Safety vs. Performance: How Multi-Objective Learning Reduces Barriers to Market Entry

TL;DR

A multi-objective high-dimensional regression framework that captures reputational damage is defined, and the number of data points that a new company needs to enter the market is characterized, demonstrating how multi-objective considerations can fundamentally reduce barriers to entry.

Abstract

Emerging marketplaces for large language models and other large-scale machine learning (ML) models appear to exhibit market concentration, which has raised concerns about whether there are insurmountable barriers to entry in such markets. In this work, we study this issue from both an economic and an algorithmic point of view, focusing on a phenomenon that reduces barriers to entry. Specifically, an incumbent company risks reputational damage unless its model is sufficiently aligned with safety objectives, whereas a new company can more easily avoid reputational damage. To study this issue formally, we define a multi-objective high-dimensional regression framework that captures reputational damage, and we characterize the number of data points that a new company needs to enter the market. Our results demonstrate how multi-objective considerations can fundamentally reduce barriers to entry -- the required number of data points can be significantly smaller than the incumbent company's dataset size. En route to proving these results, we develop scaling laws for high-dimensional linear regression in multi-objective environments, showing that the scaling rate becomes slower when the dataset size is large, which could be of independent interest.

Figures (4)

• Figure 1: Market-entry threshold $N_E^*$ as a function of the incumbent's safety constraint $\tau_I$, when the incumbent has infinite data and entrant has no safety constraint (Theorem \ref{['thm:tradeoffwarmup']}). The plots show varying values of the scaling exponent $\nu$ where the correlation parameter $\rho = 0.5$ is held fixed (left) and varying values of $\rho$ where $\nu = 0.34$ is held fixed (right). The market-entry threshold $N_E^*$ is finite. It is also higher when the constraint $\tau_I$ is weaker, when the correlation $\rho$ is stronger, and when the scaling exponent $\nu$ is lower.
• Figure 2: Data scaling laws for multi-objective environments where a fraction $\alpha = 0.9$ of the data is labelled according to the primary objective and a fraction $1-\alpha = 0.1$ is labelled according to the secondary objective. The plots show, up to constants, the loss $\Theta(\inf_{\lambda \in (0, 1)} \mathbb{E}[L_1(\hat{\beta}(\alpha, \lambda, X))])$ (left, Theorem \ref{['thm:scalinglawoptreginformal']}) and excess loss $\Theta(\inf_{\lambda \in (0, 1)} (\mathbb{E}[L_1(\hat{\beta}(\alpha, \lambda, X)) - L_1(\beta(\alpha, 0))]))$ (right, Theorem \ref{['thm:scalinglawoptregexcessinformal']}) as a function of the total number of training data points $N$. The loss and excess loss both take the form $N^{-c}$, but where the scaling exponent $c$ takes on multiple (two or three) different values depending on the size of $N$ relative to other parameters. The scaling exponent is smaller when $N$ is larger, thus demonstrating that the scaling rate becomes slower as the dataset size $N$ increases.
• Figure 3: The market-entry threshold $N_E^*$ as a function of the incumbent dataset size $N_I$, when the new company has no safety constraint (Theorem \ref{['thm:finitedata']}). The plots show varying values of the scaling exponent $\nu$ where the correlation parameter $\rho = 0.5$ is held fixed (left) and varying values of $\rho$ where $\nu = 0.34$ is held fixed (right). When $N_I$ is sufficiently large, the market-entry threshold $N_E^*$ is asymptotically less than $N_I$ (i.e., below the dotted black line). Each curve is the union of three line segments with slope decreasing in $N_I$, demonstrating that the new company can afford to scale up their dataset at a slower rate as $N_I$ increases.
• Figure 4: The market-entry threshold $N_E^*$ as a function of the difference $D$ between the infinite-data performance loss of the incumbent and new company, when the incumbent has infinite data (Theorem \ref{['thm:alignment']}). The plots show varying values of the scaling exponent $\delta$ where the correlation parameter $\rho = 0.49$ is held fixed (left) and varying values of $\rho$ where $\delta = 2.5$ is held fixed (right). The plots are shown in log space. The market-entry threshold is finite in all cases. Each curve is the union of multiple line segments with slope increasing in magnitude as $\log D$ decreases, demonstrating that the new company needs to scale up their dataset at a faster rate as $D$ decreases.

Theorems & Definitions (104)

• Definition 1
• Example 1
• Theorem 1
• proof : Proof sketch of Theorem \ref{['thm:tradeoffwarmup']}
• Theorem 2: Informal Version of Corollary \ref{['cor:scalinglawoptreg']}
• Theorem 3: Informal Version of Corollary \ref{['cor:scalinglawoptregexcess']}
• Theorem 4
• proof : Proof sketch
• Theorem 5
• proof : Proof sketch