rcb/core.py

"""RCB: Recommendation Contextual Bandit (Li, Cheng, Dai; ICML 2026, arXiv 2406.04374).

Faithful implementation of Algorithm 1 (Cold Start Stage) and Algorithm 2
(Exploitation Stage) with the inverse proportional gap sampling (IPGS) kernel
of Eq. (6), the DBIC calibration constants N(eps), L(eps) of Theorem 1, and the
epoch/spread schedule gamma_m = 4 sqrt(K / E_{F,delta}(|T_{m-1}|)).

Notation follows the paper:
  K   number of arms (products)
  d   covariate dimension
  x_t stochastic covariate of user t   (a *random* draw, not fixed design)
  mu(x,i) = x' beta_i  (+ offset)      mean reward of arm i
  P_{i,0} = N(beta_{i,0}, Sigma_{i,0}) public Gaussian prior over beta_i
"""

import numpy as np


# ---------------------------------------------------------------- Theorem 1

def N_eps(K, d, sigma, eps, tau, phi0, C=1.0):
    """Cold-start per-arm sample size, Theorem 1 Eq. (7):

        N(eps) >= (sigma^2 d + 1) K^3 / ( phi0 (tau_{P*} + eps)^2 ).

    `C` is the implementation constant left unspecified by the paper (the
    proof, Eq. C.20, carries unresolved constants c2, c3 from Corollary 1).
    It rescales N but leaves every exponent -- the content of Theorem 1 --
    untouched.
    """
    return C * (sigma ** 2 * d + 1.0) * K ** 3 / (phi0 * (tau + eps) ** 2)


def L_eps(eps, tau_P0, rho_P0, delta_max=1.0):
    """Inverse exploration probability, Theorem 1 Eq. (7) / proof Eq. (C.11):

        L >= 1 + (Delta0 - eps) / (tau_{P0} rho_{P0} + eps),   Delta0 <= 1.

    The paper states the worst case Delta0 = 1, giving L >= 1 + (1-eps)/(tau rho + eps).
    """
    return 1.0 + (delta_max - eps) / (tau_P0 * rho_P0 + eps)


def m0_eps(N):
    """Epoch at which the Exploitation stage starts: m0 = ceil(2 + log2 N)."""
    return int(np.ceil(2.0 + np.log2(max(N, 1.0))))


def EF_ridge(n, d, sigma, phi0, c3=1.0):
    """Ridge / random-design generalization error, Corollary 1: c3 sigma^2 d / (phi0 n)."""
    return c3 * sigma ** 2 * d / (phi0 * max(n, 1.0))


# ---------------------------------------------------------------- IPGS, Eq. 6

def ipgs(mu_hat, gamma):
    """Inverse proportional gap sampling kernel of Eq. (6).

        p_t(i) = 1 / (K + gamma (mu_hat(x,b_t) - mu_hat(x,i)))   for i != b_t
        p_t(b_t) = 1 - sum_{i != b_t} p_t(i)

    `mu_hat` is the length-K vector of predicted rewards for the current x_t.
    """
    K = len(mu_hat)
    b = int(np.argmax(mu_hat))
    gaps = mu_hat[b] - mu_hat
    p = 1.0 / (K + gamma * gaps)
    p[b] = 0.0
    p[b] = 1.0 - p.sum()
    return p, b


# ------------------------------------------------------- Gaussian posteriors

class ArmPosterior:
    """Conjugate Gaussian posterior over beta_i given (x, y) pairs and noise sigma^2.

    Prior N(beta0, Sigma0); posterior precision Lambda = Sigma0^{-1} + X'X/sigma^2.
    `trust_scale` implements Assumption 4 (Evolution of Trust): the *prior*
    covariance is inflated over time so users become more diffuse / open.
    """

    def __init__(self, d, beta0, Sigma0, sigma):
        self.d = d
        self.beta0 = np.asarray(beta0, float).copy()
        self.Sigma0 = np.asarray(Sigma0, float).copy()
        self.sigma = float(sigma)
        self.XtX = np.zeros((d, d))
        self.Xty = np.zeros(d)
        self.n = 0

    def update(self, x, y):
        self.XtX += np.outer(x, x)
        self.Xty += x * y
        self.n += 1

    def _prior_cov(self, trust_scale):
        return self.Sigma0 * trust_scale

    def post(self, trust_scale=1.0):
        S0 = self._prior_cov(trust_scale)
        S0inv = np.linalg.inv(S0)
        Lam = S0inv + self.XtX / self.sigma ** 2
        Cov = np.linalg.inv(Lam)
        mean = Cov @ (S0inv @ self.beta0 + self.Xty / self.sigma ** 2)
        return mean, Cov

    def post_mean(self, trust_scale=1.0):
        """Cached: the posterior mean only moves when new data arrives or the
        Assumption-4 trust scale changes materially, so we key the cache on
        (n, trust_scale quantized to 1%). Exact to within that quantization."""
        key = (self.n, round(np.log(max(trust_scale, 1e-12)) / 0.01))
        if getattr(self, "_ck", None) != key:
            self._ck, self._cv = key, self.post(trust_scale)[0]
        return self._cv


