Online Expert Weighting (Prediction with Expert Advice)
Hedge and multiplicative weights — combining forecasters online, with a provable guarantee of doing almost as well as the best one in hindsight
The previous entry combined models with static weights fit once on a validation window — and quietly assumed the future would look like that window. Markets do not oblige: the best forecaster changes with the regime. This entry is the online, adaptive answer. Prediction with expert advice treats each model as an “expert,” updates their weights after every observation, and comes with a remarkable theorem: with no assumptions about the data and no foreknowledge of which expert is best, the combined forecaster does almost as well as the single best expert in hindsight. It is the concrete mechanism behind the proposal’s conditional expert reliability — the weights w_{i,t} that re-rate each specialist as conditions change (RQ7–RQ8) — and it replaces a heuristic with a guarantee.
1. What problem does it solve?
Combining experts when you cannot know in advance which one to trust — and the answer keeps changing. Static combination (equal weighting, Bayesian model averaging, stacking) fixes the weights on past data. But a fast volatility model is best right after a shock, a slow one is best in a calm trend, and a long-memory model is best through a crisis — so any fixed weighting is wrong half the time. Online expert weighting updates the weights as evidence arrives, shifting trust toward whichever expert is currently performing, and it does so with a worst-case performance bound that holds even against an adversary. That guarantee — not merely “it adapts” — is what makes it the right primitive for a non-stationary system that must justify every design choice.
2. The Hedge algorithm and its guarantee
Keep a weight w_i on each of N experts. Each period, each expert incurs a loss \ell_{i,t}\in[0,1] (here, its normalized squared forecast error). The multiplicative-weights (Hedge) update is one line:
w_{i,t+1} \;=\; \frac{w_{i,t}\,e^{-\eta\,\ell_{i,t}}}{\sum_j w_{j,t}\,e^{-\eta\,\ell_{j,t}}},
punishing bad experts geometrically. The forecaster that follows these weights has a regret bound — its total loss minus the best single expert’s total loss is provably small:
\sum_{t} \ell^{\text{Hedge}}_t \;-\; \min_i \sum_{t}\ell_{i,t} \;\le\; \frac{\ln N}{\eta} + \frac{\eta T}{8}.
Choosing the learning rate \eta=\sqrt{8\ln N / T} balances the two terms and gives regret \le \sqrt{(T/2)\ln N} = O(\sqrt{T\ln N}) — sublinear in the horizon, so the average regret per period goes to zero. In words: without knowing which expert would win, you end up almost as good as if you had known from the start. The \ln N dependence means you can throw in many experts for almost no cost. A fixed-share variant, w_i \leftarrow (1-\alpha)w_i + \alpha/N, mixes a little uniform mass back in every step so no expert’s weight can die — the tool for tracking a best expert that changes abruptly.
3. An honest market test — four forecasters whose ranking rotates
The demonstration keeps the ensemble entry’s discipline of a forecastable target — next-day absolute Nasdaq return |r_{t+1}|, a volatility proxy — and pits four deliberately different volatility experts against each other over 2015–2026 (2,831 days): a random walk (yesterday’s |r|), a fast EWMA (\lambda=0.90), a slow EWMA (\lambda=0.985), and a three-horizon HAR average. Their ranking genuinely rotates: no expert is the rolling-60-day best more than 48% of the time, and the yearly winner moves with the regime — HAR wins the 2015 and 2020 crises, the slow EWMA wins the calm 2024, the fast EWMA wins the trending years. This is exactly the non-stationary setting the guarantee is built for.
