Recommendation Systems

• Feedback Types

  • Explicit
    • Ratings, likes, or dislikes provided directly by users
    • Sparse, values often missing (not missing at random)
    • Higher quality but harder to collect
  • Implicit
    • Inferred from user behavior - clicks, watches, purchases, etc.
    • Sparse, positive-only feedback (absence doesn't mean dislike)
    • Easier to collect but noisier

• Collaborative Filtering

  • Users collaborate indirectly to help recommend items

  • User-based: Recommend items that similar users liked

    • $\hat{y}{ui} = \frac{\sum{u'} \text{sim}(u, u') y_{u', i}}{\sum_{u'} \text{sim}(u, u')}$

  • Item-based: Recommend items that are similar to what the user already likes

  • Similarity calculations typically use common ratings between users/items

  • Data sparsity makes similarity calculation challenging

  • Matrix Factorization

    • View the problem as matrix completion
    • Predict all missing entries in the ratings matrix
    • Optimization: $L = ||Z - Y||^2$ (for observed entries only)
    • Break up Z into low-rank matrices: $Z = U^TV$
    • U represents user embeddings, V represents item embeddings
    • Can't use SVD directly due to missing values
    • Use ALS (Alternating Least Squares): Estimate U given V, then V given U
    • Add biases to account for user/item rating tendencies:
      • $\hat{y}_{ui} = \mu + b_u + c_i + u_u^T v_i$
        • $\mu$ is global average rating
        • $b_u$ is user bias (rating tendency)
        • $c_i$ is item bias (intrinsic quality)
        • $u_u^T v_i$ is user-item interaction
      • $L = \sum_{(u,i) \in \text{observed}} (y_{ui} - \hat{y}_{ui})^2 + \lambda(\sum ||u_u||^2 + \sum ||v_i||^2)$

  • Probabilistic Matrix Factorization

    • Bayesian approach to matrix factorization
    • $p(Y=y_{ui}) = \mathcal{N}(\mu + b_u + c_i + u_u^T v_i, \sigma^2)$
    • Can add priors on user/item vectors

• Bayesian Personalized Ranking (BPR)

  • For implicit feedback data
  • Ranking approach: Model ranks items user interacted with (positive set) ahead of others (negative set)
  • $p(y = (u,i,j) | \theta) = \sigma(f(u,i;\theta) - f(u,j;\theta))$
    • $f(u,i;\theta)$ is predicted score for user u and item i
  • Use hinge or logistic loss to estimate parameters
  • Samples triplets (user, positive item, negative item) for training

• Factorization Machines (FM)

  • Generalization of matrix factorization that can incorporate side features
  • Represent user-item pairs as feature vectors:
    • $x = \text{concat}[\text{user_features}, \text{item_features}]$
  • Model:
    • $f(x) = \mu + \sum w_i x_i + \sum_{i<j} (v_i \cdot v_j) x_i x_j$
  • The dot product term captures pairwise interactions between features
  • Uses low-rank approximation to reduce parameter count
  • Can easily incorporate context (time, location, etc.) and user/item metadata
  • Loss functions:
    • Explicit feedback: MSE or MAE
    • Implicit feedback: Ranking loss (BPR)

• Cold-Start Problem

  • Difficult to generate predictions for new users or items with no history
  • Solutions:
    • Content-based approaches using item/user metadata
    • Hybrid models combining collaborative and content-based methods
    • Active learning to efficiently collect initial preferences
    • Transfer learning from related domains

• Exploration-Exploitation Tradeoff

  • Counterfactual challenge: Users might like items they never see
  • Recommendation systems must balance:
    • Exploitation: Recommend items likely to be relevant based on history
    • Exploration: Recommend items with uncertain but potentially high value
  • Approaches:
    • Multi-armed bandits (Thompson Sampling, UCB)
    • Contextual bandits for personalized exploration
    • Diversity promotion in recommendations