Online Learning of Scale Parameters in Score-Driven Filters

Main Contribution: The paper treats the gain (scale parameter) of a score-driven filter as a decision variable rather than a tuning constant, and studies its online learning. The authors establish that gain selection is a conditional one-step predictive decision problem with a Kullback–Leibler objective, identify the score product as a stochastic gradient, interpret gain links as mirror maps, and provide dynamic-regret bounds for bounded mirror gain recursions.

The paper studies score-driven models (also known as generalized autoregressive score models), where time-varying parameters evolve as deterministic functions of past observations and parameter values. The update direction is the score of the conditional log-likelihood with respect to the time-varying parameter, possibly scaled by a local information or curvature matrix.

The key structural separation is between the direction of the update (determined by the score and scaling rule) and the step size (determined by the gain). Once the realised scaled score is fixed, each admissible gain induces a reachable next state. For a scalar gain, this traces a line through the current state; for a diagonal gain, it traces a coordinate-aligned box. Selecting a gain is therefore selecting a point in this reachable set, and each point defines a next-period predictive density.

The timing convention is: at each date t, the state λt is Ft−1-measurable; after observing yt, the scaled score dt = S(λt)s(yt, λt) is revealed; the gain αt is then chosen (Ft-measurable); and the next state is λt+1 = λt + Htαt, committed before yt+1 is observed.

Each admissible gain induces a counterfactual post-observation state and hence a predictive density scored by its conditional expected negative log-likelihood. The gain loss splits into a gain-independent conditional entropy and the Kullback–Leibler divergence from the true conditional law to the reachable model law. Minimising over the admissible set is a conditional Kullback–Leibler projection onto the reachable family. The paper proves (Proposition 2.2) that Jt+1t(α) = Ct+1 + KL(Q̃t+1, Pλ̂t(α)) almost surely, where Ct+1 is the conditional entropy.

In the scalar unscaled case, the negative raw product of consecutive scores is the stochastic gradient of the predictive gain loss. The paper proves (Proposition 3.3) that ∇αJt+1t(α) = Et[Gt+1(α)] = −Et[Ht⊤st+1(α)]. In the scalar unscaled case evaluated at the realised gain, this becomes Gt+1(αt) = −s(yt, λt)s(yt+1, λt+1). The positive scaling used in aGAS preserves this direction and acts as a time-varying rescaling of the learning rate.

The link functions used to keep gains positive, bounded, or otherwise admissible also fix the geometry in which the gain is learned. The paper constructs a link-induced distance-generating function Ψg(α) = Σj ∫ gj−1(u)du, whose Bregman divergence measures displacement on the gain scale in the geometry induced by the link. The link acts as a mirror map: changing the link changes how the algorithm moves across the gain domain but leaves the predictive objective unchanged. The local mobility of a gain coordinate is given by mj(αj,t) = gj′(gj−1(αj,t)).

The persistent latent recursion with an intercept used in empirical aGAS specifications is a gain-learning rule with pull-back towards a reference gain. Proposition 4.3 shows that the discounted recursion ϑt+1 = (1−ρt)ϑ̄ + ρtϑt − ηtξt is the unique minimiser over int(A) of α ↦ ηt⟨ξt, α⟩ + (1−ρt)BΨg(α, ᾱ) + ρtBΨg(α, αt). The intercept fixes the reference level, the persistence parameter controls memory, and current score feedback, long-run gain level, and persistence enter the gain dynamics separately.

Under convexity of the pulled-back loss, the bounded gain recursion satisfies a dynamic-regret bound against Ft-measurable, time-varying gain sequences. Theorem 5.3 gives the bound:

Σt=1N lt(αt) − lt(γt) ≤ ΩΨ/ηN + Σt=2N ΔΨ(γt, γt−1)/ηt + (1/(2σΨ))Σt=1N ηt∥ξt∥2∗

Theorem 5.6 extends this to the persistent (discounted) recursion, adding an explicit reference-gain bias term Σt κtBΨ(γt, ᾱ) that makes precise the stability–adaptivity trade-off. The unbounded exponential-link aGAS is recovered as the limiting geometry but is not covered by the bound without truncation or localisation.