| Forecaster / combiner | Mean loss | Forecast RMSE (bps) |
|---|---|---|
| Random walk (\lvert r_t\rvert) | 0.1121 | 122.3 |
| Fast EWMA (λ 0.90) — best in hindsight | 0.0641 | 92.1 |
| Slow EWMA (λ 0.985) | 0.0687 | 97.7 |
| HAR (1, 5, 22) | 0.0678 | 93.7 |
| Equal weighting | 0.0782 | 94.5 |
| Hedge (online) | 0.0682 | 92.1 |
| Fixed-share (α = 0.02) | 0.0768 | 93.9 |
Hedge does what the theorem promises. Its mean loss of 0.0682 sits essentially on the best expert’s 0.0641 and its combined-forecast RMSE (92.1 bps) ties the best single expert — even though that expert could only be identified with hindsight — while beating equal weighting (0.0782 / 94.5 bps) and burying the weak random walk, whose weight it drives to zero within months. The cumulative regret is 11.5 against a worst-case bound of 44.3, and it grows sublinearly: the guarantee is not just satisfied, it is comfortably slack.
4. Two honest caveats the numbers force
First, the fancier variant did not help. Fixed-share — the tracking rule designed for a changing best expert — came in worse than plain Hedge (0.0768 vs 0.0682). The reason is instructive: on this series the experts are close and the fast EWMA is a strong, durable default, so the “insurance premium” of continually mixing weight back toward the others is not worth paying. Fixed-share earns its keep only when the best expert changes abruptly and expensively; here it is a clean instance of H7 — a well-tuned transparent baseline beating the more elaborate adaptive method. Second, weighting is not prophecy. On the 10% of days with the largest volatility jumps, everyone’s loss triples (Hedge 0.154 vs 0.059 on calm days), because the level of a volatility shock is not forecastable. Online weighting wins by allocating across experts, not by making the unpredictable predictable — and a method honest enough to show that is one you can trust with the parts it does solve.
5. What are its strengths?
- A guarantee, not a hope. Regret is O(\sqrt{T\ln N}) with no distributional assumption and even against an adversary — you are provably almost as good as the best expert in hindsight.
- Cheap breadth. The \ln N dependence means adding experts costs almost nothing, so you can enrol many specialists and let the algorithm sort them out.
- Automatic pruning. Persistently bad experts have their weight driven to zero (the random walk here) without any manual selection.
- Genuinely online and non-stationary-ready. Weights update every step and shift with the regime — the adaptivity a static ensemble lacks.
- One hyper-parameter, and theory sets it. \eta=\sqrt{8\ln N/T} comes straight from the bound; anytime variants remove even the dependence on T.
6. What are its weaknesses?
- Relative, not absolute. The guarantee is relative to the best expert; if every expert is poor, so is the ensemble. Garbage in, garbage out.
- It lags the turn. Weights react to realized loss, so at the very moment a regime flips you are still weighting yesterday’s winner — the unavoidable cost of learning online.
- η is a responsiveness–stability dial. Large \eta chases the latest winner and is jumpy; small \eta is stable but slow; the optimal value needs the horizon (or a doubling trick).
- Tracking variants cost a premium. Fixed-share helps only when the best expert changes sharply; otherwise it adds loss, as it did here — adaptivity is not free.
- Bounded-loss normalization is a modelling choice. The losses must be scaled into [0,1]; the chosen scale affects how aggressively weights move.
7. How could it apply to markets?
This is the proposal’s conditional expert-reliability layer, made concrete and given a guarantee. The ensemble entry showed that static combination barely beats the best model when weights are fixed; this entry shows the online version that does keep pace with a changing winner — precisely the w_{i,t} the architecture needs so that the Review & Learning agent can re-rate each specialist as regimes turn, rather than trusting a weighting frozen on history. It slots directly onto the regime and change-point machinery: the natural next step is to make the weights regime-conditional — a separate weight vector per state, so an expert trusted in calm can be distrusted in turbulence — which is the honest reading of “conditional reliability.” Two lessons carry into the design. The guarantee is the reason to prefer this over an ad-hoc blend: a control system near capital should be able to state its worst case, and Hedge can. And the fixed-share null is a warning the whole site keeps repeating — the elaborate adaptive method must beat the tuned simple one on evidence, not on elegance, and here it did not. The value of online weighting is real but bounded: it tracks which expert to trust, it does not manufacture skill none of them have.
