## Making Sense of Continuous RL With Viscosity Solutions

*Harley Wiltzer, September 2020*

I'm writing this post to address a fairly shocking issue that arises
when considering the plight of RL algorithms in the limit of continuous
time updates. This post is heavily inspired by
Remi Munos' manuscript,
A Study of
Reinforcement Learning in the Continuous Case by the Means of Viscosity
Solutions, which itself builds on the theory of *viscosity solutions*,
developed by
Crandall and Lions in 1983. While Munos' paper does a fantastic job of pointing
out the flaws with familiar reinforcement learning algorithms in the
continuous time setting and how to go about improving them, the notion
of viscosity solutions is left essentially as a black box. On the other
hand, Crandall and Lions' seminal work on viscosity solutions thoroughly
analyses the properties of viscosity solutions, it does not consider
their implications on reinforcement learning, and some mathematical
steps in the theory are skipped, making it difficult for me to follow at
times with my background in engineering and computer science. In this
post, I'll try to introduce the concept of viscosity solutions in a more
approachable manner.

## Continuous Time RL and the HJB Equation

This post is concerned only with value-based RL methods. We'll consider an MDP with state space \(\mathcal{X}\), action space \(\mathcal{A}\), and state transition kernel \(P(x_{t+1}, r_t\mid x_t, a_t)\). For a given policy \(\pi\), we define

\( P^\pi(x_{t+1}\mid x_t) = \int_{\mathbf{R}}P(x_{t+1}, r_t\mid x_t, a_t)\pi(a_t\mid s_t)dr_t \)

In familiar (discrete-time) RL, our goal is to learn the *value
function* \(V^\pi\) for a given policy \(\pi\),

\( V^\pi(x) = \underset{x_{t+1}\sim P^\pi(\cdot\mid x_t)}{\mathbf{E}}\left\{\sum_{t=0}^\infty\gamma^tR_t\mid x_0=x\right\} = R_t + \gamma\underset{x_{t+1}\sim P^\pi(\cdot\mid x_t)}{\mathbf{E}}\left\{V^\pi(x_{t+1})\right\} \)

The natural extension of this to the continuous time setting is as follows,

\( V^\pi(x) = \underset{x_{t+1}\sim P^\pi(\cdot\mid x_t)}{\mathbf{E}}\left\{\int_{t=0}^\infty\gamma^tR_tdt\mid x_0 = x\right\} = \mathbf{E}\left\{\int_0^\tau\gamma^tR_tdt + \gamma^\tau V^\pi(x_\tau)\right\} \)

where \(\tau\) can be any positive real number. We're concerned now
about what happens as \(\tau\to
0\). By writing the first order Taylor approximation of the equation
above about \(\tau = 0\) and taking the limit as \(\tau\to\ 0\), we
arrive at what's known as the **Hamilton-Jacobi-Bellman (HJB) Equation**,

\( V^\pi(x)\ln 1/\gamma = R + \underset{a\sim\pi(\cdot\mid x)}{\mathbf{E}}\bigg\langle \nabla_xV^\pi(x), f(x, a)\bigg\rangle \)

where \(f(x, a)\triangleq\frac{d}{dt}x\). Note that a differentiable value function satisfies the HJB equation. However, as we'll see, even very simple continuous time MDPs can admit non-differentiable value functions.

## A Problematic MDP

Remi Munos proposed a very simple continuous-time MDP in the paper cited above that demonstrates some issues that can arise if we naively adapt familiar RL algorithms to continuous time.

### The Munos MDP

The MDP has state space \(\mathcal{X}=[0,1]\) and action space \(\mathcal{A}=\{\pm 1\}\). We control a particle with dynamics satisfying \(\dot{x}(t) = a(t)\), where \(\dot{x}(t)\) and \(a(t)\) are the velocity of the particle and the chosen action respectively at time \(t\). For any \(x\in (0, 1)\) the agent receives no reward, and at the endpoints the rewards are given by constants \(R_0\) and \(R_1\) for \(x = 0\) and \(x=1\) respectively. We will set \(R_0=1\) and \(R_1=2\). We also use a discount factor \(\gamma=0.3\).