# ------------------------------------------------------------------ the algo

class RCB:
    """Two-stage RCB. Stage 1 = Algorithm 1 (MPASC then RASC); Stage 2 = Algorithm 2."""

    def __init__(self, K, d, sigma, beta0, Sigma0, N, L, offset=0.0,
                 phi0=1.0, c3=1.0, trust_mode="linear", trust_rate=1.0,
                 gamma_const=4.0, use_empirical_EF=False, ridge_lam=1e-2, rng=None):
        self.K, self.d, self.sigma = K, d, float(sigma)
        self.offset = float(offset)
        self.N = int(max(1, round(N)))
        self.L = float(L)
        self.phi0, self.c3 = float(phi0), float(c3)
        self.trust_mode, self.trust_rate = trust_mode, float(trust_rate)
        self.gamma_const = float(gamma_const)
        self.use_empirical_EF = use_empirical_EF
        self.ridge_lam = float(ridge_lam)
        self.rng = rng if rng is not None else np.random.default_rng(0)

        self.post = [ArmPosterior(d, beta0[i], Sigma0[i], sigma) for i in range(K)]
        self.beta0 = [np.asarray(b, float) for b in beta0]

        self.Ni = np.zeros(K, int)          # counts of *exploration* pulls (Algorithm 1)
        self.B = set()                      # saturated arms B_t
        self.stage = 1
        self.t = 0
        self.Tcold = None
        self.m0 = m0_eps(self.N)

        # exploitation-stage state
        self.W = [[] for _ in range(K)]     # (x, y) collected, fed to the offline oracle
        self.beta_hat = [np.asarray(b, float).copy() for b in beta0]
        self.gamma_m = 1.0
        self.cur_m = None
        self._EF_prev = None

    # --- Assumption 4: prior covariance inflation ---------------------------
    def trust_scale(self):
        t = max(self.t, 1)
        if self.trust_mode == "none":
            return 1.0
        if self.trust_mode == "linear":
            return 1.0 + self.trust_rate * t / 1000.0
        if self.trust_mode == "sqrt":
            return 1.0 + self.trust_rate * np.sqrt(t) / 1000.0
        if self.trust_mode == "log":
            return 1.0 + self.trust_rate * np.log(1.0 + t) / 1000.0
        raise ValueError(self.trust_mode)

    # --- belief helpers -----------------------------------------------------
    def prior_mean_rewards(self, x):
        """E[mu(x,i)] under the public prior P_0 -- what a myopic user believes."""
        return np.array([self.offset + x @ self.beta0[i] for i in range(self.K)])

    def trusted_mean_rewards(self, x):
        """E[mu(x,i) | S_{B_t}]: posterior mean for saturated arms, prior mean otherwise.

        This is exactly the conditional expectation appearing in Eq. (5) and in
        Assumption 1's prior-posterior gap G_t(i).
        """
        ts = self.trust_scale()
        out = np.empty(self.K)
        for i in range(self.K):
            if i in self.B:
                out[i] = self.offset + x @ self.post[i].post_mean(ts)
            else:
                out[i] = self.offset + x @ self.beta0[i]
        return out

    def dbic_gain(self, x, rec):
        """Realized per-round gap for the arm actually recommended:

            E[mu(x, I_t) | Gamma_{t-1}] - max_{j != I_t} E[mu(x, j) | Gamma_{t-1}].
        """
        m = self.trusted_mean_rewards(x)
        return m[rec] - np.max(np.delete(m, rec))

    def dbic_gain_expected(self, x, p):
        """The quantity Definition 1 / Eq. (2) actually constrains, in closed form:

            sum_i Pr(I_t = i) ( E[mu(x,i)|Gamma] - max_{j != i} E[mu(x,j)|Gamma] ),

        i.e. the expectation over the recommendation kernel, which is exactly
        "Part I Reward Gap + Part II Reward Gap" of proof Eq. (C.14). `p` is the
        recommendation distribution over arms at this round.
        """
        m = self.trusted_mean_rewards(x)
        g = np.array([m[i] - np.max(np.delete(m, i)) for i in range(self.K)])
        return float(p @ g)

    def rec_kernel(self, x, info):
        """Recover the recommendation distribution p_t(.) used at this round."""
        p = np.zeros(self.K)
        if "p" in info:
            return info["p"]
        if info["phase"] == "MPASC":
            p[int(np.argmax(self.prior_mean_rewards(x)))] = 1.0
            return p
        # RASC: 1/L on the promoted arm, 1 - 1/L on the organic arm
        unsat = [i for i in range(self.K) if i not in self.B]
        org = int(np.argmax(self.trusted_mean_rewards(x)))
        if unsat:
            pm = self.prior_mean_rewards(x)
            prom = max(unsat, key=lambda j: pm[j])
            p[prom] += 1.0 / self.L
            p[org] += 1.0 - 1.0 / self.L
        else:
            p[org] = 1.0
        return p