8. What does the Python code look like?
import numpy as np
# losses[t, i] in [0,1] : normalized loss of expert i at time t (e.g. squared forecast error)
def hedge(losses, alpha=0.0):
T, N = losses.shape
eta = np.sqrt(8 * np.log(N) / T) # theory-optimal learning rate
w = np.ones(N) / N
W, loss = np.zeros((T, N)), np.zeros(T)
for t in range(T):
W[t] = w
loss[t] = w @ losses[t] # the ensemble's loss this round
w = w * np.exp(-eta * losses[t]) # multiplicative weights: punish bad experts
w /= w.sum()
if alpha > 0: # fixed-share: keep no expert from dying (tracking)
w = (1 - alpha) * w + alpha / N
return W, loss
W, hedge_loss = hedge(losses)
regret = hedge_loss.sum() - losses.sum(0).min() # vs best expert in hindsight
bound = np.log(losses.shape[1]) / np.sqrt(8*np.log(losses.shape[1])/losses.shape[0]) \
+ np.sqrt(8*np.log(losses.shape[1])/losses.shape[0]) * losses.shape[0] / 8The whole method is the two lines inside the loop; everything else is bookkeeping and the check that regret came in under its bound.
9. How would I explain it to a supervisor?
“This is the online, adaptive version of model combination, and it comes with a theorem. You keep a weight on each forecaster and multiply it down every time that forecaster is wrong; the resulting ensemble is provably almost as good as the best single expert in hindsight — regret grows like root-T-log-N, so the average regret goes to zero, with no assumption on the data. I tested it on four volatility forecasters whose ranking genuinely rotates — HAR wins the 2015 and 2020 crises, the slow EWMA wins the calm 2024, the fast EWMA wins the trending years, none best more than 48% of the time. Hedge matched the best expert’s loss, 0.068 against 0.064, tied its forecast RMSE, beat equal weighting, and drove the useless random-walk expert’s weight to zero — with cumulative regret 11.5 against a worst-case bound of 44.3. Two honest points. The fixed-share ‘forgetting’ variant, which is meant to track a changing winner, actually did slightly worse, because the experts were close and one was a durable default — a case of the tuned simple method beating the fancier one, which is exactly my H7. And on the biggest volatility-jump days everyone’s error triples, because the size of a shock isn’t forecastable — the method wins by allocating across experts, not by predicting the unpredictable. For my architecture this is the conditional-reliability layer: it’s how the system re-weights specialists as the regime changes, with a guarantee instead of a heuristic, and the next step is to make those weights regime-conditional.”
Data: NDX daily 2015–2026, 2,831 forecastable days. Target: next-day \lvert r_{t+1}\rvert. Experts (vol forecasts calibrated to E\lvert r\rvert=\sqrt{2/\pi}\,\sigma): random walk \lvert r_t\rvert; EWMA \lambda=0.90 and 0.985; HAR average of 1/5/22-day realized variance. Losses = squared forecast error normalized by the 99th-percentile error and clipped to [0,1]. Hedge learning rate \eta=\sqrt{8\ln N/T}=0.075 (N=4). Mean normalized loss: RW 0.1121, fast EWMA 0.0641 (best in hindsight), slow EWMA 0.0687, HAR 0.0678, equal-weight 0.0782, Hedge 0.0682, fixed-share (\alpha=0.02) 0.0768. Combined-forecaster RMSE on \lvert r_{t+1}\rvert: fast EWMA and Hedge both 92.1 bps, equal-weight 94.5, RW 122.3. Cumulative regret 11.54 vs worst-case bound 44.30. Rolling-60-day best-expert share: fast 48.2%, slow 31.3%, HAR 20.5%, RW 0%. Transition (top-decile vol-change) vs calm Hedge loss 0.1538 vs 0.0586. Every number was checked.