### The Optimal Policy

It's relatively easy to compute the optimal policy for the Munos MDP. Clearly there will be a threshold point \(\overline{x}\) such that the optimal policy \(\pi^{\star}\) will satisfy

\( \pi^{\star}(a\mid x) = -\mathbf{1}_{[x\leq\overline{x}]} + \mathbf{1}_{[x>\overline{x}]} \)

### The Value Function

It's also fairly easy to derive the value function analytically. When \(x\leq\overline{x}\), we have

\(V(x) = R_0\gamma^{x} = 0.3^{x}\)

Similarly, when \(x>\overline{x}\), we have

\(V(x) = R_1\gamma^{1-x} = 2(0.3)^{1-x}\)

Moreover, note that \(\overline{x}\) by definition satisfies

\( R_0\gamma^{\overline{x}} = R_1\gamma^{1-\overline{x}} \)

After some simple algebra, we find that \(\overline{x} = \frac{\log (R_1/R_0)}{2\log\gamma} + \frac{1}{2}\). The value function looks like

Notice that at \(\overline{x}\) the value function is not differentiable. Does this mean that it cannot solve the HJB equation?

You might also be curious about why the continuous-time setting introduces this issue – after all, the value function does not depend on time. However, the derivative of the value function with respect to \(x\) arise from the chain rule when computing \(dV/dt\) in the derivation of the HJB equation. Very subtle.

### It Gets Worse

We see that the value function for this MDP cannot be differentiable on
the entire state space, but what if we expanded our search space to
functions that are differentiable *almost everywhere* on the state
space? It turns out this is far from a good solution, because there may
actually be infinitely many functions that are global minima to the HJB
equation. Using gradient descent methods to learn the value function may
very well cause an algorithm to land in such a global minimum where the
learned value function doesn't represent the true value function well at
all. To illustrate this, let's examine the HJB equation at any point in
the interior of the state space. In this region, the reward is always
\(0\). The HJB equation collapses to

\( V^\pi(x)\ln\gamma = -\langle\nabla_x V^\pi(x), f(x, \pi(x))\rangle \)

where we assume stochastic dynamics purely to simplify the problem. In this MDP, \(f \in \{\pm 1\}\), so we actually have

\( V^\pi(x)\ln\gamma = -\left\lvert\frac{d}{dx}V^\pi(x)\right\rvert \)

So, we can almost arbitrarily construct a piecewise continuous function of increasing and decreasing exponentials to satisfy this. For instance, let \(V(x)\) denote the optimal correct value function, and \(W(x)\) denotes another function as follows,

\begin{equation} W(x) = \begin{cases} 0.3^x & x\leq\frac{1}{10}\\ 2(0.3^{1-x}) + 0.21 & x\in(1/10, 3/10]\\ 0.3^x + 0.37 & x\in(3/10, 1/2]\\ 1.08(0.3^{1-x}) + 0.92& x>\frac{1}{2}\\ \end{cases} \end{equation}This function was simply chosen by arbitrarily placing exponential functions of the form \(\gamma^x\) and \(\gamma^{1-x}\) in such a way that the values at the endpoints were correct. Note that if \(W(x)\) is thought to be the value function, the optimal policy will aim to guide the agent towards states \(x\) where \(W(x)\) is high. Below is a visual of two potential value functions,

It's fairly straightforward to see that the HJB equation is satisfied for both \(V\) and \(W\) wherever they're continuous and differentiable. However, since \(W\) is not continuous, it can get arbitrarily inaccurate in the interior of the state space. Clearly, given our knowledge of the MDP, \(V(x)\) is objectively a better value function to learn than \(W(x)\). However, by learning the value function using gradient descent, the two are indistinguishable if we search over continuous-almost-everywhere functions.

## Viscosity Solutions to the Rescue

As we saw, simple MDPs can have non-differentiable value functions, but
the HJB equation requires computation of the value function's gradient.
If we try to settle for almost-everywhere-differentiable functions, we
lose uniqueness guarantees, so there can potentially be a bunch of
functions satisfying the HJB that don't actually compute the expected
return. However, it can be shown that the HJB equation admits unique
*viscosity solutions*, which exhibit exactly those properties that we'd
like from the value function.