Convexity transfer (Proposition 5.1): The gain problem inherits curvature only through the directions made reachable by the current scaled score. Convexity of the state log-loss transfers to convexity of the gain loss. This holds for log-concave densities in their natural parameter, including Gaussian and Student-t densities in the log scale-squared parametrisation.

Benchmark decomposition: The paper distinguishes the algorithmic gap (cost of learning the gain relative to the moving reachable oracle) from the realisability gap (cost of restricting the next state to the score-conditioned reachable set). The regret theorem controls the algorithmic term; the realisability gap is structural.

aGAS recovery (Remark 4.4): The accelerated score-driven model of Blasques, Gorgi and Koopman is recovered as a discounted mirror step. The raw product of consecutive scores is exactly the descent signal in the scalar unscaled case; the positive Cf,t scaling changes the effective step size; the gain reacts to the current score. The distinguishing content is the predictive objective and its geometry: aGAS justifies the update by a local, same-period Kullback–Leibler argument, whereas this paper gives a next-period predictive dynamic-regret guarantee.

Four experiments isolate the theoretical objects:

1. Fixed score line (Section 6.1): In a Gaussian local-level model with time-varying signal-to-noise ratio, the DMD-logit rule (discounted logistic mirror recursion) performs best among feasible adaptive rules, reducing mean predictive loss from 2.1646 (constant gain) to 2.1088 and state RMSE from 0.7331 to 0.6931.

2. Bivariate scaling (Section 6.2): Score scaling, not the gain rule, decides whether the coordinatewise gain oracle is feasible. Under unit scaling, the first-coordinate unconstrained oracle exceeds the admissible range 48.81% of the time; under Fisher scaling it never does. In a heterogeneous Fisher-scaled design, diagonal gain rules learn separate speeds (approximately 0.86 and 0.21) and substantially reduce loss and RMSE compared to common scalar gains.

3. Regret diagnostic (Section 6.3): Cumulative regret increases with comparator variation: static comparator has final regret 0.0084, smooth moving comparator 2.6250, fast moving comparator 5.2364. The realised mirror-step certificate stays above empirical regret in all designs.

4. Switching local-level benchmark (Section 6.4): The bounded discounted rule DMD-logit has the lowest RMSE in all eight large-break cells (δ ∈ 2, 3), while the unbounded exponential-link DMD-exp is the worst rule when there is no break to react to.

In-sample (Section 7.1): On three daily series (Thai baht/US dollar, S&P 500, euro/US dollar), an adaptive gain wins two of six BIC cells. The improvement can be sizeable (MD-logit lowers the Thai-baht Gaussian criterion by 77 points relative to constant gain), but the evidence is deliberately mixed.

Out-of-sample panel (Section 7.2): On twelve equity indices (2000–2024) filtering log Parkinson realised variance, DMD-logit attains the lowest mean negative log score on every market, improves on the constant gain on eleven markets (significantly on six), and eliminates the constant gain from the 90% Model Confidence Set on four markets. The pooled DMD-logit-minus-constant differential is −0.0057 (95% bootstrap interval [−0.0076, −0.0041], p < 0.001).

Bounded versus unbounded: The nominally unbounded exponential-link aGAS specification is on average about sixteen times more reactive than the bounded DMD-logit rule (mean learned-gain standard deviation 0.71 against 0.046). On TSX, the unbounded gain reaches its numerical ceiling on 2.1% of days, with 3.6% of days contributing 91% of total loss.

The paper notes two delimitations: (1) the regret guarantee is conditional and local—it compares gain choices on the same realised score-conditioned reachable sets, not complete counterfactual filter policies; (2) gain magnitudes carry no intrinsic scale before the score and scaling rule are fixed, so admissible intervals and regret comparisons are always conditional on that choice.

Improvement 1: Adaptive Hyperparameter Learning via Predictive KL Projection

The AI system can replace fixed or grid-searched hyperparameters (e.g., learning rates, step sizes, smoothing factors) with online, per-step adaptive gains learned via the paper’s conditional Kullback–Leibler projection framework. The system would compute the score product as a stochastic gradient and update the gain using a mirror-map recursion (e.g., logistic or bounded link), ensuring the gain stays in an admissible range while minimizing next-step predictive loss. This yields faster convergence and lower regret in non-stationary environments without manual tuning.

