Exact methods assume you can list every state and store a value for each. For any real project, that assumption dies immediately.
Consider a modest project: a dozen work packages, each in one of several statuses, a budget position, an approvals queue, a handful of live risks. The number of distinct states — every combination at once — runs into the billions. This is the curse of dimensionality: state counts multiply as factors combine, and no table could ever hold them. Continuous quantities such as cost variance make it formally infinite. Approximate value functions are how sequential decision-making survives contact with reality.
1Value as a function, not a lookup
The idea is to stop treating value as a giant lookup table and start treating it as a function of a few meaningful features. Rather than storing a number for every conceivable project state, you describe each state by a compact set of indicators — schedule variance, cost variance, open high-severity risks, approvals outstanding — and fit a function that maps those features to an estimated value. Compute the value at a manageable set of sampled states, then let the function generalise to the vast majority you never explicitly visited.
2A spectrum of approximators
The techniques range from simple to sophisticated, trading transparency against flexibility:
- Nearest neighbour and kernel smoothing estimate a new state's value from the nearby sampled states — intuitive and local.
- Linear and simplex interpolation blend values across a grid of reference points.
- Linear regression on features fits value as a weighted sum of your chosen indicators — compact, fast, and interpretable (you can read off which features drive value).
- Neural network regression learns rich, non-linear value surfaces when the relationships are complex and you have enough data to fit them.
Approximation trades exactness for reach. You give up a perfect value for every state in exchange for a usable estimate at any state — the only way sequential decisions become tractable on a project of real size.
3The feature choice is the model
The quality of an approximate value function lives or dies by its features. Choose indicators that genuinely capture what makes a project state better or worse, and a simple linear fit will serve you well. Choose poorly, and no amount of modelling sophistication rescues it. For risk practitioners this is oddly familiar territory: deciding which few metrics summarise project health is exactly the judgement behind any good dashboard.
You will never have a value for every project state, and you don't need one. Identify the handful of features that actually characterise how healthy a project is, and estimate value as a function of those. Start simple and interpretable — a weighted combination of familiar variance and exposure metrics — before reaching for anything heavier. Your effort belongs in choosing the right features, not in chasing a more elaborate approximator on top of the wrong ones.