Temperature Adaptation with a Prior Over Temperatures

Why Put a Prior on Temperature?

Exponential weighting depends on a temperature: a small temperature concentrates weights quickly on predictors with low past loss, while a large temperature changes weights more cautiously. The useful choice depends on how much curvature the loss provides. A temperature that is too small may invalidate a sharp aggregation guarantee; one that is unnecessarily large can make its oracle penalty costly.

A finite prior over candidate temperatures lets us update their weights from observed losses instead of committing to just one. The resulting guarantee pays for selecting a temperature, as well as for selecting a predictor within that temperature’s aggregate. Those are distinct costs, and both are cumulative unless explicitly divided by the sample size.

Setup and One-Temperature Aggregation

At round \(t=1,\ldots,n\), let the available predictions lie in a convex set \(\mathcal{F}\), and let \(\ell_t(f)\) denote the loss of prediction \(f\in\mathcal{F}\). Write \(L_n(f_{1:n})=\sum_{t=1}^n \ell_t(f_t)\) for the cumulative loss of a sequence of predictions. The loss may vary with \(t\). No probabilistic sampling assumption is needed for the results below.

Suppose there are \(J\) candidate predictors \(f_1,\ldots,f_J\) and a prior \(q=(q_1,\ldots,q_J)\), where each \(q_j>0\) and \(\sum_j q_j=1\). At temperature \(\tau\), define the round-\(t\) weights using losses observed before round \(t\):

The aggregate \(g_{\tau,t}\) is a convex combination, so it lies in \(\mathcal{F}\). The following guarantee explains why a temperature is useful when it is admissible.

Proof. Exp-concavity at \(\alpha\) implies exp-concavity at every \(1/\tau\leq\alpha\): the function \(\exp\{-\ell_t(f)/\tau\}\) is the concave function \(\exp\{-\alpha\ell_t(f)\}\) raised to the power \(1/(\alpha\tau)\leq1\), and the composition \(x\mapsto x^{1/(\alpha\tau)}\) is increasing and concave on the nonnegative reals. Thus, by concavity,

Let \(Z_t=\sum_j q_j\exp\{-L_t(f_j)/\tau\}\), with \(Z_0=1\). Substituting the weights into the right-hand side gives \(Z_t/Z_{t-1}\). Taking negative logarithms and summing over rounds therefore yields

For the last inequality, \(Z_n\geq q_j\exp\{-L_n(f_j)/\tau\}\) for every \(j\). Taking negative logarithms proves the claim. \(\square\)

Putting a Prior on Candidate Temperatures

Choose a finite set of temperatures \(\tau_1,\ldots,\tau_K\), and a prior \(\pi=(\pi_1,\ldots,\pi_K)\) with all \(\pi_k>0\) and \(\sum_k\pi_k=1\). For each \(k\), construct its own sequence \(g_{k,t}=g_{\tau_k,t}\) using the same predictor prior \(q\). An outer exponential-weights update tracks the losses of these temperature-specific aggregates:

Here \(\tau_{\mathrm{out}}\) is the outer temperature; it need not equal any inner temperature \(\tau_k\). The next result makes the two levels of selection explicit.

Proof. Apply exp-concavity to the convex combination \(h_t=\sum_k v_{t,k}g_{k,t}\). It gives \(\exp\{-\ell_t(h_t)/\tau_{\mathrm{out}}\}\geq \sum_k v_{t,k}\exp\{-\ell_t(g_{k,t})/\tau_{\mathrm{out}}\}\). Define \(A_t=\sum_k\pi_k\exp\{-L_t(g_k)/\tau_{\mathrm{out}}\}\), so \(A_0=1\). Substitution of \(v_{t,k}\) makes the right-hand side \(A_t/A_{t-1}\). Taking negative logarithms and summing gives \(\sum_t\ell_t(h_t)\leq-\tau_{\mathrm{out}}\log A_n\). Since \(A_n\geq \pi_k\exp\{-L_n(g_k)/\tau_{\mathrm{out}}\}\) for each \(k\), the stated comparison follows. \(\square\)

Combining this with the one-temperature result gives a hierarchical oracle inequality whenever \(\tau_k\) is admissible for the inner aggregation:

The first penalty is the cost of comparing with predictor \(j\) within temperature \(k\); the second is the cost of comparing with temperature \(k\). The outer theorem alone compares with the actual loss of each \(g_k\), whether or not its inner temperature is admissible. Admissibility is needed to replace that aggregate loss by a predictor’s loss.

Temperature Grids and Unknown Curvature

A prior over temperatures can avoid committing to a single inner temperature when the curvature is not known exactly, but the guarantee remains conditional on admissibility. If the loss has curvature \(\alpha\), an inner temperature is safe when \(\tau_k\geq1/\alpha\). Thus, only the safe members of the grid can be used in the hierarchical comparison. The outer temperature also needs to be safe: one sufficient condition is a known lower bound \(\alpha_{\min}>0\) on curvature and \(\tau_{\mathrm{out}}\geq1/\alpha_{\min}\).

For example, if curvature is known to lie between positive bounds \(\alpha_{\min}\) and \(\alpha_{\max}\), a grid spanning temperatures from \(1/\alpha_{\max}\) to at least \(1/\alpha_{\min}\) can include a safe choice across that range. The outer update can then compare with the safe candidate without being told in advance which one is most effective. A geometric grid is often convenient because it covers a wide range with relatively few points; its prior weights determine the selection costs \(\tau_{\mathrm{out}}\log(1/\pi_k)\).

This is adaptation to a finite set, not a free removal of curvature assumptions. If there is no known positive lower bound on curvature, a fixed outer temperature is not guaranteed to be admissible. A prior cannot repair an invalid exp-concavity condition. One may still use a method with a different guarantee, but the sharp exp-concave comparison derived here requires valid curvature for the outer update and for any inner aggregate invoked in the predictor-level bound.

Common Pitfall and Practical Meaning

A frequent error is to use the inner temperature-selection cost but forget that the outer mixture has its own cost. Another is to report a penalty such as \(\tau_{\mathrm{out}}\log(1/\pi_k)/n\) as though it were cumulative, then add it to a cumulative oracle loss. The safe procedure is to write the cumulative inequality first, include each penalty at its cumulative scale, and divide every term by \(n\) only when reporting average loss.

The prior also has a practical role: assigning a candidate temperature more prior mass makes it cheaper to compare against, while assigning very little mass makes that comparison more expensive. A finite grid is most useful when its range is chosen to include admissible temperatures that perform well, and when the outer temperature is itself valid. It lets the data reweight candidate temperatures online, while the oracle inequality records exactly what that flexibility costs.