Good question! As you suggest in your comment, increasing marginal returns to capacity induce monosemanticity, and decreasing marginal returns induce polysemanticity.
We observe this in our toy model. We didn’t clearly spell this out in the post, but the marginal benefit curves labelled from A to F correspond to points in the phase diagram. At the top of the phase diagram where features are dense, there is no polysemanticity because the marginal benefit curves are increasing (see curves A and B). In the feature sparse region (points D, E, F), we see polysemanticity because the marginal benefit curves are decreasing.
The relationship between increasing/decreasing marginal returns and polysemanticity generalizes beyond our toy model. However, we don’t have a generic technique to define capacity across different architectures and loss functions. Without a general definition, it’s not immediately obvious how to regularize the loss for increasing returns to capacity.
You’re getting at a key question the research brings up: can we modify the loss function to make models more monosemantic? Empirically, increasing sparsity increases polysemanticity across all models we looked at (figure 7 from the arXiv paper)*. According to the capacity story, we only see polysemanticity when there is decreasing marginal returns to capacity. Therefore, we hypothesize that there is likely a fundamental connection between feature sparsity and decreasing marginal returns. That is to say, we are suggesting that: if features are sparse and similar enough in importance, polysemanticity is optimal.
*Different models showed qualitatively different levels of polysemanticity as a function of sparsity. It seems possible that tweaking the architecture of a LLM could change the amount of polysemanticity, but we might take a performance hit for doing so.
we don’t have a generic technique to define capacity across different architectures and loss functions
Got it. I imagine that for some particular architectures, and given some particular network weights, you can numerically compute the marginal returns to capacity curves, but that it’s hard to express capacity analytically as a function of network weights since you really need to know what the particular features are in order to compute returns to capacity—is that correct?
Good question! As you suggest in your comment, increasing marginal returns to capacity induce monosemanticity, and decreasing marginal returns induce polysemanticity.
We observe this in our toy model. We didn’t clearly spell this out in the post, but the marginal benefit curves labelled from A to F correspond to points in the phase diagram. At the top of the phase diagram where features are dense, there is no polysemanticity because the marginal benefit curves are increasing (see curves A and B). In the feature sparse region (points D, E, F), we see polysemanticity because the marginal benefit curves are decreasing.
The relationship between increasing/decreasing marginal returns and polysemanticity generalizes beyond our toy model. However, we don’t have a generic technique to define capacity across different architectures and loss functions. Without a general definition, it’s not immediately obvious how to regularize the loss for increasing returns to capacity.
You’re getting at a key question the research brings up: can we modify the loss function to make models more monosemantic? Empirically, increasing sparsity increases polysemanticity across all models we looked at (figure 7 from the arXiv paper)*. According to the capacity story, we only see polysemanticity when there is decreasing marginal returns to capacity. Therefore, we hypothesize that there is likely a fundamental connection between feature sparsity and decreasing marginal returns. That is to say, we are suggesting that: if features are sparse and similar enough in importance, polysemanticity is optimal.
*Different models showed qualitatively different levels of polysemanticity as a function of sparsity. It seems possible that tweaking the architecture of a LLM could change the amount of polysemanticity, but we might take a performance hit for doing so.
Thanks for this.
Got it. I imagine that for some particular architectures, and given some particular network weights, you can numerically compute the marginal returns to capacity curves, but that it’s hard to express capacity analytically as a function of network weights since you really need to know what the particular features are in order to compute returns to capacity—is that correct?