StefanHex's Shortform

StefanHex

PSA: People use different definitions of "explained variance" / "fraction of variance unexplained" (FVU)

${F V U}_{A}$ is the formula I think is sensible; the bottom is simply the variance of the data, and the top is the variance of the residuals. The $∥$ indicates the $L_{2}$ norm over the dimension of the vector $x$ . I believe it matches Wikipedia's definition of FVU and R squared.

{F V U}_{B} = \frac{1}{N} N \sum n = 1 \frac{∥ x_{n} - x_{n, p r e d} ∥^{2}}{∥ x_{n} - μ ∥^{2}}

${F V U}_{B}$ is the formula used by SAELens and SAEBench. It seems less principled, @Lucius Bushnaq and I couldn't think of a nice quantity it corresponds to. I think of it as giving more weight to samples that are close to the mean, kind-of averaging relative reduction in difference rather than absolute.

A third version (h/t @JoshEngels) which computes the FVU for each dimension independently and then averages, but that version is not used in the context we're discussing here.

In my recent comment I had computed my own ${F V U}_{A}$ , and compared it to FVUs from SAEBench (which used ${F V U}_{B}$ ) and obtained nonsense results.

Curiously the two definitions seem to be approximately proportional—below I show the performance of a bunch of SAEs—though for different distributions (here: activations in layer 3 and 4) the ratio differs.^[1] Still, this means using ${F V U}_{B}$ instead of ${F V U}_{A}$ to compare e.g. different SAEs doesn't make a big difference as long as one is consistent.

Thanks to @JoshEngels for pointing out the difference, and to @Lucius Bushnaq for helpful discussions.

^{^}
If a predictor doesn't perform systematically better or worse at points closer to the mean then this makes sense. The denominator changes the relative weight of different samples but this doesn't have any effect beyond noise and a global scale, as long as there is no systematic performance difference.

StefanHex2y*294

{F V U}_{B} = \frac{1}{N} N \sum n = 1 \frac{∥ x_{n} - x_{n, p r e d} ∥^{2}}{∥ x_{n} - μ ∥^{2}}

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3StefanHex2y

Yep, that’s the generalisation that would make most sense

1Oliver Clive-Griffin2y

Thanks. Also, in the case of crosscoders, where you have multiple output spaces, do you have any thoughts on the best way to aggregate across these? currently I'm just computing them separately and taking the mean. But I could see imagine it perhaps being better to just concat the spaces and do fvu on that, using l2 norm of the concated vectors.

StefanHex2y20

Yeah you probably shouldn't concat the spaces due to things like "they might have very different norms & baseline variances". Maybe calculate each layer separately, then if they're all similar average them together, otherwise keep separate and quote as separate numbers in your results

StefanHex's Shortform

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StefanHex's Shortform

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