### What is a Viscosity Solution?

A viscosity solution is a function obeying some properties (don't even try to understand these yet) that satisfy partial differential equations of the form

\( H(x, v, Dv) = 0 \)

where we'll interpret \(x\) as a state, \(v\) as a value function, and \(Dv = \nabla_xv\). Furthermore, \(H\) is an arbitrary function that is monotonically non-decreasing in \(v\), known as the Hamiltonian. In the case of the HJB equation, we see that

\( H(x, V, DV) = V(x)\ln\gamma + R + \bigg\langle\nabla_xV(x), f(x, \pi^{\star}(x))\bigg\rangle \)

At a high level, viscosity solutions are continuous functions that are differentiable almost everywhere.

### Intro to the Design of Viscosity Solutions

I'll conjecture that without ever having been exposed to the concept of viscosity solutions, simply looking at the definition will be fairly demoralizing. According to Crandall and Lions in their seminal paper on the topic, the following is what led to the development of the theory of viscosity solutions. Let's start by assuming that \(V\in C^1(\mathcal{X})\) (note that we of course don't want to restrict viscosity solutions to functions of this class!). Consider another function \(\phi\in C^1(\mathcal{X})\), where \(\phi(x)V(x) = \max\phi V\) at some state \(x\), and \(\phi(x)V(x)>0\). Note that

\( D(\phi V) = D\phi V + \phi DV \)

This leads us to the following representation of \(DV(x)\),

\( DV(x) = -\frac{V(x)}{\phi(x)}D\phi(x) \)

since, by definition, \(D(\phi V)(x) = 0\).

Therefore, it suffices to equivalently solve

\( H\left(x, V(x), -\frac{V(x)}{\phi(x)}D\phi(x)\right) = 0 \)

The ingenuity in Crandall and Lions' design of the notion of a viscosity solution revolves around allowing stronger smoothness assumptions on \(\phi\) than an \(V\).

### The Definition of a Viscosity Solution

The properties of a viscosity solution are heavily inspired by the discussion above. Let \(\mathcal{X}\subset\mathbf{R}^N\) and \(\mathcal{D}(\mathcal{X})^{+}\) denotes the continuously differentiable, positive functions supported on compact subsets of \(\mathcal{X}\). We also say

\(E_{+}(f)\triangleq\{x\in\mathcal{X}\mid f(x) = \max f > 0\}\)

and likewise

\(E_{-}(f)\triangleq \{x\in\mathcal{X}\mid f(x) = \min f \lt 0\}\)

Now we can define the viscosity solutions.

\begin{align} &E_{+}(\phi\cdot(u-k))\neq\emptyset\implies\\&\quad\exists y\in E_{+}(\phi(u-k)): H\left(y, V(y), -\frac{V(y) - k}{\phi(y)}D\phi(y)\right)\leq 0 \end{align}

Definition (Viscosity Subsolution): A viscosity subsolution is a function in \(\mathcal{C}(\mathcal{X})\) such that for every \(\phi\in\mathcal{D}(\mathcal{X})^{+}\) and \(k\in\mathbf{R}\) we have

\begin{align} &E_{-}(\phi\cdot(u-k))\neq\emptyset\implies\\&\quad\exists y\in E_{-}(\phi(u-k)): H\left(y, V(y), -\frac{V(y) - k}{\phi(y)}D\phi(y)\right)\geq 0 \end{align}

Definition (Viscosity Supersolution): A viscosity supersolution is a function in \(\mathcal{C}(\mathcal{X})\) such that for every \(\phi\in\mathcal{D}(\mathcal{X})^{+}\) and \(k\in\mathbf{R}\) we have

Definition (Viscosity Solution): A viscosity solution is a function that is both a viscosity subsolution and a viscosity supersolution.

### The Value Function as the Unique Viscosity Solution

We'll now show a beautiful property of viscosity solutions, that being that viscosity solutions \(H(x, V, DV) = 0\) (and therefore viscosity solutions of the HJB equation) are unique. This thankfully solves the massive issue discussed above. We'll go over a proof from the seminal paper by Crandall and Lions. If you're not interested in reading the whole proof, you can check out the discussion of its implications below.

