Real-time decomposition of a signal into a sum of responses to labeled events

Martin Jambon, July 2017

Given events occurring along a discrete timeline, the goal is to determine the effect of each event kind. The observable is a real-valued signal \(\phi\) which is assumed to be a sum of the effects of recent events. The effects of any event are assumed negligible past a certain delay \(w\).

The solution consists in adjusting the contributions of each event to the signal knowing which events occurred recently from \(t-w+1\) to \(t\) and the latest value of the signal, \(\phi(t)\). The original predicted contributions are corrected so as to predict \(\phi(t)\) perfectly by absorbing a share of the difference with the predicted signal \(\hat{\phi}(t)\). Each share is determined roughly by the recent standard deviation of the contribution, allowing the predictions of stable contributions to eventually converge, regardless of fluctuations in other contributions.

A valuable property of this system is that only resources corresponding to active events are involved, as opposed to systems in which all the nodes are updated at every time step.

1 Motivation in the context of artificial general intelligence

The problem we are trying to solve arises while developing a cognitive system that operates in real-time and is driven by a single goal function. In this context, a goal function \(\phi\) is a real-valued signal over discrete time, whose value becomes available at each time step. It represents how well the system is doing, i.e. it would combine rewards and penalties. Some of these rewards and penalties may correspond to direct interactions of the system with its environment, such as acquiring food or losing energy. Other rewards and penalties may be generated internally by the system itself, as a way to encourage itself to pursue certain paths.

In such a cognitive system, there is a finite (but possibly growing) collection of possible actions. Each action can be viewed as a button. We call an act an instance of an action, i.e. the press of a button. An act is a pair (action, instant). An action, naturally can correspond to a modification of the system in its environment, such as an attempt to move forward. It can also directly feed back into the system’s input ports without directly affecting the environment. No matter what kind of action is triggered, it takes some amount of time to have an effect on the goal function. Here and throughout this paper, we assume that most meaningful effects of an action occur within a given time window, which can be a short as 5 steps. We choose this window roughly to capture reactions of the system to its own decisions, so it has enough time to “realize” what it just did and produce a self-reward or a self-penalty. Longer-term effects of course exist and will have to be dealt with differently.

Given such a goal function \(\phi\) and the knowledge of which actions were triggered within a time window of length \(w\), we wish to determine the impact of each action on \(\phi\).

2 General design contraints

2.1 Independence from context

The response to an action is considered the same independently from the context. In the case of artificial general intelligence (AGI), this is generally not the case. However, it is possible to create as many controls (“buttons”) as there are contexts. Instead of studying the response to an action regardless of the context, we can study the response to the pair (context, action).

2.2 Effects can be delayed

We wish to capture the “immediate” effects of an action. However, our system is such that during a time step it can only propagate information from one node to another. It take normally several steps for some input information to reach the nodes in charge of triggering actions.

So, while we are only interested in the immediate effects of actions, we need to leave sufficient time for the system to react to such effects.

2.3 Overlapping effects

Multiple actions can take place simultaneously, or close enough that their effects on the goal function overlap. Our main challenge is to decompose the signal into a combination of responses from multiple recent actions.

The simplest approach is to model \(\phi\) as a sum of responses to actions, and this is the one we’ll follow.

2.4 Incremental learning

It should be possible to start inferring the effects of new actions when they start appearing. If only one action is new and the effects of all the other actions are already well known, no relearning should be necessary, and learning the effects of the new action should be as fast as if it were the only action.

2.5 Adaptation

If the effects of actions change progressively over time, the system should be able to adapt. The adaptation rate should not slow down as the system ages.

3 Notations

Our approach doesn’t require the notion of goal function or actions. We’ll simply refer to the goal function \(\phi\) as the signal. Each act is an action occurring at a given time and will be called an event. The action is the kind of the event.

The variable \(k\) will be used typically to identify each event uniquely among the set of events \(K\).

Each event is a pair \((E_k, t_k)\) of the kind \(E_k\) and of the instant \(t_k\) at which it occurred.

