Embedding Drift in Crypto Sentiment Models: When to Retrain?

Abstract Crypto sentiment models built on learned text embeddings decay the moment they ship. The vocabulary rotates — a token launch, an ETF narrative, a new memecoin meta — and the reference embedding space drifts out of alignment with live text until classification accuracy silently collapses. The operational question is not whether to retrain but how often. We derive a closed-form answer: under a rotating-centroid model of embedding drift, the cost-optimal fixed retraining cadence is T\ = 2·√(CR / (γ·σθ²)), where CR is the fixed retrain cost, γ is the stale-model cost coefficient, and σθ is the angular drift rate of the population mean direction. A Monte Carlo sweep over twelve cadences reproduces the U-shaped cost curve and lands its minimum exactly at the closed-form T\ (0.00% gap). The accompanying small-angle law — E[cos(μ₀, μt)] ≈ 1 − σθ²·t/2 — matches simulation to within 2% for t ≤ 100 time-units before higher-order terms dominate. The catch, and the honest lesson, is that σθ is itself regime-dependent: a single global cadence is wrong whenever the market transitions between calm and mania. Hypotheses - H1 (high confidence): For a rotating-centroid embedding model, mean-direction cosine similarity decays approximately linearly in time at rate σθ²/2, accurate to within 2% while t ≲ T\. - H2 (high confidence): The closed-form cadence T\ = 2·√(CR/(γ·σθ²)) minimizes the steady-state total-cost rate; a fixed-cadence sweep will place its empirical minimum at or adjacent to T\. - H3 (low confidence, exploratory): A single globally-optimal cadence is dominated by an adaptive, drift-triggered schedule whenever the drift-rate variance across regimes is high. Data Provenance | Quantity | Type | Source | Reliability | |----------|------|--------|-------------| | σθ (angular drift rate) | Estimated parameter | Calibrated to operational regime shifts | UNRELIABLE — regime-dependent, few independent observations | | CR (retrain cost) | Measured | Compute hours × price + labeling labor | RELIABLE — directly observable | | γ (stale-model cost) | Counterfactual | What-accuracy-a-fresh-model-would-have-earned, never observed | UNIDENTIFIABLE — bounded but not pinned | | E[cos(μ₀,μt)] = 1 − σθ²t/2 | Derived | Small-angle Taylor expansion of the rotating-centroid model | RELIABLE (form) — within its validity window | | T\ = 2·√(CR/(γ·σθ²)) | Derived | Maintenance-replacement cost minimization | RELIABLE (form) / UNRELIABLE (magnitude, due to σθ, γ inputs) | | Funding-rate operational context | Real-time DB | PerpsTrader funding.db, 944,150 observations across 230 symbols | RELIABLE (motivation only, not the subject of measurement) | All model figures are SIMULATED — self-contained numpy Monte Carlo, exact script shipped below. The data source for every simulated number is a fixed random seed (reproducible to three decimals). The funding-rate statistics were collected via real-time API from the PerpsTrader venue set (from Binance, from Hyperliquid, and from Asterdex) and are used solely to motivate why crypto is an extreme regime for embedding drift, not as a measurement of the paper's subject. Motivation: Why Crypto Is an Edge Case for Drift Our production funding-rate logs span 944,150 observations across roughly 230 symbols. The per-symbol annualized funding dispersion is violent: MOVE ranges from −2166.65% to +1.37% annualized (mean −36.96%), LAYER from −621.92% to +1.37%, ME from −337.35% to +1.37%. A sentiment model trained on the lexicon of one regime will encounter a near-orthogonal vocabulary in the next. This is the structural reason embedding drift in crypto is not the gentle quarterly drift of an e-commerce review classifier — it is the kind of discontinuous, shock-driven drift that breaks static cadences. Analysis 1. The rotating-centroid drift law Embed text to the unit sphere S^{d−1} (any modern sentence embedder, after L2 normalization). Model the population mean direction μt as undergoing a Brownian rotation in a fixed plane at angular drift rate σθ (radians per unit time): μt = R(θt) · μ₀, where θt N(0, σθ² · t). The expected cosine similarity between the day-zero centroid and the live centroid is, by the small-angle expansion of E[cos θ]: E[cos(μ₀, μt)] ≈ 1 − σθ²·t / 2 (linear decay, valid while σθ·√t ≪ 1). Monte Carlo over a 128-dimensional sphere confirms this with 8,000 samples per horizon: | t | E[cos] (MC) | Closed-form | Error | |---|-------------|-------------|-------| | 5 | 0.9909 | 0.9910 | 0.01% | | 25 | 0.9565 | 0.9550 | 0.16% | | 50 | 0.9148 | 0.9100 | 0.52% | | 100 | 0.8363 | 0.8200 | 1.94% | | 150 | 0.7616 | 0.7300 | 4.14% | | 250 | 0.6379 | 0.5500 | 13.78% | The linear law is trustworthy within 2% for t ≲ 100 (roughly 60% of a typical T\); beyond that the quadratic tail of cos(θ) matters and the linear approximation underestimates true similarity. This is not a flaw — it is the declared validity window of a Taylor expansion. 2. The decision-theoretic optimal cadence Model the operatin