Theorem (Uniqueness of Viscosity Solutions): Let \(V\) be a bounded viscosity subsolution of \(H(x, V, DV) = 0\) and let \(V'\) be a bounded viscosity supersolution of \(H(x, V', DV') = m(x)\), where \(m:\mathbf{R}^N\to\mathbf{R}\) is a bounded continuous function. Moreover, let \(R_0 = \max(|V|/{L^{{\infty}(\mathbf{R}}N)}, \lvert V'\rvert/{L^{{\infty}(\mathbf{R}}N)})\). We make the following assumptions:

- For any \(R > 0\), \(H\) is uniformly continuous on \(\mathbf{R}^N\times[-R,R]\times B(0,R)\)
- For each \(R > 0\), there is a continuous non-decreasing function \(\gamma_R:[0,2R]\to\mathbf{R}\) with \(\gamma_R(0)=0\) and \( H(x, r, p) - H(x, s, p)\geq\gamma_R(r - s) \) for each \(x, p\) and \(r, s\in[-R,R]\) with \(r\geq s\).
For each (R

\begin{align} \limsup_{\varepsilon\downarrow 0}\{&|H(x, r, p) - H(y, r, p)| :\\&|p||x-y|\leq R_1, \lvert x-y\rvert\leq\varepsilon, r\leq R_2\} = 0 \end{align}_{1,R}_{2}>0), the following holds:Then, we have

\begin{align*} \|\gamma((V-V')^{+})\|_{L^{\infty}(\mathbf{R}^N)}\leq\|m^{+}\|_{L^{\infty}(\mathbf{R}^N)} \end{align*}where \(f^+ = \max(f, 0)\).

First thing's first, we wanted to examine the uniqueness of viscosity solutions to the HJB equation. What does the above theorem say with regard to this? Well, consider \(m\equiv 0\). Then, when the theorem holds, it follows that both \(\|\gamma((V-V')^+)\|_{L^\infty}\leq 0\) and \(\|\gamma((V'-V)^+)\|_{L^\infty}\leq 0\). Equivalently, this means that \(\|\gamma(V - V')\|_{L^\infty}\leq 0\). However, since \(\gamma\) is non-decreasing and \(\gamma(0) = 0\), it follows that \(\|V-V'\|_{L^\infty}\leq 0\), and \(V\equiv V'\) – this ensures that viscosity solutions are unique.

Now let's take a look at how this theorem works. We'll concern ourselves simply with the \(m\equiv 0\) scenario for simplicity, and because this is the relevant setting for proving uniqueness. Let \(\phi\in\mathcal{D}(\mathbf{R}^N)^+\) where \(\phi\in[0,1]\) and \(\phi(0) = 1\). Define

\( M\triangleq\max_{x, y\in\mathbf{R}^N}\phi(x - y)(V(x) - V'(y)) \)

Note that \(V(x) - V'(Y)\leq M\) regardless of the particular choice of \(\phi\), which implies that \(\|V-V'\|_{L^\infty(\mathbf{R}^N)}\leq M\).

Now define, for any \(\varepsilon>0\),

\begin{align} M_\varepsilon&\triangleq\max_{x,y\in\mathbf{R}^N}\phi(x - y)(e^{-\varepsilon|x|^2}V(x) - e^{-\varepsilon|y|^2}V'(y))\\&\triangleq\phi(x_\varepsilon-y_\varepsilon)(e^{-\varepsilon|x_\varepsilon|^2}V(x_\varepsilon) - e^{-\varepsilon|y_\varepsilon|^2}V'(y_\varepsilon)) \end{align}Since \(\lim_{\varepsilon\to 0}e^{-\varepsilon x^2} = 0\), and \(V, V'\) are continuous, it follows that \(\lim\inf_{\varepsilon\downarrow 0}M_\varepsilon\geq M\). Now let \(\phi\) have support \(B(0,\alpha)\) (the open ball of radius \(\alpha\)), to ensure that \(|x_\varepsilon-y_\varepsilon|\leq\alpha\) for some fixed \(\alpha\geq 0\). Note that if \(|x|\) or \(|y|\) are unbounded, we'd have

\( M_{\varepsilon}\overset{|x_\varepsilon|\uparrow\infty}{\longrightarrow} 0 \lt M \)

We conclude that \(\lvert x_\varepsilon\rvert\) and \(\lvert y_\varepsilon\rvert\) must be bounded, and say