An event kind is associated with a function that represents its linear contribution to \(\phi\). We’ll denote \(E_k(\tau)\) the value of this function at \(\tau\), where \(\tau\) is the time elapsed since the occurrence of an event of this kind. Naturally, the event has no contribution to \(\phi\) before it occurs, so we have:

\[ \forall k \in K: \forall \tau < 0: E_k(\tau) = 0 \]

Additionally, our model assumes that the effect of any event doesn’t last longer than \(w\), called window length or just window:

\[ \forall k \in K: \forall \tau \ge w: E_k(\tau) = 0 \]

The only interesting values of \(E_k\) are the \(w\) values \(E_k(0), E_k(1), \dots, E_k(w-1)\). These are the values that our algorithm will try to determine, for each event kind \(k \in K\).

The signal \(\phi\) is modeled as the sum of the effects of all events, i.e.

\[ \phi: t \rightarrow \sum_{k \in K} E_k(t-t_k) \]

Given the properties of \(E_k\) mentioned above, all events occurring outside the window (\(t_k \notin [t - w + 1, t]\)) can be ignored in the prediction of \(\phi(t)\), denoted \(\hat{\phi}(t)\).

In general, a hat \(\hat{}\) on a variable denotes a prediction of this variable.

Since estimators have a state that changes over time, we use a parenthesized superscript to specify which state we’re referring to. In the following example, we use the variable \(\hat{\mu}\) to represent the exponential moving average of the sequence \(x\). \(\hat{\mu}\) is updated as follows:

\[ \hat{\mu}^{(t+1)} = r x_t + (1-r) \hat{\mu}^{(t)} \]

which could be implemented as the in-place assignment

\[ \hat{\mu} \leftarrow r x_t + (1-r) \hat{\mu} \]

4 Solution

4.1 Outline

Informally, the solution consists in maintaining for each event kind a number that represents the expected effect of this action after some delay. Multiple event of different types can take place simultaneously, each with an expected effect which is a contribution to the signal at some future instant \(t\). The expected value of the signal is the sum of the contributions at \(t\) of the recent events. The observed value of the signal is used to correct the expected contributions into what each contribution should have been. The contributions are not corrected evenly, but according to a weight proportional to the standard deviation of the contribution. In other words, the more a contribution fluctuates, the more it is prone to being corrected. Conversely, a contribution that fluctuates less will receive a smaller correction relative to the other co-occurring contributions.

4.2 Description

4.2.1 Prediction and update

At each step \(t\) of the computation, the value of the signal is observed and denoted \(\phi(t)\).

All the events that occurred within the last \(w\) steps are assumed to contribute to the signal. The predicted signal at instant \(t\) is denoted \(\hat{\phi}(t)\) and is computed as follows:

\[ \hat{\phi}(t) = \sum_{\{ k \in K | t_k \in [t-w+1, t] \}} \hat{E}_k^{(t)}(t-t_k) \]

As soon as \(\phi(t)\) is known, each predicted term \(\hat{E}_k^{(t)}(t-t_k)\) is corrected so as to add up to the observed \(\phi(t)\):

\[ \phi(t) = \sum_{\{ k \in K | t_k \in [t-w+1, t] \}} \hat{E}^{(t+1)}(t-t_k) \]

The difference between the prediction and the actual signal is denoted \(\delta\):

\[ \delta_t = \hat{\phi}(t) - \phi(t) \]

The terms are corrected by receiving each a share of \(\delta_t\):

\[ \hat{E}^{(t+1)}(t-t_k) = \hat{E}^{(t)}(t-t_k) - v_k^{(t)}(t-t_k) \delta_k \]

where each weight \(v_k^{(t)}(t-t_k)\) is nonnegative. All weights at instant \(t\) add up to 1. They are determined so as to reflect the uncertainty on each contributing term, and so that the terms predicted consistently with high certainty will be corrected by a small amount while the more uncertain terms will be corrected by a greater amount. An estimation of the standard deviation of each term is used for this purpose:

\[ v_k^{(t)}(t-t_k) = \frac{ \hat{\sigma}_k^{(t)}(t-t_k) } { \sum_{\{ k \in K | t_k \in [t-w+1, t] \}} \hat{\sigma}_k^{(t)}(t-t_k) } \]

where \(\hat{\sigma}_k^{(t)}(t-t_k)\) is an estimate of the standard deviation of \(\hat{E}_k\) based on the earlier known values of \(\hat{E}_k\).

