How to Relax Business Problems into AI Problems

Instead of solving business problems as they are, AI organizations first turn them into optimizable AI problems. Since complex business goals are hard to solve directly, the key is to relax them into simpler goals and verify whether those assumptions hold in reality.

A representative example is Lasso regression. The original (\ell_0) sparsity constraint is a combinatorial optimization problem and thus hard to solve, but relaxing it into (\ell_1) regularization turns it into a convex problem that is far more tractable. However, the two solutions are not always the same, and assumptions such as the mutual incoherence condition or restricted isometry property must hold for the relaxed solution to be expected to be close to the original problem's solution.

The same logic applies to business. Instead of directly maximizing long-term revenue, it is first decomposed into DAU, PUR, and ARPPU, which are then further broken down into smaller metrics like retention and aha moments. In Azar's recommendation system case, the long-term revenue maximization problem was transformed, through several stages of assumptions, into a problem of predicting conversation time between users.

The unfolding proceeds roughly as follows.

• Approximate long-term revenue as revenue on day N.
• Break down revenue into DAU × PUR × ARPPU.
• Explain DAU through retention and inflow, and further connect retention to aha moments.
• View the matching of each tick as a maximum weight matching problem, and approximate it with a greedy algorithm.
• Ultimately, this comes down to training a conversation time predictor for two users.

The advantage of this approach is debuggability. In this actual case, while retention hit an all-time high, PUR declined, and this was interpreted as being caused by a broken Assumption 2. Afterward, redefining the problem by re-decomposing long-term revenue around the number of purchase-users led to improved business metrics.

The key is not to hide the assumptions. The more a business problem is turned into an AI problem, the easier it becomes to solve, but this also accumulates technical debt in equal measure. Only by explicitly managing which assumptions have broken and where relaxation needs to be redone can AI optimization actually translate into real business impact.