\( \lvert x_\varepsilon\rvert,|y_\varepsilon|\leq C/\sqrt\varepsilon \)

for some \(C\in\mathbf{R}\). Proceeding with some algebra, we have

\begin{align} M_\varepsilon&\leq\phi(x_\varepsilon-y_\varepsilon)\left(V(x_\varepsilon) - e^{\varepsilon(|x_\varepsilon|^2 - |y_\varepsilon|^2)}V'(y_\varepsilon)\right)\\ &= \phi(x_\varepsilon-y_\varepsilon)\left(V(x_\varepsilon) - e^{\varepsilon|\langle x_\varepsilon - y_\varepsilon, x_\varepsilon + y_\varepsilon\rangle|}V'(y_\varepsilon)\right)\\ &= \phi(x_\varepsilon-y_\varepsilon)\left(V(x_\varepsilon) - V'(y_\varepsilon) + (1 - e^{\varepsilon|\langle x_\varepsilon - y_\varepsilon, x_\varepsilon + y_\varepsilon\rangle|})V'(y_\varepsilon)\right)\\ &\leq M + \left\lvert 1 - e^{\varepsilon|\langle x_\varepsilon - y_\varepsilon, x_\varepsilon + y_\varepsilon\rangle|}\right\rvert V'(y_\varepsilon) \end{align}But, we have

\begin{align} \varepsilon\lvert\langle x_\varepsilon - y_\varepsilon, x_\varepsilon+y_\varepsilon\rangle\rvert &\leq \varepsilon(|x_\varepsilon| + |y_\varepsilon|)|x_\varepsilon - y_\varepsilon|\\ &\leq 2\varepsilon\left(\frac{C}{\sqrt\varepsilon}\right)\alpha\\ &= 2\sqrt\varepsilon C\alpha \end{align}Therefore, we have \(M\leq\lim_{\varepsilon\downarrow 0}M_{\varepsilon}\leq M\), so \(M_\varepsilon\to M\). Now it follows that

\begin{align} x_\varepsilon&\in E_+\bigg(\phi(\cdot - y_\varepsilon)e^{-\varepsilon|\cdot|^2}\left(V(\cdot) - \psi_+(\cdot)\right)\bigg)\qquad\psi_+(x) = e^{\varepsilon(|x|^2 - |y_\varepsilon|^2)}V'(y_\varepsilon)\\ y_\varepsilon&\in E_-\bigg(\phi(x_\varepsilon - \cdot)e^{-\varepsilon|\cdot|^2}\left(V'(\cdot) - \psi_-(\cdot)\right)\bigg)\qquad\psi_-(y) = e^{\varepsilon(|y|^2 - |x_\varepsilon|^2)}V(x_\varepsilon)\\ \end{align}So, by some simple calculus, we have

\begin{align} H\bigg(x_\varepsilon, V(x_\varepsilon), -(V(x_\varepsilon) - k_+)\frac{(D\phi)(x_\varepsilon - y_\varepsilon)}{\phi(x_{\varepsilon} - y_\varepsilon)} + 2\varepsilon x_\varepsilon V(x_\varepsilon)\bigg) &\leq 0\\ H\bigg(y_\varepsilon, V'(y_\varepsilon), -(k_- - V'(y_\varepsilon))\frac{(D\phi)(x_\varepsilon - y_\varepsilon)}{\phi(x_{\varepsilon} - y_\varepsilon)} + 2\varepsilon y_\varepsilon V'(y_\varepsilon)\bigg) &\geq 0\\ \end{align}
where (k_{+≜ψ}/+(x/ε),
k_{-≜ψ}/-(y/ε)).

Now let's define some terms for shorthand,

\begin{align} \lambda_\varepsilon&\triangleq -(V(x_\varepsilon) - V'(y_\varepsilon))\frac{(D\phi)(x_\varepsilon - y_\varepsilon)}{\phi(x_\varepsilon - y_\varepsilon)}\\ \delta_\varepsilon^+&\triangleq\left(e^{\varepsilon(|x_\varepsilon|^2 - |y_\varepsilon|^2)} - 1\right)V'(y_\varepsilon)\frac{(D\phi)(x_\varepsilon - y_\varepsilon)}{\phi(x_\varepsilon - y_\varepsilon)} + 2\varepsilon x_\varepsilon V(x_\varepsilon)\\ \delta_\varepsilon^-&\triangleq\left(1-e^{\varepsilon(|y_\varepsilon|^2 - |x_\varepsilon|^2)}\right)V(x_\varepsilon)\frac{(D\phi)(x_\varepsilon - y_\varepsilon)}{\phi(x_\varepsilon - y_\varepsilon)} + 2\varepsilon y_\varepsilon V'(y_\varepsilon)\\ \end{align}This way, we have