Note that standard deviations are estimated using exponential smoothing because of their simplicity and low memory requirements, but other methods should work well too. Until a certain number of samples is reached, they behave like the classic sample mean and sample standard deviation estimators. Beyond that short initial phase, they give more weight to recent values.

4.2.2 Details and refinements

There are two classes of special cases where the standard deviation cannot be used to determine the weight:

  1. In the presence of fewer than 2 samples, the sample standard deviation is undefined.
  2. If the initial samples are all equal, the estimated standard deviation is 0, which results in \(\hat{E}_k\) being not updated ever again.

For special case (1), a possible trick is to pretend the standard deviation is so large that all the other weights are negligible, except those in the similar situation with an undefined standard deviation. Given \(n\) such problematic terms, we can assign them each a weight of \(1/n\), and assign a weight of \(0\) to the terms whose standard deviation is defined.

For special case (2), we can impose a minimum share to each term, even if the standard deviation is zero.

Additionally, we observed that with noisy background noise, non-noisy terms converge much faster if the weights are not exactly proportional to the standard deviations, but if we slightly amplify the top weights.

The final recipe for obtaining the weights consists in the following:

  1. Express the standard deviations relative to the maximum standard deviation among all the terms, denoted \(r_k\) for term \(k\).
  2. Transform each \(r_k\) into \(f(r_k)\) to favor larger values and avoid null weights, using for example \(f : x \rightarrow \max(0.001, 0.5 x^{1.3} + 0.5 x)\), then normalize them such that they add up to 1.

4.3 Selected scenarios

A series of tests was conducted to evaluate the behavior of the system with different parameters and under different conditions.

The source code is currently available at, commit c725d8dd.

4.3.1 Shared protocol

A single run starts from time \(t=0\) and runs as many steps as necessary to reach specified conditions. This number of steps \(T\) will be our main criterion for determining how well the system performs.

These runs being non-deterministic, we repeat them 100 times, and report simple statistics on the value of \(N\) in each case.

The default setup consists in the following:

For example, if action \(A\) is triggered at some step \(t\) and action \(B\) is triggered at \(t+1\), and no action took place earlier and no action takes place after that, the values of the goal function \(\phi\) are: \[ \begin{eqnarray} \dots && \\ \phi(t-1) &=& 0\\ \phi(t) &=& 1 \\ \phi(t+1) &=& -0.5 + 0.1 \\ \phi(t+2) &=& 0.25 + 0.2 \\ \phi(t+3) &=& 0.05\\ \phi(t+4) &=& 0\\ \dots && \end{eqnarray} \]

We’ll predict the values of \(E_A\) and \(E_B\) and look at how many steps it takes converge to the expected values \(E_A(0)\) and \(E_B(0)\). The convergence criteria are the following:

\[ \mathrm{Condition}_A: \left| \hat{E}_A^{(T)}(0) - E_A(0) \right| \le \epsilon_A \]

\[ \mathrm{Condition}_B: \left| \hat{E}_B^{(T)}(0) - E_B(0) \right| \le \epsilon_B \]

with the following default error thresholds:

\[ \begin{eqnarray} \epsilon_A &=& 0.05 \\ \epsilon_B &=& 0.005 \end{eqnarray} \]

We assume convergence if a given condition remains valid for a large number (1000) of consecutive steps.

