.$

Theorem 4

At any time step $t\geq 2,$ with probability at least $\delta^{2}$, $||BK_{t+1}-BK_{t}||_{2}\leq\zeta_{t}$

Proof: It follows from Theorem 3 that $K^{\sharp}_{t+1}$ is the solution to the optimization (9) with $\lambda\equiv 0,$ whereas $K_{t+1}$ is the solution to the optimization (9). Thus

[TABLE]

Hence

[TABLE]

Theorem 5

At any time step $t\geq 2,$ with probability at least $\delta^{2}$, $\|\bm{x}_{t+1}\|_{2}\leq(\alpha+\zeta_{t})\|\bm{x}_{t}\|_{2}.$

Proof: It follows from (5) that $\|\hat{\bm{x}}_{t}\|_{2}\geq\|{\bm{x}}_{t}\|_{2}-\|(BK_{t+1}-BK_{t})\|_{2}\|{\bm{x}}_{t-1}\|_{2}.$ It follows from Theorem 1 and 4 that w.p at least $\delta^{2}$, $\alpha\|\bm{x}_{t-1}\|_{2}\geq\|\hat{\bm{x}}_{t}\|_{2},$ and $||BK_{t+1}-BK_{t}||_{2}\leq\zeta_{t}.$ Hence w.p. at least $\delta^{2},\|{\bm{x}}_{t}\|_{2}\leq(\alpha+\zeta_{t})\|{\bm{x}}_{t-1}\|_{2}.$

Corollary 4.2

If there exists such $t_{s}$ that $\zeta_{t}<1,\forall t>t_{s},$ then it follows from Theorem 5 that $\alpha+O(\frac{1}{t})<1$, then $\|{\bm{x}}_{t}\|_{2}$ will exponentially decay, thus the algorithm Alice will be a no-regret algorithm.

5 Experiments

Since under Assumption 1, 2, and 3, Alice as a pure online controller, has no need to tune the algorithm parameters, and requires no special information as initial input, the algorithm parameter settings are uniformly set as: $\gamma=1.2,\alpha=0.9,\lambda=0.001,t_{w}=1,t_{c}=1.$

5.1 Experiment 1

To make good comparisons with previous papers, this paper consider the exact the same example as Dean et al., (2018), in which Figure 1 compares the median regret of the robust adaptive method(Robust) over 500 experiments with methods like Thompson Sampling (TS, Abeille and Lazaric, (2017)), and so on. Other works also use the same example: Abbasi-Yadkori et al., (2019), Dean et al., (2017), and Tu and Recht, (2017).

The comparison experiment is carried out on the following problem: the initial state is $\bm{x}_{0}=0,$ and

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which corresponds to a marginally unstable Laplacian system where adjacent nodes are weakly connected.

Figure 2 shows the median regret comparison between Alice and the proposed algorithm in Dean et al., (2018), and Figure 2 shows the state infinity norm comparisons among Alice, the proposed algorithm in Dean et al., (2018) and LQR.

However, it is actually an "unfair" comparison, because Robust has a stabilizing controller as an initial input to the system, and the algorithm adapts the controller on this basis, and $LQR$ knows the system dynamics $A,$ hence has the privilege to predict the future losses. But Alice only knows her state-action alignment matrix $B$ without any parameter tuning or training phases. It is the first pure online control algorithm that neither uses any information about the system dynamics, nor owns a stable policy as input.

Thus, as can be seen in Figure 2, Alice suffers great regret in the initial steps compared with Robust. As can be seen in Figure 2, Alice oscillates more compared with Robust. But the state norm is smaller than Robust on average since the parameter $\gamma$ is set to be 1.2, which means that we would like the steady state norm to be $\|\bm{x}_{T}\|_{2}\leq 3.6\|\bm{\sigma}\|_{2}=3.6.$

5.2 Experiment 2

To get distinguished from the online controllers that requires a stable policy as input, we change A matrix in the experiment 1 adversarially online after $t=10$ to $A^{{}^{\prime}}$. To get a more clear sense of the convergence of the algorithm, we reduce the state noise covariance, and change the initial state far from the equilibrium as below. The convergence of the state norm can be seen in Figure 4, the red line as a comparison is also get by Alice but without changing the environment dynamics $A$ matrix online. Compared with our result, the online controllers using stable policies as initial input are vulnerable to such a case.

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5.3 Experiment 3

We further change A matrix in the experiment 1 adversarially online as time varying as below, and the convergence of the state norm can be seen in Figure 4.

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