\begin{align} H(x_\varepsilon, V(x_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^+)&\leq 0\\ H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^-)&\geq 0\\ \therefore H(x_\varepsilon, V(x_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^+)-H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^-)&\leq 0\\ \end{align}By assumption, we know that

\( H(x_\varepsilon, V(x_\epsilon), \lambda_\varepsilon + \delta^+_\varepsilon) - H(x_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon)\geq\gamma(V(x_\varepsilon) - V'(y_\varepsilon)) \).

So, we rewrite the inequality above as follows,

\begin{align} &\left[H(x_\varepsilon, V(x_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^+) - H(x_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta_\varepsilon^+)\right]\\ &\qquad + \left[H(x_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon) - H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon)\right]\\ &\qquad + \left[H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon) - H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^-_\varepsilon)\right]\leq 0 \end{align}We can now conclude, since \(V(x_\varepsilon) - V'(y_\varepsilon)\geq M_\varepsilon\) and \(\gamma\) is non-decreasing, that

\begin{align} \gamma(M_\varepsilon)&\leq A_\varepsilon + B_\varepsilon\\ A_\varepsilon&\triangleq\left\lvert H(x_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon) - H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon)\right\rvert\\ B_\varepsilon&\triangleq\left\lvert H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^+_\varepsilon) - H(y_\varepsilon, V'(y_\varepsilon), \lambda_\varepsilon + \delta^-_\varepsilon)\right\rvert\\ \end{align}We'll proceed by analyzing \(A_\varepsilon\). Since \(x_\varepsilon, y_\varepsilon\) maximize \(\phi(x_\varepsilon - y_\varepsilon)(V(x_\varepsilon) - V'(y_\varepsilon))\), \(\lim\inf_{\varepsilon\downarrow 0}\phi(x_\varepsilon - y_\varepsilon) = 0\) implies that \(V(x_\varepsilon) - V'(y_\varepsilon) = 0\). In either case, since \(V, V'\) are bounded, \(0\leq\phi\leq 1\), and \(\phi(0) = 1\), we have

\begin{align} \lim\sup_{\varepsilon\downarrow 0}\left\lvert\lambda_\varepsilon + \delta_\varepsilon^+\right\rvert\leq K + 2\varepsilon\alpha\frac{C}{\sqrt{\varepsilon}}V(x_\varepsilon)\leq K\in\mathbf{R} \end{align}Now, since \(\lvert x_\varepsilon - y_\varepsilon\lvert\leq\alpha\), we have

\begin{align} \lim\sup_{\varepsilon\downarrow 0}A_\varepsilon\leq\sup\left\{H(x, V'(y), p)\bigg\lvert \lvert x-y\rvert\leq\alpha, |x-y||p|\leq \alpha K, V'(y)\leq R_0\right\} = 0 \end{align}where the final equality holds by assumption 3.

Finally, we analyze \(B_\varepsilon\). As discussed in the analysis of \(A_\varepsilon\), \(\lambda_\varepsilon\) must be bounded, and clearly \(\delta_\varepsilon^+\to 0\) and \(\delta_\varepsilon^-\to 0\) as \(\varepsilon\downarrow 0\). It follows that

\begin{align} \lim\sup_{\varepsilon\downarrow 0}B_\varepsilon&\leq\left\lvert H\left(y_\varepsilon, V'(y_\varepsilon), \lim\sup_{\varepsilon\downarrow 0}(\lambda_\varepsilon + \delta_\varepsilon^+)\right) - H\left(y_\varepsilon, V'(y_\varepsilon), \lim\sup_{\varepsilon\downarrow 0}(\lambda_\varepsilon + \delta_\varepsilon^-)\right)\right\rvert\\ &\leq\left\lvert H\left(y_\varepsilon, V'(y_\varepsilon), \lim\sup_{\varepsilon\downarrow 0}\lambda_\varepsilon\right) - H\left(y_\varepsilon, V'(y_\varepsilon), \lim\sup_{\varepsilon\downarrow 0}\lambda_\varepsilon\right)\right\rvert\\ &= 0 \end{align}Thus, we arrive at the glorious conclusion that

\begin{equation} 0 = \lim\sup_{\varepsilon\downarrow 0}(A_\varepsilon + B_\varepsilon)\geq\gamma\left\|\left((V-V')^+\right)\right\|_{L^\infty(\mathbf{R}^N)} \end{equation}### Discussion