4.3.2 Default setup

This setup uses only the default parameters described in the previous section. Number of steps to converge to Condition\(_{A}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 19.1 \dots 147.2\\ \mathrm{median} &=& 70.0\\ \hat{\mu} &=& 73.7\\ \hat{\sigma} &=& 48.9\\ \end{eqnarray} \]

Number of steps to converge to Condition\(_{B}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 95.8 \dots 231.3\\ \mathrm{median} &=& 149.0\\ \hat{\mu} &=& 156.1\\ \hat{\sigma} &=& 53.7\\ \end{eqnarray} \]

4.3.3 Window of 1

In this setup, each action only has an immediate effect \(E_A(0)\) or \(E_B(0)\) and no delayed effect (\(E_A(1)\), \(E_B(1)\), etc.).

Additionally, the window length \(w\) is 1, which is the smallest useful window possible. This results in very fast convergence as shown below.

Number of steps to converge to Condition\(_{A}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 1.0 \dots 9.0\\ \mathrm{median} &=& 3.0\\ \hat{\mu} &=& 4.6\\ \hat{\sigma} &=& 4.1\\ \end{eqnarray} \]

Number of steps to converge to Condition\(_{B}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 1.0 \dots 12.0\\ \mathrm{median} &=& 4.0\\ \hat{\mu} &=& 5.5\\ \hat{\sigma} &=& 5.5\\ \end{eqnarray} \]

4.3.4 Large difference between contributions

In this setup, \(E_A(0)\) is 100 instead of 1, while \(E_B(0)\) remains 0.1.

Number of steps to converge to Condition\(_{A}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 49.6 \dots 353.0\\ \mathrm{median} &=& 235.0\\ \hat{\mu} &=& 215.3\\ \hat{\sigma} &=& 118.3\\ \end{eqnarray} \]

Number of steps to converge to Condition\(_{B}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 141.4 \dots 455.5\\ \mathrm{median} &=& 303.0\\ \hat{\mu} &=& 300.2\\ \hat{\sigma} &=& 120.1\\ \end{eqnarray} \]

It takes 2-4 times longer to converge in this setup than in the default setup. A crude explanation is that the relative error tolerance is now smaller, since \(\epsilon_A\) and \(\epsilon_B\) are unchanged but the gap between extreme values has increased.

4.3.5 Background noise

In this setup, we study the effects of permanent background noise on the determination of constant contribution \(E_A(0)\).

In order to produce noise, each event \(B\) is set to occur at each time step (\(P(B)=1\)) and its effects are shifted by the same random number following a centered normal distribution of parameter \(\sigma=0.5\). The control uses no noise, i.e. \(\sigma=0\).

Number of steps to reach Condition\(_A\) (control):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 26.7 \dots 179.2\\ \mathrm{median} &=& 92.0\\ \hat{\mu} &=& 97.3\\ \hat{\sigma} &=& 66.6\\ \end{eqnarray} \]

Number of steps to reach Condition\(_A\) (background noise):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 484.8 \dots 4706.0\\ \mathrm{median} &=& 1108.0\\ \hat{\mu} &=& 2113.4\\ \hat{\sigma} &=& 1940.1\\ \end{eqnarray} \]

This shows that convergence toward \(E_A(0)\) is slower in the presence of background noise but nonetheless happens reliably.

4.3.6 Interdependent events

In this setup, event \(A\) occurs with a probability of 0.5 only if \(B\) occurs too. \(A\) never occurs alone:

\[\begin{eqnarray} P(B) &=& 0.5 \\ P(A|B) &=& 0.5 \\ P(A|\neg B) &=& 0 \end{eqnarray}\]

The results are similar to the default setup in which \(A\) and \(B\) are independent.

Number of steps to converge to Condition\(_{A}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 4.8 \dots 168.3\\ \mathrm{median} &=& 73.5\\ \hat{\mu} &=& 82.4\\ \hat{\sigma} &=& 64.7\\ \end{eqnarray} \]

Number of steps to converge to Condition\(_{B}\):

\[ \begin{eqnarray} \mathrm{10^{th} \dots 90^{th}\ percentile} &=& 90.3 \dots 297.9\\ \mathrm{median} &=& 178.5\\ \hat{\mu} &=& 187.5\\ \hat{\sigma} &=& 87.8\\ \end{eqnarray} \]

5 Conclusion

It works fine but convergence is particularly fast with a window of length 1 and applications should probably try to take advantage of this.