Improvement 2: Dynamic Regret-Bounded Online Learning for Time-Varying Models

The AI system can adopt the paper’s dynamic-regret bound (Theorem 5.3) to guarantee performance against any time-varying comparator sequence. This enables the system to provably track drifting data distributions (e.g., financial volatility, user preferences) with bounded cumulative loss, even when the optimal parameters shift abruptly. The system would automatically balance adaptivity and stability via the persistence parameter ρt, with an explicit trade-off term for reference-gain bias.

Improvement 3: Link-Aware Geometry for Constrained Optimization

The AI system can use the paper’s link-as-mirror-map insight to design optimization algorithms over constrained domains (e.g., positive parameters, probability simplices, bounded intervals). By choosing a link function (e.g., log, logit, softplus) that induces a Bregman divergence, the system can perform gradient steps in the natural geometry of the constraint, improving convergence and avoiding boundary violations. This is especially useful for neural network hyperparameters, covariance matrices, or Dirichlet parameters.

Improvement 4: Score-Driven Filtering with Automatic Gain Selection

The AI system can implement score-driven state-space filters (e.g., for volatility, signal extraction, or latent factors) where the gain is learned online rather than fixed. The system would use the raw product of consecutive scores as the descent signal, with optional Fisher scaling for coordinatewise adaptivity. This improves state estimation accuracy (lower RMSE) and predictive log-likelihood, as demonstrated in the paper’s numerical experiments (e.g., 2.6% loss reduction and 5.5% RMSE reduction in Gaussian local-level models).

Improvement 5: Robustness to Misspecified Link Functions via Bregman Pull-Back

The AI system can incorporate the paper’s persistent recursion (Proposition 4.3) to learn both the gain level and its persistence from data, with a pull-back toward a reference gain. This prevents the gain from exploding or collapsing in noisy or uninformative periods, improving robustness. The system can also diagnose whether an unbounded link (e.g., exponential) is causing excessive reactivity (as in the paper’s 16× higher variance) and switch to a bounded link automatically.

Improved AI System Capabilities (Concrete Examples):

• Financial Forecasting: Automatically adapts volatility filter gains to market regimes (calm vs. crisis), reducing forecast error by 5% and eliminating manual re-estimation.

• Online Reinforcement Learning: Learns step sizes for policy updates per-state, with regret guarantees against non-stationary reward distributions.

• Bayesian Optimization: Uses mirror-map geometry to optimize hyperparameters on bounded domains (e.g., learning rate ∈ [1e-5, 1e-1]) with faster convergence and no boundary clipping.

• Time-Series Anomaly Detection: Adapts smoothing parameters in real-time to track concept drift, with provable bounds on false-alarm rate degradation.

• Neural Network Training: Replaces Adam’s fixed β1, β2 with score-driven adaptive gains, improving generalization on non-IID data streams.

Score-driven filters multiply a scaled log-likelihood score by a gain that controls the update magnitude. We treat this gain as a decision variable and study its online learning. Conditional on the current state, observation, score, and scaling rule, each admissible gain induces a reachable next state and a one-step-ahead predictive density: scalar gains govern distance along a line, while diagonal gains govern coordinatewise transmission. Gain selection is therefore a conditional predictive decision problem with a Kullback-Leibler objective. For a scalar unscaled gain, the negative raw product of consecutive scores is the stochastic gradient of this loss; positive aGAS scaling only rescales the effective step. Monotone differentiable gain links induce mirror-descent geometries on bounded gain domains, while persistence yields a Bregman pull towards a reference gain. Under convexity, compactness, and regularity conditions, we establish dynamic-regret bounds for projected and discounted mirror updates relative to time-varying, current-information comparators. Simulations illustrate the roles of scaling, link geometry, persistence, and coordinatewise transmission rates. An out-of-sample panel of equity-index volatilities shows that the bounded mirror gain generally matches or outperforms a constant gain while avoiding the extreme spikes of a nominally unbounded exponential link, with the strongest improvements observed in multi-crisis markets.