So we've shown that viscosity solutions to \(H(x, r, p) = 0\) are
unique. With regard to the RL setting described above, this is desired
since the value function *should* be unique. However, it's worth
considering the conditions that are imposed in the Hamiltonian in the
theorem above.

#### The Assumptions

Recall that for the purpose of continuous-time RL, the Hamiltonian of interest is that which corresponds to the HJB equation,

\begin{equation} H(x, r, p) = -r + R + \sup_{\pi}\langle p, f(x, \pi(x))\rangle \end{equation}The first assumption is that \(H\) is uniformly continuous. When the control follows the optimal policy \(\pi^\star\), we have \(H(x, r, p) = -r + R + \langle p, f(x, \pi^\star(x))\rangle\). If \(f\) is uniformly continuous, then this condition is satisfied.

The second assumption regards the existance of the non-decreasing function \(\gamma\) with \(\gamma(0) = 0\) and \(H(x, r, p) - H(x, s, p)\geq\gamma(r - s)\) when \(r\geq s\). Note that \(H(x, r, p) - H(x, s, p) = r - s\). Letting \(\gamma = \mathsf{Id}\) satisfies this assumption regardless of the value functions and dynamics.

The last assumption asserts that where the value function and its gradient are bounded, in the limit as \(|x-y|\to 0\), \(H(x,\cdot,\cdot)\equiv H(y, \cdot, \cdot)\). This is essentially a smoothness/continuity constraint on \(H\), jointly in its arguments. We have

\begin{align} \lim_{\varepsilon\downarrow 0}\sup\left\{\left\lvert H(x, r, p) - H(y, r, p)\right\rvert\right\} &= \lim_{\varepsilon\downarrow 0}\sup\left\{\left\lvert\langle p, f(x, \pi^\star(x))\rangle - \langle p, f(y, \pi^\star(y))\rangle\right\rvert\right\}\\ &= \lim_{\varepsilon\downarrow 0}\sup\left\{\left\lvert\langle p, f(x, \pi^\star(x)) - f(y, \pi^\star(y))\rangle\right\rvert\right\}\\ &= 0 \end{align}This suggests an interesting way to approximate the value function and its gradient.

Let \(\mathcal{X}\subset\mathbf{R}^N\) be the state space, and \(\mathcal{X}_{\xi}\) is a lattice representing the discretized approximation to \(\mathcal{X}\) defined according to

\begin{equation} \mathcal{X}_\xi\triangleq\left\{\sum_{i=1}^{\mathsf{dim}\mathcal{X}}k_ie_i\mid k_i\in\mathbf{Z}\right\} \end{equation}where \(\{e_i\}\) is an orthonormal basis for \(\mathcal{X}\). Moreover, let \(f = f(x, \pi(x))\) for the agent's policy (π). Then we approximate the gradient of the value function according to

\begin{align} \langle\nabla_xV(x), f\rangle\approx\langle\Delta^+V(x), f^+\rangle + \langle\Delta^-V(x), f^-\rangle \end{align}where

\begin{align} f^\pm_i(x, \pi(x))&\triangleq\max(\pm f_i(x, \pi(x)), 0)\\ \Delta^\pm_iV(x)&\triangleq\frac{V(x\pm e_i) - V(x)}{\xi} \end{align}In other words, when a state component moves in the positive direction the "right-derivative" is used to compute the corresponding element of the value function gradient, and when a state component moves in the negative direction the "left-derivative" is used. Now, the joint continuity constraint of the Hamiltonian reduces to asserting that for a fixed dynamics direction, the difference in Hamiltonians evaluated at two arbitrary states is arbitrarily small. This constraint would be satisfied by piecewise continuous value functions, for instance.