    # --- epoch / oracle bookkeeping ----------------------------------------
    def _epoch_of(self, t):
        return int(np.floor(np.log2(max(t, 1)))) + 1

    def _ridge(self, X, y):
        A = X.T @ X + self.ridge_lam * np.eye(self.d)
        return np.linalg.solve(A, X.T @ y)

    def _fit_oracle(self):
        """Offline oracle Off_F: per-arm ridge regression on the accumulated data.

        Also returns an estimate of E_{F,delta}(n) (Definition 2): the oracle's mean
        squared prediction error, measured out-of-fold so it is a genuine
        *generalization* error rather than an in-sample residual.
        """
        errs, ws = [], []
        for i in range(self.K):
            if len(self.W[i]) == 0:
                continue
            X = np.array([w[0] for w in self.W[i]])
            y = np.array([w[1] for w in self.W[i]]) - self.offset
            self.beta_hat[i] = self._ridge(X, y)
            if len(y) >= 4:                                    # 2-fold out-of-fold MSPE
                h = len(y) // 2
                for tr, te in ((slice(0, h), slice(h, None)), (slice(h, None), slice(0, h))):
                    b = self._ridge(X[tr], y[tr])
                    errs.append(float(np.mean((X[te] @ b - y[te]) ** 2)))
                    ws.append(X[te].shape[0])
        return float(np.average(errs, weights=ws)) if errs else None

    def _start_epoch(self, m):
        self.cur_m = m
        mspe = self._fit_oracle()
        if self.use_empirical_EF and mspe is not None:
            # Definition 2 bounds E[(mu_hat - mu)^2], the excess risk w.r.t. the true
            # *mean* reward, so the irreducible observation noise sigma^2 is removed.
            EF = max(mspe - self.sigma ** 2, 1e-8)
        else:
            n_prev = max(2 ** (m - 2), 1)      # |T_{m-1}| = 2^{m-2}
            EF = EF_ridge(n_prev, self.d, self.sigma, self.phi0, self.c3)
        self._EF_prev = EF
        self.gamma_m = self.gamma_const * np.sqrt(self.K / max(EF, 1e-12))

    # --- one round ----------------------------------------------------------
    def recommend(self, x):
        """Return (recommended arm I_t, info dict). The DBIC constraint makes a_t = I_t."""
        self.t += 1

        if self.stage == 1:
            if len(self.B) == 0:
                # STEP 1 -- MPASC: recommend the highest context-dependent prior mean
                i = int(np.argmax(self.prior_mean_rewards(x)))
                return i, {"phase": "MPASC", "explore": True}
            # STEP 2 -- RASC
            q = self.rng.random() < 1.0 / self.L
            unsat = [i for i in range(self.K) if i not in self.B]
            if q and unsat:
                # (a) promoted recommendation, Eq. (4)
                pm = self.prior_mean_rewards(x)
                i = max(unsat, key=lambda j: pm[j])
                return i, {"phase": "RASC-promote", "explore": True}
            # (b) organic recommendation, Eq. (5)
            i = int(np.argmax(self.trusted_mean_rewards(x)))
            return i, {"phase": "RASC-organic", "explore": False}

        # ---- Stage 2, Algorithm 2 ----
        m = self._epoch_of(self.t)
        if m != self.cur_m:
            self._start_epoch(m)
        mu_hat = np.array([self.offset + x @ self.beta_hat[i] for i in range(self.K)])
        p, b = ipgs(mu_hat, self.gamma_m)
        p = np.clip(p, 0.0, None)
        p = p / p.sum()
        i = int(self.rng.choice(self.K, p=p))
        return i, {"phase": "IPGS", "explore": i != b, "p": p, "b": b, "gamma": self.gamma_m}

    def update(self, x, arm, y, info):
        if self.stage == 1:
            # organic pulls deliberately do NOT increment N or S (Section 3.1)
            if info["explore"]:
                self.post[arm].update(x, y)
                self.Ni[arm] += 1
                self.W[arm].append((x, y))
                if self.Ni[arm] >= self.N:
                    self.B.add(arm)
            if len(self.B) == self.K:
                self.stage = 2
                self.Tcold = self.t
                self._start_epoch(max(self._epoch_of(self.t), self.m0))
        else:
            self.post[arm].update(x, y)
            self.W[arm].append((x, y))